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MATHEMATICSTextbook for Class XI Foreword The National Curriculum Framework (NCF), 2005, recommends that children’s life at school must be linked to their life outside the school. This principle marks a departure from the legacy of bookish learning which continues to shape our system and causes a gap between the school, home and community. The syllabi and textbooks developed on the basis of NCF signify an attempt to implement this basic idea. They also attempt to discourage rote learning and the maintenance of sharp boundaries between different subject areas. We hope these measures will take us significantly further in the direction of a child-centred system of education outlined in the National Policy on Education (1986). The success of this effort depends on the steps that school principals and teachers will take to encourage children to reflect on their own learning and to pursue imaginative activities and questions. We must recognise that given space, time and freedom, children generate new knowledge by engaging with the information passed on to them by adults. Treating the prescribed textbook as the sole basis of examination is one of the key reasons why other resources and sites of learning are ignored. Inculcating creativity and initiative is possible if we perceive and treat children as participants in learning, not as receivers of a fixed body of knowledge. These aims imply considerable change in school routines and mode of functioning. Flexibility in the daily time-table is as necessary as rigour in implementing the annual calendar so that the required number of teaching days are actually devoted to teaching. The methods used for teaching and evaluation will also determine how effective this textbook proves for making children’s life at school a happy experience, rather than a source of stress or boredom. Syllabus designers have tried to address the problem of curricular burden by restructuring and reorienting knowledge at different stages with greater consideration for child psychology and the time available for teaching. The textbook attempts to enhance this endeavour by giving higher priority and space to opportunities for contemplation and wondering, discussion in small groups, and activities requiring hands-on experience. The National Council of Educational Research and Training (NCERT) appreciates the hard work done by the Textbook Development Committee responsible for this book. We wish to thank the Chairperson of the advisory group in Science and Mathematics, Professor J.V. Narlikar and the Chief Advisor for this book Professor P.K. Jain for guiding the work of this committee. Several teachers contributed to the development of this textbook; we are grateful to their principals for making this possible. We are indebted to the institutions and organisations which have generously permitted us to draw upon their resources, material and personnel. We are especially grateful to the members of the National Monitoring Committee, appointed by the Department of Secondary and Higher Education, Ministry of Human Resource Development under the Chairpersonship of Professor Mrinal Miri and Professor G.P. Deshpande, for their valuable time and contribution. As an organisation committed to the systemic reform and continuous improvement in the quality of its products, NCERT welcomes comments and suggestions which will enable us to undertake further revision and refinement. New Delhi 20 December 2005 Director National Council of Educational Research and Training Textbook Development Committee CHAIRPERSON, ADVISORY GROUP IN SCIENCE AND MATHEMATICS J.V. Narlikar, Emeritus Professor, Chairman, Advisory Committee Inter University Centre for Astronomy & Astrophysics (IUCCA), Ganeshkhind, Pune University, Pune CHIEF ADVISOR P.K. Jain, Professor, Department of Mathematics, University of Delhi, Delhi CHIEF COORDINATOR Hukum Singh, Professor, DESM, NCERT, New Delhi MEMBERS A.K. Rajput, Associate Professor, RIE Bhopal, M.P. A.K. Wazalwar, Associate Professor, DESM NCERT, New Delhi B.S.P. Raju, Professor, RIE Mysore, Karnataka C.R. Pradeep, Assistant Professor, Department of Mathematics, Indian Institute of Science, Bangalore, Karnataka. Pradeepto Hore, Sr. Maths Master, Sarla Birla Academy Bangalore, Karnataka. S.B. Tripathy, Lecturer, Rajkiya Pratibha Vikas Vidyalaya, Surajmal Vihar, Delhi. S.K.S. Gautam, Professor, DESM, NCERT, New Delhi Sanjay Kumar Sinha, P.G.T., Sanskriti School Chanakyapuri, New Delhi. Sanjay Mudgal, Lecturer, CIET, New Delhi Sneha Titus, Maths Teacher, Aditi Mallya School Yelaharika, Bangalore, Karnataka Sujatha Verma, Reader in Mathematics, IGNOU, New Delhi. Uaday Singh, Lecturer, DESM, NCERT, New Delhi. MEMBER-COORDINATOR V.P. Singh, Associate Professor, DESM, NCERT, New Delhi Acknowledgements The Council gratefully acknowledges the valuable contributions of the following participants of the Textbook Review Workshop: P. Bhaskar Kumar, P.G.T., Jawahar Navodaya Vidyalaya, Ananthpur, (A.P.); Vinayak Bujade, Lecturer, Vidarbha Buniyadi Junior College, Sakkardara Chowk Nagpur, Maharashtra; Vandita Kalra, Lecturer, Sarvodaya Kanya Vidyalaya Vikashpuri District Centre, New Delhi; P.L. Sachdeva Deptt. of Mathematics, Indian Institute of Science, Bangalore, Karnataka; P.K.Tiwari Assistant Commissioner (Retd.), Kendriya Vidyalaya Sangathan; Jagdish Saran, Department of Statistics, University of Delhi; Quddus Khan, Lecturer, Shibli National P.G. College Azamgarh (U.P.); Sumat Kumar Jain, Lecturer, K.L. Jain Inter College Sasni Hathras (U.P.); R.P. Gihare, Lecturer (BRC), Janpad Shiksha Kendra Chicholi Distt. Betul (M.P.); Sangeeta Arora, P.G.T., A.P.J. School Saket, New Delhi; P.N. Malhotra, ADE (Sc.), Directorate of Education, Delhi; D.R. Sharma, P.G.T., J.N.V. Mungespur, Delhi; Saroj, P.G.T. Government Girls Sr. Secondary School, No. 1, Roop Nagar, Delhi, Manoj Kumar Thakur, P.G.T., D.A.V. Public School, Rajender Nagar, Sahibabad, Ghaziabad (U.P.) and R.P. Maurya, Reader, DESM, NCERT, New Delhi. Acknowledgements are due to Professor M. Chandra, Head, Department of Education in Science and Mathematics for her support. The Council acknowledges the efforts of the Computer Incharge, Deepak Kapoor; Rakesh Kumar, Kamlesh Rao and Sajjad Haider Ansari, D.T.P. Operators; Kushal Pal Singh Yadav, Copy Editor and Proof Readers, Mukhtar Hussain and Kanwar Singh. The contribution of APC–Office, administration of DESM and Publication Department is also duly acknowledged. Contents 1. Foreword iii Sets 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1 1 1 5 6 7 9 12 12 13 14 18 21 Introduction Sets and their Representations The Empty Set Finite and Infinite Sets Equal Sets Subsets Power Set Universal Set Venn Diagrams Operations on Sets Complement of a Set Practical Problems on Union and Intersection of Two Sets 2. Relations and Functions 2.1 Introduction 2.2 Cartesian Product of Sets 2.3 Relations 2.4 Functions 30 30 30 34 36 3. Trigonometric Functions 3.1 Introduction 3.2 Angles 3.3 Trigonometric Functions 3.4 Trigonometric Functions of Sum and Difference of Two Angles 3.5 Trigonometric Equations 49 49 49 55 63 74 4. Principle of Mathematical Induction 4.1 Introduction 4.2 Motivation 4.3 The Principle of Mathematical Induction 86 86 87 88 5. Complex Numbers and Quadratic Equations 5.1 Introduction 5.2 Complex Numbers 5.3 Algebra of Complex Numbers 5.4 The Modulus and the Conjugate of a Complex Number 5.5 Argand Plane and Polar Representation 5.6 Quadratic Equations 97 97 97 98 102 104 108 6. Linear Inequalities 6.1 Introduction 6.2 Inequalities 6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation 6.4 Graphical Solution of Linear Inequalities in Two Variables 6.5 Solution of System of Linear Inequalities in Two Variables 116 116 116 7. Permutations and Combinations 7.1 Introduction 7.2 Fundamental Principle of Counting 7.3 Permutations 7.4 Combinations 134 134 134 138 148 8. Binomial Theorem 8.1 Introduction 8.2 Binomial Theorem for Positive Integral Indices 8.3 General and Middle Terms 160 160 160 167 9. Sequences and Series 9.1 Introduction 9.2 Sequences 9.3 Series 9.4 Arithmetic Progression (A.P.) 9.5 Geometric Progression (G.P.) 9.6 Relationship Between A.M. and G.M. 9.7 Sum to n terms of Special Series 177 177 177 179 181 186 191 194 10. Straight Lines 10.1 Introduction 10.2 Slope of a Line 10.3 Various Forms of the Equation of a Line 10.4 General Equation of a Line 10.5 Distance of a Point From a Line viii 118 123 127 203 203 204 212 220 225 11. Conic Sections 11.1 Introduction 11.2 Sections of a Cone 11.3 Circle 11.4 Parabola 11.5 Ellipse 11.6 Hyperbola 236 236 236 239 242 247 255 12. Introduction to Three Dimensional Geometry 12.1 Introduction 12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space 12.3 Coordinates of a Point in Space 12.4 Distance between Two Points 12.5 Section Formula 268 268 13. Limits and Derivatives 13.1 Introduction 13.2 Intuitive Idea of Derivatives 13.3 Limits 13.4 Limits of Trigonometric Functions 13.5 Derivatives 281 281 281 284 298 303 14. Mathematical Reasoning 14.1 Introduction 14.2 Statements 14.3 New Statements from Old 14.4 Special Words/Phrases 14.5 Implications 14.6 Validating Statements 321 321 321 324 329 335 339 15. Statistics 15.1 Introduction 15.2 Measures of Dispersion 15.3 Range 15.4 Mean Deviation 15.5 Variance and Standard Deviation 15.6 Analysis of Frequency Distributions 347 347 349 349 349 361 372 ix 269 269 271 273 16. Probability 16.1 Introduction 16.2 Random Experiments 16.3 Event 16.4 Axiomatic Approach to Probability 383 383 384 387 394 Appendix 1: Infinite Series A.1.1 Introduction A.1.2 Binomial Theorem for any Index A.1.3 Infinite Geometric Series A.1.4 Exponential Series A.1.5 Logarithmic Series 412 412 412 414 416 419 Appendix 2: Mathematical Modelling A.2.1 Introduction A.2.2 Preliminaries A.2.3 What is Mathematical Modelling 421 421 421 425 Answers 433 Supplementary Material 466 x xi . 100 Feet Road Hosdakere Halli Extension Banashankari III Stage Bangalore 560 085 Phone : 080-26725740 Navjivan Trust Building P. be lent. in any form or by any means. by way of trade. New Delhi 110 016 and printed at Sam Industrial Enterprises.Navjivan Ahmedabad 380 014 Phone : 079-27541446 CWC Campus Opp. Noida 201 301 Head. hired out or otherwise disposed of without the publisher’s consent. stored in a retrieval system or transmitted. A-17/18. Sri Aurobindo Marg. Sector 60. photocopying. re-sold. 2006 OFFICES OF THE PUBLICATION DEPARTMENT. ALL RIGHTS RESERVED PD 410T BS © National Council of Educational Research and Training. This book is sold subject to the condition that it shall not. Any revised price indicated by a rubber stamp or by a sticker or by any other means is incorrect and should be unacceptable.O.ISBN 81-7450-486-9 First Edition February 2006 Phalguna 1927 Reprinted October 2006 Kartika 1928 November 2007 Kartika 1929 No part of this publication may be reproduced. recording or otherwise without the prior permission of the publisher. Dhankal Bus Stop Panihati Kolkata 700 114 Phone : 033-25530454 CWC Complex Maligaon Guwahati 781 021 Phone : 0361-2674869 Publication Team Printed on 80 GSM paper with NCERT watermark Published at the Publication Department by the Secretary. Publication Department : Peyyeti Rajakumar Chief Production Officer : Shiv Kumar Chief Editor : Shveta Uppal Chief Business Manager : Gautam Ganguly Assistant Editor : Bijnan Sutar Production Assistant : Mukesh Gaur Cover and Layout Arvinder Chawla . mechanical.00 NCERT Campus Sri Aurobindo Marg New Delhi 110 016 Phone : 011-26562708 108. electronic. in any form of binding or cover other than that in which it is published. National Council of Educational Research and Training. NCERT Rs 105. The correct price of this publication is the price printed on this page. 9 (ii) The rivers of India (iii) The vowels in the English alphabet. i. but is the oldest and the youngest. 7.e. a cricket team. we come across collections. In this Chapter. such as.. Georg Cantor (1845-1918) 1. a. e. i. prime numbers. 5. The study of geometry.5 and 7 (vi) The solution of the equation: x2 – 5x + 6 = 0. etc. HARDY bl 1. sequences. 2. 2 and 3. etc. a pack of cards. namely. Sets are used to define the concepts of relations and functions.2 Sets and their Representations In everyday life. points. for example. Today this concept is being used in almost every branch of mathematics. probability.H. More specially. namely. — G. 3. we often speak of collections of objects of a particular kind. u (iv) Various kinds of triangles (v) Prime factors of 210. there is must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein. requires the knowledge of sets.Chapter 1 he SETS In these days of conflict between ancient and modern studies. 1. a crowd of people. He first encountered sets while working on “problems on trigonometric series”. of natural numbers. etc. we discuss some basic definitions and operations involving sets.1 Introduction © no N C tt E o R be T re pu The concept of set serves as a fundamental part of the present day mathematics. In mathematics also. viz. We note that each of the above example is a well-defined collection of objects in .3. we examine the following collections: (i) Odd natural numbers less than 10. o. The theory of sets was developed by German mathematician Georg Cantor (1845-1918). 2 MATHEMATICS the sense that we can definitely decide whether a given particular object belongs to a given collection or not. If a is an element of a set A. 4. it is not a well-defined collection. the elements are being separated by commas and are enclosed within braces { }. 2. In the set P of prime factors of 30. For example. in the set V of vowels in the English alphabet. We shall say that a set is a well-defined collection of objects. etc. (ii) Sets are usually denoted by capital letters A. c. the river Ganga does belong to this colleciton. (iii) The elements of a set are represented by small letters a. 3. C. . If ‘b’ is not an element of a set A. because the criterion for determining a mathematician as most renowned may vary from person to person. 6. Again the collection of five most renowned mathematicians of the world is not well-defined. X. 21. Thus. viz. we write b ∉ A and read “b does not belong to A”. all the elements of a set are listed. y. x. a ∈ V but b ∉ V. 7. 14. we can say that the river Nile does not belong to the collection of rivers of India. Thus. and is N Z Q R Z+ Q+ he We give below a few more examples of sets used particularly in mathematics. For example. elements and members of a set are synonymous terms. we write a ∈ A. Thus. the set of all even positive integers less than 7 is described in roster form as {2. B. (i) In roster form. Y. Some more examples of representing a set in roster form are given below : (a) The set of all natural numbers which divide 42 is {1. : : : : : : the set of all natural numbers the set of all integers the set of all rational numbers the set of real numbers the set of positive integers the set of positive rational numbers. Z. The following points may be noted : (i) Objects. © no N C tt E o R be T re pu bl R + : the set of positive real numbers. 6}. 3 ∈ P but 15 ∉ P. we say that “ a belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. b. On the other hand. 42}. z. The symbols for the special sets given above will be referred to throughout this text. etc. There are two methods of representing a set : (i) Roster or tabular form (ii) Set-builder form. 2. Hence. H.SETS 3 $Note In roster form. could be used) which is followed by a colon “ : ”. Here. C can also be represented in set-builder form as follows: A= {x : x is a natural number which divides 42} B= {y : y is a vowel in the English alphabet} C= {z : z is an odd natural number} © no N C tt E o R be T re pu bl (ii) is of letters forming the word ‘SCHOOL’ is { S. Example 1 Write the solution set of the equation x2 + x – 2 = 0 in roster form. C. L} or {H. If we denote the sets described in (a).. 7. we write V = {x : x is a vowel in English alphabet} It may be observed that we describe the element of the set by using a symbol x (any other symbol like the letters y. the set In set-builder form. In this description the braces stand for “the set of all”. – 2 Therefore. The set of odd natural numbers is represented by {1. B.}. C. C. Thus. and no other letter possess this property. the numbers 4. After the sign of colon. u}. Example 2 Write the set {x : x is a positive integer and x2 < 40} in the roster form. e. we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. all the elements are taken as distinct. . o. L. x = 1. Solution The given equation can be written as (x – 1) (x + 2) = 0. For example. The above description of the set V is read as “the set of all x such that x is a vowel of the English alphabet”. The dots tell us that the list of odd numbers continue indefinitely. O. the set A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that x is a natural number and x lies between 3 and 10. Denoting this set by V. S}. 8 and 9 are the elements of the set A. 42}. . o. e. 7. 5. in the set {a. i. For example. 5. . For example. each of them is a vowel in the English alphabet. the colon stands for “such that”. i. (b) (c) The set of all vowels in the English alphabet is {a. i.e. 3. then A. e. all the elements of a set possess a single common property which is not possessed by any element outside the set. 14. respectively. (b) and (c) above in roster form by A. B. O.. the above set can also be represented as {1. . he Note It may be noted that while writing the set in roster form an element is not $ generally repeated. the solution set of the given equation can be written in roster form as {1. z. all the elements possess a common property. – 2}. namely. i. 3. the order of listing elements has no relevance. u}. etc. the order in which the elements are listed is immaterial. 21. 6. 6. 3. (iii) A team of eleven best-cricket batsmen of the world. 2. 6. Hence.1 1. (vi) A collection of novels written by the writer Munshi Prem Chand. EXERCISE 1. Also. 9. 4. L} (a) { x : x is a positive integer and is a divisor of 18} (ii) { 0 } (b) { x : x is an integer and x2 – 9 = 0} (iii) {1.3. 2. 9. (ii) matches (c) as x + 1 = 1 implies x = 0. 25. we can write A = {x : x = n2. . 4. 3. 2 . . 6. . so (i) matches (d). 9. (ii) The collection of ten most talented writers of India. the given set in the roster form is {1. 16. . (v) The collection of all natural numbers less than 100. Similarly. in the set-builder form the given set is © no N C tt E o R be T re pu n ⎧ ⎫ . 3. N. (iv) The collection of all boys in your class. where n is a natural number and 1 ≤ n ≤ 6 ⎬ ⎨x : x = n +1 ⎩ ⎭ Example 5 Match each of the set on the left described in the roster form with the same set on the right described in the set-builder form : (i) {P. 18} (c) {x : x is an integer and x + 1= 1} (iv) {3. } in the set-builder form. 6. Also. x2 – 9 = 0 implies x = 3. where n ∈ N} is 1 2 3 4 5 6 Example 4 Write the set { . Finally. –3} (d) {x : x is a letter of the word PRINCIPAL} Solution Since in (d). 2. }in set-builder form. there are 9 letters in the word PRINCIPAL and two letters P and I are repeated. 5. So. bl Solution We see that each member in the given set has the numerator one less than the denominator. –3 and so (iv) matches (b). 2 3 4 5 6 7 he Example 3 Write the set A = {1. C. Which of the following are sets ? Justify your answer. R. 18 are all divisors of 18 and so (iii) matches (a). 4. (i) The collection of all the months of a year beginning with the letter J. 5. (vii) The collection of all even integers. A. . . the numerator begin from 1 and do not exceed 6. .4 MATHEMATICS Solution The required numbers are 1. I. . Solution We may write the set A as A = {x : x is the square of a natural number} Alternatively. 1. 6}. . 9. . 1. Thus.SETS © no N C tt E o R be T re pu 4. 5. Let A = {1. 6. .T.E. bl is 3. (ii) B = {x : x is an integer. 1 9 <x< } 2 2 (iii) C = {x : x is an integer. 5.A (iv) 4. A (v) 2. We now write another set B as follows: .9. . 4.16.C. . 25. Match each of the set on the left in the roster form with the same set on the right described in set-builder form: (i) {1.8. .I. . (ix) A collection of most dangerous animals of the world. 3. . 7. 4. A (iii) 0. . . – 6. .} (v) {1. 3. . 125. (viii) The collection of questions in this Chapter.A (ii) 8 .3 The Empty Set Consider the set A = { x : x is a student of Class XI presently studying in a school } We can go to the school and count the number of students presently studying in Class XI in the school. 2.A (vi) 10.A. .. .100} List all the elements of the following sets : (i) A = {x : x is an odd natural number} he 2. 3} (b) {x : x is an odd natural number less than 10} (iii) {M. 5 5. . the set A contains a finite number of elements. 6. . 9} (d) {x : x is a letter of the word MATHEMATICS}.4.32} (iii) {5. 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5. . 12} (ii) {2.S} (c) {x : x is natural number and divisor of 6} (iv) {1. .A Write the following sets in roster form: (i) A = {x : x is an integer and –3 < x < 7} (ii) B = {x : x is a natural number less than 6} (iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8} (iv) D = {x : x is a prime number which is divisor of 60} (v) E = The set of all letters in the word TRIGONOMETRY (vi) F = The set of all letters in the word BETTER Write the following sets in the set-builder form : (i) (3.H. 625} (iv) {2. 3. 2. x2 ≤ 4} (iv) D = {x : x is a letter in the word “LOYAL”} (v) E = {x : x is a month of a year not having 31 days} (vi) F = {x : x is a consonant in the English alphabet which precedes k }.4. 6} (a) {x : x is a prime number and a divisor of 6} (ii) {2. 6 MATHEMATICS B = { x : x is a student presently studying in both Classes X and XI } We observe that a student cannot study simultaneously in both Classes X and XI. because there is no natural number between 1 and 2. we write all the elements of the set within braces { }. According to this definition. By number of elements of a set S.Then C is the empty set. 5}. B = {a. Then S is finite.4 Finite and Infinite Sets Let A = {1. 3. We give below a few examples of empty sets. e. x is a natural number}. B is an empty set while A is not an empty set. Then G is infinite. Consider the set of natural numbers. The empty set is denoted by the symbol φ or { }. Definition 2 A set which is empty or consists of a definite number of elements is called finite otherwise. (ii) B = {x : x2 – 2 = 0 and x is rational number}. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. we mean the number of distinct elements of the set and we denote it by n (S). b. but it is some natural number which may be quite a big number. B and C given above are finite sets and n(A) = 5. Then B is the empty set because the equation x2 – 2 = 0 is not satisfied by any rational value of x. When we represent a set in the roster form. we do not know the number of elements in C. 2. 4. because 2 is the only even prime number. Then A is the empty set. c. If n (S) is a natural number. x is odd }. d. (iv) D = { x : x2 = 4. the set B contains no element at all. So. It is not possible to write all the elements of an infinite set within braces { } because the numbers of elements of such a set is not finite. 1. Then W is finite. g} and C = { men living presently in different parts of the world} We observe that A contains 5 elements and B contains 6 elements. (i) Let A = {x : 1 < x < 2. The sets A. Then D is the empty set. Consider some examples : (i) Let W be the set of the days of the week. How many elements does C contain? As it is. (iii) Let G be the set of points on a line. Thus. © no N C tt E o R be T re pu bl is he Definition 1 A set which does not contain any element is called the empty set or the null set or the void set. then S is non-empty finite set. we represent . We say that the set of natural numbers is an infinite set. the set is called infinite. (ii) Let S be the set of solutions of the equation x2 –16 = 0. because the equation x2 = 4 is not satisfied by any odd value of x. (iii) C = {x : x is an even prime number greater than 2}. n(B) = 6 and n(C) = some finite number. 2. the two sets have exactly the same elements.} is the set of odd natural numbers. {. {1. We consider the following examples : (i) Let A = {1. since 2. Definition 3 Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. 1. it is finite. For example. Otherwise. hence.5 Equal Sets Given two sets A and B.} is the set of natural numbers. because the elements of this set do not follow any particular pattern. . 2. Hence. 2}. Then A = B. the sets A = {1.SETS 7 some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots. he Note All infinite sets cannot be described in the roster form. it is finite. . 1. Hence. then the sets A and B are said to be equal. 3} are equal. since each .3. 2 . . 3} and B = {2. (ii) Given set = {2}. 0. Note A set does not change if one or more elements of the set are repeated. 3. . 7. 3 . 2.–3. . . (iii) Given set = φ. All these sets are infinite. © no N C tt E o R be T re pu bl is Example 6 State which of the following sets are finite or infinite : (i) {x : x ∈ N and (x – 1) (x –2) = 0} (ii) {x : x ∈ N and x2 = 4} (iii) {x : x ∈ N and 2x –1 = 0} (iv) {x : x ∈ N and x is prime} (v) {x : x ∈ N and x is odd} Solution (i) Given set = {1. the sets are said to be unequal and we write A ≠ B. . 2}. For example. 3. Then A and P are equal. if every element of A is also an element of B and if every element of B is also an element of A. (ii) Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. {1. 5. . the given set is infinite. Hence the given set is infinite (v) Since there are infinite number of odd numbers. Clearly. 4. –2. 3 and 5 are the only prime factors of 30 and also these are less than 6. . 4} and B = {3.} is the set of integers.. (iv) The given set is the set of all prime numbers and since set of prime numbers is infinite. . 2. 3. –1.1. $ For example. the $ set of real numbers cannot be described in this form. Hence. 1. it is finite. O. C. hence C ≠ D. B = {1. O. Also C = {5} but –5 ∈ D. Since 0 ∈ A and 0 ∉ B. B ≠ D and B ≠ E. Thus. Solution (i) We have. 2. if any. Therefore B ≠ C. Since E = {5}. Further. O. X = {A. EXERCISE 1. . (i) X. 3. 2. 2}. 2}. Y. we find that. 5} and E = {5}. it follows that. Thus. A ≠ B. 2. 3. A and B are not equal sets. the only pair of equal sets is C and E. 100} (iv) The set of positive integers greater than 100 (v) The set of prime numbers less than 99 State whether each of the following set is finite or infinite: (i) The set of lines which are parallel to the x-axis (ii) The set of letters in the English alphabet (iii) The set of numbers which are multiple of 5 . B = {L. . Y} = B (ii) A = {–2. B = {x : x > 15 and x < 5}. L. 3. the set of letters in “ALLOY” and B. Then X and B are equal sets as repetition of elements in a set do not change a set.99. That is why we generally do not repeat any element in describing a set. E = {x : x is an integral positive root of the equation x2 – 2x –15 = 0}. .2 1. L. Y}. the set of letters in “LOYAL”. he Example 7 Find the pairs of equal sets. (ii) A = {n : n ∈ Z and n2 ≤ 4} and B = {x : x ∈ R and x2 – 3x + 2 = 0}. A ≠ E. 1. Since B = φ but none of the other sets are empty. x < 5 and x > 7 } (iv) { y : y is a point common to any two parallel lines} Which of the following sets are finite or infinite (i) The set of months of a year (ii) {1. 0. A ≠ C. bl is Solution Since 0 ∈ A and 0 does not belong to any of the sets B. D ≠ E. . D = {x: x2 = 25}. Which of the following are examples of the null set (i) Set of odd natural numbers divisible by 2 (ii) Set of even prime numbers (iii) { x : x is a natural numbers. C = E. –1. L}. give reasons: A = {0}.} (iii) {1. . X = {A. D and E. © no N C tt E o R be T re pu Example 8 Which of the following pairs of sets are equal? Justify your answer.8 MATHEMATICS element of A is in B and vice-versa. A ≠ D. A. . C = {x : x – 5 = 0 }. L. D = {–5. where “⇔” is a symbol for two way implications. . 1} bl 4. 15. In this case. It is often convenient to use the symbol “⇒” which means implies. 12. 4. i. B = { 10. . A and B are the same sets so that we have A ⊂ B and B ⊂ A ⇔ A = B. H = { 0. Definition 4 A set A is said to be a subset of a set B if every element of A is also an element of B. then we shall also have B ⊂ A. If A is not a subset of B. Since the empty set φ has no elements. 10} B = { x : x is positive even integer and x ≤ 10} (iv) A = { x : x is a multiple of 10}. 2. a}. 2} E = {–1. 9 Consider the sets : X = set of all students in your school. select equal sets : A = { 2. d } B = { d. 12}. and is usually read as if and only if (briefly written as “iff”). 25. 4}. If it so happens that every element of B is also in A. we agree to say that φ is a subset of every set. 4. A ⊂ A. we say that Y is a subset of X. D = { 3.e. It is possible that every element of B may or may not be in A. B = { 1. G = {1. 8. a } (ii) A = { 4. The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’. 4. Using this symbol. 8. It follows from the above definition that every set A is a subset of itself. 8. 30. We note that every element of Y is also an element of X. F = { 0. The fact that Y is subset of X is expressed in symbols as Y ⊂ X. In other words. b. –1}. We now consider some examples : . We may note that for A to be a subset of B. 8. c. we can write the definiton of subset as follows: A ⊂ B if a ∈ A ⇒ a ∈ B We read the above statement as “A is a subset of B if a is an element of A implies that a is also an element of B”.0) In the following. we write A ⊄ B. then a ∈ B. } Are the following pair of sets equal ? Give reasons. 12. he © no N C tt E o R be T re pu 1. 6. Y = set of all students in your class. 16. 16 } B = { 8. 4. 20. 3. 14}.SETS 6. (iv) The set of animals living on the earth (v) The set of circles passing through the origin (0.6 Subsets is 5. b. C = { 4. all that is needed is that every element of A is in B. 18} (iii) A = {2. c. B = {x : x is solution of x2 + 5x + 6 = 0} (ii) A = { x : x is a letter in the word FOLLOW} B = { y : y is a letter in the word WOLF} From the sets given below. (i) A = {2. 1}. state whether A = B or not: (i) A = { a. A ⊂ B if whenever a ∈ A. . 3}.. 1. 5. i. 0. 5} and B = {x : x is an odd natural number less than 6}.} The set of integers Z = {. 9}. C φ ⊂ B as φ is a subset of every set. B Solution (i) (ii) (iii) (iv) (ii) A . . . C = {1. A ⊄ B as 3 ∈ A and 3 ∉ B A ⊂ C as 1. For example. 5. Here A ∈ B as A = {1} and B ⊂ C. 3} is a proper subset of B = {1. . i. © no N C tt E o R be T re pu Example 9 Consider the sets φ. . Is B a subset of A? No. Insert the symbol ⊂ or ⊄ between each of the following pair of sets: (i) φ . If A ⊂ B and A ≠ B . B = {1. (iii) Let A = {1. 3. 1. 4. 1. d}. 2. . then A is called a proper subset of B and B is called superset of A. –1. B and C be three sets. e.. 3. 3 ∈ A also belongs to C B ⊂ C as each element of B is also an element of C.6. If not. b. 3. –3. (iv) Let A = { a. 3 }. is it true that A ⊂ C?. A = { 1. C (iv) B . Let A = {1}. . Then A is not a subset of B. . . But A ⊄ C as 1 ∈ A and 1 ∉ C. q ∈ Z and q ≠ 0} q . e. (ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56. p. give an example. (Why?). A = {1.} The set of rational numbers Q = { x : x = p .6. If a set A has only one element. B (iii) A . o. We give below the names of some of these subsets. . 7. If A ∈ B and B ⊂ C. u} and B = { a. (Why?) Example 11 Let A. 4}. . also B is not a subset of A. 5. The set of natural numbers N = {1. 9}. 3. u} and B = { a. . Then A ⊂ B and B ⊂ A and hence A = B. Note that an element of a set can never be a subset of itself. we call it a singleton set.{ a } is a singleton set. 2. o. –2. .10 MATHEMATICS bl is he (i) The set Q of rational numbers is a subset of the set R of real numbes.1 Subsets of set of real numbers As noted in Section 1. b. then B is a subset of A and we write B ⊂ A. 3}. there are many important subsets of R. Thus. and we write Q ⊂ R. . 2. 3. c. . Is A a subset of B ? No. 2. . 2. 2} and C = {{1}. c. . B = {{1}. d}. Solution No. Example 10 Let A = { a. Let A and B be two sets. b).e. For example. if A = (–3.. we note that an interval contains infinitely many points. 2 2 3 7 The set of irrational numbers. –5 < x ≤ 7}. ∞ ) describes the set of real numbers in relation to a line extending from – ∞ to ∞.1 Here. various types of intervals described above as subsets of R. i. [ a. Q ⊂ R. where p and q are integers and q is not zero”. 5 1 is Members of T include 2 . The set [ 0. N ⊄ T.1.e. denoted by T. Some of the obvious relations among these subsets are: N ⊂ Z ⊂ Q. the set {x : x ∈ R. Thus [ a. On real number line. 5) and B = [–7. © no N C tt E o R be T re pu bl 1. Then the set of real numbers { y : a < y < b} is called an open interval and is denoted by (a. 5) can be written in setbuilder form as {x : –3 ≤ x < 5}. For example . b ) = {x : a ≤ x < b} is an open interval from a to b. is composed of all other real numbers. Members of Q include –5 (which can be 1 7 11 5 . 9]. . Thus T = {x : x ∈ R and x ∉ Q}. Fig 1.SETS 11 p which is read “ Q is the set of all numbers x such that x equals the quotient q . 7] and the interval [–3. T ⊂ R. ∞) defines the set of non-negative real numbers. These notations provide an alternative way of designating the subsets of set of real numbers.2 Intervals as subsets of R Let a. including a but excluding b. b ∈ R and a < b. b themselves do not belong to this interval. b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a. The set ( – ∞. written in set-builder form. ( a. all real numbers that are not rational. are shown in the Fig 1. b ] = {x : a ≤ x ≤ b} We can also have intervals closed at one end and open at the other. All the points between a and b belong to the open interval (a. can be written in the form of interval as (–5. i. 3 (which can be expressed as ) and – . b ]. 0 ) defines the set of negative real numbers. then A ⊂ B.. b) but a. he expressed as – ) .6. 5 and π . while set ( – ∞. The interval which contains the end points also is called closed interval and is denoted by [ a. then P( A ) = { φ. 4.12 MATHEMATICS The number (b – a) is called the length of any of the intervals (a. For example. b) or (a. 2}.8 Universal Set Usually. .5 } (ii) { a. and all its subsets by the letters A. for that matter. the universal set consists of all the people in the world. Also. { 2 } and { 1. { 1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces : (i) { 2. 2 }. { b. { 1. EXERCISE 1. The universal set is usually denoted by U. as in above. B. in human population studies. In P(A).{ 1 }. b]. We know that φ is a subset of every set . we are interested in the set of natural numbers and its subsets such as the set of all prime numbers. This basic set is called the “Universal Set”.3 1. . . . 2 }. c } . 2}. 2}. [a. viz. The set of all these subsets is called the power set of { 1. {x : x is a triangle in the same plane} (vii) {x : x is an even natural number} . 1. .7 Power Set he Consider the set {1. © no N C tt E o R be T re pu bl is Definition 5 The collection of all subsets of a set A is called the power set of A. for the set of all integers. 2}. { 1. 4 } . We see that {1} and { 2 }are also subsets of {1. the universal set can be the set of rational numbers or. . d } (iii) {x : x is a student of Class XI of your school}. if A is a set with n(A) = m. in all. 3. For another example. . the set of all even numbers. For example. while studying the system of numbers. then it can be shown that n [ P(A)] = 2m. b. C. . 2 }. 2}. It is denoted by P(A). b). the set R of real numbers. the set { 1. in a particular context. . etc. c. . {x : x is a rectangle in the plane} (vi) {x : x is an equilateral triangle in a plane} . Let us write down all the subsets of the set {1. 2 } is a subset of {1. φ is a subset of {1. So. {x : x is an integer} . . 3. Thus. [a.{x : x is a circle in the same plane with radius 1 unit} (v) {x : x is a triangle in a plane} . So.{x : x student of your school} (iv) {x : x is a circle in the plane} . Thus. 1. we have to deal with the elements and subsets of a basic set which is relevant to that particular context. 2. φ. we know that every set is a subset of itself. four subsets. . { 1 }. b]. 2 } has. . .2 }} Also. if A = { 1. note that n [ P (A) ] = 4 = 22 In general. { 2 }. every element is a set. and so forth. 2. 4} ∈ A (iii) {{3.5.7. 2. 5) What universal set(s) would you propose for each of the following : (i) The set of right triangles. Venn diagrams are named after the English logician. – 12 < x < –10} (iii) {x : x ∈ R. b. In Venn diagrams. 3} (iv) φ How many elements has P(A). b.10} (iv) {1.4. 2. Examine whether the following statements are true or false: (i) { a.2. 4. b } ⊄ { b. 6. 3.4. { 3. 2. 12] (iii) (6. 3.3) Fig 1. 3. The universal set is represented usually by a rectangle and its subsets by circles. 6} (ii) φ (iii) {0. John Venn (1834-1883). 8. (ii) The set of isosceles triangles. if A = φ? Write the following as intervals : (i) {x : x ∈ R. c.5. Which of the following statements are incorrect and why? (i) {3. the elements of the sets are written in their respective circles (Figs 1. 2.SETS 4. 3 } ⊂ { 1.2. B and C (i) {0. These diagrams consist of rectangles and closed curves usually circles. 5 } (iv) { a } ⊂ { a. 4.3.6. 3 ≤ x ≤ 4} Write the following intervals in set-builder form : (i) (– 3. 6} and C = {0. 13 1. he © no N C tt E o R be T re pu 5. b} (iii) {1. a } (ii) { a.8. – 4 < x ≤ 6} (ii) {x : x ∈ R.7. which of the following may be considered as universal set (s) for all the three sets A.2 . 2.1. 8}. 0 ≤ x < 7} (iv) {x : x ∈ R. 5}. 9. 4} ⊂ A (ii) {3. 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1. 5.8} bl 2. B = {2. 2. 4. Given the sets A = {1. 7.3. 5 }. 0) (ii) [6 . e } ⊂ { x : x is a vowel in the English alphabet} (iii) { 1. 5} ∈ A (viii) {1.9 Venn Diagrams Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. 5} ⊂ A (vii) {1.2 and 1. 6. 2. 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A Write down all the subsets of the following sets (i) {a} (ii) {a. 4 }. c } (v) { a } ∈ { a. 12] (iv) [–23. c } (vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36} Let A = { 1.9. 1. is 3.6. u } and B = { a. . Again. Ashok}. © no N C tt E o R be T re pu bl In earlier classes.. 10} is the universal set of which A = {2. 1. we will refer all our sets as subsets of some universal set. David. 10} is the universal set of which A = {2.. U = {1. 4. 6. Fig 1.10.. . Solution We have. i.3 The reader will see an extensive use of the Venn diagrams when we discuss the union.4. and also B ⊂ A.2. For example. Example 13 Let A = { a. A ∪ B = { a. e. X ∪ Y = {Ram.3.6. We will now define certain operations on sets and examine their properties.6. Ashok} be the set of students from Class XI who are in the school football team. Geeta. 4. Find A ∪ B. then A ∪ B = A.10} is a subset.10 Operations on Sets he Illustration 2 In Fig 1.8.10} and B = {4. intersection and difference of sets. we get the number 18.1 Union of sets Let A and B be any two sets. 10. Geeta. we get 65. the common elements being taken only once. 12}. Let Y = {Geeta.3. Similarly. when we perform the operation of addition on the pair of numbers 5 and 13. i. Show that A ∪ B = A Solution We have.8. Find X ∪ Y and interpret the set. 6} are subsets. Example 12 Let A = { 2. Each one of these operations was performed on a pair of numbers to get another number. u } = A. e. performing the operation of multiplication on the pair of numbers 5 and 13.2. Akbar. i. Akbar} be the set of students of Class XI. u }. o. Symbolically. 6.4. Henceforth. David.. 8} and B = { 6.. subtraction.2. The union of A and B is the set which consists of all the elements of A and all the elements of B. 12} Note that the common elements 6 and 8 have been taken only once while writing A ∪ B. 10. if B ⊂ A. we write A ∪ B and usually read as ‘A union B’. U = {1. This example illustrates that union of sets A and its subset B is the set A itself. o. .3. The symbol ‘∪’ is used to denote the union.. 8. who are in school hockey team. 8. we have learnt how to perform the operations of addition. Solution We have A ∪ B = { 2.e. This is the set of students from Class XI who are in the hockey team or the football team or both.14 MATHEMATICS Illustration 1 In Fig 1. i. there are some operations which when performed on two sets give rise to another set. Example 14 Let X = {Ram.. multiplication and division on numbers. is 1. 5 indicates the Fig 1. The shaded portion in Fig 1. Find A ∩ B. The symbol ‘∩’ is used to denote the intersection. 5. A ∪ B = { x : x ∈A or x ∈B } The union of two sets can be represented by a Venn diagram as shown in Fig 1. 7 } = B. 7. The intersection of two sets A and B is the set of all those elements which belong to both A and B.5 intersection of A and B. 2. Hence A ∩ B = { 6. 5. . 8 are the only elements which are common to both A and B. 3. 5. 9. Find X ∩ Y. we write. Example 16 Consider the sets X and Y of Example 14. In symbols. 3. Solution We have A ∩ B = { 2. Example 17 Let A = {1. Example 15 Consider the sets A and B of Example 12. 8. φ is the identity of ∪) (iv) A ∪ A = A (Idempotent law) (v) U ∪ A = U (Law of U) © no N C tt E o R be T re pu bl (iii) A ∪ φ = A 1. 4. 6.4 represents A ∪ B.2 Intersection of sets The intersection of sets A and B is the set of all elements which are common to both A and B. we can define the union of two sets as follows: Definition 6 The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). Symbolically. We note that B ⊂ A and that A ∩ B = B. we write A ∩ B = {x : x ∈ A and x ∈ B}. Definition 7 The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically.SETS 15 is he Thus.4 (ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law ) (Law of identity element. 8 }. we write A ∩ B = {x : x ∈ A and x ∈ B} The shaded portion in Fig 1.10. Solution We see that element ‘Geeta’ is the only element common to both. Solution We see that 6.4. 7 }. X ∩ Y = {Geeta}. Hence. Some Properties of the Operation of Union (i) A ∪ B = B ∪ A (Commutative law) Fig 1. Find A ∩ B and hence show that A ∩ B = B. 10} and B = { 2. 3. 4. A and B are disjoint sets. A B because there are no elements which are common to A and B.6 In the above diagram. Fig 1.6 Some Properties of Operation of Intersection (i) A ∩ B = B ∩ A (Commutative law). The disjoint sets can be represented by means of Venn diagram as shown in the Fig 1. Then A and B are disjoint sets. (iii) φ ∩ A = φ. U ∩ A = A (Law of φ and U). 3.7 (i) to (v) . 8 } and B = { 1. 5. (i) (iii) (ii) (iv) (v) Figs 1. e. 6. (iv) A ∩ A = A (Idempotent law) (v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. let A = { 2. ∩ distributes over ∪ This can be seen easily from the following Venn diagrams [Figs 1. 7 }. (ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).. then U A and B are called disjoint sets.16 MATHEMATICS © no N C tt E o R be T re pu bl is he If A and B are two sets such that A ∩ B = φ. For example.7 (i) to (v)]. A ∩ B and B – A are mutually disjoint sets. 7. u } and B = { a. 2. find . Find V – B and B – V Remark The sets A – B. c}. we can rewrite the definition of difference as A – B = { x : x ∈ A and x ∉ B } The difference of two sets A and B can be represented by Venn diagram as shown in Fig 1.e. We note that A – B ≠ B – A. 4. c} (iii) A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6} (iv) A = {x : x is a natural number and 1 < x ≤ 6 } B = {x : x is a natural number and 6 < x < 10 } (v) A = {1. since the elements e. u} B = {a. 5.8. 3. 6}. Example 18 Let A = { 1. then what is A ∪ B ? 4. 3} (ii) A = [ a. since the elements 1. u}. 6}. If A and B are two sets such that A ⊂ B. C = {5.9 EXERCISE 1.4 1. b. i. A – B = { 1. Using the setbuilder notation. 8 }. 4}. Find A – B and B – A. Let A = { a. bl Fig 1. Symbolically. o }. the intersection of any of these two sets is the null set as shown in Fig 1. The shaded portion represents the difference of the two sets A and B. b. o. 5. 3.10. i.8 © no N C tt E o R be T re pu Solution We have. 3. We note that V – B ≠ B – V. 8. 3. 8 }and D = { 7. B = φ 2. 3. is Example 19 Let V = { a. o belong to V but not to B and B – V = { k }. 5 }. 6. 5} Y = {1. 6. e. Find the union of each of the following pairs of sets : (i) X = {1. V – B = { e. 4. 3}. i.3 Difference of sets The difference of the sets A and B in this order is the set of elements which belong to A but not to B. he Solution We have. we write A – B and read as “ A minus B”. i. o. 4.SETS 17 1. k.9. 2. 2. 2.. since the element k belongs to B but not to V. 9. since the element 8 belongs to B and not to A. Is A ⊂ B ? What is A ∪ B ? 3. If A = {1. B = { 2. e. B = {a. 10 }. B = {3. 5 belong to A but not to B and B – A = { 8 }. Fig 1. b }. d } and Y = { f. 4} and {x : x is a natural number and 4 ≤ x ≤ 6 } (ii) { a.. 6. Thus. 12. A = {x : x ∈ U and x is not a divisor of 42 }. 14. C = { 2. 15. The set of these three prime numbers. 3. 4. (iv) { 2. 14 } and { 3.11 Complement of a Set Let U be the universal set which consists of all prime numbers and A be the subset of U which consists of all those prime numbers that are not divisors of 42. i. Similarly. 11. (i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D (v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D Find the intersection of each pair of sets of question 1 above. d. and is denoted by . and 7 ∈ U but 7 ∉ A. e. find (i) X – Y (ii) Y – X (iii) X ∩ Y If R is the set of real numbers and Q is the set of rational numbers. 7. 7} is called the Complement of A with respect to U. i. D = {5. i. 20 }. 10 } and { 3. 6. the set {2. e. 3 ∈ U but 3 ∉ A. 20 }. find (i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D (iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C) (vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) ( A ∩ B ) ∩ ( B ∪ C ) (x) ( A ∪ D) ∩ ( B ∪ C) If A = {x : x is a natural number }. d }are disjoint sets. 12. 11 }. 15}and D = {15. o. 7. g}. f } (iii) {x : x is an even integer } and {x : x is an odd integer} If A = {3. 3 and 7 are the only elements of U which do not belong to A. 10. 15. 6. 1.e. c. 4. (i) { 2. If A = { 3. 3. MATHEMATICS bl 18 10. find (i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D Which of the following pairs of sets are disjoint (i) {1. 16 }. 6. find (i) A – B (ii) A – C (iii) A – D (iv) B – A (v) C – A (vi) D – A (vii) B – C (viii) B – D (ix) C – B (x) D – B (xi) C – D (xii) D – C If X= { a. © no N C tt E o R be T re pu 9. We see that 2 ∈ U but 2 ∉ A. 2. 21}. 11} are disjoint sets. o. 11. 13}. 5. 10. 18. c. u } and { a. 12. 16. 6} are disjoint sets. 5 } and { 3. e. 3. d. 12. 9. 11. 15} are disjoint sets. 8. B = {7. is 5. b. B = { 4. he 7. because 2 is divisor of 42.8. C = {11. 17}. B = {x : x is an even natural number} C = {x : x is an odd natural number}andD = {x : x is a prime number }. 6. 8. 9. Now 2. 7. 13. (ii) { a. then what is R – Q? State whether each of the following statement is true or false. u } and { c. 10. b. (iii) { 2. Justify your answer. b. 9. is also a Again in Example 20 above. 3. 2. then ( A ∪ B )′ = A′ ∩ B′. 4. 7. B′ = { 1. A′ is clearly the set of all boys in the class. 6 } Also A ∪ B = { 2. 5 }. Find A′. © no N C tt E o R be T re pu Solution Since A is the set of all girls. 8. then its complement A′ $ subset of U. 5}. 6}. Then the complement of A is the set of all elements of U which are not the elements of A.10 }. 5. 9} = A It is clear from the definition of the complement that for any subset of the universal set U. Note If A is a subset of the universal set U. Find A′. 5. 8. 4. A = {2. Symbolically. 7}. Find A′. A ∪ B and hence show that ( A ∪ B )′ = A′ ∩ B′. 5. so that (A ∪ B )′ = { 1. 10} and A = {1. A′ = {x : x ∈ U and x ∉ A }. 6. 3. 4. 4. ( A ∩ B )′ = A′ ∪ B′ . A′ ∩ B′. 6}. we want to find the results for ( A ∪ B )′ and A′ ∩ B′ in the followng example. 3. Example 22 Let U = {1. we see that A′ = {x : x ∈ U and x ∉ A }. as the difference between a universal set U and the set A. we write A′ to denote the complement of A with respect to U. 2. he Definition 8 Let U be the universal set and A a subset of U. Hence A′ = { 2. we have ( A′ )′ = A Now. Thus. 8. 3. 6. bl Example 21 Let U be universal set of all the students of Class XI of a coeducational school and A be the set of all girls in Class XI. 4. is Solution We note that 2. 7. 3. Obviously A′ = U – A We note that the complement of a set A can be looked upon. 5. 10 } Hence (A′ )′ = {x : x ∈ U and x ∉ A′} = {1. 6. 6 } ( A ∪ B )′ = { 1. 2. 4. B′ . 10 are the only elements of U which do not belong to A. alternatively.SETS 19 A′. Hence A′ ∩ B′ = { 1. Similarly. This leads to the following definition. 8. Thus. 6 } = A′ ∩ B′ It can be shown that the above result is true in general. 4. 3. If A and B are any two subsets of the universal set U. 7. Example 20 Let U = {1. 3} and B = {3. These two results are stated in words as follows : . 9. Solution Clearly A′ = {1. 6 }. So we have A′ = {2. we have A′ = { 2. 4. 5. 6. 9}. 10. 6 }. e. B = { 2. 5. Complement laws: (i) A ∪ A′ = U he 20 (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B )′ = A′ ∪ B′ © no N C tt E o R be T re pu bl 3. f. 4. 7. c. e. 5. 8} and B = { 2. 4. These are named after the mathematician De Morgan.5 1. 6. 5. g. 4}. e. 8. 5. f. (iv) A′ ∪ B′ Let U be the set of all triangles in a plane. A = { 1. 9 }.10 (ii) A ∩ A′ = φ is Some Properties of Complement Sets 1. 8 } and C = { 3. 6. 8. EXERCISE 1. (ii) A′ ∩ B′. 2. 3. b. The shaded portion represents the complement of the set A. 2. 3. 9 }. b. (iii) (A ∩ B)′. The complement A′ of a set A can be represented by a Venn diagram as shown in Fig 1. 7. Laws of empty set and universal set φ′ = U and U′ = φ. g} (iii) C = {a. 2. If A is the set of all triangles with at least one angle different from 60°. 4. A = {2. what is A′? . Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′ If U = { a. h}. 4. g} (iv) D = { f. 2. write down the complements of the following sets: (i) {x : x is an even natural number} (ii) { x : x is an odd natural number } (iii) {x : x is a positive multiple of 3} (iv) { x : x is a prime number } (v) {x : x is a natural number divisible by 3 and 5} (vi) { x : x is a perfect square } (vii) { x : x is a perfect cube} (viii) { x : x + 5 = 8 } (ix) { x : 2x + 5 = 9} (x) { x : x ≥ 7 } (xi) { x : x ∈ N and 2x + 1 > 10 } If U = {1. 3. 3. 6. Verify that (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′ Draw appropriate Venn diagram for each of the following : (i) (A ∪ B)′. a} Taking the set of natural numbers as the universal set. Let U = { 1. 4. c. d. find the complements of the following sets : (i) A = {a. 6. Law of double complementation : (A′ )′ = A 4. 4. c} (ii) B = {d.MATHEMATICS The complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. These are called De Morgan’s laws. 6. 2. g. De Morgan’s law: Fig 1. 3. 5. 7}. h. These laws can be verified by using Venn diagrams. we get n [ A ∩ ( B ∪ C ) ] = n ( A ∩ B ) + n ( A ∩ C ) – n [ ( A ∩ B ) ∩ (A ∩ C)] = n ( A ∩ B ) + n ( A ∩ C ) – n (A ∩ B ∩ C) Therefore n ( A ∪ B ∪ C ) = n (A) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) –n(A ∩C)+n(A ∩B ∩C) This proves (3). then (ii) n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B ) . we have n ( A ∪ B ∪ C ) = n (A) + n ( B ∪ C ) – n [ A ∩ ( B ∪ C ) ] [ by (2) ] = n (A) + n ( B ) + n ( C ) – n ( B ∩ C ) – n [ A ∩ ( B ∪ C ) ] [ by (2) ] Since A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ). (1) The elements in A ∪ B are either in A or in B but not in both as A ∩ B = φ. (iii) A ∩ A′ = . he 1. we have learnt union. In general. (3) In fact. . . (2) Note that the sets A – B. . . A ∩ B and B – A are disjoint and their union is A ∪ B (Fig 1.SETS 7. if A and B are finite sets. In this Section. (ii) φ′ ∩ A = . ... X has 28 elements and Y has 32 elements. So.11 © no N C tt E o R be T re pu bl Let A and B be finite sets. Example 23 If X and Y are two sets such that X ∪ Y has 50 elements.. . then n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) –n(A ∩C)+n(A ∩B ∩C) . If A ∩ B = φ. then (i) n ( A ∪ B ) = n ( A ) + n ( B ) . which verifies (2) (iii) If A. . how many elements does X ∩ Y have ? . (1) follows immediately. (iv) U′ ∩ A = .The formulae derived in this Section will also be used in subsequent Chapter on Probability (Chapter 16). we will go through some practical problems related to our daily life.. . intersection and difference of two sets. 21 Fill in the blanks to make each of the following a true statement : (i) A ∪ A′ = .. is In earlier Section.11). B and C are finite sets. Therefore n ( A ∪ B) = n ( A – B) + n ( A ∩ B ) + n ( B – A ) = n ( A – B) + n ( A ∩ B ) + n ( B – A ) + n ( A ∩ B ) – n ( A ∩ B) = n ( A ) + n ( B ) – n ( A ∩ B)..12 Practical Problems on Union and Intersection of Two Sets Fig 1. n ( Y ) = 32. n ( M ) = 12 and n ( M ∩ P ) = 4 We wish to determine n ( P ). the word ‘or’ gives us a clue of union and the word ‘and’ gives us a clue of intersection. and X ∩ Y is the set of students who like to play both games. Given n ( X) = 24. n ( X ) = 28. we get 35 = 24 + 16 – n (X ∩ Y) . 24 like to play cricket and 16 like to play football. Using the result n ( M ∪ P ) = n ( M ) + n ( P ) – n ( M ∩ P ). n ( X ∪ Y ) = 35. n (X ∩ Y) = ? By using the formula n ( X ∪ Y ) = n ( X ) + n ( Y ) – n ( X ∩ Y ).12 ) This gives 50 = n ( X ∪ Y ) = n (X – Y) + n (X ∩ Y) + n ( Y – X) = ( 28 – k ) + k + (32 – k ) Hence k = 10. © no N C tt E o R be T re pu Example 24 In a school there are 20 teachers who teach mathematics or physics. n ( Y – X ) = 32 – k (by Venn diagram in Fig 1. we obtain 20 = 12 + n ( P ) – 4 Thus n ( P ) = 12 Hence 12 teachers teach physics. How many students like to play both cricket and football ? Solution Let X be the set of students who like to play cricket and Y be the set of students who like to play football. In the statement of the problem. Then X ∪ Y is the set of students who like to play at least one game. Example 25 In a class of 35 students. have n ( M ∪ P ) = 20 . n ( Y ) = 16. then n ( X – Y ) = 28 – k .12 = 28 + 32 – 50 = 10 Alternatively. We. we find that n(X∩Y) = n(X)+n(Y)–n(X∪Y) Fig 1. therefore. Of these. Also. How many teach physics ? Solution Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics. suppose n ( X ∩ Y ) = k. 12 teach mathematics and 4 teach both physics and mathematics.22 MATHEMATICS bl is he Solution Given that n ( X ∪ Y ) = 50. n (X ∩ Y) = ? Using the formula n ( X ∪ Y ) = n ( X ) + n ( Y ) – n ( X ∩ Y ). each student likes to play at least one of the two games. Here n ( U) = 200. n (A) = 100.13. n (U) = 400. bl is Now he Solution Let U denote the set of surveyed students and A denote the set of students taking apple juice and B denote the set of students taking orange juice. 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. we have A = ( A – B ) ∪ ( A ∩ B ). 120 had been exposed to the chemical C1. 50 to chemical C2. Then © no N C tt E o R be T re pu Example 27 There are 200 individuals with a skin disorder. n (B) = 150 and n (A ∩ B) = 75. 100 were listed as taking apple juice. n (A) = n( A – B ) + n( A ∩ B ) (Since A – B) and A ∩ B are disjoint. the number of individuals exposed to chemical C1 but not to chemical C2 is 90.e. i. n ( A ) = 120. Find the number of individuals exposed to (i) Chemical C1 but not chemical C2 (ii) Chemical C2 but not chemical C1 (iii) Chemical C1 or chemical C2 Solution Let U denote the universal set consisting of individuals suffering from the skin disorder. 23 n (X ∩ Y) = 5 5 students like to play both games. Find how many students were taking neither apple juice nor orange juice. n (B) = n (B – A) + n ( A ∩ B) (Since B – A and A ∩B are disjoint.) or n ( B – A ) = n ( B ) – n ( A ∩ B ) = 50 – 30 = 20 Fig 1. (ii) From the Fig 1. and so. we have B = ( B – A) ∪ ( A ∩ B).13.. n (A′ ∩ B′) = n (A ∪ B)′ = n (U) – n (A ∪ B) = n (U) – n (A) – n (B) + n (A ∩ B) = 400 – 100 – 150 + 75 = 225 Hence 225 students were taking neither apple juice nor orange juice. and 30 to both the chemicals C1 and C2. n ( B ) = 50 and n ( A ∩ B ) = 30 (i) From the Venn diagram given in Fig 1. A denote the set of individuals exposed to the chemical C1 and B denote the set of individuals exposed to the chemical C2.) or n ( A – B ) = n ( A ) – n ( A ∩ B ) = 120 –30 = 90 Hence.13 .SETS Thus. Example 26 In a survey of 400 students in a school. 20 speak Spanish and 10 speak both Spanish and French. 1}. 1}. Example 29 List all the subsets of the set { –1. 1 }. If X and Y are two sets such that n ( X ) = 17. It follows that X = Y. and S ∩ T has 11 elements. The subset of A having no element is the empty set φ. How many like tennis only and not cricket? How many like tennis? In a committee. 37 like coffee. 0}. R } Let Y be the set of letters in “ TRACT”. X ∪ Y has 60 elements and X ∩ Y has 10 elements. n(A∪B) =n(A)+n(B)–n (A∩B) = 120 + 50 – 30 = 140. T has 32 elements.. { 1 }. . Solution Let X be the set of letters in “CATARACT”.24 MATHEMATICS Thus. 3. 0. {–1. i. he EXERCISE 1. 1} and {–1. How many people can speak both Hindi and English? If S and T are two sets such that S has 21 elements. 1 }. The subsets of A having one element are { –1 }. 0. A. {–1. n ( Y ) = 23 and n ( X ∪ Y ) = 38. { 0 }. all the subsets of A are φ. C. T } = { T. 50 people speak French. So. A. 7. © no N C tt E o R be T re pu 5. 0}. how many elements does S ∪ T have? If X and Y are two sets such that X has 40 elements. 10 like both cricket and tennis.{0. {0}. A. R. {0. 250 can speak Hindi and 200 can speak English. 52 like tea and each person likes at least one of the two drinks. 1} . X has 8 elements and Y has 15 elements . The subset of A having three elements of A is A itself. find n ( X ∩ Y ). (iii) The number of individuals exposed either to chemical C1 or to chemical C2. Then Y = { T. R. {1}. 4.6 6. 8. Solution Let A = { –1. 0. T. If X and Y are two sets such that X ∪ Y has 18 elements. The subsets of A having two elements are {–1. {–1}. how many elements does Y have? In a group of 70 people. 1}. How many speak at least one of these two languages? bl 1. 40 like cricket. {–1. How many people like both coffee and tea? In a group of 65 people.e. C } Since every element in X is in Y and every element in Y is in X. is 2. how many elements does X ∩ Y have? In a group of 400 people. Miscellaneous Examples Example 28 Show that the set of letters needed to spell “ CATARACT ” and the set of letters needed to spell “ TRACT” are equal. the number of individuals exposed to chemical C2 and not to chemical C1 is 20. Then X = { C. show that P ( A ∩ B ) = P ( A ) ∩ P ( B ). n (M ) = 400. Thus. what is the least number that must have liked both products? Solution Let U be the set of consumers questioned. then b ∈ A ∪ B. M be the set of persons who owned car A and S be the set of persons who owned car B. Given that n ( U ) = 1000. the least number of consumers who liked both products is 170. X ∈ P ( A ) and X ∈ P ( B ) which implies X ∈ P ( A ) ∩ P ( B). Then n ( S ∪ M ) = n ( S ) + n ( M ) – n ( S ∩ M ) = 200 + 400 – 50 = 550 But S ∪ M ⊂ U implies n ( S ∪ M ) ≤ n ( U ). A ⊂ B. maximum values of n ( S ∪ T ) is 1000. a ∈ A ∩ B. Then Y ∈ P ( A) and Y ∈ P ( B ). Therefore. Y ⊂ A ∩ B. Given that n ( U ) = 500. This gives P ( A ) ∩ P ( B ) ⊂ P ( A ∩ B) Hence P ( A ∩ B ) = P ( A ) ∩ P ( B ). So. Hence. n ( S ∪ T ) is maximum when n ( S ∩ T ) is least. A = B he Example 31 For any sets A and B. So. 50 owned both A and B cars. Is this data correct? Solution Let U be the set of car owners investigated. So a ∈ B. So. how many received medals in exactly two of the three sports ? . Thus. b ∈ A. if b ∈ B. 400 owned car A and 200 owned car B. Similarly. B ⊂ A. But S ∪ T ⊂ U implies n ( S ∪ T ) ≤ n ( U ) = 1000. So. © no N C tt E o R be T re pu Example 32 A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B. the least value of n ( S ∩ T ) is 170.SETS 25 Example 30 Show that A ∪ B = A ∩ B implies A = B Solution Let a ∈ A. S be the set of consumers who liked the product A and T be the set of consumers who like the product B. the given data is incorrect. Then X ⊂ A ∩ B. This gives P ( A ∩ B ) ⊂ P ( A ) ∩ P ( B ). Therefore. Since A ∪ B = A ∩ B. n ( S ) = 200 and n ( S ∩ M ) = 50. Example 34 A college awarded 38 medals in football. n ( S ) = 720. Therefore. bl is Solution Let X ∈ P ( A ∩ B ). Therefore. Then a ∈ A ∪ B. If these medals went to a total of 58 men and only three men got medals in all the three sports. Y ⊂ A and Y ⊂ B. b ∈ A ∩ B. X ⊂ A and X ⊂ B. Example 33 Out of 500 car owners investigated. 15 in basketball and 20 in cricket. Let Y ∈ P ( A ) ∩ P ( B ). This is a contradiction. Since A ∪ B = A ∩ B . So. n ( T ) = 450 So n(S∪T)=n(S)+n(T)–n(S∩T) = 720 + 450 – n (S ∩ T) = 1170 – n ( S ∩ T ) Therefore. which implies Y ∈ P ( A ∩ B ). Show that A = B Is it true that for any sets A and B.14 Here. Show that the following four conditions are equivalent : (i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A Show that if A ⊂ B. then C – B ⊂ C – A. b denotes the number of men who got medals in football and cricket only. respectively. Then n ( F ) = 38. prove it. Show that B = C. D = { 6 }. B. d = n ( F ∩ B ∩ C ) = 3 and a + d + b + d + c + d = 18 Therefore a + b + c = 9. 7.26 MATHEMATICS © no N C tt E o R be T re pu bl is he Solution Let F. then A ⊄ C (v) If x ∈ A and A ⊄ B . n ( C ) = 20 n (F ∪ B ∪ C ) = 58 and n (F ∩ B ∩ C ) = 3 Therefore. . If it is true. 5. If it is false. 3. Thus. which is the number of people who got medals in exactly two of the three sports. then A ⊂ C (iv) If A ⊄ B and B ⊄ C . P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your answer. B = { 2. then x ∈ B (vi) If A ⊂ B and x ∉ B .14 n ( F ∩ B ∩ C ). }. Miscellaneous Exercise on Chapter 1 1. 4. B and C denote the set of men who received medals in football. 6. give an example. 2. then x ∉ A Let A. then x ∈ B (ii) If A ⊂ B and B ∈ C . (i) If x ∈ A and A ∈ B . . a denotes the number of men who got medals in football and basketball only. In each of the following. which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x2 – 8x + 12 = 0 }. n (F ∪ B ∪ C ) = n ( F ) + n ( B ) + n ( C ) – n (F ∩ B ) – n (F ∩ C ) – n (B ∩ C ) + Fig 1. C = { 2. basketball and cricket. n ( B ) = 15. . determine whether the statement is true or false. Decide. 6 }. 6. 4. 4. Assume that P ( A ) = P ( B ). . then A ∈ C (iii) If A ⊂ B and B ⊂ C . and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. among the following sets. c denotes the number of men who got medals in basket ball and cricket only and d denotes the number of men who got medal in all the three. gives n ( F ∩ B ) + n ( F ∩ C ) + n ( B ∩ C ) = 18 Consider the Venn diagram as given in Fig 1. 8. is 9. it was found that 25 people read newspaper H. B = B ∩ ( B ∪ X ) and use Distributive law ) Find sets A. Show that A ∩ B = A ∩ C need not imply B = C. show that A = B. In a survey it was found that 21 people liked product A. Intervals are subsets of R. A set which does not contain any element is called empty set. the set is called infinite set. It is denoted by P(A). Two sets A and B are said to be equal if they have exactly the same elements. Show that for any sets A and B. 27 Summary This chapter deals with some basic definitions and operations involving sets. A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B ) Using properties of sets. A set which consists of a definite number of elements is called finite set. B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ. 12 people liked products C and A. How many students are there in the group? In a survey of 60 people. 26 read newspaper I. 50 know English and 25 know both. 26 liked product B and 29 liked product C. (ii) the number of people who read exactly one newspaper. 13. Find: (i) the number of people who read at least one of the newspapers. Each of the students knows either Hindi or English. Let A and B be sets. 11. show that (i) A ∪ ( A ∩ B ) = A (ii) A ∩ ( A ∪ B ) = A. 14 people liked products B and C and 8 liked all the three products. otherwise. 11 read both H and T. 9 read both H and I. 150 students were found to be taking tea and 225 taking coffee. © no N C tt E o R be T re pu 15. 26 read newspaper T. 100 students know Hindi. B and C such that A ∩ B.SETS 12. he 10. if every element of A is also an element of B. If 14 people liked products A and B. 8 read both T and I. 100 were taking both tea and coffee. Find how many liked product C only. These are summarised below: A set is a well-defined collection of objects. 3 read all three newspapers. (Hints A = A ∩ ( A ∪ X ) . 16. A power set of a set A is collection of all subsets of A. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X. 14. . Find how many students were taking neither tea nor coffee? In a group of students. In a survey of 600 students in a school. bl 8. A set A is said to be subset of a set B. he publishd several papers showing various properties of abstract sets. The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A. He published in a paper in 1874 that the set of real numbers could not be put into one-to-one correspondence wih the integers. It was the famous Englih Philosopher Bertand Russell (18721970 ) who showed in 1902 that the assumption of existence of a set of all sets leads to a contradiction. If A ∩ B ≠ φ. For any two sets A and B..Halmos writes about it in his book ‘Naïve Set Theory’ that “nothing contains everything”. From 1879 onwards..MATHEMATICS The union of two sets A and B is the set of all those elements which are either Historical Note bl is he in A or in B. (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′ If A and B are finite sets such that A ∩ B = φ. The Russell’s Paradox was not the only one which arose in set theory. Till then the entire set theory was based on the assumption of the existence of the set of all sets. His study of set theory came when he was studying trigonometric series of the form a1 sin x + a2 sin 2x + a3 sin 3x + . But Kronecker (1810-1893) castigated him for regarding infinite set the same way as finite sets. The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B. at the turn of the century. then n (A ∪ B) = n (A) + n (B). This led to the famous Russell’s Paradox. then n (A ∪ B) = n (A) + n (B) – n (A ∩ B) © no N C tt E o R be T re pu 28 The modern theory of sets is considered to have been originated largely by the German mathematician Georg Cantor (1845-1918). His papers on set theory appeared sometimes during 1874 to 1897. Paul R. presented the set theory as principles of logic. Another German mathematician Gottlob Frege. . Cantor’s work was well received by another famous mathematician Richard Dedekind (1831-1916). Many paradoxes were produced later by several mathematicians and logicians. these days most of the concepts and results in mathematics are expressed in the set theoretic language. Another one was proposed by Abraham Fraenkel in 1922. the first axiomatisation of set theory was published in 1908 by Ernst Zermelo. A modification of these axioms was done by Kurt Gödel in his monograph in 1940. In fact.SETS © no N C tt E o R be T re pu bl is he As a consequence of all these paradoxes. Despite all these difficulties. Cantor’s set theory is used in present day mathematics. Later in 1937 Paul Bernays gave a set of more satisfactory axiomatisation. This was known as Von Neumann-Bernays (VNB) or GödelBernays (GB) set theory. 29 — — . John Von Neumann in 1925 introduced explicitly the axiom of regularity. 2. A = {red. (blue. Leibnitz (1646–1716) special relations which will qualify to be functions. How many pairs of coloured objects can be made from these two sets? Proceeding in a very orderly manner. c. W. Let us recall from our earlier classes that an ordered pair of elements taken from any two sets P and Q is a pair of elements written in small Fig 2. (blue.e.Chapter 2 2. b). line l is parallel to line m.1 Introduction is Mathematics is the indispensable instrument of all physical research. c). Thus. we can see that there will be 6 distinct pairs as given below: (red. we will learn how to link pairs of objects from two sets and then introduce relations between the two objects in the pair. s). In mathematics also. set A is a subset of set B. s). (blue. where b. we notice that a relation involves pairs of objects in certain order.2 Cartesian Products of Sets Suppose A is a set of 2 colours and B is a set of 3 objects. (red. In our daily life.. c and s represent a particular bag. blue}and B = {b. b). father and son. respectively. In all these.1 . Finally. i. – BERTHELOT he RELATIONS AND FUNCTIONS © no N C tt E o R be T re pu bl Much of mathematics is about finding a pattern – a recognisable link between quantities that change. The concept of function is very important in mathematics since it captures the idea of a mathematically precise correspondence between one quantity with the other. we will learn about G . In this Chapter. s}. we come across many relations such as number m is less than number n. teacher and student. we come across many patterns that characterise relations such as brother and sister. coat and shirt. we get 6 distinct objects (Fig 2. (red.1). c). (DL. (KA. b2). b3. 01 If the three states.RELATIONS AND FUNCTIONS 31 brackets and grouped together in a particular order. (red.02). b2) will be distinct from the point in the position (b2. Delhi. with the DL MP KA restriction that the code begins with an element from set A. (blue.03). (p.01). 01) will not be the same as the code (01. if and only if the corresponding first elements are equal and the second elements are also equal. (MP. i. (KA. DL). (KA. 03 Madhya Pradesh and Karnataka.2)? The available pairs are:(DL. 03}representing codes for the licence plates of vehicles issued 02 by DL. i. consider the two sets: A = {DL. b3). A × B = {( a1. the code (DL.c).01). b2.b). MP. b3). (a1. (KA. p ∈ P and q ∈ Q . i. P × Q = { (p.02).02). KA represent Delhi. where DL.03). MP and KA . b4)}. consider the two sets A= {a1. since there are 3 elements in each of the sets A and B. b1). (MP. b4} (Fig 2. Also note that the order in which these elements are paired is crucial. (MP. respectively and B = {01. The 8 ordered pairs thus formed can represent the position of points in the plane if A and B are subsets of the set of real numbers and it is obvious that the point in the position (a1. It can easily be seen that there will be 9 such pairs in the Cartesian product. (KA. b1). b2). (a2.3).s). (DL. This leads to the following definition: © no N C tt E o R be T re pu bl is he Definition 1 Given two non-empty sets P and Q. Fig 2..b).01).2 which are the pairs available from these sets and how many such pairs will there be (Fig 2. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q. a1).q). Madhya Pradesh and Karnataka were making codes for the licence plates of vehicles.01). b4). (MP.03)}.03). Fig 2. Again. (a2. then P × Q will also be empty set.c).02). KA}. (MP. (DL.e. (a1. (DL. (a2. q ∈ Q } If either P or Q is the null set. P × Q = φ From the illustration given above we note that A × B = {(red. a2} and B = {b1. (red.03).e. (MP. (a1. (a2..3 Remarks (i) Two ordered pairs are equal.. As a final illustration. This gives us 9 possible codes.03) and the product of set A and set B is given by A × B = {(DL.01). (blue.01).02). (blue.02.s)}. For example. .e.02). MP.q) : p ∈ P. (KA. 3). . (3.4).4). (1. (r. (r. c} and Q = {r}.3).3). (2.5). 4). form the sets P × Q and Q × P.6)} Therefore.4)} and (A × C) = {(1.6)}. i. (c. a). if n(A) = p and n(B) = q.1). 4).4). (3. B = {3.4). (iii) If A and B are non-empty sets and either A or B is an infinite set. c ∈ A}.3). (3.5).6)}. (ii) Now (A × B) = {(1. b. a). (2. (2. (2.4). we obtain (A × B) ∪ (A × C) = {(1.4). (1.5).6}. (b.6). (3. (1. 6}. (A × B) ∩ (A × C) = {(1. (B ∪ C) = {3. then there will be pq elements in A × B. b. 5.32 MATHEMATICS (ii) If there are p elements in A and q elements in B. (2. we conclude that P × Q ≠ Q × P.5. the pair (a. b.4).3}. (1. Find (ii) (A × B) ∩ (A × C) (i) A × (B ∩ C) (iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C) Solution (i) By the definition of the intersection of two sets. (3. Therefore x + 1 = 3 and y – 2 = 1. P × Q = {(a.5). (3.. bl Solving we get is Solution Since the ordered pairs are equal. (2. (2.4). However. r) is not equal to the pair (r. (2. 4. (2.3). then n(A × B) = pq. (iii) Since. (3. (1. (2. (3. (3. r).4). (2. c)} Since. the number of elements in each set will be the same.3).4). (1.6).3). find the values of x and y.6). (3.6).4} and C = {4. Here (a. r). (2. A × (B ∩ C) = {(1. by the definition of equality of ordered pairs. (1. Therefore. (B ∩ C) = {4}. (3. © no N C tt E o R be T re pu Example 2 If P = {a. r)} and Q × P = {(r. Example 3 Let A = {1. (iv) Using the sets A × B and A × C from part (ii) above.e.4). c) is called an ordered x = 2 and y = 3.5). triplet. the corresponding elements are equal.5).3).2. b). (1. we have A × (B ∪ C) = {(1.5). b.4).4)}. (1. (3. (3.5). Are these two products equal? Solution By the definition of the cartesian product. c) : a.6). he (iv) A × A × A = {(a.6). Example 1 If (x + 1. (3. (2. (3.4).5).3). (2. 4)}. (2. y – 2) = (3. then so is A × B. 1. How many subsets will A × B have? List them. 2}. (z. If A × B = {(a. 6. If ⎜ + 1. 7. (m. 6} and D = {5. (1. form the set P × P × P. y) such that x ∈ A and y ∈ B. find the values of x and y. then P × Q = {(m. 2}. 4}. 3. x). (2. n} and Q = { n. y – ⎟ = ⎜ . y ∈ R} which represents the coordinates of all the points in two dimensional space and the cartesian product R × R × R represents the set R × R × R ={(x. m}. z) : x. . find A × A × A. pairs (x.1. rewrite the given statement correctly. 2}. State whether each of the following statements are true or false. Example 5 If R is the set of all real numbers.(p.1). Find A and B. 3. q). (2. 8. find A and B. Write A × B.2)}. q). 3⎠ ⎝ 3 3⎠ ⎝3 2. r}.2). 7. (m. B = {3. 4. ⎟ . Let A and B be two sets such that n(A) = 3 and n(B) = 2. 5}.2. r). 2} and B = {3. y). (1. P × P × P = {(1.1). 8}. 8} and H = {5. then A × (B ∩ φ) = φ. If the statement is false. (y. 6. (2. (1. If the set A has 3 elements and the set B = {3. r)}. 9. A = set of first elements = {p. 2).2.1). what do the cartesian products R × R he and R × R × R represent? is Solution The Cartesian product R × R represents the set R × R={(x.1). (2. Let A = {1. Solution We have. (b. 1}. If (x. EXERCISE 2. where x.1 2⎞ ⎛5 1⎞ ⎛x 1. z ∈ R} which represents the coordinates of all the points in three-dimensional space. find A and B. (ii) If A and B are non-empty sets. 4}. (iii) If A = {1.1. y) : x. y and z are distinct elements.(n. n). If A = {–1. (ii) A × C is a subset of B × D. © no N C tt E o R be T re pu Solution bl Example 6 If A × B ={(p. 4. 2}. 4}. 1).(a .2.2). (i) If P = {m. m} B = set of second elements = {q.2. C = {5. B = {1. x). 1) are in A × B. m)}. Let A = {1. (b. y)}. y.2).RELATIONS AND FUNCTIONS 33 Example 4 If P = {1.1. 4. then A × B is a non-empty set of ordered 5. then find the number of elements in (A×B). Verify that (i) A × (B ∩ C) = (A × B) ∩ (A × C). If G = {7. find G × H and H × G. 2. y. 4 second element y of each ordered pair (x. y ∈ Q}. Definition 4 The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The Cartesian product A × A has 9 elements among which are found (–1.4). (2.1).Bhanu).4. Definition 2 A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. Ali).3). codomain and range of R. Definition 3 The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. b. Find the set A and the remaining elements of A × A. Define a relation R from A to A by R = {(x. 4.34 MATHEMATICS 10. 2. Ali). x ∈ P. (b. Divya)}. The cartesian product of P and Q has 15 ordered pairs which can be listed as P × Q = {(a.5). Then R = {(a. 6}. Bhanu. y) as R= { (x. y) : y = x + 1 } (i) Depict this relation using an arrow diagram. Chandra)} A visual representation of this relation R (called an arrow diagram) is shown in Fig 2. The second element is called the image of the first element. Bhanu). c} and Q = {Ali. (c. Binoy). R = {(1. The whole set B is called the codomain of the relation R. Binoy.. (a. (ii) Write down the domain. Divya}. (ii) An arrow diagram is a visual representation of a relation. Remarks (i) A relation may be represented algebraically either by the Roster method or by the Set-builder method. 5.2). Binoy). The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.3 Relations © no N C tt E o R be T re pu bl is he Consider the two sets P = {a.. 3. Example 7 Let A = {1.6)}. Note that range ⊆ codomain. (3. .y): x is the first letter of the name y. (c. 2. (4. (5. (b. (a.. Chandra. . We can now obtain a subset of P × Q by introducing a relation R between the first element x and the Fig 2. Solution (i) By the definition of the relation. 0) and (0. 2 1. 4)}. 3. (1. (2. x ∈ P. If n(A ) = p and n(B) = q. 3. Therefore.. the number of subsets of A×B is 24. A × B = {(1. 6}. 2} and B = {3. 2.14}. (25. 2). y ∈ Q} (ii) In roster form. 4). (ii) in roster form.. 6} and the codomain = {1. Find the number of relations from A to B. 2. 9.The corresponding arrow diagram is shown in Fig 2. Solution It is obvious that the relation R is is What is its domain and range? © no N C tt E o R be T re pu bl “x is the square of y”. –3.. 2. 5. Fig 2.6 shows a relation between the sets P and Q. then n (A × B) = pq and the total number of relations is 2pq. the number of relations from A into B will be 24. Fig 2. y) : 3x – y = 0. 5. 4. (4. y ∈ A}.5. where x. Note that the element 1 is not related to any element in set P. 3). codomain and range. –5. 25}. Write this relation (i) in set-builder form. (25. the range = {2.5 35 he RELATIONS AND FUNCTIONS Example 8 The Fig 2. Solution We have. 3.} Similarly. (4. Write down its domain. –2). 5}. 3. $ Note The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A × B. 5). 3. R = {(x. .. 4}. Let A = {1. 4. 3). Since n (A×B ) = 4. Define a relation R from A to A by R = {(x. Remark A relation R from A to A is also stated as a relation on A. 3). Example 9 Let A = {1. (i) In set-builder form. 5. 4. (ii) We can see that the domain ={1. –5)} The domain of this relation is {4. EXERCISE 2. The range of this relation is {– 2. (2. y): x is the square of y. –3).6 (9. The set Q is the codomain of this relation. 2. R = {(9. 36 2. b) ∈ f. 2. a – b is an integer}. .4 Functions In this Section. x3) : x is a prime number less than 10} in roster form. Definition 5 A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. 9. Find the domain and range of R. 6. It is one of the most important concepts in mathematics. In other words. 4. 2}. A = {1. 2. then f (a) = b. 3. 2. Let R be the relation on A defined by {(a. y) : y = x + 5. Find the number of relations from A to B. b ∈ Z. we study a special type of relation called function. If f is a function from A to B and (a. 3. x ∈ A. b is exactly divisible by a}. Determine the domain and range of the relation R defined by R = {(x. he 3. Write R in roster form.b): a. (ii) Find the domain of R 6. 1. x is a natural number less than 4. 4.7 © no N C tt E o R be T re pu (i) Write R in roster form bl 5. Let A = {x. 4. y ∈N}. which produces new elements out of some given elements. 9}. Write down the domain and the range. a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element. x + 5) : x ∈ {0. b ∈A. 2. What is its domain and range? Let A = {1. Depict this relationship using roster form. x. (i) in set-builder form (ii) roster form. We can. Write the relation R = {(x. There are many terms such as ‘map’ or ‘mapping’ used to denote a function. Let R be the relation on Z defined by R = {(a. visualise a function as a rule. y. (iii) Find the range of R. The Fig2. Define a relation R from A to B by R = {(x. y ∈ B}. 5}}. 8. 5} and B = {4. Write this relation is Fig 2. y): the difference between x and y is odd. MATHEMATICS Define a relation R on the set N of natural numbers by R = {(x. b): a . 3. z} and B = {1.7 shows a relationship between the sets P and Q. 6}. 7. where b is called the image of a under f and a is called the preimage of b under f. we will see many more relations some of which are functions and others are not. 4 are the elements of domain of R having their unique images.(3.. f (4) = .. the relation in Example 9 is also not a function. (5. this relation is a function. f (2) = .3). Similarly.(3.1).2)}. y ∈ N}. What is the domain. we can easily see that the relation in Example 7 is not a function because the element 6 has no image. The codomain is also N. this relation R is a function.(3. this relation is a function. Again. (4.4)} (iii) R = {(1. Definition 6 A function which has either R or one of its subsets as its range is called a real valued function.. Looking at the previous examples. is Example 10 Let N be the set of natural numbers and the relation R be defined on N such that R = {(x. (4.. Since every element has one and only one image.2).2).RELATIONS AND FUNCTIONS 37 he The function f from A to B is denoted by f: A B. © no N C tt E o R be T re pu Example 11 Examine each of the following relations given below and state in each case.7)} Solution (i) (ii) (iii) Since 2. f (6) = . this relation is not a function. f (3) = .. 3..3)..4). Since every natural number n has one and only one image.. Further. (ii) R = {(2. the relation in Example 8 is not a function because the elements in the domain are connected to more than one images.(2. if its domain is also either R or a subset of R.4). Example 12 Let N be the set of natural numbers. (4.. f (7) = .6).1). (Why?) In the examples given below. codomain and range of R? Is this relation a function? bl Solution The domain of R is the set of natural numbers N. Solution The completed table is given by x 1 2 3 4 y f (1) = 3 f (2) = 5 f (3) = 7 5 6 7 f (4) = 9 f (5) = 11 f (6) = 13 f (7) =15 . x 1 2 3 4 5 6 7 y f (1) = . Using this definition. y) : y = 2x. Since the same first element 2 corresponds to two different images 2 and 4.. Define a real valued function f : N N by f (x) = 2x + 1. (6. x. giving reasons whether it is a function or not? (i) R = {(2.(2. complete the table given below... f (5) = . The range is the set of even natural numbers. it is called a real function..5).. 9 . Define the real valued function f : R → R by y = f(x) = x for each x ∈ R. Fig 2.4. Here the domain and range of f are R. The graph is a straight line as shown in Fig 2. Such a function is called the identity function. Here domain of f is R and its range is {c}. It passes through the origin.38 MATHEMATICS © no N C tt E o R be T re pu bl is he 2.1 Some functions and their graphs (i) Identity function Let R be the set of real numbers.8. Fig 2.8 (ii) Constant function Define the function f: R → R by y = f (x) = c. x ∈ R where c is a constant and each x ∈ R. . a1. x ∈ R. The functions defined by f(x) = x3 – x2 + 2.(Why?) x –4 –3 –2 –1 0 1 2 3 © no N C tt E o R be T re pu y = f(x) = x2 bl Example 13 Define the function f: R → R by y = f(x) = x2..10 1 2 3 4 1 4 9 16 2 = {x : x ∈ R}. if f(x)=3 for each x∈R. then its graph will be a line as shown in the Fig 2.. whereas the function h defined by h(x) = x 3 + 2x is not a polynomial function. The graph of f is given Fig 2. For example.9.10 . and g(x) = x4 + 2 x are some examples he (iii) 2 is of polynomial functions. a2.an∈R.+ an xn. Complete the Table given below by using this definition. Range of f by Fig 2.. where n is a non-negative integer and a0. What is the domain and range of this function? Draw the graph of f. 4 Solution The completed Table is given below: x 2 y = f (x) = x –4 –3 –2 –1 0 16 9 4 1 0 Domain of f = {x : x∈R}.. y = f (x) = a0 + a1x + a2x2 + .RELATIONS AND FUNCTIONS 39 The graph is a line parallel to x-axis.. Polynomial function A function f : R → R is said to be polynomial function if for each x in R. .5 0....5 – 0.. .x3): x∈R}.5 .25 0.67 –1 – 2 0. . f(–3) = –27. f(1) = 1. What is the domain and range of this function? Example 15 Define the real valued function f : R – {0} → R defined by f (x) = x y = 1 x –2 –1. The graph of f is given in Fig 2.67 0. –0..... f(–2) = –8. Solution The completed Table is given by x y= –2 1 x –1. . x∈R. .40 MATHEMATICS Example 14 Draw the graph of the function f :R → R defined by f (x) = x3. . f = {(x. 1 1.5 – 0.5 2 4 2 1 0. f(3) = 27....5 1 1. etc... where g(x) ≠ 0. f(–1) = –1. f(2) = 8.25 0. Therefore.11 (iv) Rational functions are functions of the type f (x) .. .5 2 . x x ∈ R –{0}.5 . where f(x) and g(x) are g (x) polynomial functions of x defined in a domain. 1 .. Fig 2.. Complete the Table given below using this definition.5 –1 .5 –1 –0.11.. © no N C tt E o R be T re pu bl is he Solution We have f(0) = 0. x < 0 The graph of the modulus function is given in Fig 2.12.. FigFig 2. But for negative values of x.x ≥ 0 f (x) = ⎨ ⎩ − x.RELATIONS AND FUNCTIONS 41 © no N C tt E o R be T re pu bl is he The domain is all real numbers except 0 and its range is also all real numbers except 0.if x < 0 ⎩ is called the signum function.122. The domain of the signum function is R and the range is .if x = 0 ⎪ −1.if x > 0 ⎪ f (x) = ⎨0 . (vi) Signum function The function f:R→R defined by Fig 2. For each non-negative value of x. The graph of f is given in Fig 2. ⎧ x. the value of f(x) is the negative of the value of x.12 (v) The Modulus function The function f: R→R defined by f(x) = |x| for each x ∈R is called modulus function.e.13. f(x) is equal to x.13 ⎧1. i. From the definition of [x]. multiply two real functions and divide one real function by another. we define (f + g): X → R by (f + g) (x) = f (x) + g (x). .15 2. x ‘ 0and 0for x = 0 he the set {–1. Such a function is called the greatest integer function. The graph of the function is shown in Fig 2. The graph of the signum function is given by the Fig 2.15. x ∈R assumes the value of the greatest integer. Then. Fig 2. is Fig 2. subtract a real function from another.42 MATHEMATICS © no N C tt E o R be T re pu (vii) Greatest integer function The function f: R → R defined by f(x) = [x]. for all x ∈ X.4.14 bl x f(x) = x . 0.2 Algebra of real functions In this Section. multiply a real function by a scalar (here by a scalar we mean a real number). we shall learn how to add two real functions. less than or equal to x. 1}. we can see that [x] = –1 for –1 ≤ x < 0 [x] = 0 for 0 ≤ x < 1 [x] = 1 for 1 ≤ x < 2 [x] = 2 for 2 ≤ x < 3 and so on. (i) Addition of two real functions Let f : X → R and g : X → R be any two real functions. where X ⊂ R.14. bl (v) Quotient of two real functions Let f and g be two real functions defined from © no N C tt E o R be T re pu f X→R where X ⊂ R. ⎛ f ⎞ x2 1 ( x ) = . ⎝g⎠ Solution We have. Then the product α f is a function from X to R defined by (α f ) (x) = α f (x). x ∈X. where X ⊂ R. 2 2 (f + g) (x) = x + 2x + 1. is (iv) Multiplication of two real functions The product (or multiplication) of two real functions f:X → R and g:X → R is a function fg:X → R defined by (fg) (x) = f(x) g(x). (f –g) (x) = x – 2x – 1. This is also called pointwise multiplication. (f –g) (x).x ≠ − (fg) (x) = x (2x + 1) = 2x + x . ⎜ ⎟ 2x + 1 ⎝g⎠ 2 2 Example 17 Let f(x) = 3 2 x and g(x) = x be two functions defined over the set of non- ⎛ f ⎞ negative real numbers. (fg) (x) and ⎜ ⎟ (x). for all x ∈ X.x ≠0 and ⎜ g ⎟ ⎝ ⎠ x . ⎛ f ⎞ f ( x) . (fg) (x). ⎝g⎠ Solution We have (f + g) (x) = (fg) x = x + x. The quotient of f by g denoted by g is a function defined by . Then. (f – g) (x). Find (f + g) (x). he (iii) Multiplication by a scalar Let f : X→R be a real valued function and α be a scalar. ⎜ ⎟ ( x ) . provided g(x) ≠ 0. (f – g) (x) = x( x ) = 3 x2 x –x. 1 ⎛ f ⎞ – x ( x ) = = x 2.Find ⎛ f ⎞ (f + g) (x). for all x ∈ X.RELATIONS AND FUNCTIONS 43 (ii) Subtraction of a real function from another Let f : X → R and g: X → R be any two real functions. we define (f – g) : X → R by (f–g) (x) = f(x) –g(x). x ∈ X ⎜ ⎟ ( x) = g ( x) ⎝g⎠ 2 Example 16 Let f(x) = x and g(x) = 2x + 1 be two real functions. we mean a real number. Here by scalar. x is a real number. x ∈ R. (ii) f (x) = x2 + 2.1)} (ii) {(2. Find the domain and range of the following real functions: bl 1. x ∈ R.5).6). . (4. (10. Miscellaneous Examples Example 18 Let R be the set of real numbers.1). A function f is defined by f(x) = 2x –5. x > 0. (iii) f (–3).1).. Therefore. The function ‘t’ which maps temperature in degree Celsius into temperature in is 3. when t(C) = 212.1). (8. (17. f(–2) = 8.. he 2. (1. (6. © no N C tt E o R be T re pu degree Fahrenheit is defined by t(C) = Find (i) t(0) 5.. (8. 9 − x2 . is called linear function. f(–10) = 0 and so on. (iii) f (x) = x.3). . etc... (5. and f(–1) = 9.3 (ii) f(x) = (i) f(x) = – x 4.44 MATHEMATICS EXERCISE 2. Which of the following relations are functions? Give reasons. where m and c are constants.2).16. f(1) = 11. Define the real function f: R→R by f(x) = x + 10 and sketch the graph of this function. f(2) = 12. (ii) t(28) (iii) Find the range of each of the following functions.. (12.7)} (iii) {(1. determine its domain and range.1).4).1).3). 9C + 32. Remark The function f defined by f(x) = mx + c . (i) {(2.16 . (14.5). (2.. Fig 2. x is a real number. 5 t(–10) (iv) The value of C. Write down the values of (i) f (0).1). (i) f (x) = 2 – 3x. (ii) f (7). (14. If it is a function. Solution Here f(0) = 10. f(10) = 20. Above function is an example of a linear function.5)}. shape of the graph of the given function assumes the form as shown in Fig 2. (11. 1). 4}. Example 22 The function f is defined by ⎧1 − x. x > 0. f(x) = 1 – x. x < 0.b): a. (ii) (a.17 . © no N C tt E o R be T re pu Solution Since f is a linear function. Thus. Find f(x). x > 0 ⎩ Draw the graph of f (x). b – c ∈ Z. f (3) = 4 f(4) = 5 and so on for f(x) = x + 1. Show that (i) (a. c) ∈ R implies that a – b ∈ Z.17 Fig 2. (b. So. (0. and f(1) = 2. a) ∈ R (iii) (a.b ∈ Q and a – b ∈ Z}.3). f (x) = mx + c.RELATIONS AND FUNCTIONS 45 Example 19 Let R be a relation from Q to Q defined by R = {(a. (–1. Also. since (1. a) ∈ R (iii) (a.b) ∈ R and (b. f(– 3) =1 – (– 3) = 4. the graph of f is as shown in Fig 2. x=0 f (x) = ⎨ ⎪ x +1. (0. Hence the domain of f is R – {1. a – c = (a – b) + (b – c) ∈ Z.c) ∈R he (i) Since. Example 21 Find the domain of the function f ( x) = x2 + 3x + 5 x2 − 5x + 4 2 Solution Since x –5x + 4 = (x – 4) (x –1). –1).a) ∈ R for all a ∈ Q (ii) (a. f(– 2) = 1 – (– 2) = 3 f(–1) = 1 – (–1) = 2. the function f is defined for all real numbers except at x = 4 and x = 1. This gives m = 2 and f(x) = 2x – 1. a – a = 0 ∈ Z. (a. if follows that (a. this gives f(– 4) = 1 – (– 4) = 5. etc. f (1) = m + c = 1 and f (0) = c = –1. b) and (b. f (2) = 3.c) ∈ R implies that (a. So. 1). b – a ∈ Z. – 1) ∈ R. Solution Here. (2. –3)} be a linear function from Z into Z.b) ∈ R implies that (b. x < 0 ⎪ 1 . Therefore. Therefore.c) ∈ R is Solution bl Example 20 Let f = {(1. a) ∈ R.b) ∈ R implies that a – b ∈ Z. 0 ≤ x ≤ 3 1. The relation f is defined by f (x) = ⎨ ⎪⎩3x.b) ∈ R. Let R be a relation from N to N defined by R = {(a.1 – 1) bl 2 2. Let 2 ⎪⎩⎝ 1 + x ⎠ ⎪⎭ of f. Let f = {(1. Find f f + g. (2. B = {1.5). x 2 – 8 x + 12 is f (1. Let f. (2.–1). Find the domain and the range of the real function f defined by f (x) = x – 1 . Find the domain of the function f (x) = x2 + 2 x + 1 . 10. for some integers a. 0 ≤ x ≤ 2 The relation g is defined by g ( x) = ⎨ ⎪⎩3x . ⎧⎪⎛ ⎫⎪ x2 ⎞ = ∈ R f x . (–1.c) ∈ R implies (a. b ∈N and a = b }.9. g : R→R be defined. Justify your answer in each case.11. respectively by f(x) = x + 1.46 MATHEMATICS Miscellaneous Exercise on Chapter 2 2 ⎪⎧ x .3 ≤ x ≤10 3.a) ∈ R (iii) (a. (2. Let A ={1. 5. Are the following true? (i) (a. Determine the range 6.a) ∈ R. b) : a. g(x) = 2x – 3.1).1). Find the domain and the range of the real function f defined by f (x) = ( x − 1) . (3. b.3.5). (0. Determine a. (1.b) ∈ R.4}. for all a ∈ N (ii) (a. .2.9). f – g and g .5. © no N C tt E o R be T re pu 4.15.11)} Are the following true? (i) f is a relation from A to B (ii) f is a function from A to B. (4. : x ⎨⎜ ⎟ ⎬ be a function from R into R.16} and f = {(1. 7.c) ∈ R. b. 2 9. 8. find he ⎧⎪ x 2 . implies (b. 2 ≤ x ≤ 10 Show that f is a function and g is not a function.3). Justify your answer in each case. (b. –3)} be a function from Z to Z defined by f(x) = ax + b.1) – f (1) . If f (x) = x . The image of an element x under a relation R is given by y. If n(A) = p and n(B) = q. b): a ∈ A.11. y) ∈ R. We write f: A→B.10. Cartesian product A × B of two sets A and B is given by If (a. where f(x) = y. y). . The domain of R is the set of all first elements of the ordered pairs in a relation R. Summary is In this Chapter. z): x. b ∈ B} In particular R × R = {(x. y ∈ R} © no N C tt E o R be T re pu and R × R × R = (x.12. 12.The main features of this Chapter are as follows: A × B = {(a. A is the domain and B is the codomain of f. Is f a function from Z to Z? Justify your answer. y): x. where (x. A×φ=φ In general. Find the range of f.13} and let f : A→N be defined by f (n) = the highest prime factor of n. a + b) : a. y.RELATIONS AND FUNCTIONS 47 he 11. b ∈ Z}. then n(A × B) = pq. b) = (x. then a = x and b = y. y. we studied about relations and functions. Function A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. A × B ≠ B × A. The range of the relation R is the set of all second elements of the ordered pairs in a relation R. Let A = {9. Let f be the subset of Z × Z defined by f = {(ab. Relation A relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B. z ∈ R} bl Ordered pair A pair of elements grouped together in a particular order. 1698. Historical Note bl ⎛f⎞ f ( x) ⎜ ⎟ (x) = g ( x) . and of numbers. x ∈ X. © no N C tt E o R be T re pu 48 — — .MATHEMATICS The range of the function is the set of images. in a letter to Leibnitz. A real function has the set of real numbers or one of its subsets both as its domain and as its range. g(x) ≠ 0 ⎝ g⎠ is (f. He considered a function in terms of “mathematical job” – the “employee” being just a curve. Leibnitz used the word in the non-analytical sense. x ∈ X. x ∈ X = f (x) . Johan Bernoulli. where k is a real number. or constant quantities”. Algebra of functions For functions f : X → R and g : X → R.g) (x) The word FUNCTION first appears in a Latin manuscript “Methodus tangentium inversa. x ∈ X (f – g) (x) = f (x) – g(x). On July 5. for the first time deliberately assigned a specialised use of the term function in the analytical sense. for an analytical expression any way compounded of a variable quantity. seu de fuctionibus” written by Gottfried Wilhelm Leibnitz (1646-1716) in 1673.g (x). Leibnitz replied showing his approval. x ∈ X (kf) (x) = k f (x) ). Function is found in English in 1779 in Chambers’ Cyclopaedia: “The term function is used in algebra. At the end of that month. we have he (f + g) (x) = f (x) + g(x). – MILNE 3. designing electric circuits.1 . The original ray is Vertex Fig 3. In this Chapter. by engineers and others. It was studied by sea captains for navigation. we will generalise the concept of trigonometric ratios to trigonometric functions and study their properties. predicting the heights of tides in the ocean. trigonometry is used in many areas such as the science of seismology. The subject was originally developed to solve geometric problems involving triangles. describing the state of an atom. surveyor to map out the new lands. 3.Chapter 3 TRIGONOMETRIC FUNCTIONS A mathematician knows how to solve a problem. we have studied the trigonometric Arya Bhatt ratios of acute angles as the ratio of the sides of a right (476-550) angled triangle. In earlier classes.2 Angles Angle is a measure of rotation of a given ray about its initial point.1 Introduction The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. Currently. he can not solve it. We have also studied the trigonometric identities and application of trigonometric ratios in solving the problems related to heights and distances. analysing a musical tone and in many other areas. If the direction of rotation is anticlockwise. One sixtieth of a degree is called a minute. 1′ = 60″ Some of the angles whose measures are 360°. th 1 3. Fig 3.2 measuring angles. – 30°. Thus. 1° = 60′. The definition of an angle suggests a unit. then the angle is negative (Fig 3. viz. 420°. The measure of an angle is the amount of rotation performed to get the terminal side from the initial side.2. This is often convenient for large angles. The point of rotation is called the vertex. For example. written as 1′.180°. and a minute is divided into 60 seconds . – 420° are shown in Fig 3. viz. the angle is said to be positive and if the direction of rotation is clockwise.2. and one sixtieth of a minute is called a second. 270°. written as 1°.3 . There are several units for Fig 3.1 Degree measure If a rotation from the initial side to terminal side is of 360 a revolution. degree measure and radian measure.1). we can say that a rapidly spinning wheel is making an angle of say 15 revolution per second. We shall describe two other units of measurement of an angle which are most commonly used. A degree is divided into 60 minutes.50 MATHEMATICS called the initial side and the final position of the ray after rotation is called the terminal side of the angle. the angle is said to have a measure of one degree.3. written as 1″. one complete revolution from the position of the initial side as indicated in Fig 3. an arc of length r will subtend an angle of 1 radian. in a circle of radius r. Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.2. Since in a circle of radius r.2 Radian measure There is another unit for measurement of an angle. –1 radian. OA is the initial side and OB is the terminal side. It is well-known that equal arcs of a circle subtend equal angle at the centre. one complete revolution of the initial side subtends an angle of 2π radian. r . an arc of length l will subtend an angle whose measure is radian.4 (i) to (iv) We know that the circumference of a circle of radius 1 unit is 2π. 2 2 (ii) (iii) (iv) Fig 3. 1 (i) 1 1 radian and –1 radian. an arc of length l subtends an angle θ radian at the centre. we have θ = l or l = r θ.4(i) to (iv). Thus. if in r a circle of radius r. called the radian measure. More generally. an arc of length r subtends an angle whose measure is 1 l radian. The figures show the angles whose measures are 1 radian. Thus.TRIGONOMETRIC FUNCTIONS 51 3. In the Fig 3. 1 O 1 A 0 − −1 −2 − Q Fig 3.3 Relation between radian and real numbers Consider the unit circle with centre O. π π radian = 0.2. radian measures and real numbers can be considered as one and the same.5). it follows that 2π radian = 360° or π radian = 180° The above relation enables us to express a radian measure in terms of degree measure and a degree measure in terms of radian measure. Thus.52 MATHEMATICS P 2 3. Let the point A represent the real number zero.4 Relation between degree and radian Since a circle subtends at the centre an angle whose radian measure is 2π and its degree measure is 360°. Using approximate value of π as 22 . Consider the line PAQ which is tangent to the circle at A. Let A be any point on the circle. If we rope the line AP in the anticlockwise direction along the circle. Consider OA as initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the centre of the circle. then every real number will correspond to a radian measure and conversely. we have 7 1 radian = Also 1° = 180 = 57° 16′ approximately.2.01746 radian approximately.5 3. AP represents positive real number and AQ represents negative real numbers (Fig 3. and AQ in the clockwise direction. 180 The relation between degree measures and radian measure of some common angles are given in the following table: Degree 30° 45° 60° 90° 180° 270° 360° Radian π 6 π 4 π 3 π 2 π 3π 2 2π . we can say that omitted.4 cm (use π 22 ). Solution We know that 180° = π radian. Hence 6 radians = 1080 7 180 degree 6 degree = π 22 = 343 7 degree 11 = 343° + 38′ + Hence = 343° + 2 minute 11 7 60 minute [as 1° = 60′] 11 [as 1′ = 60″] = 343° + 38′ + 10. Hence Therefore 40° 20′ = 40 121 1 π 121π degree = radian = radian. 6 radians = 343° 38′ 11″ approximately. Thus. π 180 and Radian measure = π Degree measure 180 Degree measure = 180 Radian measure π Example 1 Convert 40° 20′ into radian measure. the word ‘radian’ is frequently π π 45 are written with the understanding that π and 4 4 are radian measures. 540 Example 2 Convert 6 radians into degree measure. we mean the angle whose radian measure is β. Example 3 Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.TRIGONOMETRIC FUNCTIONS 53 Notational Convention Since angles are measured either in degrees or in radians. Note that when an angle is expressed in radians.9″ = 343°38′ 11″ approximately. Solution We know that π radian = 180°. we mean the angle whose degree measure is θ and whenever we write angle β. Thus. 7 . we adopt the convention that whenever we write angle θ°. 3 180 540 3 40° 20′ = 121π radian. which gives r1 22 13π 22π r1 = r2 . Find the radian measures corresponding to the following degree measures: (i) 25° (ii) – 47°30′ (iii) 240° (iv) 520° . Therefore. in 40 minutes.14)..e. Hence. Therefore. Then l = r1θ1 = r2θ2. r = 36 36 13 2 Hence r1 : r2 = 22 : 13. the minute hand of a watch completes one revolution. Given that θ1 = 65° = π 13π 65 = radian 36 180 θ2 = 110° = and π 22π 110 = radian 36 180 Let l be the length of each of the arc.14 cm = 6.4×3×7 = = 35.5 cm long. EXERCISE 3. we have θ 37.1 1.5 4π cm = 2π cm = 2 3.7 cm π 22 Example 4 The minute hand of a watch is 1. θ = × 360° 3 3 4π radian. How far does its tip move in 40 minutes? (Use π = 3.28 cm. Solution Let r1 and r2 be the radii of the two circles. by r = r= 60π π radian = 180 3 l .4 cm and θ = 60° = Hence. the required distance travelled is given by 3 l = r θ = 1.54 MATHEMATICS Solution Here l = 37. 3 Example 5 If the arcs of the same lengths in two circles subtend angles 65°and 110° at the centre. find the ratio of their radii. Solution In 60 minutes. i.4×3 37. the minute hand turns through or 2 2 of a revolution. e. 7 5π 7π (iv) 3 6 A wheel makes 360 revolutions in one minute. the length of a chord is 20 cm.6). length of arc AP = x (Fig 3. 3. We define cos x = a and sin x = b Since ΔOMP is a right triangle. 7. Find the length of minor arc of the chord. Consider a unit circle with centre at origin of the coordinate axes. 6. b) be any point on the circle with angle AOP = x radian. for every point on the unit circle. 11 16 22 ). arcs of the same length subtend angles 60° and 75° at the centre. Find the degree measures corresponding to the following radian measures (Use π 4.TRIGONOMETRIC FUNCTIONS 2. we have a2 + b2 = 1 or cos2 x + sin2 x = 1 Since one complete revolution subtends an angle of 2π radian at the centre of the circle. find the ratio of their radii. we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle.. 55 (ii) – 4 (iii) 22 ). we have OM2 + MP2 = OP2 or a2 + b2 = 1 Thus.3 Trigonometric Functions In earlier classes. ∠AOB = π . 2 Fig 3. Let P (a.6 . Find the angle in radian through which a pendulum swings if its length is 75 cm and th e tip describes an arc of length (i) 10 cm (ii) 15 cm (iii) 21 cm radius 100 cm by an arc of length 22 cm (Use π 5. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions. i. 7 In a circle of diameter 40 cm. Through how many radians does it turn in one second? Find the degree measure of the angle subtended at the centre of a circle of (i) 3. If in two circles. –1). the values of sine and cosine functions do not change.. 2 cos x cos x . C and D are. π. we again come back to same point P. cos x vanishes when x is an odd 2 2 2 π . where n is any integer. sin x 1 π .. 0). (1. B. Thus. (–1. if x = 0.e. ± 2π . = 2 cos x cosec x = sec x tan x cot x sin x π . ± 3π. x ≠ n π.± . . n ∈ Z Further.± . x ≠ (2n +1) . where n is any integer. The coordinates of the points A. where n is any integer.56 MATHEMATICS 3π π . = sin x = . π =0 2 cosπ = − 1 cos π =1 2 sinπ = 0 sin 3π 3π =0 sin = –1 2 2 cos 2π = 1 sin 2π = 0 Now. ± π. when x is an integral multiple of π cos and cos x = 0. n ∈ Z . x ≠ (2n + 1) . we also observe that if x increases (or decreases) by any integral multiple of 2π. respectively. i.. .. 1).. if x = ± multiple of π 3π 5π .. we have ∠AOC = π and ∠AOD = cos 0° = 1 sin 0° = 0. x ≠ nπ. if we take one complete revolution from the point P. for quadrantal angles. where n is any integer. i. All angles which are integral multiples of are called 2 2 quadrantal angles. where n is any integer π cos x = 0 implies x = (2n + 1) . cos (2nπ + x) = cos x . Thus 2 sin x = 0 implies x = nπ. sin (2nπ + x) = sin x . where n is any integer. 0) and (0.. (0. Therefore. sin x = 0. Thus. 2 We now define other trigonometric functions in terms of sine and cosine functions: 1 .e. cos x and tan x. b) be a point on the unit circle with centre at the origin such that ∠AOP = x.3. we have the following table: 0° π 6 π 4 π 3 π 2 π 3π 2 2π sin 0 1 2 1 2 3 2 1 0 –1 0 cos 1 3 2 1 2 1 2 0 –1 0 1 tan 0 1 3 1 3 not defined 0 not defined 0 The values of cosec x. 3.1 Sign of trigonometric functions Let P (a. sec x and cot x are the reciprocal of the values of sin x. we have discussed the values of trigonometric ratios for 0°. respectively. –b) (Fig 3. – 1 ≤ a ≤ 1 and Fig 3. 60° and 90°.TRIGONOMETRIC FUNCTIONS 57 We have shown that for all real x. 45°. sin2 x + cos2 x = 1 It follows that 1 + tan2 x = sec2 x (why?) 1 + cot2 x = cosec2 x (why?) In earlier classes. then the coordinates of the point Q will be (a. Thus.7). b) on the unit circle. If ∠AOQ = – x. The values of trigonometric functions for these angles are same as that of trigonometric ratios studied in earlier classes.7 . Therefore cos (– x) = cos x and sin (– x) = – sin x Since for every point P (a. 30°. in the 2 π < x <π) a is negative and b is positive. we have the following table. 1]. Therefore.2 Domain and range of trigonometric functions From the definition of sine and cosine functions. and negative for (π < x < π < x < 2π. – 1 ≤ y ≤ 1. . Further. We have learnt in previous classes that in the first quadrant (0 < x < second quadrant ( π ) a and b are both positive. Likewise. we have – 1 ≤ cos x ≤ 1 and –1 ≤ sin x ≤ 1 for all x. we observe that they are defined for all real numbers.58 MATHEMATICS – 1 ≤ b ≤ 1. we observe that for each real number x. sin x is positive for 0 < x < π. domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1.3.e. in the third quadrant 2 3π 3π ) a and b are both negative and in the fourth quadrant ( < x < 2π) a is 2 2 positive and b is negative. negative for < x < and also 2 2 2 3π < x < 2π. Similarly. In fact. I II III IV sin x + + – – cos x + – – + tan x + – + – cosec x + + – – sec x + – – + cot x + – + – 3. cos x is positive for 0 < x < positive for π π 3π . i. we can find the signs of other trigonometric 2 functions in different quadrants.. – 1 ≤ sin x ≤ 1 and – 1 ≤ cos x ≤ 1 Thus. TRIGONOMETRIC FUNCTIONS 59 1 Since cosec x = sin x . In the 2 3π . we have the following table: third quadrant. as x increases from 0 to increases from 0 to 1. the domain π . x ≠ (2n + 1) We further observe that in the first quadrant. n ∈ Z} and range is the set 2 {y : y ∈ R. the statement tan x increases from 0 to ∞ (infinity) for 0<x< π π simply means that tan x increases as x increases for 0 < x < and 2 2 . Similarly. as x increases from π . sin x decreases from 1 to 0. n ∈ Z} and range is the set {y : y ∈ R. y ≥ 1 or y ≤ – 1}. In fact. as x increases from π to I quadrant II quadrant sin increases from 0 to 1 decreases from 1 to 0 cos decreases from 1 to 0 decreases from 0 to – 1 increases from –1 to 0 increases from 0 to 1 tan increases from 0 to ∞ increases from –∞to 0 increases from 0 to ∞ cot decreases from ∞ to 0 decreases from 0 to–∞ decreases from ∞ to 0 decreases from 0to –∞ sec increases from 1 to ∞ increases from –∞to–1 decreases from –1to–∞ decreases from ∞ to 1 cosec decreases from ∞ to 1 increases from 1 to ∞ III quadrant IV quadrant decreases from 0 to –1 increases from –1 to 0 increases from –∞to 0 increases from –∞to–1 decreases from–1to–∞ Remark In the above table. sin x increases from –1 to 0 as x increases from 2 Similarly. sin x decreases from 0 to –1and finally. we can discuss the behaviour of other trigonometric functions. The domain of y = tan x is the set {x : x ∈ R and of y = sec x is the set {x : x ∈ R and x ≠ (2n + 1) π . n ∈ Z} and the range is the set of all real numbers. in 2 3π to 2π. sin x 2 π to π. the fourth quadrant. n ∈ Z} and range is the set of all real numbers. The domain of 2 y = cot x is the set {x : x ∈ R and x ≠ n π. the domain of y = cosec x is the set { x : x ∈ R and x ≠ n π. y ≤ – 1or y ≥ 1}. Similarly. 2π) and assumes arbitrarily large negative values as 2 x approaches to 2π. to say that 2 cosec x decreases from –1 to – ∞ (minus infinity) in the fourth quadrant means that assumes arbitraily large positive values as x approaches to 3π .8 Fig 3.9 Fig 3. We have already seen that values of sin x and cos x repeats after an interval of 2π. Hence. The symbols ∞ and – ∞ simply specify certain types of behaviour cosec x decreases for x ∈ ( of functions and variables.60 MATHEMATICS π .10 Fig 3. We Fig 3.11 . values of cosec x and sec x will also repeat after an interval of 2π. e. sin2 x = 1 – cos2 x or sin2 x = 1 – Solution Since cos x = 16 9 = 25 25 4 5 Since x lies in third quadrant. i. sin x is negative. Hence. find the values of other five 5 trigonometric functions. Using this knowledge and behaviour of trigonometic functions. values of tan x will repeat after an interval of π. Since cot x is reciprocal of tan x. The graph of these functions are given above: Example 6 If cos x = – 3 . we have sec x = 5 3 sin2 x + cos2 x = 1. Therefore Hence sin x = ± sin x = – 4 5 which also gives cosec x = – 5 4 .. we can sketch the graph of these functions.13 shall see in the next section that tan (π + x) = tan x.12 61 Fig 3. x lies in the third quadrant. its values will also repeat after an interval of π.TRIGONOMETRIC FUNCTIONS Fig 3. Now 3 5 . 3 Solution We know that values of sin x repeats after an interval of 2π. find the values of other five 12 trigonometric functions. we have tan x = sin x 4 cos x 3 = and cot x = = . 3 3 3 2 . cos x 3 sin x 4 Example 7 If cot x = – 5 . Solution Since cot x = – 12 5 . sin x 12 Example 8 Find the value of sin 31π . we have tan x = – 5 12 Now sec2 x = 1 + tan2 x = 1 + Hence sec x = ± 144 169 = 25 25 13 5 Since x lies in second quadrant. sec x will be negative.62 MATHEMATICS Further. Therefore sec x = – 13 . 5 which also gives cos x 5 13 Further. x lies in second quadrant. we have sin x = tan x cos x = (– and cosec x = 12 5 12 ) (– )= 5 13 13 1 13 = . Therefore sin 31π π π 3 = sin (10π + ) = sin = . Therefore. x lies in fourth quadrant. cos (–1710°) = cos (–1710° + 5 360°) = cos (–1710° + 1800°) = cos 90° = 0. The basic results in this connection are called trigonometric identities. 2 2. 4 13 . EXERCISE 3. We have seen that 1. tan 10.2 Find the values of other five trigonometric functions in Exercises 1 to 5. x lies in third quadrant. sin (– x) = – sin x 2. 5 3. 1. 12 4. Solution We know that values of cos x repeats after an interval of 2π or 360°. x lies in second quadrant.4 Trigonometric Functions of Sum and Difference of Two Angles In this Section. we shall derive expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions.TRIGONOMETRIC FUNCTIONS 63 Example 9 Find the value of cos (–1710°). tan x = – . cos (– x) = cos x We shall now prove some more results: . sec x = Find the values of the trigonometric functions in Exercises 6 to 10. x lies in third quadrant. cot (– 11π ) 3 15π ) 4 3. cot x = 3 . x lies in second quadrant. cosec (– 1410°) 19π 3 9. 5 5 5. cos x = – 1 . sin (– 8. 6. sin 765° 7. sin x = 3 . 14 Consider the triangles P1OP3 and P2OP4.14). MATHEMATICS cos (x + y) = cos x cos y – sin x sin y Consider the unit circle with centre at the origin. Let x be the angle P4OP1and y be the angle P1OP2. Therefore. we get P 1P 32 = [cos x – cos (– y)]2 + [sin x – sin(–y]2 = (cos x – cos y)2 + (sin x + sin y)2 = cos2 x + cos2 y – 2 cos x cos y + sin2 x + sin2 y + 2sin x sin y = 2 – 2 (cos x cos y – sin x sin y) Also. They are congruent (Why?). sin (x + y)]. sin x). P3 [cos (– y). P1P3 and P2P4 are equal. P 3 and P 4 will have the coordinates P 1 (cos x. 0) (Fig 3. P2 [cos (x + y). sin (– y)] and P4 (1. P 1 . Also let (– y) be the angle P4OP3. By using distance formula. Fig 3. P 2P 42 (Why?) = [1 – cos (x + y)] 2 + [0 – sin (x + y)]2 = 1 – 2cos (x + y) + cos2 (x + y) + sin2 (x + y) = 2 – 2 cos (x + y) . Therefore. Then (x + y) is the angle P4OP2.64 3. P 2 . we get the following results: = cos ( π cos ( + x ) = – sin x 2 π – x) = – cos x cos (π π sin ( + x ) = cos x 2 π – x) = sin x sin (π . 4. in the Identity 7. Hence cos (x + y) = cos x cos y – sin x sin y 4 . cos (x – y) = cos x cos y + sin x sin y Replacing y by – y in identity 3.TRIGONOMETRIC FUNCTIONS 65 Since P 1 P 3 = P2P4. we have P1P32 = P2P42. sin (x – y) = sin x cos y – cos x sin y If we replace y by –y. we get cos (x + (– y)) = cos x cos (– y) – sin x sin (– y) or cos (x – y) = cos x cos y + sin x sin y 5. sin (x + y) = sin x cos y + cos x sin y We know that sin ( π π sin (x + y) = cos (x y ) = cos ( x) y 2 2 π π x ) cos y + sin ( x) sin y 2 2 = sin x cos y + cos x sin y 8. By taking suitable values of x and y in the identities 3. 2 2 2 π – x ) = cos x 2 Using the Identity 5. we have 6. we get the result. 2 –2 (cos x cos y – sin x sin y) = 2 – 2 cos (x + y). sin ( π π π x ) = cos x = cos x. we get 2 π π π x ) = cos cos x + sin sin x = sin x. cos ( π – x ) = sin x 2 If we replace x by cos ( π and y by x in Identity (4). 2 2 2 7. 9. Therefore. 7 and 8. If none of the angles x. y and (x + y) is an odd multiple of π . then 2 tan x + tan y 1 – tan x tan y Since none of the x. cos( x y ) cos x cos y sin x sin y Dividing numerator and denominator by cos x cos y. If none of the angles x. y and (x + y) is an odd multiple of tan (x + y) = π . cot x. then cot x cot y – 1 cot ( x + y) = cot y +cot x . tan ( x – y) = tan x – tan y 1 + tan x tan y If we replace y by – y in Identity 10. Now tan (x + y) = sin( x y ) sin x cos y cos x sin y = . 10. we have sin x cos y cos x sin y cos x cos y cos x cos y tan (x + y) = cos x cos y sin x sin y cos x cos y cos x cos y tan x tan y = 1 – tan x tan y 11. it follows that cos x. y and (x + y) is a multiple of π. sec x and cosec x can be obtained from the results of sin x and cos x. we get tan (x – y) = tan [x + (– y)] = tan x tan ( y ) tan x tan y = 1 tan x tan ( y ) 1 tan x tan y 12.66 MATHEMATICS π + x) = – cos x cos (π π – x) = cos x cos (2π π + x) = – sin x sin (π π – x) = – sin x sin (2π Similar results for tan x. 2 cos y and cos (x + y) are non-zero. cot ( x + y) = cos ( x y ) cos x cos y – sin x sin y sin ( x y ) sin x cos y cos x sin y Dividing numerator and denominator by sin x sin y. where n is an integer 2 2 1+ tan x 15. We have cos 2 x sin 2 x cos 2x = cos x – sin x = cos 2 x sin 2 x 2 2 Dividing numerator and denominator by cos2 x. we find that sin x sin y and sin (x + y) are non-zero. cos 2x = cos2 x – sin2 x = 1 – sin2 x – sin2 x = 1 – 2 sin2 x. cot (x – y) = cot x cot y – 1 cot y cot x cot x cot y + 1 if none of angles x. x n π . y and x–y is a multiple of π cot y – cot x If we replace y by –y in identity 12. we get cos 2x = 1 – tan 2 x π . Now. we have cot (x + y) = 13. sin 2x = 2 sinx cos x = 2tan x π x n π . Again sin 2x = 2sin x cos x cos 2 x sin 2 x .TRIGONOMETRIC FUNCTIONS 67 Since. we get cos 2x = cos2x – sin2 x = cos2 x – (1 – cos2 x) = 2 cos2x – 1 Again. none of the x. where n is an integer 2 1 + tan x 2 We have sin (x + y) = sin x cos y + cos x sin y Replacing y by x. cos 2x = cos x – sin x = 2 cos x – 1 = 1 – 2 sin x = 1 + tan 2 x 2 2 2 2 We know that cos (x + y) = cos x cos y – sin x sin y Replacing y by x. we get sin 2x = 2 sin x cos x. we get the result 1 – tan 2 x 14. y and (x + y) is multiple of π. we get tan 2 x 2 tan x 1 tan 2 x 17. tan 3 x 2tan x tan x 1 – tan 2 x tan 2 x tan x = 1 – tan 2 x tan x 1 – 2tan x . sin 3x = 3 sin x – 4 sin3 x We have. cos 3x = 4 cos3 x – 3 cos x We have. 3 tan x – tan 3 x π if 3x n π . sin 3x = sin (2x + x) = sin 2x cos x + cos 2x sin x = 2 sin x cos x cos x + (1 – 2sin2 x) sin x = 2 sin x (1 – sin2 x) + sin x – 2 sin3 x = 2 sin x – 2 sin3 x + sin x – 2 sin3 x = 3 sin x – 4 sin3 x 18. where n is an integer 2 1 – 3 tan x 2 We have tan 3x =tan (2x + x) 19. tan 2x = 2tan x 1tan 2 x 2tan x π if 2 x n π . where n is an integer 2 1 – tan x 2 We know that tan x tan y tan (x + y) = 1 – tan x tan y Replacing y by x .68 MATHEMATICS Dividing each term by cos2 x. we get sin 2x = 16. tan x 1 – tan 2 x . cos 3x = cos (2x +x) = cos 2x cos x – sin 2x sin x = (2cos2 x – 1) cos x – 2sin x cos x sin x = (2cos2 x – 1) cos x – 2cos x (1 – cos2 x) = 2cos3 x – cos x – 2cos x + 2 cos3 x = 4cos3 x – 3cos x. ... Therefore θ θ x and y 2 2 Substituting the values of x and y in (3)... (2) .TRIGONOMETRIC FUNCTIONS 20. we get sin (x + y) + sin (x – y) = 2 sin x cos y sin (x + y) – sin (x – y) = 2cos x sin y Let x + y = θ and x – y = φ... we get cos (x + y) + cos(x – y) = 2 cos x cos y and cos (x + y) – cos (x – y) = – 2 sin x sin y Further sin (x + y) = sin x cos y + cos x sin y and sin (x – y) = sin x cos y – cos x sin y Adding and subtracting (5) and (6). .. (7) and (8). 2tan x tan x – tan 3 x 3 tan x – tan 3 x 1 – tan 2 x – 2tan 2 x 1 – 3tan 2 x (i) cos x + cos y = 2cos x+ y x– y cos 2 2 (ii) cos x – cos y = – 2sin x+ y x– y sin 2 2 (iii) sin x + sin y = 2sin x+ y x– y cos 2 2 (iv) sin x – sin y = 2cos x+ y x– y sin 2 2 We know that cos (x + y) = cos x cos y – sin x sin y and cos (x – y) = cos x cos y + sin x sin y Adding and subtracting (1) and (2). we get θ θ cos θ + cos φ = 2 cos cos 2 2 θ θ – cos θ – cos φ = – 2 sin sin 2 θ θ sin θ + sin φ = 2 sin cos 2 2 . (3) (4) (5) (6) ... .. . (1) . (4)..... (8) 69 .. (7) . we can prove the following results: 2 cos x cos y = cos (x + y) + cos (x – y) –2 sin x sin y = cos (x + y) – cos (x – y) 2 sin x cos y = sin (x + y) + sin (x – y) 2 cos x sin y = sin (x + y) – sin (x – y).H. sin x – sin y = 2 cos . 2 Example 11 Find the value of sin 15°.70 MATHEMATICS θ θ sin θ – sin φ = 2 cos sin 2 2 Since θ and φ can take any real values. =3–4× Solution We have sin 15° = sin (45° – 30°) = sin 45° cos 30° – cos 45° sin 30° = 1 3 1 1 3 –1 . 2 2 2 2 As a part of identities given in 20. = 3sin sec 4sin cot 6 3 6 4 =3× 1 × 2 – 4 sin × 1 = 3 – 4 sin 6 2 6 1 = 1 = R.S.S. Example 10 Prove that 5 3sin sec 4sin cot 1 6 3 6 4 Solution We have 5 L. Thus. cos x – cos y = – 2 sin . (i) (ii) (iii) (iv) x y x y x y x y cos sin . 2 2 2 2 2 2 Example 12 Find the value of tan 13 .H. we can replace θ by x and φ by y. 12 . 2 2 2 2 x y x y x y x y cos sin . we get cos x + cos y = 2 cos sin x + sin y = 2 sin Remark 21. Solution We have L. Example 15 Prove that cos x cos x 2 cos x 4 4 Solution Using the Identity 20(i).S. Example 14 Show that tan 3 x tan 2 x tan x = tan 3x – tan 2 x – tan x Solution We know that 3x = 2x + x Therefore.H.TRIGONOMETRIC FUNCTIONS Solution We have tan 13 = tan 12 tan = tan 12 12 4 6 1 tan tan 4 6 = = 1 1 tan tan 4 6 1 3 3 1 2 3 1 3 1 3 Example 13 Prove that sin ( x y ) tan x tan y sin ( x y ) tan x tan y . we have 71 . tan 3x = tan (2x + x) or tan 3x tan 2 x tan x 1– tan 2 x tan x or tan 3x – tan 3x tan 2x tan x = tan 2x + tan x or tan 3x – tan 2x – tan x = tan 3x tan 2x tan x or tan 3x tan 2x tan x = tan 3x – tan 2x – tan x. we get sin ( x y ) tan x tan y sin ( x y ) tan x tan y . sin (x y ) sin x cos y cos x sin y sin (x y ) sin x cos y cos x sin y Dividing the numerator and denominator by cos x cos y. H.H. = = 7 x 5x 7 x 5x sin x 2cos sin 2 2 2cos L.72 MATHEMATICS L.S.S. sin 2 x 2sin x cos x . cos x cos x 4 4 4 x 4 x 4 x – ( 4 x) 2cos cos 2 2 = 2 cos 1 cos x = 2 × cos x = 2 4 2 cos x = R.S. cos 7 x cos 5 x cot x sin 7 x – sin 5 x Example 16 Prove that Solution Using the Identities 20 (i) and 20 (iv).H. we get 7 x 5x 7 x 5x cos cos x 2 2 cot x = R. sin 5 x sin x 2sin 3 x sin 5 x 2sin 3x sin x cos5 x cos x cos5 x cos x 2sin 3 x cos 2 x 2sin 3 x sin 3 x (cos 2 x 1) – – 2sin 3x sin 2x sin 3x sin 2x 1 cos 2 x 2sin 2 x = tan x = R.H.S.H. Example 17 Prove that sin 5 x 2sin 3x sin x tan x cos5 x cos x Solution We have L.S.S.H. 2sin 6 6 6 4 4 3 5. cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x) 16. cos x cos (2π x) cot x cot (2π x) 1 2 2 10. cos2 2x – cos2 6x = sin 4x sin 8x 14. 20. Find the value of: (i) sin 75° (ii) tan 15° 2 3. 18. sin 5x sin 3x cos 5x cos 3x sin x sin 3x cos x cos 3x tan 4 x tan 2 x cos 4 x cos 3x cos 2 x sin 4 x sin 3x sin 2 x cot 3x . sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x 11. cos ( x) cos ( x) cot 2 x sin ( x) cos x 2 3π 3π 9. π tan x 2 4 1 tan x π 1 tan x tan x 4 8. cos x cos y sin x sin y sin ( x y ) 4 4 4 4 7. cos 9 x cos 5x sin 17 x sin 3x sin x sin y cos x cos y tan sin x sin 3x 2 2 sin x cos x sin 2 x cos10 x x y 2 2 sin x 17. 21. sin2 x + 2 sin 4x + sin 6x = 4 cos2 x sin 4x 15. 2sin2 7 3 cos 2 + cosec2 6 6 3 2 5 2 3 cosec 3tan 2 6 2cos 2 2sec2 10 4.TRIGONOMETRIC FUNCTIONS 73 EXERCISE 3. cot Prove the following: 6. sin2 6x – sin2 4x = sin 2x sin 10x 13. sin2 π 1 + cos2 – tan2 – 3 6 4 2 2. 3 3 cos x cos x 2 sin x 4 4 12.3 Prove that: 1. 19. In this Section. The solutions of a trigonometric equation for which 0 ≤ x < 2π are called principal solutions. Thus. cos 6x = 32 cos6 x – 48cos4 x + 18 cos2 x – 1 3. 3 3 3 2 3 2 Therefore. We shall use ‘Z’ to denote the set of integers. sin 3 2 2π π π 3 π 3 sin π sin and sin . tan π – = – tan = – 6 6 6 3 3 and π π 1 tan 2π tan 6 6 3 Thus tan 5π 11π 1 tan . 3 3 Example 19 Find the principal solutions of the equation tan x Solution We know that. The expression involving integer ‘n’ which gives all solutions of a trigonometric equation is called the general solution. principal solutions are x π 2π and .5 Trigonometric Equations Equations involving trigonometric functions of a variable are called trigonometric equations. principal solutions are . We have already learnt that the values of sin x and cos x repeat after an interval of 2π and the values of tan x repeat after an interval of π. 6 6 We will now find the general solutions of trigonometric equations. cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1 23. We have already Therefore. tan 4 x 4tan x (1 tan 2 x) 1 6 tan 2 x tan 4 x 24.74 MATHEMATICS 22. tan . The following examples will be helpful in solving trigonometric equations: Example 18 Find the principal solutions of the equation sin x Solution We know that. cos 4x = 1 – 8sin2 x cos2 x 25. π π 1 π 1 . 6 6 3 5π 11π and . 1 3 . we shall find the solutions of such equations. where n ∈ Z π . then cos x – cos y = 0 i. where n ∈ Z Proof If cos x = cos y. Combining these two results. = nπ. 2 We shall now prove the following results: cos x =0 gives x = (2n + 1) Theorem 1 For any real numbers x and y. sin x = sin y implies x = nπ + (–1)n y..e.TRIGONOMETRIC FUNCTIONS 75 seen that: sin x =0 gives x = nπ. Therefore = (2n + 1) or Theorem 2 For any real numbers x and y. cos x = cos y. Thus sin x y 2 x y =0 or –2 sin sin x y 2 x y 2 sin x y 2 =0 x y i.e. then 2 . where n ∈ Z Proof If sin x = sin y. where n ∈ Z Hence x = 2nπ ± y. where n ∈ Z. implies x = 2nπ ± y. where n ∈ Z π . where n∈Z Hence x = (2n + 1)π + (–1)2n + 1 y or x = 2nπ +(–1)2n y. where n ∈ Z. where n ∈ Z Therefore 2 = nπ or =0 Theorem 3 Prove that if x and y are not odd mulitple of tan x = tan y implies x = nπ + y. where n ∈ Z 2 x = 2nπ – y or x = 2nπ + y. we get x = nπ + (–1)n y. then sin x – sin y = 0 or 2cos which gives cos x y 2 x y = 0 or sin x y 2 x y 2 sin x y 2 =0 =0 x y = nπ.e. x = (2n + 1) π – y or x = 2nπ + y. where n ∈ Z. where n ∈ Z 2 2 2 i. then tan x – tan y = 0 or sin x cos y cos x sin y cos x cos y 0 which gives sin (x – y) = 0 (Why?) Therefore x – y = nπ. 1 π cos 2 3 π . 3 π Example 22 Solve tan 2 x cot x . 3 4π 3 is one such value of x for which sin x . i. where n ∈ Z Example 20 Find the solution of sin x = – Solution We have sin x = – Hence sin x = sin 3 2 2 . π π 4π sin π sin 3 3 3 4π . One may take any 3 2 other value of x for which sin x 3 . 3 π π Solution We have. cos x Therefore x 2nπ 1 2 . where n ∈ Z. which gives 3 x nπ ( 1) n Note = sin 3 4π . tan 2 x cot x π = tan x 2 3 3 .e. where n ∈ Z. The solutions obtained will be the same 2 although these may apparently look different. x = nπ + y.76 MATHEMATICS Proof If tan x = tan y. Example 21 Solve cos x Solution We have.. 6 Example 23 Solve sin 2x – sin 4x + sin 6x = 0. sin4 x 0 or cos 2 x cos Hence 4 x nπ or 2 x 2nπ i. 2 6 77 .e. where n∈Z 6 5π . x nπ or Solution The equation can be written as sin 6 x sin 2 x sin 4 x 0 or 2 sin 4 x cos 2 x sin 4 x 0 i.e.TRIGONOMETRIC FUNCTIONS or 5π tan2 x tan x 6 Therefore 2 x nπ x 5π . sin 4 x ( 2 cos 2 x 1) 0 1 2 or cos 2 x Therefore sin 4x = 0 i. 4 6 Example 24 Solve 2 cos2 x + 3 sin x = 0 Solution The equation can be written as 2 1 sin 2 x 3 sin x 0 or 2 sin 2 x 3 sin x 2 0 or (2sinx 1) (sinx 2) 0 Hence sin x = But 1 or sin x = 2 2 sin x = 2 is not possible (Why?) Therefore sin x = 1 7π = sin . where n∈Z 3 π nπ or x nπ . where n∈Z. x π 3 π .e. where n∈Z. . cos y = 5 find the value of sin (x + y). cos 3x + cos x – cos 2x = 0 7. sin 2x + cos x = 0 8. where n ∈ Z. we get 5 13 . where x and y both lie in second quadrant. cosec x = – 2 Find the general solution for each of the following equations: 5. cos x is negative. sin y = the values of sin x.4 Find the principal and general solutions of the following equations: 1. Therefore 4 Hence cos x = Now sin2y = 1 – cos2y = 1 – i. Solution We know that sin (x + y) = sin x cos y + cos x sin y Now cos2 x = 1 – sin2 x = 1 – 9 25 = . hence sin y is positive. cos 4 x = cos 2 x 6. tan x 2. Therefore..78 MATHEMATICS Hence. sec2 2x = 1– tan 2x 9. (1) 16 25 4 cos x = . 6 EXERCISE 3. sin x + sin 3x + sin 5x = 0 Miscellaneous Examples Example 25 If sin x = 3 . cot x 3 4. sec x = 2 3 3. the solution is given by x nπ ( 1)n 7π . Since y lies in second quadrant. sin y. 5 Since x lies in second quadrant. 12 13 . sin y = 5 5 13 144 169 25 169 . Substituting .e. cos x and cos y in (1). 4 8 2 tan x Now tan 2 x or π 2tan π 8 tan 4 1 tan 2 π 8 Let y = tan 2y π . 8 79 . = x 1 9x 2cos 2 x cos 2cos cos 3 x 2 2 2 = 1 x x 9x 9x cos 2 x cos 2 x cos 3x cos 3 x 2 2 2 2 2 = 1 5x 3x 15x 3x 1 5x 15x cos cos cos cos = cos cos 2 2 2 2 2 2 2 2 5 x 15 x 5 x 15 x 2 2 2 2 1 2sin sin = 2 2 2 5x 5x = sin 5x sin sin 5x sin = R.S. 65 65 65 Prove that x 9x 5x cos 2 x cos cos 3x cos sin 5 x sin . 2 2 Example 27 Find the value of tan Solution Let x π π . 2 2 2 We have L.TRIGONOMETRIC FUNCTIONS sin( x y ) Example 26 Solution 3 12 4 5 5 13 5 13 = 36 20 56 . Then 1 = 1 y2 8 1 tan 2 x π . Then 2 x .H.H.S. Hence 8 8 π 2 1. cos x is negative. 2 2 4 Also Therefore. π < x < . Now sec2 x = 1 + tan2 x = 1 Therefore cos2 x = Now 2 sin 2 Therefore sin2 or sin x 2 x 2 16 9 16 or cos x = – 25 4 5 25 16 (Why?) x 4 9 = 1 – cos x = 1 . 4 2 2 2 2 Solution Since π x π x 3π . 8 tan x x x 3 3π Example 28 If tan x = .80 MATHEMATICS or y2 + 2y – 1 = 0 Therefore Since y= 2 2 2 2 1 2 π π lies in the first quadrant. 2 x 2 is positive and cos x 2 is negative. sin 3π . y = tan is positve. cos and tan . 5 5 2 = = 9 10 3 (Why?) 10 Again 2cos2 Therefore cos2 x 2 x 2 = 1+ cos x = 1 = 1 10 4 5 1 5 . find the value of sin . S. 2 cos 2.TRIGONOMETRIC FUNCTIONS or cos Hence tan Example 29 Solution x 2 sin = cos x 2 x 1 = 2 x 10 (Why?) 10 = – 3.H. 2 2 Miscellaneous Exercise on Chapter 3 Prove that: π 9π 3π 5π cos cos cos 0 13 13 13 13 1. (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0 81 . 3 3 2 We have L. 2 2 = 1 2π 2π 3 cos 2 x cos 2 x cos 2 x 2 3 3 = 1 2π 3 cos 2 x 2cos 2 x cos 2 3 = 1 π 3 cos 2 x 2cos 2 x cos π 2 3 = 1 π 3 cos 2 x 2cos 2 x cos 2 3 = 1 3 3 cos 2 x cos 2 x = R.H. 10 1 3 2 π π 3 Prove that cos2 x + cos2 x cos 2 x . = 1 cos 2 x 2 2π 2π 1 cos 2 x 1 cos 2 x 3 3 .S. x in quadrant II 4 Summary If in a circle of radius r. cos x = . x in quadrant II 3 10. (cos x – cos y)2 + (sin x – sin y)2 = 4 sin2 6. an arc of length l subtends an angle of l=rθ Radian measure = π Degree measure 180 180 Degree measure = π cos2 x + sin2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x cos (2nπ + x) = cos x sin (2nπ + x) = sin x sin (– x) = – sin x cos (– x) = cos x Radian measure θ radians. cos x 2 and tan x 2 x 2 cos 3x in each of the following : 4 8. (sin 7 x sin 5 x) (sin 9 x sin 3 x ) tan 6 x (cos 7 x cos 5 x) (cos 9 x cos 3 x ) 7. sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x 4. sin 3x + sin 2x – sin x = 4sin x cos Find sin x 2 . x in quadrant III 3 1 . sin x = 2 1 9. then . (cos x + cos y)2 + (sin x – sin y)2 = 4 cos2 x y 2 5.82 MATHEMATICS x y 2 3. tan x = . y and (x y) is a multiple of π. y and (x tan (x + y) = y) is an odd multiple of π . then 2 tan x tan y 1 tan x tan y tan x tan y tan (x – y) = 1 tan x tan y If none of the angles x.TRIGONOMETRIC FUNCTIONS cos (x + y) = cos x cos y – sin x sin y cos (x – y) = cos x cos y + sin x sin y π cos ( 2 x ) = sin x π sin ( 2 x ) = cos x sin (x + y) = sin x cos y + cos x sin y sin (x – y) = sin x cos y – cos x sin y π + x = – sin x 2 cos (π – x) = – cos x π sin + x = cos x 2 sin (π – x) = sin x cos (π + x) = – cos x sin (π + x) = – sin x cos (2π – x) = cos x sin (2π – x) = – sin x cos If none of the angles x. then cot x cot y 1 cot (x + y) = cot y cot x cot (x – y) = cot x cot y 1 cot y cot x cos 2x = cos2 x – sin2 x = 2cos2 x – 1 = 1 – 2 sin2 x 1 – tan 2 x 1 tan 2 x 83 . where n ∈ Z.84 MATHEMATICS sin 2x = 2 sin x cos x 2 tan x 1 tan 2 x 2tanx tan 2x = 1 tan 2 x sin 3x = 3sin x – 4sin3 x cos 3x = 4cos3 x – 3cos x 3tan x tan 3 x tan 3x = 1 3tan 2 x (i) cos x + cos y = 2cos x y x y cos 2 2 (ii) cos x – cos y = – 2sin (iii) sin x + sin y = 2 sin x y x y sin 2 2 x y x y cos 2 2 x y x y sin 2 2 (i) 2cos x cos y = cos ( x + y) + cos ( x – y) (iv) sin x – sin y = 2cos (ii) – 2sin x sin y = cos (x + y) – cos (x – y) (iii) 2sin x cos y = sin (x + y) + sin (x – y) (iv) 2 cos x sin y = sin (x + y) – sin (x – y). cos x = cos y. where n ∈ Z. sin x = 0 gives x = nπ. implies x = 2nπ ± y. tan x = tan y implies x = nπ + y. . where n ∈ Z. 2 sin x = sin y implies x = nπ + (– 1)n y. cos x = 0 gives x = (2n + 1) π . where n ∈ Z. where n ∈ Z. He is credited with the determination of the height of a great pyramid in Egypt by measuring shadows of the pyramid and an auxiliary staff (or gnomon) of known height. and comparing the ratios: H h = tan (sun’s altitude) S s Thales is also said to have calculated the distance of a ship at sea through the proportionality of sides of similar triangles. Problems on height and distance using the similarity property are also found in ancient Indian works. 36°. Hersehel (1813) The names of Thales (about 600 B. known as the sine of an angle.TRIGONOMETRIC FUNCTIONS Historical Note The study of trigonometry was first started in India. were suggested by the astronomer Sir John F. 72°. Exact expression for sines or cosines of 18°.. arc cos x. Aryabhatta (476). and the introduction of the sine function represents the main contribution of the siddhantas (Sanskrit astronomical works) to the history of mathematics. Bhaskara I (600) and Bhaskara II (1114) got important results. The ancient Indian Mathematicians. the predecessor of the modern trigonometric functions. cos–1 x. A sixteenth century Malayalam work Yuktibhasa (period) contains a proof for the expansion of sin (A + B). etc. it was immediately adopted throughout the world. The symbols sin–1 x.) is invariably associated with height and distance problems. etc... The Greeks had also started the study of trigonometry but their approach was so clumsy that when the Indian approach became known. are given by Bhaskara II. 54°. — — 85 . Brahmagupta (598). etc. In India. Bhaskara I (about 600) gave formulae to find the values of sine functions for angles more than 90°.C. for arc sin x.W. All this knowledge first went from India to middle-east and from there to Europe. where collecting and analysing data is the norm.. Peano (ii) Any number divisible by two is an even number. (c) Socrates is mortal.e. G . we can say the word induction means the generalisation from particular cases or facts. It is frequently used in mathematics and is a key aspect of scientific reasoning. valid deductive steps are derived and a proof may or may not be established. – LAPLACE One key basis for mathematical thinking is deductive reasoning. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity. (b) All men are mortal. borrowed from the study of logic. deduction is the application of a general case to a particular case. To make this simple mathematical example. inductive reasoning depends on working with each case. In contrast to deduction. Thus. and example of deductive reasoning. in simple language. i. An informal. often called a conjecture or a theorem in mathematics.Chapter 4 he PRINCIPLE OF MATHEMATICAL INDUCTION is Analysis and natural philosophy owe their most important discoveries to © no N C tt E o R be T re pu 4. therefore. we could write: (i) Eight is divisible by two.1 Introduction bl this fruitful means. which is called induction. is an argument expressed in three statements: (a) Socrates is a man. (iii) Eight is an even number. (1858-1932) therefore. deduction in a nutshell is given a statement to be proven. and developing a conjecture by observing incidences till we have observed each and every case. Thus. If statements (a) and (b) are true. then the truth of (c) is established. . 3.. the set of natural numbers N is a special ordered subset of the real numbers.1.. when n = 4 and so on and suppose that in some manner we are led to believe that the formula 1 + 2 + 3+. This is the underlying principle of mathematical induction. To prove such statements the well-suited principle that is used–based on the specific technique. To understand the basic principles of mathematical induction. of course. it is sufficient to know that (a) The first tile falls. the value 1 + 2 + 3 + 4. Illustration Suppose we wish to find the formula for the sum of positive integers 1.PRINCIPLE OF MATHEMATICAL INDUCTION 87 In algebra or in other discipline of mathematics. Fig 4. it follows that any subset of R that is an inductive set must contain N. there are certain results or statements that are formulated in terms of n. 4. suppose a set of thin rectangular tiles are placed as shown in Fig 4. and (b) In the event that any tile falls its successor necessarily falls.2 Motivation © no N C tt E o R be T re pu bl is he In mathematics. all the tiles will fall. where n is a positive integer. We know.1 When the first tile is pushed in the indicated direction.. 2 How can this formula actually be proved? We can. a formula which will give the value of 1 + 2 + 3 when n = 3. What is needed is some kind of chain reaction which will . verify the statement for as many positive integral values of n as we like. In fact. but this process will not prove the formula for all values of n.+ n = n ( n + 1) is the correct one. N is the smallest subset of R with the following property: A set S is said to be an inductive set if 1∈ S and x + 1 ∈ S whenever x ∈ S... is known as the principle of mathematical induction.. we use a form of complete induction called mathematical induction. 2. that is. To be absolutely sure that all the tiles will fall.n. Since N is the smallest subset of R which is an inductive set. This is sometimes referred to as the inductive step. i. The assumption that the given statement is true for n = k in this inductive step is called the inductive hypothesis.e. we suppose that P (k) is true for some . + (2n – 1) = n2. the sum of the first n odd natural numbers is the square of n. i. P(1) is true. We wish to prove that P(n) is true for all n. then it is also true for n = k + 1... Obviously 1 = 12.e.88 MATHEMATICS have the effect that once the formula is proved for a particular positive integer the formula will automatically follow for the next positive integer and the next indefinitely. This step is called the basic step. For example.. So. and (ii) If the statement is true for n = k (where k is some positive integer). sum of the first three odd natural numbers is the square of third natural number and so on. P(4). but only that if it is true for n = k. only prove that conditional proposition: If the statement is true for n = k. Property (i) is simply a statement of fact. P(1) is true.. to prove that the property holds . Let us write P(n): 1 + 3 + 5 + 7 + . It does not assert that the given statement is true for n = k. Property (ii) is a conditional property. step 1 will start from n = 4 and we shall verify the result for n = 4. + (2n – 1) = n2 . In this case. a formula will be discovered that appears to fit a pattern like 1 = 12 =1 4 = 22 = 1 + 3 9 = 32 = 1 + 3 + 5 16 = 42 = 1 + 3 + 5 + 7. then it is also true for n = k +1.. The first step in a proof that uses mathematical induction is to prove that P (1) is true..Thus. i.e.e. from this pattern it appears that 1 + 3 + 5 + 7 + . Then. It is worth to be noted that the sum of the first two odd natural numbers is the square of second natural number.. Here. There may be situations when a statement is true for all n ≥ 4.3 The Principle of Mathematical Induction © no N C tt E o R be T re pu bl is he Suppose there is a given statement P(n) involving the natural number n such that (i) The statement is true for n = 1.. Such a reaction may be considered as produced by the method of mathematical induction. etc. The next step is called the inductive step. then the statement is also true for n = k + 1. i. i.e. P(n) is true for all natural numbers n. truth of P(k) implies the truth of P (k + 1). 4. frequently in mathematics. i.. the statement P(n) is true for all natural numbers n. Since P (k) is true. Hence P(n) is true for all natural numbers n... P (k + 1) is true and the inductive proof is now completed.PRINCIPLE OF MATHEMATICAL INDUCTION 89 Therefore... + (2k – 1) + {2(k +1) – 1} . 6 Solution Let the given statement be P(n). n( n + 1) (2n + 1) . n(n + 1)(2n + 1) 6 © no N C tt E o R be T re pu P(n) : 12 + 22 + 32 + 42 +…+ n2 = bl 12 + 22 + 32 + 42 +…+ n2 = is Example 1 For all n ≥ 1. P(1): 1 = k (k + 1) (2k + 1) 6 We shall now prove that P(k + 1) is also true. we have 1 + 3 + 5 + 7 + .. Now... we have (12 +22 +32 +42 +…+k2 ) + (k + 1) 2 12 + 22 + 32 + 42 +…+ k2 = = k (k + 1)(2k + 1) + (k + 1)2 6 = k ( k + 1) (2k + 1) + 6( k + 1) 2 6 . = 6 6 Assume that P(k) is true for some positive integer k. Hence. prove that he positive integer k and we need to prove that P (k + 1) is true. + (2k – 1) = k2 . i. For n = 1. (1) [Using (1)] ( k + 1) (2k 2 + 7 k + 6) = 6 (k + 1)(k + 1 + 1){2(k + 1) + 1} 6 Thus P(k + 1) is true. whenever P (k) is true.. (1) Consider 1 + 3 + 5 + 7 + .. = .e.. (2) [Using (1)] = k2 + (2k + 1) = (k + 1)2 1(1 + 1) (2 × 1 + 1) 1× 2 × 3 =1 which is true. from the principle of mathematical induction..e. e.2 2.. by principle of mathematical induction.3 3. Thus.2 2. P(n) is true for every positive integer n..3 3. .. Hence P(1) is true.4 k 1 = k + 1 + (k + 1) ( k + 2) [Using (1)] . + = 1... We have 1 1 1 1 1 + + + ..4 k (k + 1) (k + 1) (k + 2) ⎡ 1 1 1 1 ⎤ 1 + + . we get 2. + = 1..2 2. which is true. i.e.... Multiplying both sides of (1) by 2..2 2 1 +1 Assume that P(k) is true for some natural number k. bl Therefore.2 + 2.. + n(n + 1) = n + 1 We note that P(1): 1 1 1 = = . (1) We need to prove that P(k + 1) is true whenever P(k) is true.4 + ..3 3. Solution Let P(n): 2n > n When n =1.. 21 >1. P(n) is true for n = 1.4 k (k + 1) k + 1 . prove that © no N C tt E o R be T re pu 1 1 1 1 n + + + . P(k + 1) is true when P(k) is true.e. + + 1. Assume that P(k) is true for any positive integer k.90 MATHEMATICS Example 2 Prove that 2n > n for all positive integers n. 2k > k We shall now prove that P(k +1) is true whenever P(k) is true. 1 1 1 1 k + + + . (1) Example 3 For all n ≥ 1.. i.4 n(n + 1) n + 1 . + + = ⎢ + ⎥ k (k + 1) ⎦ ( k + 1) ( k + 2) ⎣1. Hence. 1. 2k > 2k he 2 k + 1 > 2k = k + k > k + 1 is i.3 3.2 2.3 + 3... Solution We can write 1 1 1 1 n P(n): 1. we wish to prove that P(k + 1) is true whenever P(k) is true. Consider the identity (1 + x)k + 1 = (1 + x)k (1 + x) Given that x > –1. by the principle of mathematical induction..PRINCIPLE OF MATHEMATICAL INDUCTION 91 (k 2 + 2k + 1) k +1 k +1 k (k + 2) + 1 ( k + 1)2 = = = = = (k + 1) (k + 2) (k + 1) (k + 2) ( k + 1) ( k + 2 ) k + 2 ( k + 1) + 1 Thus P(k + 1) is true whenever P(k) is true. x > – 1 is true. © no N C tt E o R be T re pu Now. we have (1 + x) k + 1 ≥ (1 + kx)(1 + x) i. by using (1 + x)k ≥ (1 + kx)..e. by principle of mathematical induction the statement is true for every positive integer n... i. bl is Solution We can write P(n) : 7n – 3n is divisible by 4. P(n): (1 + x)n ≥ (1 + nx).3k – 3(k + 1) = 7(7k – 3k) + (7 – 3)3k = 7(4d) + (7 – 3)3k = 7(4d) + 4. so (1+x) > 0. we see that 7(k + 1) – 3(k + 1) is divisible by 4.e. Now 7(k + 1) – 3(k + 1) = 7(k + 1) – 7. P(k + 1) is true when P(k) is true. We can write 7k – 3k = 4d.. he Example 4 For every positive integer n. i.. (1) .3k + 7.3k = 4(7d + 3k) From the last line.. We note that P(n) is true when n = 1. for x > – 1. Solution Let P(n) be the given statement. P(k) : 7k – 3k is divisible by 4. (3) . Thus P(n) is true for n = 1 Let P(k) be true for some natural number k. since ( 1+x) ≥ (1 + x) for x > –1 Assume that P(k): (1 + x)k ≥ (1 + kx). where d ∈ N. Thus.. prove that 7n – 3n is divisible by 4. Therefore . where x > – 1. Therefore. (1 + x)k + 1 ≥ (1 + x + kx + kx2). (2) . Example 5 Prove that (1 + x)n ≥ (1 + nx). for all natural number n. . Hence. We note that P(1): 71 – 31 = 4 which is divisible by 4.e. P(n) is true for all natural numbers. We want to prove that P(k + 1) is true for x > –1 whenever P(k) is true. and so we obtain (1 + x)k + 1 ≥ (1 + x + kx) i.. for all n ∈ N.5k+1 – 5 = 2. .5k + 30 = 7 × 24q – 6 (5k – 5) = 7 × 24q – 6 (4p) [(5k – 5) is a multiple of 4 (why?)] = 7 × 24q – 24p = 24 (7q – p) = 24 × r.7k + 3.5k – 5 = 24q.5k –5 = 7 × 24q – 21.H. of (1) is divisible by 24.5n – 5 is divisible by 24. by principle of mathematical induction.. r = 7q – p. which is divisible by 24. Hence.5k + 35 + 15. We note that P(n) is true for n = 1.7n + 3. P(n) is true for all n ∈ N..5k .5k – 5 – 3. is some natural number. © no N C tt E o R be T re pu Assume that P(k) is true i. (1) Now.7k + 3. (1 + x)k + 1 ≥ [1 + (1 + k)x] Thus. since 2. the statement in (2) is established. we wish to prove that P(k + 1) is true whenever P(k) is true.S.5k – 5 = 7 × 24q – 6.5k + 5] + 15. 5 – 5 = 7 [24q – 3. when q ∈ N . We have 2. by the principle of mathematical induction.5 – 5 = 24. 51 – 5 = 7 [2.7k .92 MATHEMATICS he Here k is a natural number and x2 ≥ 0 so that kx2 ≥ 0. Thus P(k + 1) is true whenever P(k) is true. P(n) is true for all natural numbers. bl Solution Let the statement P(n) be defined as P(n) : 2.5k .7 + 3.5n – 5 is divisible by 24. (2) The expression on the R.7n + 3. 2. Hence. Therefore (1 + x + kx + kx2) ≥ (1 + x + kx).. . is Example 6 Prove that 2.e.e. 71 + 3.5k + 5] + 3.7k+1 + 3. n∈N 3 13 3 is i. We note that P(n) is true for n = 1 since (ab)1 = a1b1.. (ab)k = akbk We shall now prove that P(k + 1) is true whenever P(k) is true. 12 + 22 + . + k 2 + ( k + 1) > 2 k3 2 + ( k + 1) 3 = 1 3 [k + 3k2 + 6k + 3] 3 = 1 1 [(k + 1)3 + 3k + 2] > (k + 1)3 3 3 [by (1)] Therefore. P(n) : 12 + 22 + . + n2 > 2 We note that P(n) is true for n = 1 since 1 > he n3 ...e.e.. by mathematical induction P(n) is true for all n ∈ N.. + n2 > Assume that P(k) is true P(k) : 12 + 22 + . P(k + 1) is also true whenever P(k) is true.n∈N 3 Solution Let P(n) be the given statement.... (1) . We have 12 + 22 + 32 + . Example 8 Prove the rule of exponents (ab)n = anbn by using principle of mathematical induction for every natural number..PRINCIPLE OF MATHEMATICAL INDUCTION 93 Example 7 Prove that n3 . we have (ab)k + 1 = (ab)k (ab) .. P(n) : (ab)n = anbn.. Let P(k) be true.... i. + k2 + (k + 1)2 ( ) = 12 + 22 + . Solution Let P(n) be the given statement i..e.. Hence.e. Now. + k2 > k3 3 .(1) bl i. © no N C tt E o R be T re pu We shall now prove that P(k + 1) is true whenever P(k) is true. 2 is 1. a1) (bk . + = 1. 4 (2n − 1)3n +1 + 3 . . P(k + 1) is also true whenever P(k) is true.11 (3n − 1) (3n + 2) (6n + 4) .22 + . 1 1 1 1 n(n + 3) + + + .3 2. 4 ⎡ n(n + 1)(n + 2) ⎤ 6. 2 4 8 2 2 10.3.. 1 + 3 + 32 + . + n = 1 − n . Hence.3 + 3. 1.22 + 3. by principle of mathematical induction. 1 1 1 1 n + + + .2..4 +…+ n(n+1) (n+2) = 5..5 n(n + 1)(n + 2) 4(n + 1)(n + 2) . P(n) is true for all n ∈ N..2.+n. 4.8 8.3 + 3.2 + 2...3n = n(n + 1) (n + 2) (n + 3) .7 +…+ (2n–1) (2n+1) = 9.n) = (n + 1) . 1. 3 ⎣ ⎦ n(4n 2 + 6n − 1) .. 1 1 1 1 1 + + + . ⎝ 2 ⎠ 3 © no N C tt E o R be T re pu 1 1 1 2n 3. 11. + (1 + 2 + 3 + .3 + 2.33 +…+ n. + = 2.. 7. 1. 1 + 2 + 3 + … +n = ⎜ ⎟ .5 + 5.3. b1) = ak+1 .3 + 2. 1.2n = (n–1) 2n + 1 + 2.1 Prove the following by using the principle of mathematical induction for all n ∈ N: (3n − 1) .5 5. EXERCISE 4.2 + 2. 1 + (1 + 2) + (1 + 2 + 3) + . 1.32 + 3. 3 8....94 MATHEMATICS = (ak bk) (ab) [by (1)] = (ak ..4.(n+1) = ⎢ ⎥.. + 3n – 1 = 3 3 3 bl 2 ⎛ n( n + 1) ⎞ 2.4 3.4 +…+ n.. bk+1 he Therefore. Each such statement is assumed as P(n) associated with positive integer n. n (n + 1) (n + 5) is a multiple of 3. 21. x2n – y2n is divisible by x + y. inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. ⎜1 + ⎟ ⎜1 + ⎟ ⎝ 1⎠ ⎝ 4⎠ 15. In contrast to deduction.⎜1 + ⎟ = (n + 1) . for which the correctness .5 5. r −1 ⎛ 7 ⎞ ⎛ (2n + 1) ⎞ 2 ⎜1 + ⎟ . 3 bl 1 1 1 1 n 16.7 7. 24. ⎜1 + ⎟ ⎜1 + ⎟ ⎜1 + ⎟ . 23. 8 19. 41n – 14n is a multiple of 27. 1.PRINCIPLE OF MATHEMATICAL INDUCTION 12... 95 18.10 + .. + = 3.7 + 7. 102n – 1 + 1 is divisible by 11.9 (2n + 1)(2n + 3) 3(2n + 3) . © no N C tt E o R be T re pu 17. n2 ⎠ ⎝ 9⎠ ⎝ he ⎛ 3⎞ ⎛ 5⎞ 13..4 + 4. Summary One key basis for mathematical thinking is deductive reasoning. 32n+2 – 8n – 9 is divisible by 8. 1 + 2 + 3 +…+ n < 1 (2n + 1)2. The principle of mathematical induction is one such tool which can be used to prove a wide variety of mathematical statements. 20. + (3n − 2)(3n + 1) = (3n + 1) . 12 + 32 + 52 + …+ (2n–1)2 = n(2n − 1)(2n + 1) . (2n + 7) < (n + 3)2. ⎝ 1⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ ⎝ n ⎠ 1 1 1 1 n + + + . 22. Thus. in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts. is ⎛ 1⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ 14. a + ar + ar2 +…+ arn-1 = a ( r n − 1) . ⎜1 + ⎟ = ( n + 1) ..... proof by mathematical induction is not the invention of a particular individual at a fixed moment. the truth of P (k+1) is established. De Morgan contributed many accomplishments in the field of mathematics on many different subjects. Peano undertook the task of deducing the properties of natural numbers from a set of explicitly stated assumptions. The French mathematician Blaise Pascal is credited with the origin of the principle of mathematical induction. G. Later the principle was employed to provide a proof of the binomial theorem.The principle of mathematical induction is a restatement of one of the Peano’s axioms. He was the first person to define and name “mathematical induction” and developed De Morgan’s rule to determine the convergence of a mathematical series. The name induction was used by the English mathematician John Wallis. now known as Peano’s axioms. It is said that the principle of mathematical induction was known by the Pythagoreans. Then assuming the truth of P(k) for some positive integer k.MATHEMATICS for the case n = 1 is examined. he Historical Note bl is Unlike other concepts and methods. © no N C tt E o R be T re pu 96 — — . where D = b2 – 4ac < 0. 5.Chapter 5 is he COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. For example. we have studied linear equations in one and two variables and quadratic equations in one variable. which is not possible in the system of real numbers. the main objective is to solve the equation ax2 + bx + c = 0. So. This means that i is a solution of the equation x2 + 1 = 0. 4 + i ⎜ ⎟ are complex numbers. – GAUSS © no N C tt E o R be T re pu In earlier classes. a is called the real part.2 Complex Numbers W. then Re z = 2 and Im z = 5. Hamilton (1805-1865) Let us denote −1 by the symbol i.1 Introduction bl Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. we need to extend the real number system to a larger system so that we can find the solution of the equation x2 = – 1. Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d. (– 1) + i 3 . For example. R. if z = 2 + i5. A number of the form a + ib. Then. is defined to be a ⎛ −1 ⎞ complex number. We have seen that the equation x2 + 1 = 0 has no real solution as x2 + 1 = 0 gives x2 = – 1 and square of every real number is non-negative. denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z. where a and b are real numbers. . In fact. ⎝ 11 ⎠ For the complex number z = a + ib. we have i 2 = −1 . 2 + i3. (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i and (2 – i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i .1 Addition of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. called the additive inverse or negative of z. Solution We have 5. called the additive identity or the zero complex number. 4 4 is which. for every complex number z. z 3 . we get 4x = 3. the sum z1 + z2 is defined as follows: z1 + z2 = (a + c) + i (b + d). z + 0 = z. (1) he 4x + i (3x – y) = 3 + i (–6) Equating the real and the imaginary parts of (1).. For example. on solving simultaneously. (z1 + z2) + z3 = z1 + (z2 + z3). Then.e.3. For example. (v) The existence of additive inverse To every complex number z = a + ib. the difference z1 – z2 is defined as follows: z1 – z2 = z1 + (– z2). (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8 The addition of complex numbers satisfy the following properties: (i) The closure law The sum of two complex numbers is a complex number. (iv) The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0). i. 3x – y = – 6. which is again a complex number.3 Algebra of Complex Numbers 3 33 and y = . 5. In this Section. We observe that z + (–z) = 0 (the additive identity). such that.2 Difference of two complex numbers Given any two complex numbers z1 and z2. z1 + z2 is a complex number for all complex numbers z1 and z2. we shall develop the algebra of complex numbers. then find the values of x and y. give x = . where x and y are real numbers. © no N C tt E o R be T re pu bl 5. z 2 ..3. (ii) The commutative law For any two complex numbers z 1 and z 2 . we have the complex number – a + i(– b) (denoted as – z)..98 MATHEMATICS Example 1 If 4x + i(3x – y) = 3 + i (– 6). z1 + z2 = z2 + z1 (iii) The associative law For any three complex numbers z 1 . (i) The closure law The product of two complex numbers is a complex number. the product z1 z2 is a complex number for all complex numbers z1 and z2. z2. the quotient z is defined by 2 z1 1 = z1 z2 z2 For example.4 Division of two complex numbers Given any two complex numbers z1 and z2. (iii) The associative law For any three complex numbers z 1 . for every complex number z. z3. which we state without proofs. let z1 = 6 + 3i and z2 = 2 – i Then z1 ⎛ 1 ⎞ = ⎜ (6 + 3i ) × ⎟ = 6 + 3i ) z2 ⎝ 2−i ⎠ ( ⎛ − ( −1) ⎞ 2 ⎜ ⎟ +i 2 2 ⎟ 2 ⎜ 22 + ( −1) 2 + − 1 ( ) ⎝ ⎠ . = 1 (the multiplicative identity).1 = z.3. (iv) The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1). we have the complex number a –b 1 +i 2 or z–1 ). Then.3 Multiplication of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. z 2 .COMPLEX NUMBERS AND QUADRATIC EQUATIONS 99 © no N C tt E o R be T re pu bl is he 5. (a) z1 (z2 + z3) = z1 z2 + z1 z3 (b) (z1 + z2) z3 = z1 z3 + z2 z3 5. z1 z2 = z2 z1. called the multiplicative identity such that z. z (vi) The distributive law For any three complex numbers z1. the product z1 z2 is defined as follows: z1 z2 = (ac – bd) + i(ad + bc) For example. z 3 . z1 where z2 ≠ 0 . called the multiplicative inverse 2 2 (denoted by a +b a +b z of z such that 2 1 z. (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28 The multiplication of complex numbers possesses the following properties. (ii) The commutative law For any two complex numbers z1 and z2. (z1 z2) z3 = z1 (z2 z3). (v) The existence of multiplicative inverse For every non-zero complex number z = a + ib or a + bi(a ≠ 0. b ≠ 0).3. etc. for any integer k. Note that .5 Power of i we know that 2 i = ( −1) i = i . We already know that −a = 3 i only. i −1 = × = i i −1 Also. we have he ( ) i5 = i 2 i4 = i2 3 i− 2 = 1 1 = = − 1.e. a i. i = 1. b < 0? Let us examine.. we would mean i only. the square roots of – 1 are i.100 MATHEMATICS ⎛2+i⎞ 1 1 = ( 6 + 3i ) ⎜ ⎟ = ⎡⎣12 − 3 + i ( 6 + 6 ) ⎦⎤ = ( 9 + 12i ) 5 ⎝ 5 ⎠ 5 5. This result also holds true when either a > 0. i = –1. i. 2 −1 i is ( ) i 3 = i 2i = ( −1) i = −i . i = i. a −1 = −3 = 3i . Now. What if a < 0. b > 0. by the symbol −1 . Similarly ( 3 i) = ( 3) 2 2 i2 = 3 (– 1) = – 3 (− 3 i) = (− 3) 2 Therefore. b < 0 or a < 0. 1 1 i i 1 1 = × = = i. the square roots of –3 are Again. we can see that i and –i both are the solutions of the equation x2 + 1 = 0 or 2 x = –1. – i. the symbol 2 i2 = – 3 3 i and − 3 i . if a is a positive real number.6 The square roots of a negative real number Note that i2 = –1 and ( – i)2 = i2 = – 1 Therefore.3. a × b = ab for all positive real number a and b.3. 1 i i = − i. −3 is meant to represent Generally. However. i −4 = 4 = = 1 3 −i i 1 1 i i 4k 4k + 1 4k + 2 4k + 3 In general. i =–i bl i −3 = © no N C tt E o R be T re pu 5. 2 2 ( ) i6 = i2 = ( −1) = 1 3 2 = ( − 1) = −1 . Example 2 Express the following in the form of a + bi: ⎛1 ⎞ (i) ( −5i ) ⎜ i ⎟ ⎝8 ⎠ Solution (i) (ii) ( − i ) ( 2i ) ⎛ 1 ⎞ ⎜− i⎟ ⎝ 8 ⎠ 3 ( −5i ) ⎛⎜ 1 ⎞ −5 2 −5 5 5 i⎟ = i = −1) = = + i 0 ( 8 8 8 8 ⎝8 ⎠ 3 1 1 2 ⎛ 1 ⎞ × i5 = i (ii) ( −i ) ( 2i ) ⎜ − i ⎟ = 2 × 8×8×8 256 ⎝ 8 ⎠ ( ) 2 i= 1 i. 256 . clearly. we can prove the following identities: (i) ( z1 − z2 )2 = z12 − 2 z1 z2 + z22 (ii) ( z1 + z2 )3 = z13 + 3 z12 z2 + 3z1 z22 + z23 (iii) ( z1 − z2 )3 = z13 − 3z12 z2 + 3z1 z22 − z23 (iv) z12 – z22 = ( z1 + z2 ) ( z1 – z2 ) In fact. 5.3. is Proof We have. then.7 Identities We prove the following identity ( z1 + z2 )2 = z12 + z22 + 2 z1 z2 . for all complex numbers z1 and z2.COMPLEX NUMBERS AND QUADRATIC EQUATIONS ( −1) ( −1) i 2 = −1 −1 = (by assuming a× b = 101 ab for all real numbers) = 1 = 1. can be proved to be true for all complex numbers. a × b ≠ ab if both a and b are negative real numbers. a × b = ab = 0. which is a contradiction to the fact that i 2 = −1 . many other identities which are true for all real numbers. if any of a and b is zero. = (z1 + z2) z1 + (z1 + z2) z2 + z2 z1 + z1 z2 + (Distributive law) z22 (Commutative law of multiplication) © no N C tt E o R be T re pu = z12 + z1 z2 + z1 z2 + z22 (Distributive law) bl = z12 = z12 + 2 z1 z2 + z22 Similarly. he Therefore. (z1 + z2)2 = (z1 + z2) (z1 + z2). Further. e. the modulus of z. the following results can easily be derived. − 3 + −2 he Example 4 Express − 3 + −2 2 3 − i in the form of a + ib bl 5. denoted as z . (5 – 3i)3 = 53 – 3 × 52 × (3i) + 3 × 5 (3i)2 – (3i)3 = 125 – 225i – 135 + 27i = – 10 – 198i. we have (i) (iii) z1 z2 = z1 z2 z1 z2 = z1 z2 (ii) (iv) z z1 = 1 provided z ≠ 0 2 z2 z2 ⎛ z1 ⎞ z1 z1 ± z2 = z1 ± z2 (v) ⎜ z ⎟ = z provided z2 ≠ 0. 2 − 5i = 22 + ( − 5) 2 = 29 . z = a – ib.4 The Modulus and the Conjugate of a Complex Number Let z = a + ib be a complex number. i. For example.e. i... 2 − 5 i = 2 + 5 i . 2 ⎝ 2⎠ . | z | = a 2 + b 2 and the conjugate © no N C tt E o R be T re pu to be the non-negative real number of z. is defined a 2 + b2 . Then. Solution We have. ( )( ( ) (2 ) ) = (− 3 + 2 i) (2 3 − i) 2 i = ( −6 + 2 ) + 3 (1 + 2 2 ) i 3 −i = −6 + 3i + 2 6i − 2 is Solution We have. denoted by | z |. is the complex number a – ib. 3 + i = 3 − i .102 MATHEMATICS Example 3 Express (5 – 3i)3 in the form a + ib. −3i − 5 = 3i – 5 Observe that the multiplicative inverse of the non-zero complex number z is given by and z–1 = a −b a − ib 1 +i 2 = 2 = 2 2 = a +b a +b a 2 + b2 a + ib z z= z or z z 2 2 Furthermore. 3 + i = 32 + 12 = 10 . For any two compex numbers z1 and z2 . 1 Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. 5 + 2i 5 + 2i 1 + 2i = × 1 − 2i 1 − 2i 1 + 2i = −35 (ii) i = 1 i 35 = = 5 + 5 2i + 2i − 2 1− ( 2i ) 2 3 + 6 2i 3(1 + 2 2i ) = = 1 + 2 2i . ( 5i ) ⎛⎜ − 3 ⎞ i⎟ ⎝ 5 ⎠ 2. 1. the multiplicative inverse of 2 − 3i is given by Example 6 Express the following in the form a + ib (i) 5 + 2i 1 − 2i (ii) i–35 Solution (i) We have. i 9 + i 19 3. he Therefore. 1+ 2 3 1 (i ) 2 17 = i i 1 i × = 2 =i −i −i i EXERCISE 5. i −39 .COMPLEX NUMBERS AND QUADRATIC EQUATIONS 103 Example 5 Find the multiplicative inverse of 2 – 3i. Solution Let z = 2 – 3i Then z = 2 + 3i and z 2 = 22 + ( − 3)2 = 13 z z–1 = 2 z 2 + 3i 2 3 = + i 13 13 13 = 2 + 3i 2 + 3i 2 3 = = + i 2 13 13 13 2 − (3i ) 2 © no N C tt E o R be T re pu = 1 2 + 3i = 2 − 3i (2 − 3i) (2 + 3i) bl z–1 = is The above working can be reproduced in the following manner also. Express the following expression in the form of a + ib : 12. 5. ⎜ −2 − i ⎟ 3 ⎠ ⎝ 3 he 104 13. D. – 5 –2i and Fig 5. respectively. .1 1 – 2i which correspond to the ordered pairs (2. (0. 3(7 + i7) + i (7 + i7) 6. 3). and F. ⎜ + 3i ⎟ 3 ⎝ ⎠ 3 1 ⎞ ⎛ 10. respectively in the Fig 5. The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane. we get a unique point in the XYplane and vice-versa with reference to a set of mutually perpendicular lines known as the x-axis and the y-axis.5 Argand Plane and Polar Representation We already know that corresponding to each ordered pair of real numbers (x. have been represented geometrically by the points A. E. – i 5 + 3i 14. – 2 + 3i. 0 + 1i. 4 – 3i is Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13. and (1. y). ⎢⎜ + i ⎟ + ⎜ 4 + i ⎟ ⎥ − ⎜ − + i ⎟ 3 ⎠⎦ ⎝ 3 ⎠ ⎣⎝ 3 3 ⎠ ⎝ ⎛1 ⎞ 9.MATHEMATICS 4. (2. –2). © no N C tt E o R be T re pu (3 + i 5 ) (3 − i 5 ) ( 3 + 2 i) − ( 3 − i 2 ) bl 11. 4). y) in the XY-plane and vice-versa. ( –5. (1 – i) – ( –1 + i6) 5⎞ ⎛1 2⎞ ⎛ ⎜ + i ⎟ −⎜4 +i ⎟ 2⎠ ⎝5 5⎠ ⎝ 8. 0). y) can be represented geometrically as the unique point P(x. 1). ( – 2.1. – 2). B. 5. C. 2 + 0i. The complex number x + iy which corresponds to the ordered pair (x. Some complex numbers such as 2 + 4i. (1 – i) 4 ⎡⎛ 1 7 ⎞ ⎛ 1 ⎞⎤ ⎛ 4 ⎞ 7. The x-axis and y-axis in the Argand plane are called. the point (x. in the Argand plane.3).2 0 + i b.2).COMPLEX NUMBERS AND QUADRATIC EQUATIONS 105 Obviously. Geometrically. – y) is the mirror image of the point (x. y) on the real axis (Fig 5. 0) © no N C tt E o R be T re pu bl is he (Fig 5. respectively. the points P (x. y) and the origin O (0. the modulus of the complex number x + iy = x 2 + y 2 is the distance between the point P(x. Fig 5. – y). respectively.3 . the real axis and the imaginary axis. The points on the x-axis corresponds to the complex numbers of the form a + i 0 and the points on the y-axis corresponds to the complex numbers of the form Fig 5. y) and Q (x. The representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are. However. there corresponds only one value of θ in 0 ≤ θ < 2π.1 Polar representation of a complex number Let the point P represent the nonzero complex number z = x + iy. We consider the origin as the pole and the positive Fig 5. We may note that the point P is uniquely determined by the ordered pair of real numbers (r.5 ( 0 ≤ θ < 2π ) Fig 5. called principal argument of z and is denoted by arg z. We have.We shall take the value of θ such that – π < θ ≤ π.4 direction of the x axis as the initial line. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis (Fig 5. can be such an interval. y = r sin θ and therefore. unless specified otherwise.5 and 5. For any complex number z ≠ 0. called the polar coordinates of the point P.106 MATHEMATICS bl is he 5.6 (– π < θ ≤ π ) . (Figs. The latter is said to be the polar form of the complex number. any other interval of length 2π. z = r (cos θ + i sin θ). θ). x = r cos θ.5.4).6) Fig 5. for example – π < θ ≤ π. 5. Here r = x 2 + y 2 = z is the © no N C tt E o R be T re pu modulus of z and θ is called the argument (or amplitude) of z which is denoted by arg z. 8 . sin θ = θ=π– π 2π = 3 3 2π 2π ⎞ ⎛ + i sin ⎟ Thus. cos θ = he r 2 cos 2 θ + sin 2 θ = 4 Fig 5. Let – 4 = r cos θ. r = 8 2 1 3 2 2 cos θ = − . sin θ = .8).e.. required polar form is z = 2 ⎜ cos + i sin ⎟ 3 3⎠ ⎝ is π 1 3 .7 bl π π⎞ ⎛ Therefore.. Solution Let 1 = r cos θ. r >0) Therefore. we get ( 2 2 2 16 + 48 = r cos θ + sin θ which gives Hence ) r = 64. 4 3 = r sin θ By squaring and adding. the required polar form is 8 ⎜ cos 3 3 ⎠ ⎝ Fig 5. 3 = r sin θ By squaring and adding.COMPLEX NUMBERS AND QUADRATIC EQUATIONS 107 Example 7 Represent the complex number z = 1 + i 3 in the polar form.7. we get ( ) i. i.e. which gives θ = 2 3 2 The complex number z = 1 + i 3 is represented as shown in Fig 5. r = 4 = 2 (conventionally. −16 into polar form. 1+ i 3 © no N C tt E o R be T re pu Example 8 Convert the complex number Solution The given complex number −16 1+ i 3 = −16 1 − i 3 × 1+ i 3 1− i 3 ( ) = –16 (1 – i 3 ) – 4 1 – i 3 = – 4 + i4 = ( ) 1+ 3 1 – (i 3 ) –16 1 – i 3 = 2 3 (Fig 5. b2 – 4ac = 12 – 4 × 1 × 1 = 1 – 4 = – 3 . Let us consider the following quadratic equation: ax 2 + bx + c = 0 with real coefficients a. z = – 3 + i Convert each of the complex numbers given in Exercises 3 to 8 in the polar form: 3.” Example 9 Solve x2 + 2 = 0 Solution We have.2 he Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.6 Quadratic Equations bl is We are already familiar with the quadratic equations and have solved them in the set of real numbers in the cases where discriminant is non-negative..e. © no N C tt E o R be T re pu Also. let us assume that the b2 – 4ac < 0.” As a consequence of this theorem. z = – 1 – i 3 2. we know that we can find the square root of negative real numbers in the set of complex numbers. the following theorem known as the Fundamental theorem of Algebra is stated below (without proof). which is of immense importance. – 3 7. 3 + i 8. b.. x2 + 2 = 0 or x2 = – 2 i.e. x = ± −2 = ± 2 i Example 10 Solve x2 + x + 1= 0 Solution Here. “A polynomial equation has at least one root. Now. Therefore. 1 – i 4. – 1 – i 6. the solutions to the above equation are available in the set of complex numbers which are given by x= −b ± b 2 − 4ac −b ± 4ac − b 2 i = 2a 2a $ Note At this point of time. the following result.108 MATHEMATICS EXERCISE 5. is arrived at: “A polynomial equation of degree n has n roots. – 1 + i 5. i. some would be interested to know as to how many roots does an equation have? In this regard. ≥ 0. 1. c and a ≠ 0. i 5. 3 is −1 ± −19 −1 ± 19i = . x2 – x + 2 = 0 © no N C tt E o R be T re pu Solve each of the following equations: 1. Solution We have . x2 + 3x + 5 = 0 bl EXERCISE 5. 2x2 + x + 2 = 0 x2 + x + 1 2 =0 3x 2 − 2 x + 3 3 = 0 8. x2 + x 2 +1 = 0 Miscellaneous Examples (3 − 2i)(2 + 3i) Example 12 Find the conjugate of (1 + 2i )(2 − i) . the discriminant of the equation is 12 − 4 × 5 × 5 = 1 – 20 = – 19 Therefore. . 9. the solutions are given by x = Example 11 Solve 109 5x2 + x + 5 = 0 he Solution Here. conjugate of (1 + 2i )(2 − i ) is 25 + 25 i . 2x2 + x + 1 = 0 4. x2 + 3x + 9 = 0 6. x2 + 3 = 0 2. the solutions are 3. (3 − 2i)(2 + 3i) (1 + 2i )(2 − i) = 6 + 9i − 4i + 6 12 + 5i 4 − 3i × = 2 − i + 4i + 2 4 + 3i 4 − 3i = 63 16 48 − 36i + 20i +15 63 −16i − i = = 25 25 16 + 9 25 (3 − 2i )(2 + 3i ) 63 16 Therefore. 10. – x2 + x – 2 = 0 5.COMPLEX NUMBERS AND QUADRATIC EQUATIONS −1 ± −3 −1 ± 3i = 2 ×1 2 Therefore. 2 5 2 5 7. Solution We have. we get r = θ = Therefore Hence. 1− i (ii) 1 1+ i 1+ i 1+ i 1+ i 1 − 1 + 2i × = =i= 0 + i = 1 − i 1− i 1+ i 1+1 Solution (i) We have.110 MATHEMATICS Example 13 Find the modulus and argument of the complex numbers: 1+ i . 2 1+ i 4 a + ib Example 14 If x + iy = a − ib .. let us put 0 = r cos θ. – = r sin θ 2 2 Proceeding as in part (i) above. (a + ib)(a + ib) a 2 − b 2 + 2abi a 2 − b2 2ab + 2 2i x + iy = (a − ib)(a + ib) = = 2 2 2 2 a +b a +b a +b . he (i) 1 = r sin θ Squaring and adding. r = 1 so that 2 π 2 1+ i π is 1 and the argument is . cosθ = .e. r = 1 i. sin θ = 1 (ii) We have 1 1− i 1− i 1 i = = = − 1 + i (1 + i ) (1 − i) 1 + 1 2 2 Let 1 1 = r cos θ. sin θ = 2 2 2 −π 4 1 1 −π is . θ = is cos θ = 0. the modulus of bl Therefore. argument is . Now. the modulus of 1 1 −1 . prove that x2 + y2 = 1. 1− i 2 © no N C tt E o R be T re pu Hence. (a 2 − b 2 )2 4a 2b 2 (a 2 + b 2 ) 2 + = 2 2 2 =1 x + y = (x + iy) (x – iy) = (a 2 + b 2 ) 2 (a 2 + b 2 )2 (a + b ) 2 he 2 Example 15 Find real θ such that is 3 + 2i sinθ is purely real.. i. n ∈ Z. Example 16 Convert the complex number z = Solution We have. z = = Now. 1− 2i sinθ © no N C tt E o R be T re pu 3 + 2i sinθ (3 + 2i sinθ)(1 + 2i sinθ) = (1 − 2i sinθ)(1 + 2i sinθ) 1 − 2i sinθ bl Solution We have.COMPLEX NUMBERS AND QUADRATIC EQUATIONS So that. cos + i sin 3 3 i −1 1 3 + i 2 2 ( ) 2(i − 1) 1 − 3i 2 i + 3 − 1 + 3i × = = 1+ 3 1 + 3i 1 − 3i 3 −1 = r cosθ . sin θ = 0 1 + 4sin 2θ Thus θ = nπ. x – iy = 111 a2 − b2 2ab − 2 2i 2 2 a +b a +b Therefore. Therefore 8sinθ = 0.e. 2 3 +1 = r sin θ 2 3 −1 + 2 3 +1 i 2 . 3+ 6i sinθ + 2i sinθ – 4sin 2θ 3 − 4sin 2θ 8i sinθ + = = 1+ 4sin 2θ 1 + 4sin 2θ 1 + 4sin 2θ We are given the complex number to be real. put i −1 π π in the polar form. ⎝ i ⎠ ⎥⎦ ⎢⎣ 2. 2 ⎞ ⎛ 3 − 4i ⎞ ⎛ 1 − 3. 27 x 2 − 10 x + 1 = 0 he 2 2 ⎛ 3 −1⎞ ⎛ 3 +1⎞ 2 ⎛⎜ r = ⎜⎜ + ⎟⎟ ⎜⎜ ⎟⎟ = ⎝ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 7.112 MATHEMATICS Squaring and adding. Evaluate: ⎢i + ⎜ ⎟ ⎥ . 2 6. θ = © no N C tt E o R be T re pu bl 5π 5π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ 12 12 ⎠ ⎝ Miscellaneous Exercise on Chapter 5 3 ⎡ 18 ⎛ 1 ⎞25 ⎤ 1. x2 − 2x + 3 =0 2 . (ii) ) 1 + 3i 1 – 2i Solve each of the equation in Exercises 6 to 9. sinθ = 2 2 2 2 π π 5π (Why?) + = 4 6 12 Hence. the polar form is is Therefore. Reduce ⎜ ⎟⎜ ⎟ to the standard form . 3 x − 4 x + 20 =0 3 8. prove that Re (z1 z2) = Re z1 Re z2 – Imz1 Imz2. For any two complex numbers z1 and z2. c − id c +d 5. If x − iy = 1 + 7i (i) (2 − i) 2 . we obtain cosθ = Hence. ⎝ 1 − 4i 1 + i ⎠ ⎝ 5 + i ⎠ 2 2 a − ib 2 2 2 a +b prove that x + y = 2 2 . r = 2 which gives ( 3) + 1⎞⎟ ⎠ = 2× 4 = 2 4 4 2 3 −1 3 +1 . Convert the following in the polar form: ( 4. Find the number of non-zero integral solutions of the equation 1 – i 19. 21x 2 − 28 x + 10 = 0 z1 + z2 +1 10. x y β–α 17. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i. If (x + iy)3 = u + iv. Find the modulus and argument of the complex number 1+ i 1− i 15. 16. Find ⎛ z1 z2 ⎞ (i) Re ⎜ z ⎟ . If a + ib = . If (a + ib) (c + id) (e + if) (g + ih) = A + iB. 1 − 3i 14. 2x2 + 1 1+ 2i . prove that a2 + b2 = 2 x 2 + 1 2 . = 2x . then show that u v + = 4( x 2 – y 2 ) . If z1 = 2 – i. then show that (a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B2 m ⎛1+ i ⎞ 20. 1 2 ( ) bl ⎛ 1 ⎞ ⎟. ⎝1 – i ⎠ x . Find the modulus of 1 − i − 1 + i . . ⎝ 1 ⎠ he ( x 2 + 1) 2 ( x + i )2 11. If α and β are different complex numbers with β = 1 . © no N C tt E o R be T re pu 13. Let z1 = 2 – i. If ⎜ ⎟ = 1 . ⎝ z1 z1 ⎠ (ii) Im ⎜ is 12. then find 1 – αβ 18.COMPLEX NUMBERS AND QUADRATIC EQUATIONS 113 9. find z – z +1 . then find the least positive integral value of m. z2 = 1 + i. z2 = –2 + i. a ≠ 0. denoted by z . there exists the ⎛ a2 −b ⎞ +i 2 ⎟ = 1 + i0 =1 multiplicative inverse of z such that (a + ib) ⎜ 2 2 a + b2 ⎠ ⎝a +b © no N C tt E o R be T re pu 114 For any integer k. c ∈ R. Then (i) z1 + z2 = (a + c) + i (b + d) (ii) z1 z2 = (ac – bd) + i (ad + bc) a −b 1 +i 2 or z –1. b ≠ 0). The value of θ. called the 2 2 . such that – π < θ ≤ π. sinθ = . where a. i4k = 1. The polar form of the complex number z = x + iy is r (cosθ + i sinθ). a is called the real part and b is called the imaginary part of the complex number. denoted by a +b a +b z 2 bl complex number is For any non-zero complex number z = a + ib (a ≠ 0. r= x 2 + y 2 (the modulus of z) and cosθ = A polynomial equation of n degree has n roots. he Let z1 = a + ib and z2 = c + id. is called the principal argument of z. i4k + 3 = – i The conjugate of the complex number z = a + ib. is called a complex number. where a and b are real numbers. (θ is known as the r r argument of z. b2 – 4ac < 0. where x y . . b. i4k + 2 = – 1. The solutions of the quadratic equation ax2 + bx + c = 0. is given by z = a – ib. are given by x = −b ± 4ac − b 2 i 2a .MATHEMATICS Summary A number of the form a + ib. i4k + 1 = i. xy = 40. it has.R. Albert Girard (about 1625) accepted square root of negative numbers and said that this will enable us to get as many roots as the degree of the polynomial equation.COMPLEX NUMBERS AND QUADRATIC EQUATIONS 115 Historical Note x + y = 10.” Cardan (1545) considered the problem of solving © no N C tt E o R be T re pu bl He obtained x = 5 + −15 and y = 5 – −15 as the solution of it. Hamilton (about 1830) regarded the complex number a + ib as an ordered pair of real numbers (a. therefore. Euler was the first to introduce the symbol i for −1 and W. which was discarded by him by saying that these numbers are ‘useless’. But the credit goes to the Indian mathematician Mahavira (850) who first stated this difficulty clearly. “There is no square root of a negative quantity. no square root”. “He mentions in his work ‘Ganitasara Sangraha’ as in the nature of things a negative (quantity) is not a square (quantity)’. — — . is he The fact that square root of a negative number does not exist in the real number system was recognised by the Greeks. written in 1150. another Indian mathematician. also writes in his work Bijaganita. b) thus giving it a purely mathematical definition and avoiding use of the so called ‘imaginary numbers’. for it is not a square. Bhaskara. Now a natural question arises: ‘Is it always possible to translate a statement problem in the form of an equation? For example. Since. we will study linear inequalities in one and two variables. Here we get certain statements involving a sign ‘<’ (less than). In this case... The study of inequalities is very useful in solving problems in the field of science. statistics. (2) .Chapter 6 he LINEAR INEQUALITIES is Mathematics is the art of saying many things in many different ways. then the total amount spent by her is Rs (40x + 20y) and we have 40x + 20y ≤ 120 . optimisation problems. 6. If x denotes the number of packets of rice. The cost of one register is Rs 40 and that of a pen is Rs 20. psychology. which is available in packets of 1kg. The price of one packet of rice is Rs 30. Your classroom can occupy atmost 60 tables or chairs or both. he may not be able to spend the entire amount of Rs 200. the height of all the students in your class is less than 160 cm. ‘>’ (greater than). we have studied equations in one variable and two variables and also solved some statement problems by translating them in the form of equations.2 Inequalities Let us consider the following situations: (i) Ravi goes to market with Rs 200 to buy rice. economics.. (ii) Reshma has Rs 120 and wants to buy some registers and pens. (Why?) Hence 30x < 200 . mathematics. the number of pens which Reshma buys. ‘≤’ (less than or equal) and ≥ (greater than or equal) which are known as inequalities. which he buys.1 Introduction © no N C tt E o R be T re pu bl In earlier classes. (1) Clearly the statement (i) is not an equation as it does not involve the sign of equality.. etc. he has to buy rice in packets only. then the total amount spent by him is Rs 30x. In this Chapter. – MAXWELL 6. if x denotes the number of registers and y. e. (14) 2 Inequalities (5)... © no N C tt E o R be T re pu Some more examples of inequalities are: ax + b < 0 . it is an inequality while statement (4) is an equation... (9). (13) ax2 + bx + c > 0 . (4) he Statement (3) is not an equation. (12).. 7 > 5 are the examples of numerical inequalities while bl x < 5.LINEAR INEQUALITIES 117 Since in this case the total amount spent may be upto Rs 120.. and (13) are slack inequalities.. Inequalities (13) and (14) are not linear (in fact... ‘>’... (12) ax + bx + c ≤ 0 . 3 < 5. these are quadratic inequalities in one variable x when a ≠ 0)... (6) ax + b ≤ 0 . while inequalities from (9) to (12) are linear inequalities in two variables x and y when a ≠ 0.. Definition 1 Two real numbers or two algebraic expressions related by the symbol is ‘<’... (11). (11) ax + by ≥ c .. (5) ax + b > 0 . In this Chapter... (9) ax + by > c . Note that the statement (2) consists of two statements and 40x + 20y < 120 40x + 20y = 120 . y ≤ 4 are some examples of literal inequalities. (3) .. Inequalities from (5) to (8) are linear inequalities in one variable x when a ≠ 0. . i. we shall confine ourselves to the study of linear inequalities in one and two variables only. 3 < 5 < 7 (read as 5 is greater than 3 and less than 7).. y > 2. (8) ax + by < c ... (10) ax + by ≤ c . (2) and (3) above are inequalities. (8). (6).. ‘≤’ or ‘≥’ form an inequality.. b ≠ 0. (7) ax + b ≥ 0 . 3 < x < 5 (read as x is greater than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities. Statements such as (1). x ≥ 3. (10) and (14) are strict inequalities while inequalities (7). H.S. Before that we should go through some more properties of numerical inequalities and follow them as rules while solving the inequalities.H. Therefore.S. = 30 (7) = 210 < 200. For x = 3. which make above inequality a true statement. we have For x = 0.5. L. we find that the values of x.S.. We have found the solutions of the above inequality by trial and error method which is not very efficient. = 30 (2) = 60 < 200. the sign of inequality is reversed (i. = 30 (0) = 0 < 200 (R. Rule 2 Both sides of an equation may be multiplied (or divided) by the same non-zero number.5. = 30 (5) = 150 < 200. ≤’ becomes ‘≥’ and so on) whenever we multiply (or divide) both sides of an inequality by a negative number. In the case of solving inequalities. For x = 1.S. For x = 4. For x = 7.118 MATHEMATICS 6.e.H.2. In the above situation.S. We must have some better or systematic techniques for solving inequalities.). 16 > 14.S.2. For x = 5.H.6} is called its solution set. L. L. we again follow the same rules except with a difference that in Rule 2. which is true. which is true. Obviously. x cannot be a negative integer or a fraction. ‘<‘ becomes ‘>’.H.2.S.S.H. = 30 (3) = 90 < 200. = 30 (4) = 120 < 200.H.H. are 0. For x = 2. L. which makes the above inequality a true statement. These values of x. Thus. L. L. which is true.). which is true. = 30 (1) = 30 < 200 (R.H. are called solutions of inequality and the set {0.1.S.3.H. = 30 (6) = 180 < 200. any solution of an inequality in one variable is a value of the variable which makes it a true statement.6.) of this inequality is 30x and right hand side (RHS) is 200.e. L. this method is time consuming and sometimes not feasible. You will recall that while solving linear equations.. which is true. . Left hand side (L. which is false. we followed the following rules: Rule 1 Equal numbers may be added to (or subtracted from) both sides of an equation.3.1. It is evident from the facts that 3 > 2 while – 3 < – 2. L.4. which is true. – 8 < – 7 while (– 8) (– 2) > (– 7) (– 2) .3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation © no N C tt E o R be T re pu bl is he Let us consider the inequality (1) of Section 6.S. which is true. 30x < 200 Note that here x denotes the number of packets of rice.S.H.4. i. For x = 6. Obviously. viz. e.3. – 2. 1. Henceforth. set of integers and in the set of real numbers.. the solutions of the given inequality are . Solution We have. But when both sides are multiplied or divided by a negative number. 1. we state the following rules for solving an inequality: Rule 1 Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality. (ii) When x is an integer. . 6 The solution set of the inequality is {. 2). in this case the following values of x make the statement true. 0. 3. (ii) x is a real number. 2.. –1. 2. 6. – 1. the solutions of the inequality are given by x < 2.LINEAR INEQUALITIES 119 Thus. 4. 5.–3. 3. 5x –3 < 3x + 1 or 5x –3 + 3 < 3x +1 +3 (Rule 1) or 5x < 3x +4 or 5x – 3x < 3x + 4 – 3x (Rule 1) or 2x < 4 or x < 2 (Rule 2) (i) When x is an integer. 5. he Rule 2 Both sides of an inequality can be multiplied (or divided) by the same positive number.6}. © no N C tt E o R be T re pu bl 30 x 200 < (Rule 2).e... 4.. 4. all real numbers x which are less than 2. 0..2.. –2. 6} or Example 2 Solve 5x – 3 < 3x +1 when (i) x is an integer. 3. 1. 0. –2. i. the solutions of the given inequality are .5. x < 20 / 3... unless stated otherwise. – 3.4.. The solution set of the inequality is {1. let us consider some examples. 1 (ii) When x is a real number. 5. We have considered solutions of inequalities in the set of natural numbers. we shall solve the inequalities in this Chapter in the set of real numbers. is Example 1 Solve 30 x < 200 when (i) x is a natural number. – 3. the solution set of the inequality is x ∈ (– ∞.–1. i. 30 30 (i) When x is a natural number. then the sign of inequality is reversed. Solution We are given 30 x < 200 (ii) x is an integer. Therefore. – 4.. 2. Now. . i. the solution set is (–2. Show the graph of the solutions on number line.e. x ∈ [8. i. 3 6 is Solution We have © no N C tt E o R be T re pu bl 5 – 2x x ≤ –5 3 6 or 2 (5 – 2x) ≤ x – 30. ∞).120 MATHEMATICS Example 3 Solve 4x + 3 < 6x +7. Fig 6.e.1. 4x + 3 < 6x + 7 or 4x – 6x < 6x + 4 – 6x or – 2x < 4 or x > – 2 i. Solution We have 7x + 3 < 5x + 9 or 2x < 6 or x < 3 The graphical representation of the solutions are given in Fig 6. x ≥ 8 Thus.1 Example 6 Solve 3x − 4 x + 1 ≥ −1 . Hence..e. ∞). 2 4 Solution We have 3x − 4 x + 1 ≥ −1 2 4 or or 3x − 4 x − 3 ≥ 2 4 2 (3x – 4) ≥ (x – 3) .. 5 – 2x x ≤ – 5. all real numbers x which are greater than or equal to 8 are the solutions of the given inequality. Show the graph of the solutions on number line. Example 5 Solve 7x + 3 < 5x + 9. are the solutions of the given inequality. or 10 – 4x ≤ x – 30 or – 5x ≤ – 40. all the real numbers which are greater than –2. Example 4 Solve he Solution We have. . respectively.. 19) . 13). x can take the values 11. (3) Since x is an odd number. bl Solution Let x be the marks obtained by student in the annual examination. 15). 17).. the student must obtain a minimum of 70 marks to get an average of at least 60 marks. Fig 6. such that their sum is less than 40. Example 8 Find all pairs of consecutive odd natural numbers. Then © no N C tt E o R be T re pu 62 + 48 + x ≥ 60 3 or 110 + x ≥ 180 or x ≥ 70 Thus. x < 19 From (1) and (3). we should have x > 10 and x + ( x + 2) < 40 Solving (2).. and 17. 13.2.. we get 10 < x < 19 . (13.2 is Example 7 The marks obtained by a student of Class XI in first and second terminal examination are 62 and 48. (1) .. Solution Let x be the smaller of the two consecutive odd natural number. the required possible pairs will be (11. both of which are larger than 10. so that the other one is x +2.e. 15. Find the minimum marks he should get in the annual examination to have an average of at least 60 marks. So.. (17. (15.LINEAR INEQUALITIES or 6x – 8 ≥ x – 3 or 5x ≥ 5 or x ≥ 1 121 he The graphical representation of solutions is given in Fig 6. (2) . Then. we get 2x + 2 < 40 i. x x x + + < 11 2 3 10. To receive Grade ‘A’ in a course. find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course. 3x – 7 > 5x – 1 7. 19. 4. 18. . 3(x – 1) ≤ 2 (x – 3) 8. (ii) x is an integer. when (i) x is a natural number. Find the minimum marks he should get in the third test to have an average of at least 60 marks. when (i) x is an integer. 2. Solve 5x – 3 < 7. 15. 37 – (3x + 5) > 9x – 8 (x – 3) 3( x − 2) 5(2 − x) ≤ 5 3 13. If Sunita’s marks in first four examinations are 87. 3 (1 – x) < 2 (x + 4) 21. 3x – 2 < 2x + 1 x (5 x – 2) (7 x – 3) ≥ – 2 3 5 Ravi obtained 70 and 75 marks in first two unit test. Find all pairs of consecutive even positive integers. 22. (ii) x is a real number. (ii) x is a real number. 3 (2 – x) ≥ 2 (1 – x) he 1. when (i) x is an integer. (2 x −1) (3 x − 2) (2 − x ) ≥ − 3 4 5 Solve the inequalities in Exercises 17 to 20 and show the graph of the solution in each case on number line 17. bl is Solve the inequalities in Exercises 5 to 16 for real x. x x > +1 3 2 © no N C tt E o R be T re pu 9. 2 (2x + 3) – 10 < 6 (x – 2) 12.1 (ii) x is an integer. 1 ⎛ 3x ⎞ 1 ⎜ + 4 ⎟ ≥ ( x − 6) 2⎝ 5 ⎠ 3 14. when (i) x is a natural number. 94 and 95. 3. 92. Solve 24x < 100. one must obtain an average of 90 marks or more in five examinations (each of 100 marks). Solve 3x + 8 >2. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11. 5. 5x – 3 > 3x – 5 20. 11. 24.122 MATHEMATICS EXERCISE 6. 4x + 3 < 5x + 7 6. 23. Solve – 12x > 30. x (5 x − 2) (7 x − 3) < − 4 3 5 16. both of which are larger than 5 such that their sum is less than 23. 4 A point in the Cartesian plane will either lie on a line or will lie in either of the half planes I or II. Now. A vertical line will divide the plane in left and right half planes and a non-vertical line will divide the plane into lower and upper half planes (Figs. a ≠ 0. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest.4). If the perimeter of the triangle is at least 61 cm. find the minimum length of the shortest side. respectively. We know that a line divides the Cartesian plane into two parts. Fig 6. A man wants to cut three lengths from a single piece of board of length 91cm. we have seen that a graph of an inequality in one variable is a visual representation and is a convenient way to represent the solutions of the inequality.LINEAR INEQUALITIES 123 is he 25.. We shall now examine the relationship. we will discuss graph of a linear inequality in two variables.4 Graphical Solution of Linear Inequalities in Two Variables © no N C tt E o R be T re pu bl In earlier section. (x + 3) and 2x are the lengths of the second and third piece. 6. Thus. of the points in the plane and the inequalities ax + by < c or ax + by > c. Each part is known as a half plane.. (1) . x + (x + 3) + 2x ≤ 91 and 2x ≥ (x + 3) + 5]. b ≠ 0 . then x . if any. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. Let us consider the line ax + by = c. 26.3 Fig 6. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second? [Hint: If x is the length of the shortest board.3 and 6. 6. Consider a point P (α. Q(α. any point in the half plane II satisfies ax + by > c.5 Thus. from Fig 6. and conversely. we deduce that all points satisfying ax + by > c lies in one of the half planes II or I according as b > 0 or b < 0. it is just sufficient to take any point (a. Hence. Fig 6. all the points lying in the half plane II above the line ax + by = c satisfies the inequality ax + by > c. Thus. so that aα + bβ = c. Consider case (ii). Now. 2. then the inequality represents the half plane and shade the region . β) be a point on line ax + by = c and an arbitrary point Q(α. and conversely. If it satisfies. let (α. b) (not online) and check whether it satisfies the inequality or not.Take an arbitrary point Q (α .. In case (i). In order to identify the half plane represented by an inequality. all points (x. and conversely any point satisfying the inequality ax + by > c lies in half plane II.124 MATHEMATICS © no N C tt E o R be T re pu bl is he There are three possibilities namely: (i) ax + by = c (ii) ax + by > c (iii) ax + by < c. Thus. γ ) lies in the half plane II. Conversely. clearly. y) satisfying (i) lie on the line it represents and conversely.β) on the line ax + by = c. graph of the inequality ax + by > c will be one of the half plane (called solution region) and represented by shading in the corresponding half plane. let us first assume that b > 0.5. Note 1 The region containing all the solutions of an inequality is called the $ solution region. γ) satisfying ax + by > c so that aα + bγ > c aα + b γ > aα + bβ (Why?) ⇒ γ>β (as b > 0) ⇒ This means that the point (α. γ) in the half plane II (Fig 6. we interpret.e. γ > β (Why?) or b γ > bβ or aα + b γ > aα + bβ (Why?) or aα + b γ > c i. γ ) satisfies the inequality ax + by > c. b > 0. we can similarly prove that any point satisfying ax + by > c lies in the half plane I.5). In case b < 0. and see what will be the solutions of (1) in this case.3).LINEAR INEQUALITIES 125 he which contains the point.H. 5.4). the solutions of (1) are (0. (3) In order to draw the graph of the inequality (1).. 4). which make the statement (1) true. 0). 2)..6 inequality or not. So draw a broken or dotted line in the solution region. 1. 2 and 3 are: (1. 0) This is shown in Fig 6. (0. For this purpose. since the number of articles cannot be a fraction or a negative number. (1. 0). the point (0.2). the set of such pairs will be the solution set of the inequality (1). 6 only.S.. So draw a dark line in the solution region. then the points on the line ax + by = c are also included in the solution region. (0. 0) is preferred. 5) and (0. the corresponding values of y can be 0. To start with.. (1. then the points on the line ax + by = c are not to be included in the solution region. 3. 2). . other solutions of (1). (0. 4. In this case. in half plane I and check whether values of x and y satisfy the Fig 6. (2. we find the pairs of values of x and y.6. For convenience. otherwise.. In fact. of (1) is 40x + 20y = 40 (0) + 20y = 20y. 4. 3). 2. (2. Let us now solve this inequality keeping in mind that x and y can be only whole numbers. (0. let us consider the (corresponding) equation and draw its graph. when x = 1. Let us now extend the domain of x and y from whole numbers to real numbers. If an inequality is of the form ax + by > c or ax + by < c. let x = 0. You will see that the graphical method of solution will be very convenient in this case. (1. (1. 40x + 20y = 120 . 0). Similarly. (2. (2) For x = 0.2. 1). the inequality represents that half plane which does not contain the point within it. 6). © no N C tt E o R be T re pu bl is In Section 6. (0. In this case. 1). we have 20y ≤ 120 or y ≤ 6 . we take one point say (0. we obtained the following linear inequalities in two variables x and y: 40x + 20y ≤ 120 .. If an inequality is of the type ax + by ≥ c or ax + by ≤ c. 3. 1). (3. (1) while translating the word problem of purchasing of registers and pens by Reshma. 0). Then L. Thus. is We observe that x = 0.8 excluding the points on the line is the solution region of the inequality.7) of the inequality. We select a point (not on the line). the solution region is the shaded region on the right hand side of the line x = 2.Example 9 Solve 3x + 2y > 6 graphically. 0). the line is also a part of the graph. Clearly. the shaded half plane II Fig 6.9.9 . say (0. we see that: 3 (0) – 6 ≥ 0 or – 6 ≥ 0 which is false. we say that the half plane I is the graph (Fig 6. half plane I is not the solution region of the given inequality. Solution Graph of 3x – 6 = 0 is given in the Fig 6. we note that 3 (0) + 2 (0) > 6 or 0 > 6 . he MATHEMATICS bl 126 Fig 6. Fig 6. Hence. Hence.7 © no N C tt E o R be T re pu Solution Graph of 3x + 2y = 6 is given as dotted line in the Fig 6. We shall now consider some examples to explain the above procedure for solving a linear inequality involving two variables. which is false. Example 10 Solve 3x – 6 ≥ 0 graphically in two dimensional plane. Thus. any point on the line does not satisfy the given strict inequality. Thus. which lies in one of the half planes (Fig 6.8. y = 0 satisfy the inequality. In other words. Clearly half plane II is not the part of the graph.8) and determine if this point satisfies the given inequality. We select a point. the graph of the given inequality is half plane I including the line itself. solutions of inequality (1) will consist of all the points of its graph (half plane I including the line). Since the points on the line also satisfy the inequality (1) above. This line divides the xy-plane in two half planes I and II. 0) and substituting it in given inequality. Thus. say (0. we draw the graph of the equation x – y = 3 as Fig 6. Solution Graph of y = 2 is given in the Fig 6.11 shown in Fig 6. Then we note that inequality (2) represents the shaded region above ..11. Thus.2 Fig 6. We note that solution of inequality (1) is represented by the shaded region above the line x + y = 5. On the same set of axes. 0) in lower half plane I and putting y = 0 in the given inequality. you have learnt how to solve linear inequality in one or two variables graphically. x + y < 5 2. (1) x–y≤3 . 2x + y ≥ 6 3. We will now illustrate the method for solving a system of linear inequalities in two variables graphically through some examples. we see that 1 × 0 < 2 or 0 < 2 which is true. – 3x + 2y ≥ – 6 8. y < – 2 10.10..5 Solution of System of Linear Inequalities in Two Variables In previous Section.. 127 he LINEAR INEQUALITIES © no N C tt E o R be T re pu Solve the following inequalities graphically in two-dimensional plane: 1. y + 8 ≥ 2x 5. Let us select a point. 3x + 4y ≤ 12 4.. (2) Solution The graph of linear equation x+y=5 is drawn in Fig 6.11.10 is Example 11 Solve y < 2 graphically. (0. x – y ≤ 2 6. x+y≥5 . Hence. including the points on the line.bl EXERCISE 6. 2x – 3y > 6 7. x > – 3. Example 12 Solve the following system of linear inequalities graphically. the solution region is the shaded region below the line y = 2. 6. every point below the line (excluding all the points on the line) determines the solution of the given inequality. 3y – 5x < 30 9. the double shaded region. (2) y≥0 .. Clearly. he Example 13 Solve the following system of inequalities graphically 5x + 4y ≤ 40 ... Fig 6. (3) Fig 6. (2) y≥3 .. (1) x≥2 . etc. number of hours worked... Clearly.13).128 MATHEMATICS the line x – y = 3. including the points on the line. in such cases. x = 2 and y = 3 Then we note that the inequality (1) represents shaded region below the line 5x + 4y = 40 and inequality (2) represents the shaded region right of line x = 2 but inequality (3) represents the shaded region above the line y = 3. In many practical situations involving system of inequalities the variable x and y often represent quantities that cannot have negative values. (3) Solution We draw the graph of the line 8x + 3y = 100 The inequality 8x + 3y ≤ 100 represents the shaded region below the line. number of units produced.12) including all the point on the lines are also the solution of the given system of the linear inequalities.. x ≥ 0. number of articles purchased... common to the above two shaded regions is the required solution region of the given system of inequalities.12 Example 14 Solve the following system of inequalities 8x + 3y ≤ 100 . (1) x≥0 .13 . Hence. shaded region (Fig 6. y ≥ 0 and the solution region lies only in the first quadrant... for example.. including the points on the line 8x +3y =100 (Fig 6. is bl © no N C tt E o R be T re pu Solution We first draw the graph of the line 5x + 4y = 40. 2x + y ≥ 6.. The inequality (1) and (2) represent Fig 6. 3x + 4y < 12 4. EXERCISE 6. x + 2y ≤ 10. every point in the shaded region in the first quadrant represent a solution of the given system of inequalities (Fig 6. y ≥ 0. y ≥ 2 3. x ≥ 0. including the points on the line and the axes. 3x + 4y ≤ 60.. x – 2y ≤ 3. x ≤ 15. 3x + 4y ≥ 12. 2x + y ≥ 4. x ≥ 1. 2x + y ≥ 8. 3x + 2y ≤ 150.. x + y ≤ 9. x +3y ≤ 30. y ≥ 2 2. 3x + 2y ≤ 12. x ≥ 0 15. x + 2y ≥ 10 8. x + y ≥ 1. x ≥ 0. x + y ≥ 4.14). x + y ≤ 3. x ≥ 3. y ≥ 1 13. 2x – y >1. x – 2y < – 1 6.. x + y ≤ 6. y ≥ 2x. x ≥ 0 . 2x – y > 0 5. x – y ≤ 0. y > x. y ≥ 0.. 4x + 3y ≤ 60. represents the solution of the given system of inequalities.LINEAR INEQUALITIES 129 Since x ≥ 0. y ≥ 0 . Since x ≥ 0. including the point on the respective lines.. x.14 the region below the two lines. x ≥ 1. (1) 2x + y ≤ 8 . (4) © no N C tt E o R be T re pu bl Solution We draw the graphs of the lines x + 2y = 8 and 2x + y = 8.3 Solve the following system of inequalities graphically: 1. x ≥ 3. every point in the shaded region in the first quadrant.. y ≥ 0. x + 4y ≤ 80.. y ≥ 0 11. 5x + 4y ≤ 20. x + y ≥ 4 7. (3) x>0 y>0 . is he Example 15 Solve the following system of inequalities graphically x + 2y ≤ 8 . (2) . 2x – 3y ≤ 6 12. y ≥ 2 10. y ≥ 0 14. x ≥ 0 9. we have 3x – 7 < 5 + x or x<6 . x ≥ 2 ..15. 2≤x<6 . we have 11 – 5 x ≤ 1 or – 5 x ≤ – 10 i. 2 bl Example 17 Solve – 5 ≤ he Solution In this case. Fig 6. are shown by bold line in Fig 6. (2) Solution From inequality (1)... (1) .e. which are common to both.. (3) Also. which we will solve simultaneously.15 Thus... We have – 8 ≤ 5x –3 < 7 or –5 ≤ 5x < 10 or –1 ≤ x < 2 .130 MATHEMATICS Miscellaneous Examples Example 16 Solve – 8 ≤ 5x – 3 < 7. i. we see that the values of x. from inequality (2)... (4) If we draw the graph of inequalities (3) and (4) on the number line.. © no N C tt E o R be T re pu which can be written as is Solution We have – 5 ≤ 5 – 3x ≤ 8.. – 8 ≤ 5x – 3 and 5x – 3 < 7. we have two inequalities.e. or –10 ≤ 5 – 3x ≤ 16 or 5≥x≥– 5 – 3x ≤8 2 or – 15 ≤ – 3x ≤ 11 11 3 –11 ≤ x ≤5 3 Example 18 Solve the system of inequalities: 3x – 7 < 5 + x 11 – 5 x ≤ 1 and represent the solutions on the number line. solution of the system are real numbers x lying between 2 and 6 including 2. 9 9 9 × (30) < (F – 32) < × (35) 5 5 bl Putting is Solution It is given that 30 < C < 35.LINEAR INEQUALITIES 131 Example 19 In an experiment. we get 9 30 < 5 (F – 32) < 35. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%? Solution Let x litres of 30% acid solution is required to be added. Example 20 A manufacturer has 600 litres of a 12% solution of acid. 30x + 7200 > 15x + 9000 30x + 7200 < 18x + 10800 15x > 1800 and 12x < 3600 x > 120 and x < 300. where C and F represent temperature in degree 9 he Celsius and degree Fahrenheit. Then Total mixture = (x + 600) litres Therefore 30% x + 12% of 600 > 15% of (x + 600) and 30% x + 12% of 600 < 18% of (x + 600) or 30 x 12 15 + (600) > (x + 600) 100 100 100 and 30 x 12 18 + (600) < (x + 600) 100 100 100 or and or or i. respectively. or C= 5 (F – 32). Thus.e. 120 < x < 300 . the required range of temperature is between 86° F and 95° F. What is the range of temperature in degree Fahrenheit if conversion formula is given by C = 5 (F – 32). a solution of hydrochloric acid is to be kept between 30° and 35° Celsius. © no N C tt E o R be T re pu or 54 < (F – 32) < 63 or 86 < F < 95. If 80 ≤ IQ ≤ 140 for a group of 12 years old children.132 MATHEMATICS Thus. 2 ≤ 3x – 4 ≤ 5 5x + 1 > – 24. If we have 640 litres of the 8% solution. 10. A solution is to be kept between 68° F and 77° F. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content? 14. how many litres of the 2% solution will have to be added? 13. 1. find the range of their mental age. 6 – x > 11 – 2x 5 (2x – 7) – 3 (2x + 3) ≤ 0 . 8. . What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by F= 9 C + 32 ? 5 12. −12 < 4 − 7x ≤ 18 2 4. 2x + 19 ≤ 6x + 47 . 9. 2 is Solve the inequalities in Exercises 1 to 6. 6 ≤ – 3 (2x – 4) < 12 ( 3x +11 ) ≤ 11 . IQ of a person is given by the formula IQ = MA × 100. the number of litres of the 30% solution of acid will have to be more than 120 litres but less than 300 litres. 2. CA where MA is mental age and CA is chronological age. A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. bl Solve the inequalities in Exercises 7 to 10 and represent the solution graphically on number line. © no N C tt E o R be T re pu 7. 7 ≤ 3( x − 2 ) ≤0 5 he 3. −15 < 3x ≤2 −5 6. 11. Miscellaneous Exercise on Chapter 6 – 3≤ 4 − 5. 3 (x + 2) > 2 – x 3x – 7 > 2 (x – 6) . 5x – 1 < 24 2 (x – 1) < x + 5. The resulting mixture is to be more than 4% but less than 6% boric acid. put a dark circle on the number a and dark the line to the left (or right) of the number x. To represent x ≤ a (or x ≥ a) on a number line. To represent x < a (or x > a) on a number line. The values of x. then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies to the left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part. Both sides of an inequality can be multiplied (or divided ) by the same positive © no N C tt E o R be T re pu bl is he number. are called solutions of the inequality.LINEAR INEQUALITIES 133 Summary Two real numbers or two algebraic expressions related by the symbols <. Equal numbers may be added to (or subtracted from ) both sides of an inequality. then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by dark line that satisfies an arbitrary point in that part. If an inequality is having < or > symbol. — — . >. But when both sides are multiplied (or divided) by a negative number. ≤ or ≥ form an inequality. then the inequality is reversed. If an inequality is having ≤ or ≥ symbol. put a circle on the number a and dark line to the left (or right) of the number a. The solution region of a system of inequalities is the region which satisfies all the given inequalities in the system simultaneously. which make an inequality a true statement. 1 Introduction Suppose you have a suitcase with a number lock. these techniques will be useful in determining the number of different ways of arranging and selecting objects without actually listing them. You remember only the first digit which is 7. immediately. you may. How many different pairs of a pant and a shirt. we shall examine a principle which is most fundamental to the learning of these techniques. The number lock has 4 wheels each labelled with 10 digits from 0 to 9. because the number of possible (1654-1705) sequences may be large. Similarly.2 Fundamental Principle of Counting Let us consider the following problem. As a first step. Jacob Bernoulli this method will be tedious. . a shirt can be chosen in 2 ways. we shall learn some basic counting techniques which will enable us to answer this question without actually listing 3-digit arrangements.Chapter 7 PERMUTATIONS AND COMBINATIONS Every body of discovery is mathematical in form because there is no other guidance we can have – DARWIN 7. But. Therefore. Mohan has 3 pants and 2 shirts. can he dress up with? There are 3 ways in which a pant can be chosen. there are 2 choices of a shirt. how many sequences of 3-digits you may have to check with? To answer this question. Some how. you have forgotten this specific sequence of digits. For every choice of a pant. The lock can be opened if 4 specific digits are arranged in a particular sequence with no repetition. Here. In order to open the lock. in this Chapter. start listing all possible arrangements of 9 remaining digits taken 3 at a time. because there are 3 pants available. In fact. 7. there are 3 × 2 = 6 pairs of a pant and a shirt. 7. Fig 7. the three tiffin boxes as T1. B2. 3 tiffin boxes and 2 water bottles. Let us consider another problem of the same type. there are 6 × 2 = 12 different ways in which.2 . these six possibilities can be illustrated in the Fig.PERMUTATION S AND COMBINATIONS 135 Let us name the three pants as P1. P2 . Sabnam has 2 school bags. 7. Hence.1 Hence. W2 . In how many ways can she carry these items (choosing one each). Fig 7. S2. A school bag can be chosen in 2 different ways. a tiffin box can be chosen in 3 different ways. P 3 and the two shirts as S1. T3 and the two water bottles as W1.1.2. For each of these pairs a water bottle can be chosen in 2 different ways. After a school bag is chosen. these possibilities can be illustrated in the Fig. If we name the 2 school bags as B1. T2. Sabnam can carry these items to school. Then. there are 2 × 3 = 6 pairs of school bag and a tiffin box. the required number of ways was the number of different ways of the occurence of the following events in succession: (i) (ii) (iii) the event of choosing a school bag the event of choosing a tiffin box the event of choosing a water bottle. following which the third place can be filled in 2 different ways. Hence. following which another event can occur in n different ways. by the multiplication principle.E. following which a third event can occur in p different ways. where the repetition of the letters is not allowed. then the total number of occurrence of the events in the given order is m×n.O.136 MATHEMATICS In fact. Thus. the second place can be filled in by anyone of the remaining 3 letters in 3 different ways. the required number of words is 24. which can be formed out of the letters of the word ROSE. the multiplication principle. Solution There are as many words as there are ways of filling in 4 vacant places by the 4 letters. then the total number of occurrence to ‘the events in the given order is m × n × p. the required number of ways of wearing a pant and a shirt was the number of different ways of the occurence of the following events in succession: (i) (ii) the event of choosing a pant the event of choosing a shirt. the fourth place can be filled in 1 way. simply.” The above principle can be generalised for any finite number of events. In the second problem. following which another event can occur in n different ways. we have to choose any one of the possible orders and count the number of different ways of the occurence of the events in this chosen order. which states that “If an event can occur in m different ways. for 3 events. is 4 × 3 × 2 × 1 = 24.S. Following which. . The first place can be filled in 4 different ways by anyone of the 4 letters R. keeping in mind that the repetition is not allowed. For example. following which. the problems of the above types are solved by applying the following principle known as the fundamental principle of counting. the principle is as follows: ‘If an event can occur in m different ways. the events in each problem could occur in various possible orders. with or without meaning. But. Here.” In the first problem. Example 1 Find the number of 4 letter words. in both the cases. or. the number of ways in which the 4 places can be filled. Here. we start filling in unit’s place. the number of ways is 5 × 4 = 20. 3 flags. following which. 3 flags. how many different signals can be generated. there will be as many 3 flag signals as there are ways of filling in 3 vacant places in succession by the 5 flags. if five different flags are available. the required number of words = 4 × 4 × 4 × 4 = 256. i. 4 flags or 5 flags. 4. by the multiplication principle. 3. Hence. 2. Solution A signal can consist of either 2 flags. Now.. . following which the ten’s place can be filled by any of the 5 digits in 5 different ways as the digits can be repeated. Hence.e.PERMUTATION S AND COMBINATIONS 137 Note If the repetition of the letters was allowed. By Multiplication rule. Similarly. Example 3 How many 2 digit even numbers can be formed from the digits 1. let us count the possible number of signals consisting of 2 flags. how many words can be formed? One can easily understand that each of the 4 vacant places can be filled in succession in 4 different ways. 4 flags and 5 flags separately and then add the respective numbers. the required number of two digits even numbers is 2 × 5. in this case. because the options for this place are 2 and 4 only and this can be done in 2 ways. Example 4 Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff. The upper vacant place can be filled in 4 different ways by anyone of the 4 flags. by the multiplication principle. Example 2 Given 4 flags of different colours. the required number of signals = 4 × 3 = 12. 10. the lower vacant place can be filled in 3 different ways by anyone of the remaining 3 different flags. There will be as many 2 flag signals as there are ways of filling in 2 vacant places in succession by the 5 flags available. Therefore. 5 if the digits can be repeated? Solution There will be as many ways as there are ways of filling 2 vacant places in succession by the five given digits. if a signal requires the use of 2 flags one below the other? Solution There will be as many signals as there are ways of filling in 2 vacant places in succession by the 4 flags of different colours. The required number of words = 6 × 5 × 4 = 120 (by using multiplication principle). REOS. How many possible outcomes are there? Given 5 flags of different colours. if we have to determine the number of 3-letter words. .1 1. which can be formed out of the letters of the word NUMBER.. In other words. we are counting the permutations of 6 different letters taken 3 at a time. 6 if the digits can be repeated? How many 4-letter code can be formed using the first 10 letters of the English alphabet. Each arrangement is called a permutation of 4 different letters taken all at a time. 2. 3. If the repetition of the letters was allowed. Here. we need to count the arrangements NUM. . 4 and 5 assuming that (i) (ii) 2. MUN. with or without meaning. how many different signals can be generated if each signal requires the use of 2 flags.. we are actually counting the different possible arrangements of the letters such as ROSE. the order of writing the letters is important. etc.. How many 3-digit numbers can be formed from the digits 1. we find that The number of 4 flag signals = 5 × 4 × 3 × 2 = 120 and the number of 5 flag signals = 5 × 4 × 3 × 2 × 1 = 120 Therefore. EXERCISE 7. in this list. 3.138 MATHEMATICS The number of ways is 5 × 4 × 3 = 60. each arrangement is different from other. 6..3 Permutations In Example 1 of the previous Section. if no letter can be repeated? How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once? A coin is tossed 3 times and the outcomes are recorded. one below the other? 7. etc. 4. 2. Now. Here. 5. NMU. 5.. repetition of the digits is allowed? repetition of the digits is not allowed? How many 3-digit even numbers can be formed from the digits 1. Continuing the same way. the required number of words would be 6 × 6 × 6 = 216. the required no of signals = 20 + 60 + 120 + 120 = 320.. . where the repetition of the letters is not allowed. 4. NUB. 3. × (n – 1) × n = n ! 1=1! 1×2=2! 1× 2 × 3 = 3 ! 1 × 2 × 3 × 4 = 4 ! and so on. 7.3. . which is denoted by n Pr. ....PERMUTATION S AND COMBINATIONS 139 Definition 1 A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. Thus.... .. We read this symbol as ‘n factorial’. .( n – r + 1). places by r vacant places the n objects. × (n – 1) × n is denoted as n!. the product 1 × 2 × 3 × . (n – r + 1) This expression for nPr is cumbersome and we need a notation which will help to reduce the size of this expression.1 Permutations when all the objects are distinct Theorem 1 The number of permutations of n different objects taken r at a time. .. In the following text we will learn what actually n! means. Proof There will be as many permutations as there are ways of filling in r vacant . . In the following sub Section. the second place can be filled in (n – 1) ways. i. the number of ways of filling in r vacant places in succession is n(n – 1) (n – 2) . the rth place can be filled in (n – (r – 1)) ways.e. following which the third place can be filled in (n – 2) ways. where 0 < r ≤ n and the objects do not repeat is n ( n – 1) ( n – 2). 7. we shall obtain the formula needed to answer these questions immediately. We define 0 ! = 1 We can write 5 ! = 5 × 4 ! = 5 × 4 × 3 ! = 5 × 4 × 3 × 2 ! = 5 × 4 × 3 × 2 × 1! Clearly. The first place can be filled in n ways. (n – (r – 1)) or n ( n – 1) (n – 2) . . 1 × 2 × 3 × 4 . .3.. Therefore. for a natural number n n ! = n (n – 1) ! = n (n – 1) (n – 2) ! [provided (n ≥ 2)] = n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)] and so on.2 Factorial notation The notation n! represents the product of first n natural numbers. . following which. The symbol n! (read as factorial n or n factorial ) comes to our rescue. 140 MATHEMATICS Example 5 Evaluate (i) 5 ! Solution and (ii) 7 ! (iii) 7 ! – 5! (i) 5 ! = 1 × 2 × 3 × 4 × 5 = 120 (ii) 7 ! = 1 × 2 × 3 × 4 × 5 × 6 ×7 = 5040 (iii) 7 ! – 5! = 5040 – 120 = 4920. 10! 2! 10! 2 n! Example 7 Evaluate r ! n r ! . when n = 5. find x. 1 1 x Solution We have 8! 9 8! 10 9 8! Therefore So 1 x 1 or 9 10 9 x = 100. r = 2) We have 5! 5! 4 5 10 .2 1. = 2 ! 5 2 ! 2! 3! 2 1 1 x Example 8 If 8! 9! 10! . r = 2. 10 x 9 10 9 EXERCISE 7. Evaluate (i) 8 ! (ii) 4 ! – 3 ! . Solution 5! We have to evaluate 2! 5 2 ! (since n = 5. 7! Example 6 Compute (i) 5! Solution (i) We have and (ii) 12! (ii) 10! (2!) 7 6 5! 7! = = 7 × 6 = 42 5! 5! 12 11 10! 12! = = 6 × 11 = 66. . .. n r 1 n r n r 1 . n r ! . Is 3 ! + 4 ! = 7 ! ? 5. n! Evaluate n r ! .3 Derivation of the formula for n Pr n Pr = n! n r ! . the formula (1) is applicable for r = 0 also. r = 2 141 1 1 x 4. we can have In particular.PERMUTATION S AND COMBINATIONS 8! 3.3 2 1 n Pr = n! n r ! .. 3 × 2 × 1. n! n! 0! Counting permutations is merely counting the number of ways in which some or all objects at a time are rearranged. n Pn n P0 = 1 = n! n! n ! ( n 0)! Therefore. (n – r + 1) Multiplying numerator and denomirator by (n – r) (n – r – 1) .. when (i) n = 6. Arranging no object at all is the same as leaving behind all the objects and we know that there is only one way of doing so. 7. n! n r ! . r = 5.. we get n Thus Pr n n 1 n 2 . . Compute 6! 2! 2. .3 2 1 n r n r 1 . If 6! 7! 8! . where 0 < r n This is a much more convenient expression for nPr than the previous one. (1) . Thus n Pr n! . when r = n.. .. find x (ii) n = 9.0 r n . Thus.3. 0 ≤ r ≤ n Let us now go back to the stage where we had determined the following formula: n Pr = n (n – 1) (n – 2) .. The number of ways in which a Chairman and a Vice-Chairman can be chosen from amongst a group of 12 persons assuming that one person can not hold more than one position. 4! 3 4 12 . RO1O 2 T RO 2 O1 T ROOT T O1 O2 R T O2 O1R TOOR . the required number of words would be 44 = 256. Let us treat..e. i. If the repetition is allowed. 2! Permutations when O1. O1 and O2. the letters of the word are not all different. In this case. the 2 Os as different. which are of the same kind. if O1 and O2 are the same O at both places. There are 2 Os. where repetition is allowed. Corresponding to this permutation. Here repetition is not allowed.the required number of words would be 63 = 216. Here. In Example 1. 10! 7. in this case. If repetition is allowed. Consider one of these permutations say. we are solving some of the problems of the pervious Section using the formula for nPr to illustrate its usefulness. RO1O 2T. is n r.4 Permutations when all the objects are not distinct objects Suppose we have to find the number of ways of rearranging the letters of the word ROOT. the repetition is not allowed. clearly 12 P2 12! 11 12 = 132. O2 are the same O. The number of permutations of 4-different letters. say. O2 are different. The number of 3-letter words which can be formed by the letters of the word 6! 6 NUMBER = P3 3! = 4 × 5 × 6 = 120.142 MATHEMATICS Theorem 2 The number of permutations of n different objects taken r at a time. Here. taken all at a time is 4!. Proof is very similar to that of Theorem 1 and is left for the reader to arrive at. the required number of words = 4P4 = 4! = 24. the required number of permutations = Permutations when O1. in this case also. Therefore.we have 2 ! permutations RO1O2T and RO2O1T which will be exactly the same permutation if O1 and O 2 are not treated as different.3. temporarily. in this case. T2 . T1 . The number of permutations of 9 different letters. in which I appears 2 times and T appears 3 times. say. taken all at a time is 9 !. Here if I1. In this case there are 9 letters. Consider one such permutation. I2 are not same . Temporarily. T3. I2. I1 NT1 SI2 T2 U E T3 .PERMUTATION S AND COMBINATIONS R O1T O2 R O2 T O1 ROTO T O1 R O2 T O 2 R O1 TORO R T O1 O2 R T O2 O1 RTOO T R O1 O 2 T R O 2 O1 TROO O1 O2 R T O2 O1 T R OORT O1 R O 2 T O2 R O1 T OROT O1 T O 2 R O2 T O1 R OTOR O1 R T O 2 O2 R T O1 ORTO O1 T R O 2 O2 T R O1 OTRO O1 O2 T R O2 O1 T R OOTR 143 Let us now find the number of ways of rearranging the letters of the word INSTITUTE. let us treat these letters different and name them as I1. T3 are not same. T2.144 MATHEMATICS and T1. Hence. 2! × 3! permutations will be just the same permutation corresponding to this chosen permutation I1NT1SI2T2UET3.. 1. first. 2 L’s and rest are all different.. the required 4 digit numbers = 9 P4 = 9! 9! = = 9 × 8 × 7 × 6 = 3024. In fact. there are 9 objects (letters) of which there are 4A’s. the required number of arrangements = 9! 5 6 7 8 9 = 7560 4! 2! 2 Example 10 How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed? Solution Here order matters for example 1234 and 1324 are two different numbers. if the repetition of the digits is not allowed? Solution Every number between 100 and 1000 is a 3-digit number. T2 . . total number of different permutations will be 9! 2! 3! We can state (without proof) the following theorems: Theorem 3 The number of permutations of n objects. have to . Therefore. there will be as many 4 digit numbers as there are permutations of 9 different digits taken 4 at a time. 5.. then I1. 3.. 4. if any. where p objects are of the n! same kind and rest are all different = p! . p1! p2! .. Therefore. I 2 can be arranged in 2! ways and T1. Therefore. 2. Solution Here. p k! Example 9 Find the number of permutations of the letters of the word ALLAHABAD. 9 – 4 ! 5! Example 11 How many numbers lying between 100 and 1000 can be formed with the digits 0. are of different kind is n! . Therefore. pk are of kth kind and the rest. We. we have a more general theorem. p2 are of second kind. Theorem 4 The number of permutations of n objects. where p1 objects are of one kind. T3 can be arranged in 3! ways. . . But. 092. these permutations will include those also where 0 is at the 100’s place. . so n = 10. we fix 0 at the 100’s place and rearrange the remaining 5 digits taking 2 at a time. So = 6 P3 5 P2 The required number 6! 5! 3! 3! = 4 × 5 × 6 – 4 ×5 = 100 Example 12 Find the value of n such that n (i) n P5 42 P3 . 042. For example. [as (n – 1) (n – 2) (n – 3) ≠ 0. n> 4 P4 3 n –1 Solution (i) Given that or P5 42 n P3 n (n – 1) (n – 2) (n – 3) (n – 4) = 42 n(n – 1) (n – 2) Since n>4 n so n(n – 1) (n – 2) ≠ 0 Therefore. etc are such numbers which are actually 2-digit numbers and hence the number of such numbers has to be subtracted from 6 P3 to get the required number. by dividing both sides by n(n – 1) (n – 2). n (ii) Given that P4 5 P4 3 n –1 Therefore 3n (n – 1) (n – 2) (n – 3) = 5(n – 1) (n – 2) (n – 3) (n – 4) or or 3n = 5 (n – 4) n = 10. This number is 5P2 . This number would be 6P3 . n 4 n (ii) P4 5 = .PERMUTATION S AND COMBINATIONS 145 count the permutations of 6 digits taken 3 at a time. To get the number of such numbers. . n > 4] . we get or (n – 3 (n – 4) = 42 n2 – 7n – 30 = 0 or or or n2 – 10n + 3n – 30 (n – 10) (n + 3) = 0 n – 10 = 0 or n + 3 = 0 or n = 10 or n = – 3 As n cannot be negative. E taken all at a time . This number would be 6 P6 = 6!. we can for the time being. 3. in which there are 3 vowels. by the multiplication principle the required number of permutations = 6 ! × 3 ! = 4320. This single object together with 5 remaining letters (objects) will be counted as 6 objects. Out of 9 discs. Therefore. if 5 4Pr = 6 5P r–1 . Hence. the required number 8 ! – 6 ! × 3 ! = 6 ! (7×8 – 6) = 2 × 6 ! (28 – 3) = 50 × 6 ! = 50 × 720 = 36000 Example 15 In how many ways can 4 red. Then we count permutations of these 6 objects taken all at a time. which can be done in 8! ways. Then. A. Corresponding to each of these permutations. Example 14 Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that (i) all vowels occur together (ii) all vowels do not occur together.146 MATHEMATICS Example 13 Find r. U and E. (ii) If we have to count those permutations in which all vowels are never together. 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable ? Solution Total number of discs are 4 + 3 + 2 = 9. Since the vowels have to occur together. we shall have 3! permutations of the three vowels A. namely. 4 are of the first kind . Solution We have 5 4 Pr 6 5 Pr 1 or 5 4! 5! 6 4 r ! 5 r 1! or 5! 6 5! 4 r ! 5 r 1 5 r 5 r 1! or or or or or Hence (6 – r) (5 – r) = 6 r2 – 11r + 24 = 0 r2 – 8r – 3r + 24 = 0 (r – 8) (r – 3) = 0 r = 8 or r = 3. we have to subtract from this number. Solution (i) There are 8 different letters in the word DAUGHTER. we first have to find all possible arrangments of 8 letters taken all at a time. r = 8. the number of permutations in which the vowels are always together. U. assume them as a single object (AUE). PERMUTATION S AND COMBINATIONS 147 (red), 3 are of the second kind (yellow) and 2 are of the third kind (green). Therefore, the number of arrangements 9! = 1260 . 4! 3! 2! Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (i) do the words start with P (ii) do all the vowels always occur together (iii) do the vowels never occur together (iv) do the words begin with I and end in P? Solution There are 12 letters, of which N appears 3 times, E appears 4 times and D appears 2 times and the rest are all different. Therefore The required number of arrangements (i) Let us fix P at the extreme left position, we, then, count the arrangements of the remaining 11 letters. Therefore, the required number of words starting with P are (ii) 12! 1663200 3! 4! 2! 11! 138600 . 3! 2! 4! There are 5 vowels in the given word, which are 4 Es and 1 I. Since, they have to always occur together, we treat them as a single object EEEEI for the time being. This single object together with 7 remaining objects will account for 8 objects. These 8 objects, in which there are 3Ns and 2 Ds, can be rearranged in 8! 3! 2! ways. Corresponding to each of these arrangements, the 5 vowels E, E, E, 5! ways. Therefore, by multiplication principle the 4! required number of arrangements E and I can be rearranged in = (iii) 8! 5! 16800 3! 2! 4! The required number of arrangements = the total number of arrangements (without any restriction) – the number of arrangements where all the vowels occur together. 148 (iv) MATHEMATICS = 1663200 – 16800 = 1646400 Let us fix I and P at the extreme ends (I at the left end and P at the right end). We are left with 10 letters. Hence, the required number of arrangements = 10! = 12600 3! 2! 4! EXERCISE 7.3 1. How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated? 2. How many 4-digit numbers are there with no digit repeated? 3. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated? 4. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even? 5. From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person can not hold more than one position? 6. Find n if n – 1 P3 : nP4 = 1 : 9. 7. Find r if (i) 5 Pr 2 6 Pr1 (ii) 5 Pr 6 Pr 1 . 8. How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once? 9. How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time, (ii) all letters are used at a time, (iii) all letters are used but first letter is a vowel? 10. In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together? 11. In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S, (ii) vowels are all together, (iii) there are always 4 letters between P and S? 7.4 Combinations Let us now assume that there is a group of 3 lawn tennis players X, Y, Z. A team consisting of 2 players is to be formed. In how many ways can we do so? Is the team of X and Y different from the team of Y and X ? Here, order is not important. In fact, there are only 3 possible ways in which the team could be constructed. PERMUTATION S AND COMBINATIONS 149 Fig. 7.3 These are XY, YZ and ZX (Fig 7.3). Here, each selection is called a combination of 3 different objects taken 2 at a time. In a combination, the order is not important. Now consider some more illustrations. Twelve persons meet in a room and each shakes hand with all the others. How do we determine the number of hand shakes. X shaking hands with Y and Y with X will not be two different hand shakes. Here, order is not important. There will be as many hand shakes as there are combinations of 12 different things taken 2 at a time. Seven points lie on a circle. How many chords can be drawn by joining these points pairwise? There will be as many chords as there are combinations of 7 different things taken 2 at a time. Now, we obtain the formula for finding the number of combinations of n different objects taken r at a time, denoted by nCr .. Suppose we have 4 different objects A, B, C and D. Taking 2 at a time, if we have to make combinations, these will be AB, AC, AD, BC, BD, CD. Here, AB and BA are the same combination as order does not alter the combination. This is why we have not included BA, CA, DA, CB, DB and DC in this list. There are as many as 6 combinations of 4 different objects taken 2 at a time, i.e., 4C 2 = 6. Corresponding to each combination in the list, we can arrive at 2! permutations as 2 objects in each combination can be rearranged in 2! ways. Hence, the number of permutations = 4C2 × 2!. On the other hand, the number of permutations of 4 different things taken 2 at a time = 4P2. Therefore 4 P2 = 4C2 × 2! or 4! 4 C2 4 2 ! 2! Now, let us suppose that we have 5 different objects A, B, C, D, E. Taking 3 at a time, if we have to make combinations, these will be ABC, ABD, ABE, BCD, BCE, CDE, ACE, ACD, ADE, BDE. Corresponding to each of these 5C3 combinations, there are 3! permutations, because, the three objects in each combination can be 150 MATHEMATICS rearranged in 3 ! ways. Therefore, the total of permutations = 5 C3 3! Therefore 5 P3 5 C 3 3! or 5! 5C 3 5 3! 3! These examples suggest the following theorem showing relationship between permutaion and combination: Theorem 5 n Pr n C r r! , 0 < r ≤ n. Proof Corresponding to each combination of n Cr we have r ! permutations, because r objects in every combination can be rearranged in r ! ways. Hence, the total number of permutations of n different things taken r at a time is nC r × r!. On the other hand, it is n Pr . Thus n Pr n Cr r! , 0 r n . n! n Remarks 1. From above n r ! C r r ! , i.e., n In particular, if r n , C n n Cr n! r ! n r ! . n! 1. n! 0! 2. We define nC 0 = 1, i.e., the number of combinations of n different things taken nothing at all is considered to be 1. Counting combinations is merely counting the number of ways in which some or all objects at a time are selected. Selecting nothing at all is the same as leaving behind all the objects and we know that there is only one way of doing so. This way we define nC0 = 1. 3. As n! 1 n C 0 , the formula 0! n 0 ! n Cr n! is applicable for r = 0 also. r ! n r ! Hence n 4. n Cn r Cr n! r ! n r ! , 0 ≤ r ≤ n. n! = n r ! r ! = n Cr , n r ! n n r ! n! PERMUTATION S AND COMBINATIONS 151 i.e., selecting r objects out of n objects is same as rejecting (n – r) objects. 5. C a = nCb ⇒ a = b or a = n – b, i.e., n = a + b n Theorem 6 n C r n C r 1 Proof We have n n 1 Cr C r n C r 1 n! n! r! n r ! r 1 ! n r 1 ! = n! n! + r r 1 ! n r ! r 1 ! n r 1 n r ! = n! 1 1 r 1! n r ! r n r 1 = n! n r 1 r n 1! n 1 C = r r 1 ! n r ! r n r 1 r! n 1 r ! Example 17 If n C9 n C 8 , find n C17 . Solution We have n C9 n C 8 i.e., n! n! 9! n 9 ! n 8 ! 8! or 1 1 9 n8 Therefore n or n – 8 = 9 or n = 17 C17 17 C17 1 . Example 18 A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women? Solution Here, order does not matter. Therefore, we need to count combinations. There will be as many committees as there are combinations of 5 different persons 5 taken 3 at a time. Hence, the required number of ways = C 3 5! 4 5 10 . 3! 2! 2 Now, 1 man can be selected from 2 men in 2C 1 ways and 2 women can be selected from 3 women in 3C 2 ways. Therefore, the required number of committees 152 MATHEMATICS 2! 3! 2 3 = C1 C2 1! 1! 2! 1! 6 . Example 19 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (i) (ii) (iii) (iv) (v) four cards are of the same suit, four cards belong to four different suits, are face cards, two are red cards and two are black cards, cards are of the same colour? Solution There will be as many ways of choosing 4 cards from 52 cards as there are combinations of 52 different things, taken 4 at a time. Therefore The required number of ways = 52 C4 52! 49 50 51 52 4! 48! 2 3 4 = 270725 (i) There are four suits: diamond, club, spade, heart and there are 13 cards of each suit. Therefore, there are 13C 4 ways of choosing 4 diamonds. Similarly, there are 13 C 4 ways of choosing 4 clubs, 13 C4 ways of choosing 4 spades and 13C4 ways of choosing 4 hearts. Therefore The required number of ways = 13C 4 + 13C4 + 13 C4 + 13 C4. 13! = 4 4! 9! 2860 (ii) There are13 cards in each suit. Therefore, there are 13C 1 ways of choosing 1 card from 13 cards of diamond, 13 C 1 ways of choosing 1 card from 13 cards of hearts, 13C 1 ways of choosing 1 card from 13 cards of clubs, 13C1 ways of choosing 1 card from 13 cards of spades. Hence, by multiplication principle, the required number of ways = 13C 1 × 13 C1 × 13C 1× 13 C1 = 134 (iii) There are 12 face cards and 4 are to be selected out of these 12 cards. This can be done in 12 C4 ways. Therefore, the required number of ways = 12! 495 . 4! 8! PERMUTATION S AND COMBINATIONS 153 (iv) There are 26 red cards and 26 black cards. Therefore, the required number of ways = 26C2 × 26C2 2 26! 2 = 325 = 105625 2! 24! (v) 4 red cards can be selected out of 26 red cards in 4 black cards can be selected out of 26 black cards in 26 C4ways. C 4 ways. 26 Therefore, the required number of ways = 26C 4 + 26C4 = 2 26! = 29900. 4! 22! EXERCISE 7.4 1. 2. 3. 4. 5. 6. 7. 8. 9. If nC 8 = nC2, find nC2. Determine n if (i) 2n C3 : nC3 = 12 : 1 (ii) 2n C3 : n C3 = 11 : 1 How many chords can be drawn through 21 points on a circle? In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls? Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour. Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination. In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers? A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected. In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student? Miscellaneous Examples Example 20 How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE ? Solution In the word INVOLUTE, there are 4 vowels, namely, I,O,E,Uand 4 consonants, namely, N, V, L and T. 154 MATHEMATICS The number of ways of selecting 3 vowels out of 4 = 4C3 = 4. The number of ways of selecting 2 consonants out of 4 = 4C2 = 6. Therefore, the number of combinations of 3 vowels and 2 consonants is 4 × 6 = 24. Now, each of these 24 combinations has 5 letters which can be arranged among themselves in 5 ! ways. Therefore, the required number of different words is 24 × 5 ! = 2880. Example 21 A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls ? Solution (i) Since, the team will not include any girl, therefore, only boys are to be selected. 5 boys out of 7 boys can be selected in 7C5 ways. Therefore, the required number of ways (ii) 7! 6 7 7 = C5 5! 2! 2 21 Since, at least one boy and one girl are to be there in every team. Therefore, the team can consist of (a) 1 boy and 4 girls (b) 2 boys and 3 girls (c) 3 boys and 2 girls (d) 4 boys and 1 girl. 1 boy and 4 girls can be selected in 7C 1 × 4C4 ways. 2 boys and 3 girls can be selected in 7 C2 × 4 C3 ways. 3 boys and 2 girls can be selected in 7 C3 × 4 C2 ways. 4 boys and 1 girl can be selected in 7C 4 × 4C1 ways. Therefore, the required number of ways = 7C1 × 4C4 + 7 C2 × 4 C3 + 7C3 × 4C2 + 7C 4 × 4C 1 = 7 + 84 + 210 + 140 = 441 (iii) Since, the team has to consist of at least 3 girls, the team can consist of (a) 3 girls and 2 boys, or (b) 4 girls and 1 boy. Note that the team cannot have all 5 girls, because, the group has only 4 girls. 3 girls and 2 boys can be selected in 4 C3 × 7 C2 ways. 4 girls and 1 boy can be selected in 4C 4 × 7C1 ways. Therefore, the required number of ways = 4 C3 × 7C2 + 4C4 × 7C 1 = 84 + 7 = 91 PERMUTATION S AND COMBINATIONS 155 Example 22 Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what will be the 50th word? Solution There are 5 letters in the wordAGAIN, in which A appears 2 times. Therefore, the required number of words = 5! 60 . 2! To get the number of words starting with A, we fix the letter A at the extreme left position, we then rearrange the remaining 4 letters taken all at a time. There will be as many arrangements of these 4 letters taken 4 at a time as there are permutations of 4 different things taken 4 at a time. Hence, the number of words starting with 4! = 12 as after placing G 2! at the extreme left position, we are left with the letters A, A, I and N. Similarly, there are 12 words starting with the next letter I. Total number of words so far obtained = 24 + 12 + 12 =48. A = 4! = 24. Then, starting with G, the number of words The 49th word is NAAGI. The 50th word is NAAIG. Example 23 How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4? Solution Since, 1000000 is a 7-digit number and the number of digits to be used is also 7. Therefore, the numbers to be counted will be 7-digit only. Also, the numbers have to be greater than 1000000, so they can begin either with 1, 2 or 4. The number of numbers beginning with 1 = 6! 4 5 6 = 60, as when 1 is 3! 2! 2 fixed at the extreme left position, the remaining digits to be rearranged will be 0, 2, 2, 2, 4, 4, in which there are 3, 2s and 2, 4s. Total numbers begining with 2 = 6! 3 4 5 6 = 180 2! 2! 2 and total numbers begining with 4 6! 4 5 6 = 120 3! the three boys can be seated only at the cross marked places. it will be understood that in any number. Note If one or more than one digits given in the list is repeated. This can be done in 5! ways. How many words. For each such arrangement.. with or without meaning. the digits can be used as many times as is given in the list. e. this will include those numbers also. There are 6 cross marked places and the three boys can be seated in 6P 3 ways.156 MATHEMATICS Therefore. respectively. The number of such arrangements 6! (by fixing 0 at the extreme left position) = 60. But. which have 0 at the extreme left position. × G × G × G × G × G ×. can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together? 3. How many words. If the different permutations of all the letter of the word EXAMINATION are . = 5! × 6P3 = 5! × Miscellaneous Exercise on Chapter 7 1. each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ? 2. the required number of numbers = 60 + 180 + 120 = 360. A committee of 7 has to be formed from 9 boys and 4 girls. 3! 2! Therefore. Hence. Alternative Method 7! The number of 7-digit arrangements. clearly. 3! 2! 420 . with or without meaning. Example 24 In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together? Solution Let us first seat the 5 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls ? (ii) atleast 3 girls ? (iii) atmost 3 girls ? 4. the total number of ways 6! 3! = 4 × 5 × 2 × 3 × 4 × 5 × 6 = 14400. by multiplication principle. the required number of numbers = 420 – 60 = 360. in the above example 1 and 0 can be used only once whereas 2 and 4 can be used 3 times and 2 times.g. . 10 are to be chosen for an excursion party. 7. following which another event can occur in n different ways. How many such arrangements are possible ? From a class of 25 students. Part I and Part II. a question paper consists of 12 questions divided into two parts i. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. 8. (n r )! where 0 ≤ r ≤ n. is denoted by n Pr and is given by nPr = n! .PERMUTATION S AND COMBINATIONS 5. then the total number of occurrence of the events in the given order is m × n. selecting at least 3 from each part. 9. how many words are there in this list before the first word starting with E ? How many 6-digit numbers can be formed from the digits 0. 1. The number of permutations of n different things taken r at a time. In how many ways can a student select the questions ? Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king. How many words with two different vowels and 2 different consonants can be formed from the alphabet ? In an examination. 5. The number of permutations of n objects taken all at a time. containing 5 and 7 questions. 10. There are 3 students who decide that either all of them will join or none of them will join. 7 and 9 which are divisible by 10 and no digit is repeated ? The English alphabet has 5 vowels and 21 consonants. respectively.. A student is required to attempt 8 questions in all. taken r at a time. In how many ways can the excursion party be chosen ? In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together ? Summary Fundamental principle of counting If an event can occur in m different ways. 11. 157 listed as in a dictionary. where repeatition is allowed.. 3. is nr . where repetition is not allowed. where p1 objects . 6.×n n! = n × (n – 1) ! The number of permutations of n different things. n! = 1 × 2 × 3 × .e. the subject matter of permutations and combinations had its humble beginnings in China in the famous book I–King (Book of changes). 1 2 k The number of combinations of n different things taken r at a time. are all different is p ! p !. This was around 1140. Bhaskaracharya (born 1114) treated the subject matter of permutations and combinations under the name Anka Pasha in his famous work Lilavati. In the 6th century B. p2 objects are of the second kind.. Outside India. It is difficult to give the approximate time of this work. (around 850) is perhaps the world’s first mathematician credited with providing the general formulae for permutations and combinations.158 MATHEMATICS are of first kind. determined the number of combinations of known planets taken two at a time. if any. p k objects are of the kth n! kind and rest. Bhaskaracharya gives several important theorems and results concerning the subject. The credit. Sushruta Samhita. goes to the Jains who treated its subject matter as a self-contained topic in mathematics. p ! . under the name Vikalpa.C.. 0 ≤ r ≤ n. etc. in his medicinal work. Mahavira. Sushruta.C. three at a time and so on. two at a time. denoted by C r . Greeks and later Latin writers also did some scattered work on the theory of permutations and combinations. Among the Jains. is given by nCr = n n! . It appears that Rabbi ben Ezra did not know . . Some Arabic and Hebrew writers used the concepts of permutations and combinations in studying astronomy. in his work Chhanda Sutra. In addition to the general formulae for n C r and nP r already provided by Mahavira.. r !( n r )! Historical Note The concepts of permutations and combinations can be traced back to the advent of Jainism in India and perhaps even earlier. two at a time. etc. gives the method of determining the number of combinations of a given number of letters.. however. Rabbi ben Ezra. a Sanskrit scholar around third century B.... taken one at a time. taken one at a time. Pingala. asserts that 63 combinations can be made out of 6 different tastes. the emperor had ordered all books and manuscripts in the country to be burnt which fortunately was not completely carried out.. since in 213 B.C. for instance. PERMUTATION S AND COMBINATIONS the formula for nC r. The first book which gives a complete treatment of the subject matter of permutations and combinations is Ars Conjectandi written by a Swiss. — — 159 . This book contains essentially the theory of permutations and combinations as is known today. Levi Ben Gerson. another Hebrew writer came up with the formulae for nPr . However. posthumously published in 1713. Jacob Bernoulli (1654 – 1705). he was aware that nCr = nCn– r for specific values n and r. In 1321. nPn and the general formula for nC r. STEINMETZ 8. (iii) In each term of the expansion. we could evaluate the numerical values of numbers like (98)2 = (100 – 2)2. . number of terms is 3 whereas the index of (a + b) 2 is 2.P. – C. It gives an easier way to expand (a + b)n. etc. However.. For example. for higher powers like (98)5. we observe that (i) The total number of terms in the expansion is one more than the index. (101)6. we study binomial theorem for positive integral indices only. we have learnt how to find the squares and cubes of binomials like a + b and a – b. the calculations become difficult by using repeated multiplication.2 Binomial Theorem for Positive Integral Indices Blaise Pascal (1623-1662) Let us have a look at the following identities done earlier: (a+ b)0 = 1 a+b≠0 (a+ b)1 = a + b (a+ b) 2 = a 2 + 2ab + b 2 (a+ b) 3 = a 3 + 3a 2b + 3ab2 + b 3 (a+ b) 4 = (a + b) 3 (a + b) = a 4 + 4a3b + 6a2b 2 + 4ab3 + b 4 In these expansions. 8. etc. in the expansion of (a + b) 2 . (ii) Powers of the first quantity ‘a’ go on decreasing by 1 whereas the powers of the second quantity ‘b’ increase by 1. In this Chapter.Chapter 8 BINOMIAL THEOREM Mathematics is a most exact science and its conclusions are capable of absolute proofs.1 Introduction In earlier classes. (999)3 = (1000 – 1) 3. where n is an integer or a rational number. the sum of the indices of a and b is the same and is equal to the index of a + b. Using them. in the successive terms. This difficulty was overcome by a theorem known as binomial theorem. It can be seen that the addition of 1’s in the row for index 1 gives rise to 2 in the row for index 2.1 Do we observe any pattern in this table that will help us to write the next row? Yes we do.2 looks like a triangle with 1 at the top vertex and running down the two slanting sides. after the name of French mathematician Blaise Pascal.BINOMIAL THEOREM 161 We now arrange the coefficients in these expansions as follows (Fig 8. 2 and 2.1): Fig 8. The row for index 5 is 1 5 10 10 5 1 Using this row and our observations (i). gives rise to 3 and 3 in the row for index 3 and so on. The addition of 1. . we get (2x + 3y) 5 = (2x) 5 + 5(2x) 4 (3y) + 10(2x) 3 (3y) 2 +10 (2x)2 (3y) 3 + 5(2x)(3y)4 +(3y) 5 = 32x5 + 240x4y + 720x3 y2 + 1080x2y3 + 810xy4 + 243y5. Let us expand (2x + 3y)5 by using Pascal’s triangle. It is also known as Meru Prastara by Pingla. Also.2 by writing a few more rows. 1 in the row for index 2. Expansions for the higher powers of a binomial are also possible by using Pascal’s triangle. Fig 8.2 Pascal’s Triangle The structure given in Fig 8. 1 is present at the beginning and at the end of each row. (ii) and (iii). We can extend the pattern given in Fig 8. This can be continued till any index of our interest. This array of numbers is known as Pascal’s triangle. if we need the expansions involving still larger powers. Also. For example. we have (a + b) 7 = 7C 0 a7 + 7C1 a6b + 7C 2a5b 2 + 7C3a 4b 3 + 7C4a 3b4 + 7C5 a2b 5 + 7C 6ab6 + 7C7b 7 An expansion of a binomial to any positive integral index say n can now be visualised using these observations.3 Pascal’s triangle Observing this pattern. will become more difficult. The process.162 MATHEMATICS Now. We thus try to find a rule that will help us to find the expansion of the binomial for any power without writing all the rows of the Pascal’s triangle. 0 ≤ r ≤ n and r!(n – r )! n is a non-negative integer. that come before the row of the desired index. This is a slightly lengthy process. We know that n Cr n! . We are now in a position to write the expansion of a binomial to any positive integral index. we can now write the row of the Pascal’s triangle for any index without writing the earlier rows. 7 Thus. using this row and the observations (i). For this. (ii) and (iii). n C0 = 1 = nCn The Pascal’s triangle can now be rewritten as (Fig 8. if we want to find the expansion of (2x + 3y) 12. This can be done by writing all the rows of the Pascal’s triangle till index 12. as you observe. for the index 7 the row would be C 0 7 C1 7C2 7C 3 7 C4 7C5 7C 6 7 C7. . we make use of the concept of combinations studied earlier to rewrite the numbers in the Pascal’s triangle.3) Fig 8. we are first required to get the row for index 12. Therefore.2.. i. (a + b) k + 1 = (a + b) (a + b) k = (a + b) (kC 0 ak + kC 1ak – 1 b + kC 2 ak – 2 b2 +.1 Binomial theorem for any positive integer n.23 + 6 C4x2.e.26. Suppose P (k) is true for some positive integer k.+ kCk-1abk + kCkbk + 1 [by actual multiplication] = kC0a k + 1 + (kC1+ kC 0) akb + (kC2 + kC1)a k – 1b 2 + . ....b n–1 + nCn bn Proof The proof is obtained by applying principle of mathematical induction. P(n) is true for every positive integer n.+ kCk – 1 a2bk – 1 + kCk abk + kC0 akb + kC1ak – 1b2 + kC2ak – 2b3+. P (1) is true.+ k+1 Ck+1 bk + 1 Now.BINOMIAL THEOREM 163 8.2 5 + 6C 6...+ k + 1Ckabk + k + 1Ck + 1 b k +1 k+1 Cr and k Ck = 1= k + 1C k + 1) Thus. (a + b)k = kC 0a k + kC1a k – 1b + kC 2ak – 2b 2 + .e..b n – 1 + nC nbn For n = 1... (1) We shall prove that P(k + 1) is also true...+ nCn – 1a.2 + 6 C2x42 2 + 6C 3x3.... Let the given statement be P(n) : (a + b) n = nC 0an + nC1 an – 1b + nC 2an – 2b 2 + . by principle of mathematical induction.. it has been proved that P (k + 1) is true whenever P(k) is true... we have P (1) : (a + b) 1 = 1 C0a 1 + 1C 1b1 = a + b Thus.. i.+ kC kbk . = x6 + 12x5 + 60x4 + 160x3 + 240x2 + 192x + 64 Thus (x + 2)6 = x6 + 12x5 + 60x4 + 160x3 + 240x2 + 192x + 64.+ kCk – 1 abk – 1 + kCk b k) [from (1)] = kC0 ak + 1 + kC1 akb + kC2ak – 1b2 +.. kC r + kCr–1 = [grouping like terms] k+ 1 C2 a k – 1 b2 +.+ nCn –1a. We illustrate this theorem by expanding (x + 2)6: (x + 2)6 = 6C 0x6 + 6C 1x5.2 4 + 6C5 x. + (kCk+ kCk–1) abk + kC kb k + 1 = k + 1C0a (by using k+1 k+ 1 + k + 1C1a kb + C 0=1... (a + b)k + 1 = k+1 C0 a k + 1 + k+1 C1 akb + k+ 1 C 2 ak – 1 b2 + . (a + b) n = nC0a n + nC1 an–1 b + nC 2an –2 b 2 + . starting with zero in the first term.. one more than the index. i. 3. we have (x–2y) 5 = 5 C0x5 – 5C 1x4 (2y) + 5C2x3 (2y)2 – 5C3x2 (2y)3 + C 4 x(2y) 4 – 5C5 (2y)5 5 = x5 –10x4 y + 40x3y2 – 80x2y3 + 80xy4 – 32y5.. (n – 1) + 1 = n in the second term and so on 0 + n = n in the last term... In the successive terms of the expansion the index of a goes on decreasing by unity.. It is n in the first term.. 1 in the second and so on ending with n in the last term. + n Cnxn = nC 0 + nC1x + nC 2x2 + nC3 x3 + .. In the expansion of (a+b) n. 4. k 0 2. + nC n (–y) n = nC0xn – nC 1xn – 1y + nC 2xn – 2y2 – nC3xn – 3y3 + . (i) Taking a = x and b = – y. (n–1) in the second term. At the same time the index of b increases by unity. + (–1)n nCn yn Thus (x–y)n = nC 0xn – nC 1xn – 1 y + nC2xn – 2 y2 + ... we obtain (x – y)n = [x + (–y)]n = nC0 xn + nC1xn – 1(–y) + n C2xn –2(–y) 2 + nC3 xn–3(–y)3 + . and so on ending with zero in the last term. n The coefficients Cr occuring in the binomial theorem are known as binomial coefficients..e.2 Some special cases In the expansion of (a + b) n. 8. There are (n+1) terms in the expansion of (a+b)n . The notation n n 0 n k0 C k a n k b k stands for n C 0a b + C1a n–1 b1 + .164 MATHEMATICS Observations n 1.. + (–1)n nC n yn Using this. b = x.. + nC nxn Thus (1 + x) n = n C0 + nC1x + nC2x2 + nC 3x3 + ... the sum of the indices of a and b is n + 0 = n in the first term.+ nC ra n–rbr + . where b 0 = 1 = an–n. Hence the theorem can also be stated as n (a b ) n n C k a n k b k .+n Cna n–nbn. (ii) Taking a = 1.. + nC nxn . we obtain (1 + x) n = nC 0(1)n + nC1(1)n – 1x + nC2 (1)n – 2 x2 + . it can be seen that the sum of the indices of a and b is n in every term of the expansion.2. Thus... 5. and then use Binomial Theorem. 3 + 4 x x x x = x8 + 12x5 + 54x2 + 108 81 .01)1000000 or 10. 3 4 3 9 27 81 + 6. Write 98 = 100 – 2 Therefore. (iii) Taking a = 1. we get 0 = n C0 – nC1 + nC2 – . for x = 1. b = – x. 2 + 4.x4 .2 + 5C 2 (100)3 22 – 5C 3 (100)2 (2)3 + 5 C4 (100) (2)4 – 5C5 (2) 5 = 10000000000 – 5 × 100000000 × 2 + 10 × 1000000 × 4 – 10 ×10000 × 8 + 5 × 100 × 16 – 32 = 10040008000 – 1000800032 = 9039207968.000? Solution Splitting 1. x ≠ 0 x Solution By using binomial theorem.BINOMIAL THEOREM 165 In particular. + (– 1)n nC nxn In particular. (98)5 = (100 – 2)5 = 5C0 (100) 5 – 5 C1 (100) 4.x6 . we obtain (1– x)n = nC0 – n C1x + nC2x2 – . Example 3 Which is larger (1. + nC n.... + (–1)n nCn 4 2 3 Example 1 Expand x .x2. x x4 Example 2 Compute (98)5.. for x = 1. we have 2n = nC0 + nC1 + nC2 + .01 and using binomial theorem to write the first few terms we have . we have 4 3 2 3 3 3 3 x = 4C0 (x2)4 + 4C 1(x2)3 + 4C 2(x2 )2 + 4C 3(x2 ) + 4C4 x x x x x 2 = x8 + 4.. Solution We express 98 as the sum or difference of two numbers whose powers are easier to calculate.. 01) + other positive terms = 1 + 1000000 × 0.nC3 + . – x 2 3... 6n – 5n = 1+52 (n C2 + n C35 + . where k is some natural number..166 MATHEMATICS (1.e.nC 2 + 53..01)1000000 > 10000 Hence Example 4 Using binomial theorem. we get (1 + 5) n = n C0 + nC1 5 + nC2 52 + . 6n – 5n leaves remainder 1.....01)1000000 = (1 + 0. + 5n-2) or 6n – 5n = 25k+1 where k = nC2 + 5 . prove that 6n –5n always leaves remainder 1 when divided by 25.nC 3 + . We have (1 + a) n = nC0 + nC1 a + nC2a 2 + . then we say that b divides a with q as quotient and r as remainder. (2x – 3)6 . 5 1.nC3 + . + 5n-2 ) or 6n – 5n = 1+ 25 ( nC2 + 5 . (1–2x) 5 2 x 2. This shows that when divided by 25. we prove that 6n – 5n = 25k + 1.01 + other positive terms = 1 + 10000 + other positive terms > 10000 (1.e. + 5n i.. in order to show that 6n – 5n leaves remainder 1 when divided by 25. + nCn 5n i.1 Expand each of the expressions in Exercises 1 to 5..01)1000000 = 1000000 C0 + 1000000 C1(0. (6)n = 1 + 5n + 52. Thus. + nCna n For a = 5.. + 5n –2. EXERCISE 8.. Solution For two numbers a and b if we can find numbers q and r such that a = bq + r. evaluate ( 3 2)4 – ( 3 – 2 )4 .e. (102)5 8. whenever n is a positive integer.1)10000 or 1000. 2 th 8 For example. 13. the second term is n C1a n–1b. the third term is nC2 an–2 b2. (101)4 9. The (r + 1)th term is also called the general term of the expansion (a + b) n. in the expansion of (x + 2y) . Find (x + 1)6 + (x – 1)6. 2 8 5th term. (96)3 7. the middle term is 1 i. x x 5 167 6 Using binomial theorem.3 3 r 0 r n C r 4n .BINOMIAL THEOREM 4. 2. Find (a + b) 4 – (a – b) 4.e. i. (99)5 10. so there will be two middle terms in the . n is even so n + 1 is odd.. Looking at the pattern of the successive terms we can say that the (r + 1)th term is n C ra n–r br .. Thus Tr+1 = nCr a n–rbr . Hence. Since n 1 1 . indicate which number is larger (1. evaluate each of the following: 6. In the binomial expansion for (a + b) n. Using Binomial Theorem. Show that 9n +1 – 8n – 9 is divisible by 64. n 14. then n +1 is even. (ii) If n is odd. Prove that 8. Hence or otherwise evaluate ( 2 + 1)6 + ( 2 – 1)6. we have (i) If n is even. Regarding the middle term in the expansion (a + b) n. x 1 3 x 1 5. 12. we observe that the first term is n C 0an. and so on. the middle term is 2 th th n 1 term. 11. It is denoted by Tr+1. Therefore. General and Middle Terms 1. then the number of terms in the expansion will be n + 1. a= 50 C 16 .. = 50C16 234 a16.. i.e.e.. the middle terms are . n 1 It is given by C nx = 2nC n (constant). th i. (n + 1)th term.(2n 1) 2n xn.168 MATHEMATICS n 1 n 1 1 term.3. 5 th term. So in the expansion term and expansion... 2 2 th th 7 1 7 1 1 . where x ≠ 0. 2 50 C 17 . we have. where n is a positive integer. i. T18 = 50C17 2 33 a 17 Given that T17 = T18 So C 16 (2)34 a 16 = 50C17 (2)33 a 17 50 Therefore 50 i. as 2n is even.e. Example 5 Find a if the 17th and 18 th terms of the expansion (2 + a) 50 are equal. r + 1 = 17. i.e. For the 17th term.. 4 th and 2 2 th th 1 2 n 1 1 . 33! 2 =1 16!34! 50! Example 6 Show that the middle term in the expansion of (1+x ) 2 n is 1. r = 16 T17 = T16 + 1 = 50 C16 (2)50 – 16 a 16 Therefore. Similarly. Solution The (r + 1)th term of the expansion (x + y)n is given by Tr + 1 = nCr xn –ryr . n! . namely.e.2 33 C16 2 50 C 17 = 34 a 17 16 a 50! 17! . the middle term is x 2 2n 3. In the expansion of x . (2x – y) 7. x 2n n This term is called the term independent of x or the constant term.5.. Find x. the coefficient of x6y3 is C3 2 3 = 9 9! 3 9 .(2n 1) n n 2 x n! Example 7 Find the coefficient of x6y3 in the expansion of (x + 2y) 9.5. 720 and 1080.7 3 . x9 – r . third and fourth terms in the binomial expansion (x + a) n are 240.2 Example 8 The second.x n! n! = 1.. we get r = 3. 2n Tn +1 = 2 nCn(1)2n – n(x)n = 2n Cnxn = (2n )! n x n! n ! = 2n (2 n 1) (2n 2) .8.3 .e.2.. a and n. 3!6! 3.BINOMIAL THEOREM 169 th 2n Solution As 2n is even.( 2n)] n x n! n! = [1.. y r .2 ..3.3.. 2 i.6.1 n x n ! n! = 1. (n + 1)th term which is given by. Thus.5.5. respectively. the middle term of the expansion (1 + x) is 1 .5 ..4.(2n 1)]2 n [1..2 = .3.3.....4. Comparing the indices of x as well as y in x6y3 and in Tr + 1 . Now Tr+1 = 9Cr x9 – r (2y) r = 9Cr 2 r ... Solution Suppose x6y3 occurs in the (r + 1)th term of the expansion (x + 2y)9 .(2n 2) (2n 1) (2 n) n x n! n! = [1.4.n] n x n!n! = [1.(2n 1)] n ! n n 2 ...3.( 2n – 1)][2.2... Solution Given that second term T2 = 240 . 2 = 672.3. a = 240 . we get n or (n 1)! a ..170 MATHEMATICS We have T2 = nC 1xn – 1 .e. respectively. we get x = 2 and a = 3.. Example 9 The coefficients of three consecutive terms in the expansion of (1 + a) n are in the ratio1: 7 : 42.. i.. i. rth and (r + 1)th terms. 6 (n 2)! x C2 xn 2 a 2 720 i. Find n. The (r – 1)th term is nCr – 2 a r – 2. from (1).. n 1 2 (n 2) Hence. 6 9 .. and from (4). the coefficients of r th and (r + 1)th terms are nC r – 1 and nC r . (2) . Solution Suppose the three consecutive terms in the expansion of (1 + a) n are (r – 1)th. so we have... (3) n–2 a = 720 Dividing (2) by (1).. we have a 9 2 x (n 2) . we get.e. n – 8r + 9 = 0 C r 1 7 Cr 1 7 ... n = 5 a 3 x 2 Solving these equations for a and x. n = 55.. (1) . (2) C 3xn–3 a3 = 1080 . n n n and Cr 2 1 .e. and its coefficient is nCr – 2... Similarly. 5x4a = 240. n C1 xn 1 a 240 a 6 x ( n 1) .. n – 7r + 1 = 0 Cr 42 n Solving equations(1) and (2). (5) From (4) and (5).. (4) Dividing (3) by (2). . a So n Similarly n and n C 1xn–1 .. Thus. Since the coefficients are in the ratio 1 : 7 : 42. (1) C 2x 2 . x5 in (x + 3) 8 2. a 5b7 in (a – 2b)12 . Find a positive value of m for which the coefficient of x2 in the expansion (1 + x) m is 6. (x2 – y) 6 5. x3 3 6 9. 1 Find the 13 term in the expansion of 9 x . Miscellaneous Examples 6 3 2 1 Example 10 Find the term independent of x in the expansion of x .BINOMIAL THEOREM 171 EXERCISE 8. 3. 12. x 0. Find n and r. 7 10 x 8. The coefficients of the (r – 1)th. (x2 – yx) 12. Find the 4th term in the expansion of (x – 2y) 12. 11. 3 x 18 th Find the middle terms in the expansions of 7. x ≠ 0.2 Find the coefficient of 1. 9 y 3 . 6. rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1 : 3 : 5. Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2 n – 1. prove that coefficients of a m and an are equal. Write the general term in the expansion of 4. In the expansion of (1 + a) m+n. 10. 3x 2 6r Solution We have Tr + 1 3 2 = Cr x 2 6 3 = Cr 2 6 6 r r 1 3x x2 6 r 1 r r 1 1 r x 3 . Hence the coefficients of ar – 1 .. i. (n r 1) ( n r ) r (r 1) r ( n r ) or r( r 1) (n r ) ( n r 1) 2 (n r ) ( n r 1) r ( r 1) r (n r ) or r(r + 1) + (n – r) (n – r + 1) = 2 (r + 1) (n – r + 1) or r2 + r + n 2 – nr + n – nr + r2 – r = 2(nr – r 2 + r + n – r + 1) . a r and ar + 1 in the expansion of (1 + a) n are in arithmetic progression.e. 1 1 (r 1)!( n r 1) ( n r) (n r 1)! (r 1) (r ) (r 1)!(n r 1)! 2 or 1 ( r 1)! ( n r 1)! 2 i. so we have. Hence 5th term is independent of x and is given by (– 1)4 6 C4 (2) 6 4 12 Example 11 If the coefficients of ar – 1. Since these coefficients are in arithmetic progression. Thus. Solution The (r + 1)th term in the expansion is nCr ar .e. nC r – 1+ nC r + 1 = 2. respectively. Thus it can be seen that ar occurs in the (r + 1)th term. prove that n2 – n(4r + 1) + 4r2 – 2 = 0.172 MATHEMATICS r = ( 1) 6 Cr (3)6 2 r 12 3 r x (2) 6 r The term will be independent of x if the index of x is zero. and its coefficient is n Cr . 12 – 3r = 0. This gives n! n! n! 2 (r 1)!( n r 1)! ( r 1)!( n r 1)! r !(n r )! i.e. ar and ar + 1 are n Cr – 1. 1 r (r 1)!(n r ) (n r 1)! 1 1 ( n – r ) ( n r 1) ( r 1) ( r ) 1 ( r 1)! ( n r 1)![ r ( n – r )] 1 1 2 . nCr . r = 4 (3) 6 8 5 . nCr and n Cr + 1. (2n – 1) being odd. as required.. We write only those terms which involve a4. n+1 Example 13 Find the coefficient of a 4 in the product (1 + 2a)4 (2 – a) 5 using binomial theorem. n th and (n + 1) th terms. The terms containing a 4 are 1 (10a 4) + (8a) (–40a 3) + (24a 2) (80a2) + (32a 3) (– 80a) + (16a 4) (32) = – 438a 4 . Solution As 2n is even so the expansion (1 + x)2 n has only one middle term which is 2n 1 2 th i.e. Now th 2n – 1 th Cn – 1 + Cn = 2n Cn 2n – 1 [As nCr – 1+ n Cr = C r]. We have (1 + 2a) 4 = 4C 0 + 4C 1 (2a) + 4C2 (2a) 2 + 4C3 (2a) 3 + 4C 4 (2a) 4 = 1 + 4 (2a) + 6(4a 2) + 4 (8a3) + 16a4. (n + 1)th term. = 1 + 8a + 24a 2 + 32a3 + 16a4 and (2 – a)5 = 5C 0 (2)5 – 5 C1 (2)4 (a) + 5 C2 (2)3 (a)2 – 5C 3 (2)2 (a) 3 + 5C4 (2) (a)4 – 5C 5 (a) 5 = 32 – 80a + 80a 2 – 40a 3 + 10a4 – a5 Thus (1 + 2a) 4 (2 – a) 5 = (1 + 8a + 24a2 + 32a 3+ 16a 4) (32 –80a + 80a2 – 40a 3 + 10a4 – a 5) The complete multiplication of the two brackets need not be carried out..e. The (n + 1)th term is 2nC nxn.. Example 12 Show that the coefficient of the middle term in the expansion of (1 + x)2n is equal to the sum of the coefficients of two middle terms in the expansion of (1 + x) 2n – 1. 2n 1 1 2n 1 1 1 i. Solution We first expand each of the factors of the given product using Binomial Theorem. respectively.BINOMIAL THEOREM 173 n2 – 4nr – n + 4r2 – 2 = 0 n2 – n (4r + 1) + 4r2 – 2 = 0 or i. a4 – r = a4. This can be done if we note that a r. the other expansion has two middle terms. The coefficient of xn is 2nC n Similarly. The coefficients of and 2 2 2n – 1 2n – 1 these terms are C n – 1 and C n.e. e. Thus rth term from the end will be term number (n + 1) – (r – 1) = (n – r + 2) of the expansion. Therefore.. 18 3 1 Example 15 Find the term independent of x in the expansion of x 3 . Solution There are (n + 1) terms in the expansion of (x + a)n . x > 0. 2 x Solution We have Tr + 1 = 18 18 Cr Cr x 1 3 2 x 18 r x 3 1 3 18 r . 3 1 . x 2 18 2 r 3 Since we have to find a term independent of x. term not having x.e. we have m C0 –3 m C1+ 9 m C2 = 559.. r 2 . 29 Example 16 The sum of the coefficients of the first three terms in the expansion of m 3 x 2 . The second term from the end is the nth term of the expansion. m being a natural number. 1 – 3m + 9 m (m 1) 559 2 m are m C0. The third term from the end is the (n – 1)th term of the expansion and n – 1 = (n + 1) – (3 – 1) and so on. i. the coefficient of a 4 in the given product is – 438. is 559. Example 14 Find the r th term from the end in the expansion of (x + a) n. i. 3 Solution The coefficients of the first three terms of x 2 x and 9 mC 2.e. Find the term of the expansion x containing x3. by the given condition.174 MATHEMATICS Thus. i. The required term is 18 C9 18 2 r 0. Observing the terms we can say that the first term from the end is the last term. (–3) mC 1 . (n + 1)th term of the expansion and n + 1 = (n + 1) – (1 – 1). And the (n – r + 2)th term is nC n – r + 1 xr – 1 a n – r + 1. and n = (n + 1) – (2 – 1). x ≠ 0. so take We get r = 9.x r 3 = r 18 1 Cr r .. So. [Hint write a n = (a – b + b) n and expand] 5. Find a. find r. Evaluate 3 2 6 3 2 2 2 6. r = – 4 is not possible. either r – 6 = 2r – 2 or r–6 = 34 – (2r – 2) [Using the fact that if nCr = nCp.. – 5940 x3. whenever n is a positive integer. respectively. r = 14. 3. 7. Find an approximation of (0. Miscellaneous Exercise on Chapter 8 1. then either r = p or r = n – p] So. If a and b are distinct integers. x12 – 3r Tr + 1 = C r x r x Since we need the term containing x3. Find n.BINOMIAL THEOREM 175 which gives m = 12 (m being a natural number). 2. Since they are equal so 34 Cr – 6 = 34 C2r – 2 Therefore.. Thus. i.e. r Now 3 12 12 – r 2 = 12C (– 3)r . 7290 and 30375. r being a natural number. 8. b and n in the expansion of (a + b) n if the first three terms of the expansion are 729.99)5 using the first three terms of its expansion. Find the value of a a 1 6 . Example 17 If the coefficients of (r – 5)th and (2r – 1) th terms in the expansion of (1 + x) 34 are equal. if the ratio of the fifth term from the beginning to the fifth term from the n 4 1 end in the expansion of 2 4 is 3 6 :1 . prove that a – b is a factor of an – bn. the required term is 12C3 (– 3) 3 x3. respectively.e. . Find the coefficient of x5 in the product (1 + 2x) 6 (1 – x) 7 using binomial theorem. a 4 2 4 a2 1 . Find a if the coefficients of x2 and x3 in the expansion of (3 + ax) 9 are equal. so put 12 – 3r = 3 i. Solution The coefficients of (r – 5)th and (2r – 1)th terms of the expansion (1 + x) 34 are 34C r – 6 and 34C 2r – 2. we get r = – 4 or r = 14. 4. r = 3. .. th n In the expansion (a + b) . 10. The term binomial coefficients was first introduced by the German mathematician.2 . Summary integral n is given by Binomial The expansion of a binomial forn anyn positive n n n – 1 n n – 2 2 Theorem. The present form of the binomial theorem for integral values of n appeared in Trate du triange arithmetic.C.+ nC n – 1 n n a.176 MATHEMATICS 4 x 2 9. br.b + C nb . 0 ≤ n ≤ 7. 2 x 2 2 10.. which is (a + b) = C 0a + C 1a b + C2a b + . then the middle term is the 1 2 n n 1 n1 term. Michael Stipel (1486-1567) in approximately 1544. 2..7 and Oughtred (1631) gave them for n = 1. Bombelli (1572) also gave the coefficients in the expansion of (a + b) n.. This triangular arrangement is also found in the work of Chinese mathematician Chu-shi-kie in 1303.. then the middle terms are and 1 terms.If n is odd. popularly known as Pascal’s triangle and similar to the MeruPrastara of Pingla was constructed by the French mathematician Blaise Pascal (1623-1662) in 1665. Find the expansion of (3x – 2ax + 3a )3 using binomial theorem. written by Pascal and published posthumously in 1665. Expand using Binomial Theorem 1 . The general term of an expansion (a + b)n is Tr + 1 = nC ra n – r.. The arrangement of these coefficients was in the form of a diagram called Meru-Prastara. x 0 .).. The arithmetic triangle. This array is called Pascal’s triangle. 2 2 th th Historical Note The ancient Indian mathematicians knew about the coefficients in the expansions of (x + y)n . n– 1 The coefficients of the expansions are arranged in an array. if n is even. provided by Pingla in his book Chhanda shastra (200B. — — .. for n = 1. 1 Introduction is Natural numbers are the product of human spirit. parents. we usually mean that the collection is ordered in such a way that it has an identified first member. Sequences have important applications in several Fibonacci (1175-1250) spheres of human activities. i. following specific patterns are called progressions. population of human beings or bacteria at different times form a sequence.2 Sequences Let us consider the following examples: Assume that there is a generation gap of 30 years. great grandparents.. For example. the word. we are asked to find the number of ancestors. that a person might have over 300 years. etc. 9. we have studied about arithmetic progression (A. Depreciated values of certain commodity occur in a sequence.Chapter 9 he SEQUENCES AND SERIES 9. The amount of money deposited in a bank. and G. third member and so on.. In previous class.M. In this Chapter.P). geometric mean. – DEDEKIND © no N C tt E o R be T re pu bl In mathematics. second member.M. besides discussing more about A. Here. the total number of generations = 300 = 10 30 . grandparents. “sequence” is used in much the same way as it is in ordinary English.. Sequences. relationship between A.e.P. sum to n terms of squares of natural numbers and sum to n terms of cubes of natural numbers will also be studied. When we say that a collection of objects is listed in a sequence. special series in forms of sum to n terms of consecutive natural numbers. arithmetic mean. over a number of years form a sequence. These numbers form what we call a sequence. …. a6 = 3. . in the sequence of odd natural numbers 1.. The various numbers occurring in a sequence are called its terms..3.33.. a10 = 1024. if it is not a finite sequence... In fact.. ... etc...178 MATHEMATICS © no N C tt E o R be T re pu bl is he The number of person’s ancestors for the first.. In this process we get 3. ... has no visible pattern..3.. the subscripts denote the position of the term.. in the example of successive quotients a1 = 3. third.. Thus. where n is a natural number. 1024. tenth generations are 2. a2. For example. the sequence of successive quotients mentioned above is an infinite sequence. …. The nth term is also called the general term of the sequence. the nth term is given by the formula. 2. where n is a natural number. A sequence containing finite number of terms is called a finite sequence. an arrangement of numbers such as 1. .5.3. …. . 32.. We denote the terms of a sequence by a1... 3.. . 4. infinite in the sense that it never ends... 16. the sequence of even natural numbers 2. 8. a2 = 3. .. Often. . a23 = 46 = 2 × 23.. . Consider for instance. 6.. For example. it is possible to express the rule..33. which yields the various terms of a sequence in terms of algebraic formula... 4.. a3 = 3. Similarly. we see that the nth term of this sequence can be written as an = 2n.. ….3.. …. …. a2 = 4.333. . . and so on. a3..33333. . the terms of the sequence of person’s ancestors mentioned above are: a1 = 2. Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. Similarly... n > 2 This sequence is called Fibonacci sequence. In some cases. an = 2n – 1.. … Here a1 = 2 = 2 × 1 a2 = 4 = 2 × 2 a3 = 6 = 2 × 3 a4 = 8 = 2 × 4 ..3. and so on. These quotients also form a sequence. A sequence is called infinite. 8.3. a3 = 8.. but the sequence is generated by the recurrence relation given by a1 = a2 = 1 a3 = a1 + a2 an = an – 2 + an – 1. The nth term is the number at the nth position of the sequence and is denoted by an. second. 5.. 1.. etc. sequence of ancestors is a finite sequence since it contains 10 terms (a fixed number). …. an. a24 = 48 = 2 × 24. .. Example 1 Write the first three terms in each of the following sequences defined by the following: (i) an = 2n + 5. Series are often represented in © no N C tt E o R be T re pu compact form. bl is called the series associated with the given sequence .. we use the functional notation a(n) for an. be a given sequence. In view of the above.. a3. 1 + 3 + 5 + 7 is a finite series with four terms.. Then. Thus. a3 = 0 .. 9 and 11. k =1 Remark When the series is used. 3. the expression a1 + a2 + a3 +. Thus. 3. 2. We now consider some examples. 4 Solution (i) Here an = 2n + 5 Substituting n = 1.3. However. a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1. it refers to the indicated sum not to the sum itself.” we will mean the number that results from adding the terms.…. using the Greek letter ∑ (sigma) as means of indicating the summation involved. we get a1 = 2(1) + 5 = 7.k}. a2 = 9.SEQUENCES AND SERIES 179 9. 2. a2. a2. we expect a theoretical scheme or a rule for generating the terms a1. we find that there is no formula for the nth prime. a2 = − .The series is finite or infinite according as the given sequence is finite or infinite.…. the required terms are 7. a3.3 Series is Let a1. (ii) an = n−3 .an.7.5.…+ an + . called sigma notation.…. In every sequence. he In the sequence of primes 2. the sum of the series is 16.… in succession. a1 = 4 4 2 4 . a3 = 11 Therefore. the series a1 + a2 + a3 + . Such sequence can only be described by verbal description. + an is abbreviated n as ∑ ak . When we use the phrase “sum of a series.. Sometimes. (ii) Here an = n−3 1− 3 1 1 = − ..an. For example. we should not expect that its terms will necessarily be given by a specific formula. . a n = n 2. an = n (n + 2) 4. a1 = 1. an = 2n . an = an – 1 + 2 for n ≥ 2. © no N C tt E o R be T re pu Solution We have bl a1 = 1. a3 = a2 + 2 = 3 + 2 = 5. EXERCISE 9. n2 + 5 . an = n+3 .5. a7 n( n – 2) . a5 = a4 + 2 = 7 + 2 = 9. 4 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are: 7. a2 = a1 + 2 = 1 + 2 = 3. a20 .180 MATHEMATICS Hence. is = 19 × (– 18) × (23) = – 7866. an = 2n − 3 6 n n +1 3. a24 9. The corresponding series is 1 + 3 + 5 + 7 + 9 +. we obtain a20 = (20 – 1) (2 – 20) (3 + 20) Example 3 Let the sequence an be defined as follows: Find first five terms and write corresponding series. Hence. a17.7 and 9. an = 2n an = (–1)n–1 5n+1 6. a4 = a3 + 2 = 5 + 2 = 7. an = 4n – 3. 10. the first five terms of the sequence are 1. a9 n2 8. 2 4 Example 2 What is the 20th term of the sequence defined by an = (n – 1) (2 – n) (3 + n) ? he Solution Putting n = 20 . an = 5.. – 1 and 0. an = (–1)n – 1n3. the first three terms are – 1 .3.1 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are: 1. SEQUENCES AND SERIES 181 Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: 11. a1 = 3, an = 3an – 1 + 2 for all n > 1 12. a1 = – 1, an = an −1 ,n≥2 n he 13. a1 = a2 = 2, an = an – 1–1, n > 2 14. The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2, n > 2. 9.4 Arithmetic Progression (A.P.) is an +1 an , for n = 1, 2, 3, 4, 5 bl Find Let us recall some formulae and properties studied earlier. © no N C tt E o R be T re pu A sequence a1, a2, a3,…, an ,… is called arithmetic sequence or arithmetic progression if an + 1 = an + d, n ∈ N, where a1 is called the first term and the constant term d is called the common difference of the A.P. Let us consider an A.P. (in its standard form) with first term a and common difference d, i.e., a, a + d, a + 2d, ... Then the nth term (general term) of the A.P. is an = a + (n – 1) d. We can verify the following simple properties of an A.P. : (i) If a constant is added to each term of an A.P., the resulting sequence is also an A.P. (ii) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P. (iii) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P. (iv) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P. Here, we shall use the following notations for an arithmetic progression: a = the first term, l = the last term, d = common difference, n = the number of terms. Sn= the sum to n terms of A.P. Let a, a + d, a + 2d, …, a + (n – 1) d be an A.P. Then l = a + (n – 1) d MATHEMATICS Sn = We can also write, Sn = n [ 2a + (n −1)d ] 2 n [a + l ] 2 he 182 Let us consider some examples. ... (1) ... (2) bl Solution We have am = a + (m – 1) d = n, and an = a + (n – 1) d = m Solving (1) and (2), we get (m – n) d = n – m, or d = – 1, and a= n+m–1 Therefore a p = a + (p – 1)d = n + m – 1 + ( p – 1) (–1) = n + m – p Hence, the pth term is n + m – p. is Example 4 In an A.P. if mth term is n and the nth term is m, where m ≠ n, find the pth term. © no N C tt E o R be T re pu ... (3) ... (4) Example 5 If the sum of n terms of an A.P. is nP + 1 n( n – 1)Q , where P and Q 2 are constants, find the common difference. Solution Let a1, a2, … an be the given A.P. Then Sn = a1 + a2 + a3 +...+ an–1 + an = nP + 1 n (n – 1) Q 2 Therefore S1 = a1 = P, S2 = a1 + a2 = 2P + Q So that a2 = S2 – S1 = P + Q Hence, the common difference is given by d = a2 – a1 = (P + Q) – P = Q. Example 6 The sum of n terms of two arithmetic progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms. Solution Let a1, a2 and d1, d2 be the first terms and common difference of the first and second arithmetic progression, respectively. According to the given condition, we have Sum to n termsof first A.P. 3n + 8 = Sum to n termsof second A.P. 7n +15 or n [ 2a1 + ( n −1 )d1 ] 3n + 8 2 = n 7 n + 15 2 + − 1 a ( n )d [ 2 2] 2 or 2a1 + (n − 1)d1 3n + 8 = 2a2 + (n − 1)d 2 7 n + 15 Now 12 th term of first A.P. a + 11d1 = 1 th 12 term of second A.P a2 + 11d 2 is [By putting n = 23 in (1)] a1 + 11d1 12 th term of first A.P. 7 = th = a2 + 11d 2 12 term of second A.P. 16 © no N C tt E o R be T re pu Therefore ... (1) bl 2a1 + 22d1 3 × 23 + 8 = 2a2 + 22d 2 7 × 23 + 15 183 he SEQUENCES AND SERIES Hence, the required ratio is 7 : 16. Example 7 The income of a person is Rs. 3,00,000, in the first year and he receives an increase of Rs.10,000 to his income per year for the next 19 years. Find the total amount, he received in 20 years. Solution Here, we have an A.P. with a = 3,00,000, d = 10,000, and n = 20. Using the sum formula, we get, 20 [600000 + 19 × 10000] = 10 (790000) = 79,00,000. 2 Hence, the person received Rs. 79,00,000 as the total amount at the end of 20 years. S20 = 9.4.1 Arithmetic mean Given two numbers a and b. We can insert a number A between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers a and b. Note that, in this case, we have A – a = b – A, i.e., A = a+b 2 We may also interpret the A.M. between two numbers a and b as their average a+b . For example, the A.M. of two numbers 4 and 16 is 10. We have, thus 2 constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16. The natural 184 MATHEMATICS he question now arises : Can we insert two or more numbers between given two numbers so that the resulting sequence comes out to be an A.P. ? Observe that two numbers 8 and 12 can be inserted between 4 and 16 so that the resulting sequence 4, 8, 12, 16 becomes an A.P. More generally, given any two numbers a and b, we can insert as many numbers as we like between them such that the resulting sequence is an A.P. Let A1, A2, A3, …, An be n numbers between a and b such that a, A1, A2, A3, …, An, b is an A.P. Here, b is the (n + 2) th term, i.e., b = a + [(n + 2) – 1]d = a + (n + 1) d. Thus, n numbers between a and b are as follows: b−a n +1 © no N C tt E o R be T re pu A1 = a + d = a + is b−a . n +1 bl d= This gives A2 = a + 2d = a + 2(b − a ) n +1 A3 = a + 3d = a + 3(b − a ) n +1 ..... ..... ..... ..... ..... ..... An = a + nd = a + ..... ..... n (b − a ) n +1 . Example 8 Insert 6 numbers between 3 and 24 such that the resulting sequence is an A.P. Solution Let A1, A2, A3, A4, A5 and A6 be six numbers between 3 and 24 such that 3, A1, A2, A3, A4, A5, A6, 24 are in A.P. Here, a = 3, b = 24, n = 8. Therefore, 24 = 3 + (8 –1) d, so that d = 3. A2 = a + 2d = 3 + 2 × 3 = 9; Thus A1 = a + d = 3 + 3 = 6; A3 = a + 3d = 3 + 3 × 3 = 12; A4 = a + 4d = 3 + 4 × 3 = 15; A5 = a + 5d = 3 + 5 × 3 = 18; A6 = a + 6d = 3 + 6 × 3 = 21. Hence, six numbers between 3 and 24 are 6, 9, 12, 15, 18 and 21. SEQUENCES AND SERIES 185 EXERCISE 9.2 1 (pq +1), where p ≠ q. 2 If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term. Find the sum to n terms of the A.P., whose kth term is 5k + 1. If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference. The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms. If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms. Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. terms is © no N C tt E o R be T re pu 6. 1 1 and qth term is , prove that the sum of first pq p q bl 5. In an A.P., if pth term is 11 , – 5, … are needed to give the sum –25? 2 is 4. How many terms of the A.P. – 6, − he 1. Find the sum of odd integers from 1 to 2001. 2. Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5. 3. In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112. 7. 8. 9. 10. 11. Prove that a b c (q − r ) + (r − p) + ( p − q) = 0 p q r 12. The ratio of the sums of m and n terms of an A.P. is m2 : n2. Show that the ratio of mth and nth term is (2m – 1) : (2n – 1). 13. If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m. 14. Insert five numbers between 8 and 26 such that the resulting sequence is an A.P. 15. If a n + bn is the A.M. between a and b, then find the value of n. a n −1 + b n −1 16. Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A. P. and the ratio of 7th and (m – 1)th numbers is 5 : 9. Find the value of m. 186 MATHEMATICS 17. A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalment by Rs 5 every month, what amount he will pay in the 30th instalment? 18. The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120° , find the number of the sides of the polygon. Let us consider the following sequences: (i) 2,4,8,16,..., (ii) 1 –1 1 –1 , , , (iii) .01,.0001,.000001,... 9 27 81 243 ... he 9.5 Geometric Progression (G. P.) In (i), we have bl is In each of these sequences, how their terms progress? We note that each term, except the first progresses in a definite order. and so on. © no N C tt E o R be T re pu 1 a –1 a –1 a –1 and so on. In (ii), we observe, a1 = , 2 = , 3 = , 4 = 9 a1 3 a2 3 a3 3 Similarly, state how do the terms in (iii) progress? It is observed that in each case, a3 a a a1 =ratio 2, 2to=the 2, term =immediately 2, 4 = 2 preceding every term except the first term bears a constant a1 a2 a3 1 it. In (i), this constant ratio is 2; in (ii), it is – and in (iii), the constant ratio is 0.01. 3 Such sequences are called geometric sequence or geometric progression abbreviated as G.P. A sequence a1, a2, a3, …, an, … is called geometric progression, if each term is non-zero and ak + 1 ak = r (constant), for k ≥ 1. By letting a1 = a, we obtain a geometric progression, a, ar, ar2, ar3,…., where a is called the first term and r is called the common ratio of the G.P. Common ratio in 1 and 0.01, respectively. 3 As in case of arithmetic progression, the problem of finding the nth term or sum of n terms of a geometric progression containing a large number of terms would be difficult without the use of the formulae which we shall develop in the next Section. We shall use the following notations with these formulae: a = the first term, r = the common ratio, l = the last term, geometric progression (i), (ii) and (iii) above are 2, – SEQUENCES AND SERIES 187 n = the numbers of terms, Sn = the sum of first n terms. is he 9.5.1 General term of a G .P. Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. Write a few terms of it. The second term is obtained by multiplying a by r, thus a2 = ar. Similarly, third term is obtained by multiplying a2 by r. Thus, a3 = a2r = ar2, and so on. We write below these and few more terms. 1st term = a1 = a = ar1–1, 2nd term = a2 = ar = ar2–1, 3rd term = a3 = ar2 = ar3–1 4th term = a4 = ar3 = ar4–1, 5th term = a5 = ar4 = ar5–1 Do you see a pattern? What will be 16th term? a16 = ar16–1 = ar15 Therefore, the pattern suggests that the nth term of a G.P. is given by bl an = ar n−1 . © no N C tt E o R be T re pu Thus, a, G.P. can be written as a, ar, ar2, ar3, … arn – 1; a, ar, ar2,...,arn – 1... ;according as G.P. is finite or infinite, respectively. The series a + ar + ar2 + ... + arn–1 or a + ar + ar2 + ... + arn–1 +...are called finite or infinite geometric series, respectively. 9.5.2. Sum to n terms of a G .P. Let the first term of a G.P. be a and the common ratio be r. Let us denote by Sn the sum to first n terms of G.P. Then Sn = a + ar + ar2 +...+ arn–1 ... (1) Case 1 If r = 1, we have Sn = a + a + a + ... + a (n terms) = na If r ≠ 1, multiplying (1) by r, we have rSn = ar + ar2 + ar3 + ... + arn Subtracting (2) from (1), we get (1 – r) Sn = a – arn = a(1 – rn) Case 2 a (1 − r n ) 1− r ... (2) a ( r n −1) r −1 Example 9 Find the 10th and nth terms of the G.P. 5, 25,125,… . Solution Here a = 5 and r = 5. Thus, a10 = 5(5)10–1 = 5(5)9 = 510 and an = arn–1 = 5(5)n–1 = 5n . This gives Sn = or Sn = Example10 Which term of the G.P., 2,8,32, ... up to n terms is 131072? Solution Let 131072 be the nth term of the given G.P. Here a = 2 and r = 4. Therefore 131072 = an = 2(4)n – 1 or 65536 = 4n – 1 This gives 48 = 4n – 1. So that n – 1 = 8, i.e., n = 9. Hence, 131072 is the 9th term of the G.P. 188 MATHEMATICS Example11 In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term. Solution Here, a3 = ar 2 = 24 ... (1) he and ... (2) Dividing (2) by (1), we get r = 2. Substituting r = 2 in (1), we get a = 6. Hence a10 = 6 (2)9 = 3072. Example12 Find the sum of first n terms and the sum of first 5 terms of the geometric 2 . Therefore 3 bl Solution Here a = 1 and r = is series © no N C tt E o R be T re pu ⎡ ⎛ 2 ⎞n ⎤ ⎢1 − ⎜ ⎟ ⎥ n ⎡ ⎤ a (1 − r n ) ⎣⎢ ⎝ 3 ⎠ ⎥⎦ 3 ⎢1− ⎛ 2 ⎞ ⎥ ⎜ ⎟ = Sn = = 2 ⎢⎣ ⎝ 3 ⎠ ⎥⎦ 1− r 1− 3 a =2 ar45 = 192 1 6+ + + ... 3 9 ⎡ ⎛ 2 ⎞5 ⎤ 211 211 3 × In particular, S5 = 3 ⎢1 − ⎜ ⎟ ⎥ = = . 243 81 ⎣⎢ ⎝ 3 ⎠ ⎥⎦ 3 3 Example 13 How many terms of the G.P. 3, , ,... are needed to give the 2 4 3069 sum 512 ? Solution Let n be the number of terms needed. Given that a = 3, r = a (1 – r n ) 1− r Since Sn = Therefore 3069 = 512 1 ) 2n = 6 ⎛1− 1 ⎞ ⎜ n ⎟ 1 ⎝ 2 ⎠ 1− 2 3(1 − 1 3069 and Sn = 2 512 SEQUENCES AND SERIES 189 1 3069 = 1− n 2 3072 or or Example 14 The sum of first three terms of a G.P. is he 1 3069 3 1 = = n = 1− 2 3072 3072 1024 2n = 1024 = 210, which gives n = 10. or 13 and their product is – 1. 12 a , a, ar be the first three terms of the G.P. Then r © no N C tt E o R be T re pu a 13 + ar + a = r 12 bl Solution Let is Find the common ratio and the terms. ⎛a⎞ ⎜ ⎟ (a ) (ar ) = – 1 ⎝r⎠ From (2), we get a3 = – 1, i.e., a = – 1 (considering only real roots) and ... (1) ... (2) Substituting a = –1 in (1), we have 1 13 – –1– r = or 12r2 + 25r + 12 = 0. r 12 3 4 or – . 4 3 4 3 –3 3 4 –4 and , – 1, for r = , Thus, the three terms of G.P. are : , – 1, for r = 3 4 4 4 3 3 Example15 Find the sum of the sequence 7, 77, 777, 7777, ... to n terms. This is a quadratic in r, solving, we get r = – Solution This is not a G.P., however, we can relate it to a G.P. by writing the terms as Sn = 7 + 77 + 777 + 7777 + ... to n terms = 7 [9 + 99 + 999 + 9999 + ...to n term] 9 = 7 [(10 − 1) + (102 − 1) + (103 −1) + (104 − 1) + ...n terms] 9 G1.…. the geometric mean of 2 and 8 is 4.P. Thus. Gn be n numbers between positive numbers a and b such that a. ⎛ b ⎞ n +1 2 G 2 = ar = a ⎜ ⎟ . ⎝a⎠ ⎝a⎠ 3 ⎛ b ⎞ n +1 G 3 = ar = a ⎜ ⎟ ..Gn. .. and so on.G3.P. © no N C tt E o R be T re pu 9. Find the number of his ancestors during the ten generations preceding his own.G2. G2.4. This leads to a generalisation of the concept of geometric means of two numbers. 4 grandparents. Solution Let G1.P.) The geometric mean of two positive numbers a and b is the number ab .M. 8 great grandparents.3 Geometric Mean (G .G3 .n terms)] 9 ⎤ 7 ⎡10 (10n − 1) ⎤ 7 ⎡10(10n − 1) n − − n⎥ .n terms) – (1+1+1+.….5.G1.G2.256 is a G. he = Solution Here a = 2. b being the (n + 2)th term. We observe that the three numbers 2.P. ⎛ b ⎞n + 1 . Let G1. ⎢ ⎥ = ⎢ 9 ⎣ 10 − 1 9 ⎦ 9⎣ ⎦ Example 16 A person has 2 parents...b is a G. Given any two positive numbers a and b.P. G2.we have 1 b = ar n +1 .190 MATHEMATICS = 7 [(10 + 102 + 103 + . r =⎜ ⎟ ⎝a⎠ or 1 Hence 2 ⎛ b ⎞ n +1 G1 = ar = a ⎜ ⎟ . the number of ancestors preceding the person is 2046. ⎝a⎠ 3 n ⎛ b ⎞ n +1 G n = ar n = a ⎜ ⎟ ⎝a⎠ Example17 Insert three numbers between 1 and 256 so that the resulting sequence is a G.G3 be three numbers between 1 and 256 such that 1. r = 2 and n = 10 Using the sum formula is a (r n −1) Sn = r −1 bl We have S10 = 2(210 – 1) = 2046 Hence. we can insert as many numbers as we like between them to make the resulting sequence in a G.8 are consecutive terms of a G. Therefore. a+b and G = ab 2 Thus. 64 between 1 and 256 so that the resulting sequences are in G. 9.. (1) Example 18 If A.M. a+b =10 2 Solution Given that A.M. (4) 2 Putting the value of a and b from (3). respectively. b = 4 Thus.M. (3) ab = 64 . (2) From (1) and (2)..6 Relationship Between A...M. 16. numbers are – 4. the numbers a and b are 4. 16 or 16. we have G1 = ar = 4.16 and – 64.. and G. (1) . we get a + b = 20 ..SEQUENCES AND SERIES 191 he Therefore 256 = r4 giving r = ± 4 (Taking real roots only) For r = 4. G3 = ar3 = 64 Similarly. = and G. Then a− b 2 ≥0 2 From (1)... find the numbers. we have a +b a + b − 2 ab − ab = 2 2 ( ) © no N C tt E o R be T re pu A–G= bl A= is Let A and G be A. = ab = 8 . Hence.M. b = 16 or a = 16. of two positive numbers a and b are 10 and 8. we obtain a = 4. and G. (5) Solving (3) and (5)... we get (a – b)2 = 400 – 256 = 144 or a – b = ± 12 . = . we can insert 4.M. for r = – 4. respectively. and G. G2 = ar2 = 16.. 4 respectively.M.P.. we obtain the relationship A ≥ G.M. of two given positive real numbers a and b. . (4) in the identity (a – b) = (a + b)2 – 4ab. Show that q2 = ps.. whose 8th term is 192 and the common ratio is 2.0015.015.P. respectively. respectively. x.P. 3 . The 4th term of a G. 10. .. How many terms of G. . 0. 33.192 MATHEMATICS EXERCISE 9.P. 2 4 8 Find the 12th term of a G.. 7 .P.. are x. q and s. .3 5 5 5 . and the first term is – 3. z are in G. 6. 15. 11 11.P.P. 9.. If the 4th. is 16 and the sum of the next three terms is 128... are p.P. Find a G. x5. is square of its second term. 21 . (a) 2 . 17. – a. a2. 13. . with a = 729 and 7th term 64.15. is729 ? bl 2.4 . .. Evaluate ∑ (2 + 3k ) .. 10th and 16th terms of a G.. x3. .P. he 1.P.. y. For what values of x. n terms (if a ≠ – 1). Determine its 7th term. n terms (if x ≠ ± 1). 16.. The sum of first three terms of a G.. The sum of first three terms of a G. is 128 ? (c) 1 1 1 1 .? 7 2 Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 7. x7. n terms. – are in G. determine S7. Which term of the following sequences: 5.3 3 . is and their product is 1.P. 8 th and 11 th terms of a G.3. 20 terms. . – a3. y and z.. 0. Find the 20th and nth terms of the G. Determine the first term.. © no N C tt E o R be T re pu 2 7 . 32. Given a G. 3.. 1. k =1 39 12... 3. Find the 10 common ratio and the terms.P. Prove that x. 3 7 . for which sum of the first two terms is – 4 and the fifth term is 4 times the third term. 0. is ? 3 9 27 19683 (b) is 4. .2 2 .P. the common ratio and the sum to n terms of the G. . .. … are needed to give the sum 120? 14. the numbers – 8.P. The 5 th. 29. are a. The number of bacteria in a certain culture doubles every hour. 19. If A. 8. c and d are in G.M. 8888… . 1 . and G.M. If A and G be A. ar. respectively.M. Insert two numbers between 3 and 81 so that the resulting sequence is G. and G. ar2. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually? 32. he 16. If the first and the nth term of a G. respectively between two positive numbers. 4th hour and nth hour ? 31. 28.P. … ARn – 1 form a G. 32. AR. rn 25. and if P is the product of n terms. how many bacteria will be present at the end of 2nd hour. 8.P. Show that the ratio of the sum of first n terms of a G. . If a. Prove that aq – r br – pcP – q = 1. 1 . 32 and 128. prove that P2 = (ab)n. 30. 4. and the second term is greater than the 4th by 18. of roots of a quadratic equation are 8 and 5. If the pth. AR2. Find the value of n so that a n+1 + b n+1 may be the geometric mean between a n + bn a and b. The sum of two numbers is 6 times their geometric mean. Find the sum of the products of the corresponding terms of the sequences 2.M. 26. 88. then obtain the quadratic equation. 24.P. 8. is 21.. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .SEQUENCES AND SERIES 193 18. respectively. bl 20. qth and rth terms of a G. If there were 30 bacteria present in the culture originally. b. 23. 888. b and c. show that numbers ( )( ) are in the ratio 3 + 2 2 : 3 − 2 2 . respectively. 2 Show that the products of the corresponding terms of the sequences a. © no N C tt E o R be T re pu (n + 1)th to (2n)th term is 27. to the sum of terms from 22. and find the common ratio.P. …arn – 1 and A. 2. Find four numbers forming a geometric progression in which the third term is greater than the first term by 9.P. are a and b.P. Find the sum to n terms of the sequence. prove that the numbers are A ± ( A + G )( A − G ) . .. (See Section 9... ......... we get = ..... Let us take them one by one.. then Sn = n (n + 1) 2 he We shall now find the sum of first n terms of some special series... + n2) – 3 (1 + 2 + 3 + .............. (k + 1)4 – k4 = 4k3 + 6k2 + 4k + 1 Putting k = 1...... 2..... (i) 1 + 2 + 3 +… + n (sum of first n natural numbers) (ii) 12 + 22 + 32 +… + n2(sum of squares of the first n natural numbers) (iii) 13 + 23 + 33 +… + n3(sum of cubes of the first n natural numbers).194 MATHEMATICS 9............... n successively. 2…... we know that ∑ k =1 + 2 + 3 + ....... + n) + n n n k =1 k =1 n3 = 3 ∑ k 2 – 3 ∑ k + n n By (i)......4) © no N C tt E o R be T re pu bl is (ii) Here Sn= 12 + 22 + 32 + … + n2 We consider the identity k3 – (k – 1)3 = 3k2 – 3k + 1 Putting k = 1........... n3 – (n – 1)3 = 3 (n)2 – 3 (n) + 1 Adding both sides.. + n = k =1 n Hence Sn = 1⎡ ∑ k 2 = 3 ⎢⎣n3 + k =1 n (n + 1) 2 3n (n + 1) ⎤ − n ⎥ = 1 (2n3 + 3n 2 + n) 2 ⎦ 6 n (n +1)(2n +1) 6 (iii) Here Sn = 13 + 23 + ........... we obtain 13 – 03 = 3 (1)2 – 3 (1) + 1 23 – 13 = 3 (2)2 – 3 (2) + 1 33 – 23 = 3(3)2 – 3 (3) + 1 ..... 3… n........ ...... we get n3 – 03 = 3 (12 + 22 + 32 + ... namely.7 Sum to n Terms of Special Series (i) Sn=1 + 2 + 3 + … + n.......+n3 We consider the identity......... ....... n 2 (n + 1) 2 [ n (n + 1)] Sn = = 4 4 2 Hence. ............................+ n3) + 6(12 + 22 + 32 + ............................+ n2) + 4(1 + 2 + 3 +..........SEQUENCES AND SERIES n n bl is he 24 – 14 = 4(1)3 + 6(1)2 + 4(1) + 1 34 – 24 = 4(2)3 + 6(2)2 + 4(2) + 1 44 – 34 = 4(3)3 + 6(3)2 + 4(3) + 1 ................... we know that n ∑k = k =1 n (n + 1) 2 n and ∑ k2 = k =1 n (n + 1) (2n + 1) 6 Putting these values in equation (1)........ Example 19 Find the sum to n terms of the series: 5 + 11 + 19 + 29 + 41… Solution Let us write or On subtraction....... (1) From parts (i) and (ii)....... we get Sn = 5 + 11 + 19 + 29 + ... + an–2 + an–1 + an . (n – 1)4 – (n – 2)4 = 4(n – 2)3 + 6(n – 2)2 + 4(n – 2) + 1 n4 – (n – 1)4 = 4(n – 1)3 + 6(n – 1)2 + 4(n – 1) + 1 (n + 1)4 – n4 = 4n3 + 6n2 + 4n + 1 Adding both sides.. we obtain n 4 ∑ k 3 = n 4 + 4 n 3 + 6n 2 + 4n – k =1 or 6n (n + 1) (2n + 1) 4n (n + 1) – –n 6 2 4Sn = n4 + 4n3 + 6n2 + 4n – n (2n2 + 3n + 1) – 2n (n + 1) – n = n4 + 2n3 + n2 = n2(n + 1)2. + an–1 + an Sn = 5 + 11 + 19 + ............+ n) + n 195 n © no N C tt E o R be T re pu = 4∑ k3 + 6∑ k2 + 4∑ k + n k =1 k =1 k =1 .......... we get (n + 1)4 – 14 = 4(13 + 23 + 33 +........ ....................... 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + ... 6 2 3 EXERCISE 9. 10. 3 × 8 + 6 × 11 + 9 × 14 + . 52 + 62 + 72 + .. 12 + (12 + 22) + (12 + 22 + 32) + .... Sn = = n n n k =1 k =1 k =1 ∑ ak = ∑ k 2 + 3∑ k n (n + 1) (2n + 1) 3n (n + 1) n(n + 1)(n + 5) + = .196 MATHEMATICS 0 = 5 + [6 + 8 + 10 + 12 + .. 1.(n – 1) terms] – an or an = 5 + (n – 1)[12 + (n − 2) × 2] 2 n n n n k =1 k =1 k =1 1 2 2 Sn = ∑ ak = ∑ ( k + 3k + 1) = ∑ k + 3∑ k + n = n(n + 2)(n + 4) n(n +1) (2n +1) 3n(n + 1) + +n = . 3. Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by 8. 7. 1 1 1 4. 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 +... the sum to n terms is given by bl Example 20 Find the sum to n terms of the series whose nth term is n (n+3). n2 + 2n . 5... 2. + 202 6. 1× 2 + 2 × 3 + 3× 4 + .... 3 × 12 + 5 × 22 + 7 × 32 + .. 3 6 2 is Hence he = 5 + (n – 1) (n + 4) = n2 + 3n + 1 Solution Given that an = n (n + 3) = n2 + 3n © no N C tt E o R be T re pu Thus..4 Find the sum to n terms of each of the series in Exercises 1 to 7. n (n+1) (n+4). (2n – 1) 2 9. d and p are different real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd) p + (b2 + c2 + d2) ≤ 0. b2 = ac Using (1) in (2)..P. c. Since a. (1) . are in G. (2) Example 23 If a..P. we get k2y = kx + z.. (5) bl aq So . c are in G. (4) (why ?) .P.... Solution Here .. p – q. (q – r). (2) he ap = a + (p –1) d aq = a + (q –1) d ar = a + (r –1) d .. y and z are in A.P. b. and a x = b y = c z .. ap ar aq − ar q − r = = aq a p − aq p − q (why ?) ar as ar − as r − s = = = aq ar aq − ar q − r © no N C tt E o R be T re pu Similarly is = . by (5) and (6) q – r r – s ... = p–q q–r 1 1 1 Example 22 If a.. which gives 2y = x + z. 1 1 1 Let a x = b y = c z = k Then a = kx . ar and as are in G.e. b..P. prove that x.P. i. therefore. aq. Hence.. c are in G.P. (1) . c and d are in G.... Solution Given that (a2 + b2 + c2) p2 – 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0 . z are in A.. (3) as = a + (s –1) d Given that ap. b.. b.P. x.P. Solution . q th. q – r and r – s are in G. y. (6) Hence.P... (1) . r th and s th terms of an A. then show that a. b = ky and c = kz.SEQUENCES AND SERIES 197 Miscellaneous Examples Example21 If p th. (r – s) are also in G. then show that (p – q). b.H. bp – c = 0. therefore.. (ap – b)2 + (bp – c)2 + (cp – d)2 = 0 Hence a.. q2 = pr.q. . Thus x = −q −q but is also root of p p dx2 + 2ex + f = 0 (Why ?). px2 + 2qx + r = 0 and d e f . are in A. ⎝ p ⎠ ⎝ p ⎠ or dq2 – 2eqp + fp2 = 0 2 2 Dividing (1) by pq and using q = pr. . bl Example 24 If p. p q r . we have.198 MATHEMATICS But L. are in A. (1) .q.P.P. we get 2e d f d 2e fp = + − + = 0. and the equations. or q p r p q pr Hence d e f . Therefore 2 ⎛ −q ⎞ ⎛ −q ⎞ d⎜ ⎟ + 2e ⎜ ⎟ + f = 0. from (1) and (2).P. = (a2p2 – 2abp + b2) + (b2p2 – 2bcp + c2) + (c2p2 – 2cdp + d2).P. which gives (ap – b)2 + (bp – c)2 + (cp – d)2 ≥ 0 ..P. then show that Solution The equation px2 + 2qx + r = 0 has roots given by x= −2q ± 4q 2 − 4rp 2p Since p . r are in G. p q r © no N C tt E o R be T re pu dx2 + 2ex + f = 0 have a common root. (2) ap – b = 0. c and d are in G..r are in G. cp – d = 0 This implies that b c d = = =p a b c is or he Since the sum of squares of real numbers is non negative.S. If the sum of all the terms is 5 times the sum of terms occupying odd places. Let S be the sum. b. is 24 and their product is 440. The pth.P. c are in A. 3n terms of an A. b. Prove that (q + p) : (q – p) = 17:15. Show that the sum of (m + n)th and (m – n)th terms of an A. Find the last term and the number of terms. is 1. find the value of n. we obtain an arithmetic progression. Let the sum of n. 10.. d are roots of x2 – 12x + q = 0. (cn + dn) are in G. is equal to twice the mth term. 9.P. find the numbers. . c ⎜ + ⎟ are in A.P. respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0 ⎛1 1⎞ ⎛1 1⎞ ⎛1 1⎞ 16. 3. S2 and S3. 21 from these numbers in that order. Find the sum of all numbers between 200 and 400 which are divisible by 7. The sum of three numbers in G. If f is a function satisfying f (x +y) = f(x) f(y) for all x. The sum of some terms of G.P. 11. If its first term is 11. If a and b are the roots of x2 – 3x + p = 0 and c.P. respectively. Find the sum of integers from 1 to 100 that are divisible by 2 or 5. The first term of a G. = = a − bx b − cx c − dx 14. Find the sum of all two digit numbers which when divided by 4.P. 15. is 56. The sum of the last four terms is 112. Prove that P2Rn = Sn. x =1 © no N C tt E o R be T re pu 8.. If we subtract 1.P. be S1. 2n. 18. b. A G. show that S3 = 3(S2 – S1) 4. Find the numbers. are a. If a.P.P. 6.P. If a ⎜ + ⎟ . c. b ⎜ + ⎟ .P. then show that a. where a. d are in G. 2. Find the common ratio of G. 7. prove that (an + bn). yields 1 as remainder. The sum of the first four terms of an A.P. c.P. c and d are in G. 12.SEQUENCES AND SERIES 199 Miscellaneous Exercise On Chapter 9 n f(1) = 3 and bl is he 1. qth and rth terms of an A. (bn + cn). If the sum of three numbers in A.P. 5. respectively. is 56. y ∈ N such that ∑ f (x) = 120 . d form a G. then find the number of terms. P the product and R the sum of reciprocals of n terms in a G. prove that a. is 315 whose first term and the common ratio are 5 and 2. The sum of the third term and fifth term is 90. then find its common ratio. consists of an even number of terms.P. c. b. 7. ⎝b c⎠ ⎝c a⎠ ⎝a b⎠ 17.P. b. 13.P. If a + bx b + cx c + dx ( x ≠ 0) . e c d e are in G. 666+… 22. Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +… 24. 21.200 MATHEMATICS 19.P. + n 2 × (n + 1) 3n + 1 .M. which costs him Rs.. 150 workers were engaged to finish a job in a certain number of days. and G. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years..P. S3 are the sum of first n natural numbers. 1 1+ 3 1+ 3 + 5 1 × 22 + 2 × 32 + . Find the estimated value at the end of 5 years. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. 4 more workers dropped out on third day and so on. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. 29. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. b.. and . b.. c. c are in A. of two positive numbers a and b. A manufacturer reckons that the value of a machine. Find the sum of the following series up to n terms: (i) 5 + 55 +555 + … (ii) ... show that 9 S22 = S3 (1 + 8S1). A farmer buys a used tractor for Rs 12000. 32. 4 workers dropped out on second day. 66 +. + n × (n + 1) 2 3n + 5 = 12 × 2 + 22 × 3 + .. 30. + n terms. 28. respectively. 23. their squares and their 25. will depreciate each year by 20%. Find the sum of the following series up to n terms: © no N C tt E o R be T re pu 13 13 + 23 13 + 23 + 33 + + + . bl cubes.. Show ( )( that a : b = m + m 2 – n 2 : m – m 2 – n 2 ). 31.P. are in A. How much will the scooter cost him? A person writes a letter to four of his friends. S2. Find the amount spent on the postage when 8th set of letter is mailed.P. If a. prove that a.. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. The ratio of the A.6 +. . 15625. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + . is m : n. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. 26. Show that 27. d are in G. If S1. How much will the tractor cost him? Shamshad Ali buys a scooter for Rs 22000.M. c. . is he 1 1 1 20. by a. The general term or the nth term of the A. then the sum expressed as a1 + a2 + a3 + . b is in A.. A sequence is said to be a geometric progression or G. Let a1. we denote the first term of a G.P. is given by an = a + (n – 1) d...P.. is given by n n Sn = ⎡⎣ 2a + ( n – 1) d ⎤⎦ = ( a + l ) . .e. . the common difference by d and the last term by l. if the ratio of any term to its preceding term is same throughout.) is a sequence in which terms increase or decrease regularly by the same constant. we denote the first term of A. 2. a3. we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1. The general or the nth term of G. the 2 sequence a.k}. Usually. Also. A. The sum Sn of the first n terms of G.P. Summary he By a sequence. is given by an= arn – 1. A series is called finite series if it has got finite number of terms. A sequence containing a finite number of terms is called a finite sequence.P. 3. Usually.SEQUENCES AND SERIES 201 It took 8 more days to finish the work. An arithmetic progression (A..P. Find the number of days in which the work was completed. we mean an arrangement of number in definite order according is to some rule..P.P.P. a2. by a and its common ratio by r. The sum Sn of the first n terms of an A.. This constant factor is called the common ratio. A sequence is called infinite if it is not a finite sequence.. This constant is called common difference of the A. © no N C tt E o R be T re pu bl is called series.P.P.. is given by . 2 2 The arithmetic mean A of any two numbers a and b is given by a+b i. be the sequence. Aryabhatta (476) was the first to give the formula for the sum of squares and cubes of natural numbers in his famous work Aryabhatiyam. written around 499. was discovered by Italian mathematician Leonardo Fibonacci (1170-1250). Noted Indian mathematicians Brahmgupta (598). It was only through the rigorous development of algebraic and set theoretic tools that the concepts related to sequence and series could be formulated suitably. According to Boethius (510).M. He also gave the formula for finding the sum to n terms of an arithmetic sequence starting with p th term. © no N C tt E o R be T re pu bl Evidence is found that Babylonians. In 1671 James Gregory used the term infinite series in connection with infinite sequence. G.202 MATHEMATICS ( ) or a (1– r ) . called Fibonacci sequence. arithmetic and geometric sequences were known to early Greek writers. the sequence a. Mahavira (850) and Bhaskara (1114-1185) also considered the sum of squares and cubes.P. Seventeenth century witnessed the classification of series into specific forms. if r ≠ 1 a rn – 1 Sn = r –1 n 1–r he The geometric mean (G. Among the Indian mathematician.) of any two positive numbers a and b is given by Historical Note is ab i. knew of arithmetic and geometric sequences. b is G.. Another specific type of sequence having important applications in mathematics. some 4000 years ago. — — .e. 0) in the XY-plane is shown in Fig 10. we initiated René Descartes the study of coordinate geometry. – 4) is at 6 units distance from the y-axis measured along the positive x-axis and at 4 units distance from the x-axis measured along the negative y-axis. in his book ‘La Géométry. A systematic study of geometry by the use of algebra was first carried out by celebrated French philosopher and mathematician René Descartes. as a logical system. – H. We may note that the point (6. To recapitulate. The resulting combination of analysis and geometry is referred now as analytical geometry. Mainly. distance between two points. is a means and even the most powerful means to make children feel the strength of the human spirit that is of their own spirit. section formulae. Let us have a brief recall of coordinate geometry done in earlier classes. plotting of points in a plane.1 Introduction We are familiar with two-dimensional coordinate geometry from earlier classes. the location of the points (6. 0) is at 3 units distance from the y-axis measured along the positive x-axis and has zero distance from the x-axis. This book introduced the notion of the equation of a curve and related analytical methods into the study of geometry. Fig 10. where we studied about (1596 -1650) coordinate axes.Chapter 10 STRAIGHT LINES G eometry. coordinate plane. FREUDENTHAL 10.1 We also studied there following important formulae: . Similarly. published in 1637. – 4) and (3.1. In the earlier classes. etc. it is a combination of algebra and geometry. All these concepts are the basics of coordinate geometry. the point (3. (x2.. In the this Chapter. the area of the triangle. In particular. distance between the points (6. 4). Main focus is on representing the line algebraically. y2) internally. 2 2 joining the points (x1. then three points A. Area of the triangle whose vertices are (x1. which are supplementary. in the ratio 1: 3 are given by x and y 1. y3) is 1 x1 y 2 y 3 x 2 y 3 y 1 x 3 y1 y 2 . –3) and B (–3. i.( 3) 3. 16) is 54 1 4( 2 16) 3(16 4) (3)(4 2) 27. . 9) internally.2 Slope of a Line A line in a coordinate plane forms two angles with the x-axis. whose vertices are (4. (3.e.1 0 1 3 1. y2) is PQ x 2 – x1 y2 – y1 2 2 For example. B and C lie on a line. y2) are IV. y2) and (x3. they are collinear. y1) and (x2. 0) is 3 6 2 0 4 2 9 16 5 units. in the ratio m: n are For example. 3 0.9 3. The coordinates of a point dividing the line segment joining the points (x1. – 4) and (3. m n mn and (x2. Distance between the points P (x1. 2 2 Remark If the area of the triangle ABC is zero. the coordinates of the mid-point of the line segment x1 x2 y1 y 2 . we shall continue the study of coordinate geometry to study properties of the simplest geometric figure – straight line. if m = n. . II. the coordinates of the point which divides the line segment joining A (1. y1) and Q (x2. for which slope is most essential. y1) m x 2 n x1 m y 2 n y1 . . the line is a vital concept of geometry and enters into our daily experiences in numerous interesting and useful ways. 10. 1 3 III. – 2) and (– 3. 2 For example. y1).204 MATHEMATICS I. Despite its simplicity. 1 Slope of a line when coordinates of any two points on the line are given We know that a line is completely determined when we are given two points on it. slope of line l = m = tan θ. 10. Let P(x 1. 3 (i) . then tan θ is called the slope or gradient of Fig 10. (2) . have inclination of 0°. The inclination of a vertical line (parallel to or coinciding with y-axis) is 90°. Let us take these two cases.3 (i) and (ii). We observe that lines parallel to x-axis. The slope of a line is denoted by m. 10. ∠MPQ = θ.2 the line l. Draw perpendicular QR to x-axis and PM perpendicular to RQ as shown in Figs. Definition 1 If θ is the inclination of a line l. Case 1 When angle θ is acute: In Fig 10. Fig 10. Obviously 0° ≤ θ ≤ 180° (Fig 10. MP x2 x1 .STRAIGHT LINES 205 The angle (say) θ made by the line l with positive direction of x-axis and measured anti clockwise is called the inclination of the line.2.. But in ΔMPQ. x1 ≠ x2. Hence. y 2 ) be two points on non-vertical line l whose inclination is θ. we proceed to find the slope of a line in terms of the coordinates of two points on the line. Obviously. Thus.. or coinciding with x-axis. otherwise the line will become perpendicular to x-axis and its slope will not be defined. m = tan θ. (1) Therefore. The inclination of the line l may be acute or obtuse. The slope of a line whose inclination is 90° is not defined...2). y 1 ) and Q(x 2. θ ≠ 90° It may be observed that the slope of x-axis is zero and slope of y-axis is not defined. we have tan θ MQ y2 y1 .3 (i). i.4). slope of the line l Fig 10.3 (ii). y1) and (x2.. x2 x1 Case II When angle θ is obtuse: In Fig 10. . suppose that non-vertical lines l1 and l2 have slopes m1 and m2. 3 (ii) m = tan θ = tan ( 180° – ∠MPQ) = – tan ∠MPQ = y2 y1 MQ y y . tan α = tan β Therefore m1 = m2.. 2 1 = x2 x1 MP x1 x2 Consequently. Conversely. and hence. respectively. Therefore. i.. Let their inclinations be α and β.206 MATHEMATICS From equations (1) and (2). α = β. Then tan α = tan β. 4 By the property of tangent function (between 0° and 180°). Now. x2 x1 10. their slopes are equal.e. we see that in both the cases the slope m of the line through the points (x1. then their inclinations are equal.2.e. Therefore. we have m y2 y1 . y2) is given by m y2 y1 . we have ∠MPQ = 180° – θ. θ = 180° – ∠MPQ. respectively. If the line l1 is parallel to l2 (Fig 10. i.e. if the slope of two lines l1 and l2 is same. α = β.2 Conditions for parallelism and perpendicularity of lines in terms of their slopes In a coordinate plane. m1 = m2. the lines are parallel. Fig 10. 4) is m 4 ( 2) 6 3 . 4). – 2) and (3. m1 Let us consider the following example. (d) Making inclination of 60° with the positive direction of x-axis.e. – 2). Solution (a) The slope of the line through (3. m2= i. α and β differ by 90°. i. tan α tan β = – 1. lines l1 and l2 are perpendicular to each other. (c) Passing through the points (3. – 2) and (3. Example 1 Find the slope of the lines: (a) Passing through the points (3. or 1 tan m1 m2 = – 1 Conversely. 73 4 (c) The slope of the line through the points (3. If the lines l1 and l2 are perpendicular (Fig 10. – 2) is m 2 ( 2) 0 0. – 2) and (–1. Thus. m1 m2 = – 1. – 2) and (7.. 4). – 2) and (7.tan β = tan (α + 90°) = – cot α = 1 m2 = m 1 i. 5 Therefore. – 2) and (– 1.STRAIGHT LINES 207 Hence. if m1 m2 = – 1. then β = α + 90°.5). two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.e. Hence. two non vertical lines l1 and l2 are parallel if and only if their slopes are equal.. 4) is . 1 3 4 2 (b) The slope of the line through the points (3. 1 or..e. Therefore. (b) Passing through the points (3. Then tan α = – cot β = tan (β + 90°) or tan (β – 90°) Fig 10. 10. Let θ and φ be the adjacent angles between the lines L1 and L2 (Fig10.2. Then θ = α2 – α1 and α1. as 1 + m1m2 ≠ 0 1 m1m2 Fig 10. they make two pairs of vertically opposite angles such that sum of any two adjacent angles is 180°. then we find that these lines are either intersecting or parallel. respectively. Therefore tan θ = tan (α2 – α1) tan 2 tan 1 m m1 2 1 tan 1 tan 2 1 m1m2 (as 1 + m1m2 ≠ 0) and φ = 180° – θ so that tan φ = tan (180° – θ ) = – tan θ = – m2 m1 . α2 ≠ 90°.6). which is not defined. We know that when two lines intersect each other. 33 0 (d) Here inclination of the line α = 60°. Let L1 and L2 be two non-vertical lines with slopes m1 and m2. 6 Now. Here we will discuss the angle between two lines in terms of their slopes. Then m1 tan α1 and m2 tan α2 .208 MATHEMATICS 4 ( 2) 6 . slope of the line is m m = tan 60° = 3. If α1 and α2 are the inclinations of lines L1 and L2. respectively. there arise two cases: .3 Angle between two lines When we think about more than one line in a plane. Therefore. 1 m1 m 2 which means θ will be acute and φ will be obtuse. Case II If m 2 m1 is negative. we get π tan 4 m 1 1 m 2 or 1 m 1 2 1 1 m 2 1 1 m 2 1 or 2 –1. 1 m1 m 2 which means that θ will be obtuse and φ will be acute.. 1 1 1 m 1 m 2 2 m which gives 1 2 1 Therefore m 3 or m . Solution We know that the acute angle θ between two lines with slopes m1 and m2 is given by Let m1 = tan θ m2 m1 1 m1m 2 . Thus. 4 2 Now..STRAIGHT LINES Case I If 209 m 2 m1 is positive. Example 2 If the angle between two lines is π 1 and slope of one of the lines is . 3 . (1) The obtuse angle (say φ) can be found by using φ =1800 – θ. (1) 1 π .. is given by tan θ m 2 m1 .. find 4 2 the slope of the other line. putting these values in (1). as 1 m1m 2 0 1 m1m 2 . the acute angle (say θ) between lines L1 and L2 with slopes m1 and m2. . respectively. then tan θ will be positive and tan φ will be negative. then tan θ will be negative and tan φ will be positive. m2 = m and θ = . 7 explains the 3 reason of two answers. 6) and (4. then two lines will coincide. slope of the other line is 1 3 or .. m1 m2 = –1. 7 Example 3 Line through the points (–2. 8) is perpendicular to the line through the points (8. 12) and (x. Solution Slope of the line through the points (– 2. If two lines having the same slope pass through a common point.210 MATHEMATICS Hence. 24).2. Find the value of x. 3 x 8 10. if A. 8 . B and C are three points in the XY-plane. Hence. i. 8) is m1 86 2 1 4 2 6 3 Slope of the line through the points (8. Fig 10. Fig 10. then they will lie on a line. 12) and (x.e.4 Collinearity of three points We know that slopes of two parallel lines are equal.8) if and only if slope of AB = slope of BC. Fig 10. which gives m2 1 12 1 or x = 4 . 24) is 24 12 12 x8 x 8 Since two lines are perpendicular. 6) and (4. three points are collinear (Fig 10. (ii) PQ is parallel to the x-axis. 8) and (T. find law of motion. or (h – x1) (y2 – y1) = (k – y1) (x2 – x1). Two positions of time and distance are recorded as.STRAIGHT LINES 211 Example 4 Three points P (h. Using the concept of slope. which passes through the origin. 2). D = 2 and when T = 3. . i. Find vertices of the triangle. 7).1 1.. Solution Let (T.e. 4). – 4) and B (8. i. we have Slope of PQ = Slope of QR. where D denotes the distance at time T. points (0. Find the distance between P (x1. The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. time and distance graph of a linear motion is given.9 6 (T 3) 3 (D 8) EXERCISE 10. whose vertices are (– 4. 5. Also. (0. (3. k).. 0). Fig 10. which is the required relation. y1) and R (x2. Find a point on the x-axis. y1) and Q (x2.9. Q (x1. 4. 5). Show that (h – x1) (y2 – y1) = (k – y1) (x2 – x1). 6) and (3. Q and R are collinear. D) be any point on the line. how distance depends upon time.e. find its area. Solution Since points P. – 5) and (– 4. –2). D) are collinear so that 8 2 D 8 or 30 T 3 or D = 2(T + 1). D = 8. 3. y2) when : (i) PQ is parallel to the y-axis. and the mid-point of the line segment joining the points P (0. y1 k y y 2 1 x1 h x2 x1 or k y1 y 2 y1 h x1 x2 x1 . y2) lie on a line. which is equidistant from the points (7. (5. 2. when T = 0. Therefore. Example 5 In Fig 10. Find the slope of a line. Draw a quadrilateral in the Cartesian plane. show that the points (4.–2). This relationship between line and points leads us to find the solution of the following problem: . 3) and (–3.3 Various Forms of the Equation of a Line We know that every line in a plane contains infinitely many points on it. Find the angle between the x-axis and the line joining the points (3. show that points (– 2. Find the value of x for which the points (x. –1) are the vertices of a right angled triangle. – 1). (4. 5) are collinear. If slope of the line is m. 11. Without using distance formula. 2) are the vertices of a parallelogram.–1) and (4. show that k – y1 = m (h – x1). If three points (h. which makes an angle of 30° with the positive direction of y-axis measured anticlockwise. 3 12.10 10. 10. 13. k) lie on a line. 5) and (–1. (3. Consider the following population and year graph (Fig 10. Find the slope of the line. If tangent of the angle between them is 1 . 9. 0). find the slope of the line AB and using it. b) and (0. 8. A line passes through (x1. 4). The slope of a line is double of the slope of another line. k). h k 14. 0). y1) and (h.1) and (4. (a.10). 7.212 MATHEMATICS 6. Without using the Pythagoras theorem. find what will be the population in the year 2010? Fig 10. – 1). show that a b 1. (2. (3. find the slopes of the lines. 3) is x = – 2. otherwise false. 3) is y = 3. equation of the line parallel to y-axis and passing through (– 2. equation of the line parallel tox-axis and passing through (– 2. Choice of sign will depend upon the position of the line according as the line is above or below the y-axis. Fig 10.11 (a)].12. For the equation of L. we will discuss the equation of a line under different conditions. 3). Fig 10. Similarly.11 Example 6 Find the equations of the lines parallel to axes and passing through (– 2.11(b)]. Suppose P (x. Therefore.3. 10. Solution Position of the lines is shown in the Fig 10.STRAIGHT LINES 213 How can we say that a given point lies on the given line? Its answer may be that for a given line we should have a definite condition on the points lying on the line. Now.1 Horizontal and vertical lines If a horizontal line L is at a distance a from the x-axis then ordinate of every point lying on the line is either a or – a [Fig 10. the equation of a vertical line at a distance b from the y-axis is either x = b or x = – b [Fig 10. y) is an arbitrary point in the XY-plane and L is the given line. The y-coordinate of every point on the line parallel to x-axis is 3. equation of the line L is either y = a or y = – a. Similarly. we wish to construct a statement or condition for the point P that is true. therefore. Of course the statement is merely an algebraic equation involving the variables x and y. when P is on L.12 . Equation (1) is indeed the equation for the given line L. y) be an arbitrary point on L (Fig 10. whose slope is m. which is the required equation. Let P (x. x x1 x 2 x1 Fig 10.e. y) on L satisfies (1) and no other point in the plane satisfies (1).3. y y 0 m x x 0 x x0 . The three points P1. P2 and P are collinear.. By slope-intercept form formula (1) above. 3) with slope – 4. Then. x 2 x1 . y0) is (– 2. its coordinates satisfy the equation y – y0 = m (x – x0) Example 7 Find the equation of the line through (– 2. Solution Here m = – 4 and given point (x0 . y) be a general point on L (Fig 10.13 Thus. y2).3 Two-point form Let the line L passes through two given points P1 (x1. y1) and P2 (x2.14 or y y 1 y 2 y1 ( x x1 ). the point (x. if and only if. therefore.14). 3). y) lies on the line with slope m through the fixed point (x0.13). y0). by the definition. 10. equation of the given line is y – 3 = – 4 (x + 2) or 4x + y + 5 = 0. i. Fig 10. the slope of L is given by m y y0 . we have slope of P1P = slope of P1P2 i.. y0) along with all points (x.2 Point-slope form Suppose that P0 (x0.e. y y1 y 2 y1 ..3. y0) is a fixed point on a non-vertical line L..214 MATHEMATICS 10.(1) Since the point P0 (x0 . Let P (x. y1) and (x2. the point (x. Solution Here x1 = 1. Case I Suppose a line L with slope m cuts the y-axis at a distance c from the origin (Fig10. –1) and (3.. x2 = 3 and y2 = 5. Case II Suppose line L with slope m makes x-intercept d.. The distance c is called the yintercept of the line L. we have y 1 or 5 1 x 1 3 1 3 x y 4 0 . c). Therefore. c). where θ is the 2 3 (ii) x-intercept is 4.(3) Note that the value of c will be positive or negative according as the intercept is made on the positive or negative side of the y-axis.15).STRAIGHT LINES 215 Thus. 10.. the equation of L is y c m ( x 0 ) or y mx c Thus. L has slope m and passes through a fixed point (0. Using two-point form (2) above for the equation of the line.4 Slope-intercept form Sometimes a line is known to us with its slope and an intercept on one of the axes. respectively. 5).15 m and y-intercept c lies on the line if and only if y mx c . y) on the line with slope Fig 10. We will now find equations of such lines. Obviously. y2) is given by y y1 y 2 y1 ( x x1) x 2 x1 . by point-slope form. Then equation of L is y m( x d ) . equation of the line passing through the points (x1. 2 . coordinates of the point where the line meet the y-axis are (0.. y1 = – 1. Thus.3. (2) Example 8 Write the equation of the line through the points (1.. Example 9 Write the equation of the lines for which tan θ = inclination of the line and (i) y-intercept is – 1 .. which is the required equation. (4) Students may derive this equation themselves by the same method as in Case I. Solution Here a = –3 and b = 2.and y-axes respectively. 2 Therefore.. 2 2 Therefore. equation of the line is x y 1 3 2 or 2x 3 y 6 0 . (ii) Here. we have y0 i. by slope-intercept form (3) above. 0a x y 1. a b Thus.16). By intercept form (5) above. 2 2 y which is the required equation. . 2 which is the required equation.216 MATHEMATICS Solution (i) Here.intercept c = – .e. Obviously L meets x-axis at the point (a. (5) Example 10 Find the equation of the line. the equation of the line is y 1 ( x 4) or 2 y x 4 0 . the equation of the line is 1 3 x or 2 y x 3 0 . which makes intercepts –3 and 2 on the x. is x y 1 a b Fig 10. respectively.. b) (Fig . b0 (x a) or ay bx ab . 10.5 Intercept . by slope-intercept form (4) above.16 .form Suppose a line L makes x-intercept a and y-intercept b on the axes.10. slope of the line is m = tan θ = 1 3 and y . 0) and y-axis at the point (0. By two-point form of the equation of the line.. we have m = tan θ = 1 and d = 4. equation of the line making intercepts a and b on x-and y-axis.3. by sin ω point-slope form. (ii) Angle which normal makes with the positive direction of x-axis. Further. whose perpendicular distance from origin O be OA = p and the angle between the positive x-axis and OA be ∠XOA = ω. p sin ω on it. the equation of the line L is . The possible positions of line L in the Cartesian plane are shown in the Fig 10. Therefore. slope of OA tan ω sin ω cos ω and point A p cos ω.6 Normal form Suppose a non-vertical line is known to us with following data: (i) Length of the perpendicular (normal) from origin to the line. the line L has slope 1 1 cos ω . Now.STRAIGHT LINES 217 10. Therefore The slope of the line L = Thus. so that the coordinates of the point A are (p cos ω. we have OM = p cos ω and MA = p sin ω. Let L be the line. our purpose is to find slope of L and a point on it. line L is perpendicular to OA. Fig 10. p sin ω).3.17 In each case. Draw perpendicular AM on the x-axis in each case.17. (6) Example 11 Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with positive direction of x-axis is 15°. the point (F. we have two points (32. the equation of the line is 3 1 3 1 x y 4 or 2 2 2 2 This is the required equation. or K . 373) in XY-plane. Hence. Express K in terms of F and find the value of F..18 3 1 x 3 1 y 8 2. By two-point form. 273) and (212.. Given that K = 273 when F = 32 and that K = 373 when F = 212. we are given p = 4 and ω = 150 (Fig10. Now cos 15° = and sin 15º = 3 1 2 2 3 1 (Why?) 2 2 By the normal form (6) above. Solution Assuming F along x-axis and K along y-axis.. x cos 150 y sin 150 4 or Fig 10. the equation of the line having normal distance p from the origin and angle ω which the normal makes with the positive direction of x-axis is given by x cos ω + y sin ω = p .18). Solution Here. when K = 0. Example 12 The Fahrenheit temperature F and absolute temperature K satisfy a linear equation.218 MATHEMATICS y p sin ω cos ω x p cos ω or sin ω x cos ω y sin ω p (sin 2ω cos 2ω) or x cos ω + y sin ω = p. (1) .. K) satisfies the equation K 273 373 273 100 F 32 or K 273 F 32 212 32 180 5 F 32 273 9 which is the required relation. 273) and (212. that the equation y = mx + c. Write the equations for the x-and y-axes. Intersecting the x-axis at a distance of 3 units to the left of origin with slope –2.. . we can take equation in the form K = mF + c . Passing through 2. 373). (3) 5 2297 and c = ... 2 3 and inclined with the x-axis at an angle of 75o. (4) gives F = – 459. Passing through the point (– 4. 1) and (2. 5. For finding these two constants. we get m= . (1) Equation (1) is satisfied by (32. we need two conditions satisfied by the equation of line. In all the examples above. we get 5 2297 F 9 9 which is the required relation.. Alternate method We know that simplest form of the equation of a line is y = mx + c. When K = 0. we are given two conditions to determine the equation of the line. find the equation of the line which satisfy the given conditions: 1.. Intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30o with positive direction of the x-axis. EXERCISE 10. (2) and 373 = 212m + c Solving (2) and (3). Therefore 273 = 32m + c .4 5 or F= 459..4.4 . contains two constants. Again assuming F along x-axis and K along y-axis. Passing through the points (–1. Equation (1) gives 0 5 F 32 273 9 or F 32 273 9 491. – 4). 3) with slope 1 . namely. Passing through (0.STRAIGHT LINES 219 When K = 0.2 In Exercises 1 to 8. K .. 6. 4. (4) Note We know. 9 9 Putting the values of m and c in (1).. 2 3. m and c. 7. 2. 0) with slope m. 2) are collinear. 10. Graph of the equation Ax + By + C = 0 is always a straight line. B and C are real constants such that A and B are not zero simultaneously. By using the concept of equation of a line. The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. express L in terms of C. 9). 11. 3) and R (4. A line perpendicular to the line segment joining the points (1. Q (–2. – 2) and (8. Find equation of the line passing through the point (2. 18. 0). find the equation of the line. Find the equation of the line passing through (–3. prove that the three points (3. 14. 6). a b 19. .220 MATHEMATICS 8. 13. Find equation of the median through the vertex R. 10. 1). 3) divides it in the ratio 1: n. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2.4 General Equation of a Line In earlier classes. 5) and perpendicular to the line through the points (2. Point R (h. Perpendicular distance from the origin is 5 units and the angle made by the perpendicular with the positive x-axis is 300. Find the equation of the line. Find equation of the line.942 when C = 20 and L= 125.134 when C = 110. Also. 2π with the 3 positive x-axis. 0) and (2. (– 2. The vertices of Δ PQR are P (2. 12. k) divides a line segment between the axes in the ratio 1: 2. find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin. we have studied general equation of first degree in two variables. 3). Assuming a linear relationship between selling price and demand. if L = 124. In an experiment. he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. 15. 9. Show that equation Find equation of the line through the point (0. 2) making an angle of the line is x y 2. 17. 5) and (–3. 16. 2) and cutting off intercepts on the axes whose sum is 9. 20. The perpendicular from the origin to a line meets it at the point (–2. where A. b) is the mid-point of a line segment between axes. The owner of a milk store finds that. 5). Ax + By + C = 0. how many litres could he sell weekly at Rs 17/litre? P (a. A B C p cos ω sin ω . A (b) Intercept form If C ≠ 0. then Ax + By + C = 0 can be written as Ax + By = 0.STRAIGHT LINES 221 Therefore. therefore. then Ax + By + C = 0 can be written as y A C x or y mx c B B . A B If C = 0.. (1) A C and c .. has zero intercepts on the axes. then Ax + By + C = 0 can be written as x y x y 1 or 1 C C a b A B where a= . both the equations are same and therefore. by the following procedures: (a) Slope-intercept form If B ≠ 0. then x = x-intercept is C . (c) Normal form Let x cos ω + y sin ω = p be the normal form of the line represented by the equation Ax + By + C = 0 or Ax + By = – C. Thus. which is a line passing through the origin and. any equation of the form Ax + By + C = 0. 10. B B If B = 0. and y-intercept is .4. where A and B are not zero simultaneously is called general linear equation or general equation of a line. (2) C C and b = ..1 Different forms of Ax + By + C = 0 The general equation of a line can be reduced into various forms of the equation of a line. A B We know that equation (2) is intercept form of the equation of a line whose x-intercept is C C and y-intercept is . B B We know that Equation (1) is the slope-intercept form of the equation of a line where m whose slope is A C ..which is a vertical line whose slope is undefined and A C . Thus. the normal form of the equation Ax + By + C = 0 is x cos ω + y sin ω = p. Proper choice of signs is made so that p should be positive. (1) Comparing (1) with y = mx + c. we have x-intercept as a = and a b 3 . sin ω B A B 2 2 and p C 2 A B 2 . Find its (i) slope.. (ii) x . Example 13 Equation of a line is 3x – 4y + 10 = 0. C C 2 sin ω cos ω 2 Now 2 p or 2 C 2 2 A B cosω Therefore 2 Ap C or p A A B 2 2 Bp C 2 1 C 2 A B 2 and sinω B 2 A B 2 ...and y-intercepts. 4 Equation 3x – 4y + 10 = 0 can be written as 3 x 4 y 10 Comparing (2) with y-intercept as b = 5 . we have slope of the given line as m = (ii) 3 .222 MATHEMATICS cos ω which gives Ap Bp and sin ω . (2) x y 10 1 .. 2 or x y 1 10 5 3 2 . Solution (i) Given equation 3x – 4y + 10 = 0 can be written as y 3 5 x 4 2 . where cos ω A A B 2 2 . . we get p = 4 and ω = 30°. (2) 1 . Example 16 Show that two lines a1 x b1 y c1 0 and a 2 x b 2 y c 2 0 .. Example15 Find the angle between the lines y 3 x 5 0 and 3y x 6 0 . (2) 2 2 Comparing (2) with x cos ω + y sin ω = p.. Solution Given lines are y 3 x 5 0 or y 3 x 5 and . we get 3 1 x y 4 or cos 30 x sin 30 y 4 .. angle between two lines is either 30° or 180° – 30° = 150°. b2 ≠ 0 are: . Hence. (1) 12 2 . Find the values of p and ω. (3) Putting the values of m1 and m2 in (3)... where b1.. Solution Given equation is 3x y 8 0 Dividing (1) by 3 2 .. we get 1 3 3 tan θ 1 1 3 3 1 3 1 2 3 3 which gives θ = 30°.. 3 The acute angle (say) θ between two lines is given by tan θ m 2 m1 1 m1m 2 . (1) 1 3 y x 6 0 or y 3 x 2 3 Slope of line (1) is m1 = 3 and slope of line (2) is m2 = .STRAIGHT LINES Example 14 Reduce the equation 223 3 x y 8 0 into normal form.. . which gives a1 a 2 a1 a . 1 or a a + b b = 0 1 2 1 2 b1 b2 Example 17 Find the equation of a line perpendicular to the line x 2 y 3 0 and passing through the point (1. – 2) is y ( 2) 2(x 1) or y = 2 x . (2) Slopes of the lines (1) and (2) are m1 = a1 and m2 = b1 a 2 .224 MATHEMATICS (i) Parallel if a1 b1 a2 ... . – 2). Now b2 (i) Lines are parallel. slope of the line perpendicular to line (1) is 2 1 2 m1 Equation of the line with slope – 2 and passing through the point (1.(1) 1 . which gives a1 a2 . (1) y a2 x c2 b2 b2 . 2 or b1 b2 b1 b2 (ii) Lines are perpendicular. if m1. Solution Given line x 2 y 3 0 can be written as y 1 3 x 2 2 Slope of the line (1) is m1 = m2 .... if m1 = m2.m2 = – 1. and (ii) Perpendicular if a1 a 2 b1 b 2 0 . respectively. Therefore. b2 Solution Given lines can be written as and y a1 x c1 b1 b1 . which is the required equation. respectively.QR . . A x1 B y 1 C .. area (ΔPQR) 1 C C C x1 0 y1 0 y 1 0 2 B A B or 2 area (ΔPQR) QR 2 area (ΔPQR) 2 1 C C C2 x1 y 1 2 B A AB C . Then. Draw a perpendicular PM from the point P to the line L (Fig10. which gives PM = 2 Also. we get .STRAIGHT LINES 225 10. 0 and R A C 0.5 Distance of a Point From a Line The distance of a point from a line is the length of the perpendicular drawn from the point to the line. (1) . and AB C QR 0 A C 0 B 2 C AB 2 2 A B Substituting the values of area (ΔPQR) and QR in (1). whose distance from the point P (x1. coordinates of the points are C Q .19). Let L : Ax + By + C = 0 be a line. the area of the triangle PQR B is given by 1 area (PQR) PM. y1) is d. If Fig10.. Thus.19 the line meets the x-and y-axes at the points Q and R. the perpendicular distance (d) of a line Ax + By+ C = 0 from a point (x1.5..226 MATHEMATICS PM or A x1 B y 1 C 2 2 A B A x1 B y1 C d . distance between the lines (1) and (2) is m c1 c2 m 1 m2 or d = c1 c2 1 m2 .. Ax + By + C1 = 0 and Ax + By + C2 = 0. i. Therefore. y1) is given by d A x1 B y1 C 2 2 A B ... 0 as shown in Fig10.20 Distance between two lines is equal to the length of the perpendicular from point A to line (2). (2) Line (1) will intersect x-axis at the point c1 A m . If lines are given in general form.. two parallel lines can be taken in the form . Thus. 10.e. Fig10.20.1 Distance between two parallel lines We know that slopes of two parallel lines are equal. . 2 2 A B Thus. (1) y = mx + c1 and y = mx + c2 . the distance d between two parallel lines y mx c1 and y mx c2 is given by d= c1 c2 1 m2 . Therefore. . –5). 5 Example 19 Find the distance between the parallel lines 3x – 4y +7 = 0 and 3x – 4y + 5 = 0 Solution Here A = 3. (i) x + 7y = 0. The distance of the given point from given line is d Ax1 By1 C A B 2 2 3. B = –4. Given point is (x1.STRAIGHT LINES then above formula will take the form d 227 C1 C 2 A 2 B2 Students can derive it themselves.3 –4 –5 – 26 32 –4 2 3 . (ii) y – 2 = 0. 3 4 6. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis. 2. Example 18 Find the distance of the point (3. C1 = 7 and C2 = 5. B = – 4 and C = – 26.intercept form and find their slopes and the y . Find the distance of the point (–1. Reduce the following equations into normal form. Reduce the following equations into slope . whose distances from the line x y 1 are 4 units.intercepts. 1) from the line 12(x + 6) = 5(y – 2). (i) 3x + 2y – 12 = 0. 4. (1) Comparing (1) with general equation of line Ax + By + C = 0. y1) = (3.. (iii) y = 0. (iii) x – y = 4. – 5) from the line 3x – 4y –26 = 0. 5 EXERCISE 10. we get A = 3. Reduce the following equations into intercept form and find their intercepts on the axes. 5. (ii) 6x + 3y – 5 = 0. the required distance is d 7–5 32 –4 2 2 .3 1. Therefore. Find the distance between parallel lines (i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0 (ii) l (x + y) + p = 0 and l (x + y) – r = 0. 3. (ii) 4x – 3y = 6. (iii) 3y + 2 = 0.. Solution Given line is 3x – 4y –26 = 0 . (i) x – 3y + 8 = 0. Find the points on the x-axis. Find the equation of the right bisector of the line segment joining the points (3. 5 x ky 3 0 and 3 x y 2 0 are concurrent. 8. 3) and (4. 10. Find angles between the lines 3 x y 1and x 3 y 1. –1) and C (1.228 MATHEMATICS 7. 3) to the line 3x – 4y – 16 = 0. The line through the points (h. if they pass through a common point. 12. Prove that the line through the point (x1... find the value of k. The perpendicular from the origin to the line y = mx + c meets it at the point (–1.e. find equation of the other line. B (4. 16.. 2 p a b Miscellaneous Examples Example 20 If the lines 2 x y 3 0 . Find the coordinates of the foot of perpendicular from the point (–1. 3) intersects each other at an angle of 60o. Find the values of m and c. (2) . Two lines passing through the point (2. 2). prove that p2 + 4q2 = k2. 4) and (–1. (1) 5x + ky – 3 = 0 . In the triangle ABC with vertices A (2. If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b. find the equation and length of altitude from the vertex A.. i. respectively. 15. point of intersection of any two lines lies on the third line. y1) and parallel to the line Ax + By + C = 0 is A (x –x1) + B (y – y1) = 0. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3. Find the value of h. 14. 9. 2). Solution Three lines are said to be concurrent. 17. 3). 13. at right angle. Find equation of the line parallel to the line 3 x 4 y 2 0 and passing through the point (–2. If p and q are the lengths of perpendiculars from the origin to the lines x cos θ y sin θ k cos 2θ and x sec θ + y cosec θ = k. 2). Here given lines are 2x + y – 3 = 0 .. then show that 1 1 1 2 2. 1) intersects the line 7 x 9 y 19 0. 18. 3). If slope of one line is 2. 11. (3) Solving (1) and (3) by cross-multiplication method. 4). Solution Given line is 4x – y = 0 In order to find the distance of the line (1) from the point P (4. = 1 4 2 4 12 3 2 units .. the point (1.. distance of line (1) from the point P (4. (1) (1.1 + k . 4)..STRAIGHT LINES 3x – y – 2 = 0 229 . 23 9 4 23 Therefore. Since above three lines are concurrent. the point of intersection of two lines is (1. 1) is .21). 4) Fig 10.1 – 3 = 0 or k = – 2. we get x y 1 or x 1... we have to find the point of intersection of both the lines. we will first find the equation of the second line (Fig 10. Now. (2) Solving (1) and (2). 1) will satisfy equation (2) so that 5. 1) measured along the line making an angle of 135° with the positive x-axis. 1).. Solution Let Q (h.. 1) along the line (2) = the distance between the points P (4. 2) in the line x – 3y + 4 = 0 .21 y – 1 = – 1 (x – 4) or x + y – 5 = 0 . 2) in the line x 3 y 4 0 . (1) . Slope of second line is tan 135° = –1. we get x = 1 and y = 4 so that point of intersection of the two lines is Q (1. Equation of the line with slope – 1 through the point P (4. Example 22 Assuming that straight lines work as the plane mirror for a point. k) is the image of the point P (1. For this purpose. y 1 .. 1) along another line. 1) and Q (1. Example 21 Find the distance of the line 4x – y = 0 from the point P (4. find the image of the point (1. 22 Therefore.. i. Slope of line x 3 y 4 0 . will satisfy the equation (1) so that 2 2 and the mid-point of PQ. we get h = 6 7 and k = . the image of the point (1.e.. .. (2) or 3h k 5 h 1 k 2 .230 MATHEMATICS Fig10. 2) in the line (1) is . 5 5 Example 23 Show that the area of the triangle formed by the lines y m1 x c1 . point h 1 k 2 3 4 0 or h 3k 3 2 2 Solving (2) and (3). .. the line (1) is the perpendicular bisector of line segment PQ (Fig 10. (3) . 5 5 6 7 Hence. y m 2 x c 2 c –c and x = 0 is 1 2 2 2 m1 m 2 .. Hence Slope of line PQ = so that k 2 1 h 1 1 3 1 .22). . c). the area of the triangle is c 2 c1 2 m c m2 c1 c c 1 m c m2 c1 c2 2 1 c2 c1 0 c1 1 2 0 1 2 2 m1 m2 2 m1 m 2 m1 m2 m1 m2 Example 24 A line is such that its segment between the lines 5x – y + 4 = 0 and 3x + 4y – 4 = 0 is bisected at the point (1. β1) and (α2. Obtain its equation... (1) y = m2 x + c2 .STRAIGHT LINES 231 Solution Given lines are y = m1 x + c1 . 23). (2) Let the required line intersects the lines (1) and (2) at the points. Solution Given lines are 5x – y + 4 = 0 . 5). (α1. m1 m2 . (2) x=0 . c1) and Q (0.. third vertex of the triangle is R m m . (3) We know that line y = mx + c meets the line x = 0 (y-axis) at the point (0. two vertices of the triangle formed by lines (1) to (3) are P (0.24). c2) (Fig 10.. respectively (Fig10.. Therefore.... Solving (1) and (2). β2). 2 1 Now. Therefore 5α1 – β1 + 4 = 0 and 3 α2 + 4 β 2 – 4 = 0 . we get x c2 c1 m1 m2 and y m1c2 m2c1 m1 m2 Fig 10.. (1) 3x + 4y – 4 = 0 . Third vertex can be obtained by solving equations (1) and (2).23 c2 c1 m1c2 m2 c1 Therefore. . β1) is 222 5 β 5 y 5 1 ( x 1) or y 5 23 (x 1) 26 α1 1 1 23 or 107x – 3y – 92 = 0. Example 25 Show that the path of a moving point such that its distances from two lines 3x – 2y = 5 and 3x + 2y = 5 are equal is a straight line. k) is any point. or α1 α 2 2 and 20 α1 3 α 2 20 . β1 5. Therefore 3h 2k 5 94 3h 2k 5 94 or 3h 2k 5 3h 2k 5 . β2) is (1.. Therefore or β1 = 5α1 + 4 and β 2 β β α1 α 2 1 and 1 2 5 . 2 2 or α1 α 2 2 and 5 α1 4 2 4 3α2 4 5. β1) and (α2. we get α1 = 26 222 20 26 4 and α 2 = and hence. Solution Given lines are 3x – 2y = 5 … (1) and … (2) 3x + 2y = 5 Let (h.232 MATHEMATICS 4 – 3α 2 . 5). whose distances from the lines (1) and (2) are equal. . 23 23 23 23 Equation of the required line passing through (1. which is the equation of required line. 4 We are given that the mid point of the segment of the required line between (α1. (3) Solving equations in (3) for α1 and α2. which gives 3h – 2k – 5 = 3h + 2k – 5 or – (3h – 2k – 5) = 3h + 2k – 5. 5) and (α1.. 10. Find the equation of a line drawn perpendicular to the line 8. where it meets the y-axis. Find perpendicular distance from the origin to the line joining the points (cosθ. 3. If three lines whose equations are y = m1x + c1. respectively. 12. which represent straight lines. (c) Passing through the origin. then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0. 5. Hence. 9. if the equation x cos θ + y sinθ = p is the normal form of the line 3 x + y + 2 = 0. Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes. Thus. Find the value of p so that the three lines 3x + y – 2 = 0. the point (h. Find the area of the triangle formed by the lines y – x = 0. 11. 2) which make an angle of 45o with the line x – 2y = 3. Find the equation of the lines through the point (3. Find the values of k for which the line (k–3) x – (4 – k2) y + k2 –7k + 6 = 0 is (a) Parallel to the x-axis. px + 2 y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point. 4. x y 1 through the 4 6 point. Find the values of θ and p. 6. . 2. which cut-off intercepts on the axes whose sum and product are 1 and – 6. path of the point 3 equidistant from the lines (1) and (2) is a straight line. Find the equations of the lines. k) satisfies the 3 5 . Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x – 7y + 5 = 0 and 3x + y = 0. x + y = 0 and x – k = 0. y = m2x + c2 and y = m3x + c3 are concurrent. Miscellaneous Exercise on Chapter 10 1. What are the points on the y-axis whose distance from the line x y 1 is 3 4 4 units. sin θ) and (cos φ. sin φ).STRAIGHT LINES Solving these two relations we get k = 0 or h = equations y = 0 or x = 233 5 . 7. (b) Parallel to the y-axis. Find the direction in which a straight line must be drawn through the point (–1. 18. 16. 2) reflects on the x-axis at point A and the reflected ray passes through the point (5. A ray of light passing through the point (1. Find the distance of the line 4x + 7y + 5 = 0 from the point (1. α ≠ 90°. y) from the lines x + y – 5 = 0 and 3x – 2y +7 = 0 is always 10. the line joining (–1. 21. 7) is divided by the line x + y = 4? 15.0 to the line Summary Slope (m) of a non-vertical line passing through the points (x1. 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror. 2) along the line 2x – y = 0. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4. 1) and (5. 3). x 2 x1 x1 x 2 If a line makes an angle á with the positive direction of x-axis. Find an equation of the legs (perpendicular sides) of the triangle. then the slope of the line is given by m = tan α. The hypotenuse of a right angled triangle has its ends at the points (1. 3) and (– 4. 23. Find equation of the path that he should follow. a b 24. 22. is given by m . In what ratio. If sum of the perpendicular distances of a variable point P (x. Show that the equation of the line passing through the origin and making an angle θ with the line y mx c is y m tan θ . 20. Prove that the product of the lengths of the perpendiculars drawn from the x y cosθ sin θ 1is b 2 . 19. y1) and (x2.0 and a 2 b2 . y2) y 2 y1 y1 y 2 . Slope of horizontal line is zero and slope of vertical line is undefined. x 1 m tan θ 14. Find the image of the point (3. x 1 x 2. 17. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0. A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. 1). 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point. Find the coordinates of A.234 MATHEMATICS 13. points a 2 b 2 . find the value of m. Show that P must move on a line. y) on the line with slope m and y-intercept c lies on the line if and only if y mx c . Equation of the vertical line having distance b from the y-axis is either x = b or x = – b. yo). Three points A. is called the general linear equation or general equation of a line. y2) is given by y y1 y 2 y1 ( x x1). if and only if its coordinates satisfy the equation y – y o = m (x – xo). if and only if slope of AB = slope of BC. is given by d C1 C 2 2 2 A B . y1) and (x2. The perpendicular distance (d) of a line Ax + By+ C = 0 from a point (x1. Two lines are perpendicular if and only if product of their slopes is –1. respectively. Any equation of the form Ax + By + C = 0. Equation of the horizontal line having distance a from the x-axis is either y = a or y = – a.1 m1 m2 0 . Equation of the line passing through the points (x1. 2 2 A B Distance between the parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0. 235 . with A and B are not zero. a b The equation of the line having normal distance from origin p and angle between normal and the positive x-axis ω is given by x cos ω y sin ω p . simultaneously.STRAIGHT LINES An acute angle (say θ) between lines L1 and L2 with slopes m1 and m2 is m2 – m1 given by tanθ 1 m m . x 2 x1 The point (x. 1 2 Two lines are parallel if and only if their slopes are equal. Then equation of the line is y = m (x – d). is x y 1. B and C are collinear. If a line with slope m makes x-intercept d. y1) is given by d Ax1 B y1 C . Equation of a line making intercepts a and b on the x-and y-axis. The point (x. y) lies on the line with slope m and through the fixed point (xo. 1 . ellipses. 11. Suppose we rotate the line m around the line l in such a way that the angle α remains constant. we shall study about some other curves. known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. – BERTRAND RUSSELL © no N C tt E o R be T re pu In the preceding Chapter 10. These curves are in fact. we have studied various forms of the equations of a line. in the subsequent sections we will see how the intersection of a plane with a double napped right circular cone results in different types of curves. etc. circles.2 Sections of a Cone Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α (Fig11.C.) design of telescopes and antennas. Now. In this Chapter. These curves have a very wide range of applications in fields such as planetary motion. Apollonius (262 B. parabolas and hyperbolas.Chapter 11 he CONIC SECTIONS Let the relation of knowledge to real life be very visible to your pupils 11. Then the surface generated is a double-napped right circular hollow cone herein after referred as Fig 11. viz. -190 B.C. reflectors in flashlights and automobile headlights..1). The names parabola and hyperbola are given by Apollonius.1 Introduction bl is and let them understand how by knowledge the world could be transformed. 4).237 Fig 11. we have the following situations: (a) When β = 90o.7). the section so obtained is called a conic section. The vertex separates the cone into two parts called nappes.2).3). (d) When 0 ≤ β < α.6). 3 © no N C tt E o R be T re pu cone and extending indefinitely far in both directions (Fig11. If we take the intersection of a plane with a cone. Thus. parabola and hyperbola When the plane cuts the nappe (other than the vertex) of the cone.1 Circle. The rotating line m is called a generator of the cone. We obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. . the plane cuts entirely across one nappe of the cone). the section is an ellipse (Fig11. 11.5). ellipse. the line l is the axis of the cone. The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex. conic sections are the curves obtained by intersecting a right circular cone by a plane. the section is a parabola (Fig11.2. Let β be the angle made by the intersecting plane with the vertical axis of the cone (Fig11. (b) When α < β < 90o. (c) When β = α. 2 bl is he CONIC SECTIONS Fig 11. the section is a circle (Fig11. (In each of the above three situations. the plane cuts through both the nappes and the curves of intersection is a hyperbola (Fig11. The point V is called the vertex. 6 Fig 11.9). (c) When 0 ≤ β < α. It is the degenerated case of a parabola. 4 is he 238 Fig 11. the section is a pair of intersecting straight lines (Fig11. . (b) When β = α. we have the following different cases: (a) When α < β ≤ 90o. then the section is a point (Fig11. 5 © no N C tt E o R be T re pu Fig 11.8).2.10). It is the degenerated case of a hyperbola.2 Degenerated conic sections When the plane cuts at the vertex of the cone.MATHEMATICS bl Fig 11. 7 11. the plane contains a generator of the cone and the section is a straight line (Fig11. © no N C tt E o R be T re pu Fig 11. we shall obtain the equations of each of these conic sections in standard form by defining them based on geometric properties. 9 Fig 11. The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle (Fig 11. . 10 Definition 1 A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.CONIC SECTIONS 239 bl is he In the following sections. 8 11.11).3 Circle Fig 11. 2) and radius 4.MATHEMATICS Fig 11. Let P(x. Therefore. Solution Here h = –3.0) and radius r. completing the squares within the parenthesis. (x + 4)2 + (y + 5)2 = 49 i. k = 2 and r = 4. we get (x2 + 8x + 16) + (y2 + 10y + 25) = 8 + 16 + 25 i. Example 1 Find an equation of the circle with centre at (0.e. the given circle has centre at (– 4. {x – (– 4)}2 + {y – (–5)}2 = 72 Therefore. the equation of the circle is x2 + y2 = r2. Solution Here h = k = 0.e. Example 2 Find the equation of the circle with centre (–3. we derive below the equation of the circle with a given centre and radius (Fig 11. Then. (x – h)2 + (y – k)2 = r2 This is the required equation of the circle with centre at (h. 11 he 240 © no N C tt E o R be T re pu bl The equation of the circle is simplest if the centre of the circle is at the origin. Therefore. . 12 is Fig 11. k) be the centre and r the radius of circle. by the definition.12). By the distance formula. y) be any point on the circle (Fig11. Given C (h. | CP | = r .k) and radius r . the equation of the required circle is (x + 3)2 + (y –2)2 = 16 Example 3 Find the centre and the radius of the circle x2 + y2 + 8x + 10y – 8 = 0 Solution The given equation is (x2 + 8x) + (y2 + 10y) = 8 Now. –5) and radius 7. we have (x – h) 2 + (y – k ) 2 = r i.e.12). However. . Find the equation of the circle passing through (0. find the centre and radius of the circles. 2x2 + 2y2 – x = 0 10. centre (–a. – 2).3) and (–1. the equation of the required circle is (x – 0. (1) (2 – h)2 + (–2 – k)2 = r2 .58 Hence. centre (–2.7. outside or on the circle x2 + y2 = 25? . Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2.58. –b) and radius 4. (3) Solving the equations (1). 12. 11. (2) and (3 – h)2 + (4 – k)2 = r2 Also since the centre lies on the line x + y = 2.1) and (6. Does the point (–2. EXERCISE 11.2) and radius 2 2.0) and making intercepts a and b on the coordinate axes.3).1) and radius 2 a2 − b2 . In each of the following Exercises 6 to 9. and (3. 5.4) and whose centre lies on the line x + y = 2. we have h+k=2 . k = 1. x2 + y2 – 4x – 8y – 45 = 0 8.. x2 + y2 – 8x + 10y – 12 = 0 9. we have .3) and radius 4 1 1 1 ) and radius 12 2 4 3.. 6.3)2 = 12. (x + 5)2 + (y – 3)2 = 36 7. he Solution Let the equation of the circle be (x – h)2 + (y – k)2 = r2. (2) and (3).CONIC SECTIONS 241 Example 4 Find the equation of the circle which passes through the points (2. Find the equation of the circle passing through the points (2.4). © no N C tt E o R be T re pu In each of the following Exercises 1 to 5. centre (1. 15. – 2) and (3. Find the equation of the circle passing through the points (4. 3.1 is bl Since the circle passes through (2.1) and whose centre is on the line x – 3y – 11 = 0.2) and passes through the point (4.5) lie inside.5.. Find the equation of a circle with centre (2. find the equation of the circle with 1. centre (0.5) and whose centre is on the line 4x + y = 16. centre ( .7)2 + (y – 1.. 14. we get h = 0. 13.3 and r2 = 12.5).. .14).14 equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis. which are equidistant from the fixed point and the fixed line is the straight line through the fixed point and perpendicular to the fixed line.4 Parabola is Fig 11. (‘Para’ means ‘for’ and ‘bola’ means ‘throwing’. he Definition 2 A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The point of intersection of parabola with the axis is called the vertex of the parabola (Fig11. i.13). We call this straight line as degenerate case of the parabola.e. The four possible such orientations of parabola are shown below in Fig11. 13 © no N C tt E o R be T re pu bl $ Note If the fixed point lies on the fixed line.15 (a) to (d). The fixed line is called the directrix of the parabola and the fixed point F is called the focus (Fig 11.1 Standard equations of parabola The Fig 11. A line through the focus and perpendicular to the directrix is called the axis of the parabola.4. 11.242 MATHEMATICS 11. then the set of points in the plane.. the shape described when you throw a ball in the air). Let the distance from the directrix to the focus be 2a. and the equation of the directrix is x + a = 0 as in Fig11.243 bl Fig 11. Then. 0).. we have (x + a) 2 (x – a) 2 + y 2 = (x + a ) 2 i. OX the x-axis and OY perpendicular to it as the y-axis. . Take O as origin. By the distance formula. Let FM be perpendicular to the directrix and bisect FM at the point O. 0) a > 0. (1) where PB is perpendicular to l.15 (a) to (d) is he CONIC SECTIONS © no N C tt E o R be T re pu We will derive the equation for the parabola shown above in Fig 11. y) be any point on the parabola such that PF = PB. y). .16. the coordinates of the focus are (a.16 Let P(x. (x – a)2 + y2 = (x + a)2 or x2 – 2ax + a2 + y2 = x2 + 2ax + a2 or y2 = 4ax ( a > 0). and directricx x = – a as below: Let F be the focus and l the directrix. Fig 11.. The coordinates of B are (– a. the mid-point O is on the parabola and is called the vertex of the parabola.15 (a) with focus at (a. we have PF = (x – a) 2 + y 2 and PB = Since PF = PB.e. By the definition of parabola. Produce MO to X. we have the following observations: 1. 3. we can derive the equations of the parabolas in: Fig 11. the study of the equations of parabolas with focus at any point and any line as directrix is beyond the scope here. .y) lies on the parabola. Similarly. When the axis of symmetry is along the x-axis the parabola opens to the (a) right if the coefficient of x is positive... focus at (a. x can assume any positive value or zero but no negative value and the curve extends indefinitely far into the first and the fourth quadrants. (3) and so P(x.15. Fig11. Fig 11.15 (c) as x2 = 4ay.0) and directrix x = – a is y2 = 4ax. From the standard equations of the parabolas. Fig 11.If the equation has a y2 term. let P(x. However. Note The standard equations of parabolas have focus on one of the coordinate $ axis. When the axis of symmetry is along the y-axis the parabola opens (c) upwards if the coefficient of y is positive. from (2) and (3) we have proved that the equation to the parabola with vertex at the origin. vertex at the origin and thereby the directrix is parallel to the other coordinate axis. Parabola is symmetric with respect to the axis of the parabola. y) satisfy the equation (2) = (x – a) 2 + y 2 = (x + a) 2 = PB = (x – a ) 2 + 4ax he PF .. . any point on the parabola satisfies y2 = 4ax. (2) Conversely. is Thus. then the axis of symmetry is along the y-axis.15 (b) as y2 = – 4ax. then the axis of symmetry is along the x-axis and if the equation has an x2 term. The axis of the parabola is the positive x-axis. (b) left if the coefficient of x is negative. since a > 0. These four equations are known as standard equations of parabolas.15 (d) as x2 = – 4ay.244 MATHEMATICS Hence. 2. © no N C tt E o R be T re pu bl Discussion In equation (2). (d) downwards if the coefficient of y is negative.. Solution The given equation involves y2.19 Thus.18 Example 5 Find the coordinates of the focus. 0) and the equation of the directrix of the parabola is x = – 2 (Fig 11. Fig 11. the equation of the directrix and latus rectum of the parabola y2 = 8x.4. so the axis of symmetry is along the x-axis.17 Fig 11. © no N C tt E o R be T re pu bl AB = Length of the latus rectum = 4a.17). But AC = FM = 2a Hence AF = 2a. Comparing with the given equation y2 = 4ax. axis. the focus of the parabola is (2. To find the Length of the latus rectum of the parabola y2 = 4ax (Fig 11.19). he By the definition of the parabola.2 Latus rectum Definition 3 Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola. is And since the parabola is symmetric with respect to x-axis AF = FB and so Fig 11. AF = AC.18). through the focus and whose end points lie on the parabola (Fig11. . Length of the latus rectum is 4a = 4 × 2 = 8. The coefficient of x is positive so the parabola opens to the right.CONIC SECTIONS 245 11. we find that a = 2. 0). ⎝ 3⎠ EXERCISE 11.0) and directrix x = – 2. Hence the equation of the parabola is of the form either y2 = 4ax or y2 = – 4ax. i. the equation is of the form x2 = 4ay or x2 = – 4ay. y2 = – 8x 4. it must open downwards. the equation of the parabola is ⎛1⎞ x2 = − 4 ⎜ ⎟ y. 3x2 = – 4y..–3).e. we have 22 = – 4a (–3).2) which lies on y-axis. thus. y2 = 12x 2.e. x2 = 6y 3. Thus the equation is of the form x2 = – 4ay. But the parabola passes through (2. 2).–3) which lies in the fourth quadrant.0) lies on the x-axis. 0) and focus at (0. bl is Solution Since the vertex is at (0. 1.. the x-axis itself is the axis of the parabola. © no N C tt E o R be T re pu Solution Since the parabola is symmetric about y-axis and has its vertex at the origin. x2 = – 16y 5.246 MATHEMATICS Example 6 Find the equation of the parabola with focus (2. Therefore. he Solution Since the focus (2. the equation of the directrix and the length of the latus rectum. Since the parabola passes through ( 2. x2 = – 9y In each of the Exercises 7 to 12. find the equation of the parabola that satisfies the given conditions: . Since the directrix is x = – 2 and the focus is (2. and passes through the point (2. we have x2 = 4(2)y. the parabola is to be of the form y 2 = 4ax with a = 2. equation of the parabola is of the form x2 = 4ay.e. a = 1 3 Therefore.2 In each of the following Exercises 1 to 6. i. y2 = 10x 6.. x2 = 8y. the y-axis is the axis of the parabola. Hence the required equation is y2 = 4(2)x = 8x Example 7 Find the equation of the parabola with vertex at (0. i. Example 8 Find the equation of the parabola which is symmetric about the y-axis. find the coordinates of the focus.–3). axis of the parabola.0) and the focus is at (0. where the sign depends on whether the parabola opens upwards or downwards. 0) 247 8.0).22).0). bl The two fixed points are called the foci (plural of ‘focus’) of the ellipse (Fig11.3) and axis is along x-axis. Focus (6. focus (–2.20). the sum of whose distances from two fixed points in the plane is a constant.21 Fig 11. Focus (0. The end points of the major axis are called the vertices of the ellipse(Fig 11. directrix x = – 6 9.21). Fig 11.0). 5 Ellipse Note The constant which is the sum of $ the distances of a point on the ellipse from the Fig 11.0) passing through (2. Vertex (0.–3). passing through (5. Vertex (0.0) 11. The mid point of the line segment joining the foci is called the centre of the ellipse. 12. focus (3.0). .CONIC SECTIONS 7.2) and symmetric with respect to y-axis. he 11. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. Thus. Vertex (0.22 We denote the length of the major axis by 2a. is Definition 4 An ellipse is the set of all points in a plane. the length of the minor axis by 2b and the distance between the foci by 2c. Vertex (0.20 © no N C tt E o R be T re pu two fixed points is always greater than the distance between the two fixed points. the length of the semi major axis is a and semi-minor axis is b (Fig11. directrix y = 3 10. 5.3. The ellipse reduces to the line segment F1F2 joining the two foci (Fig11. we have 2 b 2 + c 2 = 2a. a = b.2 Special cases of an ellipse In the equation c2 = a2 – b2 obtained above. if we keep a fixed and vary c from 0 to a.5. Case (ii) When c = a. a = b2 + c2 c= a 2 − b2 .23 b2 + c2 + bl Take a point Q at one end of the minor axis.. or a 2 = b2 + c2 .e..3 Eccentricity Fig 11. i. a . Thus.e. 11.248 MATHEMATICS Take a point P at one end of the major axis. both foci merge together with the centre of the ellipse and a2 = b2.5. i. circle is a special case of an ellipse which is dealt in Section 11. By the definition of ellipse. the resulting ellipses will vary in shape. then b = 0. i.25 Definition 5 The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by e) i.. e = c . Sum of the distances from the point Q to the foci is F1Q + F2Q = he 11.23). Sum of the distances of the point P to the foci is F1 P + F2P = F1O + OP + F2P (Since.24). F1P = F1O + OP) = c + a + a – c = 2a is Fig 11.e.24 Fig 11. b2 + c 2 = 2 b2 + c2 © no N C tt E o R be T re pu Since both P and Q lies on the ellipse. semi-minor axis and the distance of the focus from the centre of the ellipse (Fig 11.e. Case (i) When c = 0. and so the ellipse becomes circle (Fig11..1 Relationship between semi-major axis. 11.25). 27). ( x − c) 2 + y 2 = 2a ( x + c) 2 + y 2 = 2a – ( x − c) 2 + y 2 . (1) Fig 11. The two such possible orientations are shown in Fig 11. Let O be the origin and the line from O through F2 be the positive x-axis and that through F1as the negative x-axis. © no N C tt E o R be T re pu bl is he 11. 0) and F2 be (c. . Let the coordinates of F1 be (– c.26 (a) with foci on the x-axis. in terms of the eccentricity the focus is at a distance of ae from the centre. Let. Let F1 and F2 be the foci and O be the midpoint of the line segment F1F2.27 Using the distance formula. 0) (Fig 11.26 on the x-axis or y-axis.4 Standard equations of an ellipse The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are (a) Fig 11.5. We will derive the equation for the ellipse shown above in Fig 11.. Let P(x... we have ( x + c) 2 + y 2 + i. the line through O perpendicular to the x-axis be the y-axis. y) be any point on the ellipse such that the sum of the distances from P to the two x2 y2 + =1 foci be 2a so given a 2 b2 PF1 + PF2 = 2a.e.26.CONIC SECTIONS 249 Then since the focus is at a distance of c from the centre. y) satisfy the equation (2) with 0 < c < a. (2) Conversely.. let P (x. PF1 = 2 ( x + c) 2 + y 2 = ⎛ a2 − x2 ⎞ ⎟⎟ ( x + c) 2 + b 2 ⎜⎜ 2 ⎝ a ⎠ = ⎛ a2 − x2 ⎞ ( x + c) 2 + (a 2 − c 2 ) ⎜ ⎟ (since b2 = a2 – c2) 2 a ⎝ ⎠ = cx ⎞ c ⎛ ⎜a+ ⎟ =a+ x a⎠ a ⎝ 2 Similarly c PF2 = a − x a . is x2 y2 + 2 =1 a2 a − c2 . we get x2 y2 + =1 a2 b2 (Since c2 = a2 – b2) Hence any point on the ellipse satisfies © no N C tt E o R be T re pu x2 y2 + = 1. a2 b2 bl i..250 MATHEMATICS Squaring both sides. Then ⎛ x2 ⎞ ⎟ ⎜ 1 − y =b ⎜ a 2 ⎟⎠ ⎝ 2 Therefore. we get ( x − c) 2 + y 2 + ( x − c) 2 + y 2 (x + c)2 + y2 = 4a2 – 4a which on simplification gives c x a he ( x − c) 2 + y 2 = a − Squaring again and simplifying..e. major axis is along the x-axis if the coefficient of x2 has the larger denominator and it is along the y-axis if the coefficient of y2 has the larger denominator. However. we proved that the equation of an ellipse with centre of the origin and major axis along the x-axis is bl is x2 y 2 + = 1. (3) x2 y2 + = 1. From the standard equations of the ellipses (Fig11. i. Hence from (2) and (3). the ellipse lies between the lines y = – b and y = b and touches these lines. a2 b2 Therefore. y) is a point on the ellipse. y). the study of the ellipses with centre at any other point. The foci always lie on the major axis. and any line through the centre as major and the minor axes passing through the centre and perpendicular to major axis are beyond the scope here. the ellipse lies between the lines x = – a and x = a and touches these lines. (x. we have © no N C tt E o R be T re pu x2 y2 = 1 − ≤ 1. so – a ≤ x ≤ a. Note The standard equations of ellipses have centre at the origin and the $ major and minor axis are coordinate axes. y) on the ellipse. satisfies the geometric condition and so a2 b2 he P(x. Similarly. then (– x. Similarly. it follows that for every point P (x.e. y) lies on the ellipse. –y) and (– x..CONIC SECTIONS Hence PF1 + PF2 = a + So.. a 2 b2 Discussion From the equation of the ellipse obtained above. Ellipse is symmetric with respect to both the coordinate axes since if (x. we have the following observations: 1. .26 (b) as x2 y 2 + =1 .. The major axis can be determined by finding the intercepts on the axes of symmetry. That is.26). any point that satisfies c c x + a – x = 2a a a 251 . we can derive the equation of the ellipse in Fig 11. 2. b2 a 2 These two equations are known as standard equations of the ellipses. –y) are also points on the ellipse. x2 ≤ a2. 5 Latus rectum Let the length of AF2 be l. it is symmetric w.t. both the coordinate axes). 28 Then the coordinates of A are (c.e. (ae. l ). the eccentricity and the latus rectum of the ellipse x2 y 2 + =1 25 9 x2 y2 Solution Since denominator of is larger than the denominator of . To find the length of the latus rectum (ae)2 l 2 + 2 =1 a2 b ⇒ l2 = b2 (1 – e2) But e2 = c2 a 2 – b2 b2 = = 1 – a2 a2 a2 Therefore l2 = b4 b2 l = . l ) x2 y2 + = 1 . AF2 = F2B and so length of the latus rectum is 2b 2 . the length of major axis.28). a2 a Since the ellipse is symmetric with respect to y-axis (of course.i. the minor axis. i..r. the vertices. a Example 9 Find the coordinates of the foci..252 MATHEMATICS 11. we have a 2 b2 © no N C tt E o R be T re pu Since A lies on the ellipse is x2 y2 + =1 a 2 b2 bl of the ellipse he Definition 6 Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse (Fig 11.5. Fig 11.e. the major 25 9 . we have b = 2 and a = 3. the major axis is 9 4 along the y-axis. 0) and foci are (± 5. 5 a 5 Example 10 Find the coordinates of the foci. bl is eccentricity is Solution The given equation of the ellipse can be written in standard form as © no N C tt E o R be T re pu x2 y 2 + =1 4 9 x2 y2 is larger than the denominator of . Also c = a 2 – b 2 = 25 – 9 = 4 he Therefore. the vertices.3) and (0.0). the lengths of major and minor axes and the eccentricity of the ellipse 9x2 + 4y2 = 36. Solution Since the vertices are on x-axis. Comparing the given equation with 253 x2 y2 + = 1 . –3). 0). the equation will be of the form ellipse is x2 y2 + = 1 . 5 ) and (0. a2 b2 . we get a 2 b2 a = 5 and b = 3. 0). b2 a2 Also c= and e= a 2 – b2 = 9–4= 5 c 5 = 3 a Hence the foci are (0. the length of the minor axis is 4 units and the eccentricity of the 5 . vertices are (0. – 5 ).0) and (4. 0) and (5. Length of the major axis is 10 units length of the minor axis 2b is 6 units and the 2 2b 18 4 = and latus rectum is .CONIC SECTIONS axis is along the x-axis. the coordinates of the foci are (– 4. Comparing the given equation with the standard equation Since the denominator of x2 y 2 + = 1 . 3 Example 11 Find the equation of the ellipse whose vertices are (± 13. length of the major axis is 6 units. where a is the semi-major axis. vertices are (– 5. 254 MATHEMATICS Given that a = 13, c = ± 5. Therefore, from the relation c2 = a2 – b2, we get 25 = 169 – b2 , i.e., b = 12 x2 y2 + =1. 169 144 he Hence the equation of the ellipse is Example 12 Find the equation of the ellipse, whose length of the major axis is 20 and foci are (0, ± 5). Solution Since the foci are on y-axis, the major axis is along the y-axis. So, equation is x2 y 2 + =1. b2 a 2 bl of the ellipse is of the form Given that 20 = 10 2 © no N C tt E o R be T re pu a = semi-major axis = and the relation c 2 = a2 – b2 gives 52 = 102 – b2 i.e., b2 = 75 Therefore, the equation of the ellipse is x2 y2 + =1 75 100 Example 13 Find the equation of the ellipse, with major axis along the x-axis and passing through the points (4, 3) and (– 1,4). Solution The standard form of the ellipse is and (–1, 4) lie on the ellipse, we have and x2 y2 = 1. Since the points (4, 3) + a2 b2 16 9 + 2 =1 2 a b ... (1) 1 16 + 2 =1 2 a b ….(2) 2 Solving equations (1) and (2), we find that a = Hence the required equation is 247 247 2 and b = . 7 15 CONIC SECTIONS 255 x2 y2 + = 1 , i.e., 7x2 + 15y2 = 247. ⎛ 247 ⎞ 247 ⎜ ⎟ ⎝ 7 ⎠ 15 he EXERCISE 11.3 x2 y2 + =1 25 100 3. 5. x2 y2 + =1 49 36 bl 4. 2. x2 y 2 + =1 4 25 6. x2 y 2 + =1 16 9 x2 y2 + =1 100 400 © no N C tt E o R be T re pu 1. x2 y 2 + =1 36 16 is In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. 7. 36x2 + 4y2 = 144 8. 16x2 + y2 = 16 9. 4x2 + 9y2 = 36 In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: 10. 11. 12. 13. Vertices (± 5, 0), foci (± 4, 0) Vertices (0, ± 13), foci (0, ± 5) Vertices (± 6, 0), foci (± 4, 0) Ends of major axis (± 3, 0), ends of minor axis (0, ± 2) Ends of major axis (0, ± 5 ), ends of minor axis (± 1, 0) Length of major axis 26, foci (± 5, 0) Length of minor axis 16, foci (0, ± 6). Foci (± 3, 0), a = 4 b = 3, c = 4, centre at the origin; foci on the x axis. Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6). 20. Major axis on the x-axis and passes through the points (4,3) and (6,2). 14. 15. 16. 17. 18. 19. 11.6 Hyperbola Definition 7 A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. MATHEMATICS is he 256 Fig 11.29 © no N C tt E o R be T re pu bl The term “difference” that is used in the definition means the distance to the farther point minus the distance to the closer point. The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola (Fig 11.29). We denote the distance between the two foci by 2c, the distance between two vertices (the length of the transverse axis) by 2a and we define the quantity b as b = c2 – a2 Also 2b is the length of the conjugate axis (Fig 11.30). Fig 11.30 To find the constant P1F2 – P1F1 : By taking the point P at A and B in the Fig 11.30, we have BF1 – BF2 = AF2 – AF1 (by the definition of the hyperbola) BA +AF1– BF2 = AB + BF2– AF1 i.e., AF1 = BF2 So that, BF1 – BF2 = BA + AF1– BF2 = BA = 2a CONIC SECTIONS 257 11.6.1 Eccentricity c is called the eccentricity of the a hyperbola. Since c ≥ a, the eccentricity is never less than one. In terms of the eccentricity, the foci are at a distance of ae from the centre. Definition 8 Just like an ellipse, the ratio e = © no N C tt E o R be T re pu bl is he 11.6.2 Standard equation of Hyperbola The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x-axis or y-axis. The two such possible orientations are shown in Fig11.31. (b) (a) Fig 11.31 We will derive the equation for the hyperbola shown in Fig 11.31(a) with foci on the x-axis. Let F1 and F2 be the foci and O be the mid-point of the line segment F1F2. Let O be the origin and the line through O through F2 be the positive x-axis and that through F 1 as the negative x-axis. The line through O perpendicular to the x-axis be the y-axis. Let the coordinates of F1 be (– c,0) and F2 be (c,0) (Fig 11.32). Let P(x, y) be any point on the hyperbola such that the difference of the distances from P to the farther point minus the closer point be 2a. So given, PF1 – PF2 = 2a Fig 11.32 258 MATHEMATICS Using the distance formula, we have (x + c) 2 + y 2 – (x – c) 2 + y 2 = 2a (x + c) 2 + y 2 = 2a + (x – c) 2 + y 2 i.e., he Squaring both side, we get (x – c) 2 + y 2 + (x – c)2 + y2 (x + c)2 + y2 = 4a2 + 4a and on simplifying, we get cx –a= a is (x – c) 2 + y 2 On squaring again and further simplifying, we get © no N C tt E o R be T re pu x2 y 2 – =1 a2 b2 bl x2 y2 – =1 a2 c2 – a2 i.e., (Since c2 – a2 = b2) x2 y 2 – = 1 1. a 2 b2 Conversely, let P(x, y) satisfy the above equation with 0 < a < c. Then Hence any point on the hyperbola satisfies 2 y Therefore, = b 2 PF1 = + = + Similarly, PF2 = a – ⎛ x2 – a2 ⎞ ⎜ ⎟ 2 ⎝ a ⎠ (x + c) 2 + y 2 ⎛ x2 – a2 ⎞ c (x + c) 2 + b 2 ⎜ x ⎟ = a+ 2 a a ⎝ ⎠ a x c In hyperbola c > a; and since P is to the right of the line x = a, x > a, a– c c x becomes negative. Thus, PF2 = x – a. a a c x > a. Therefore, a CONIC SECTIONS 259 c cx x – + a = 2a a a Also, note that if P is to the left of the line x = – a, then Therefore PF1 – PF2 = a + In that case P F2 – PF1 = 2a. So, any point that satisfies he c ⎞ ⎛ c PF1 = – ⎜ a + x ⎟ , PF2 = a – x . a ⎠ a ⎝ x2 y 2 – = 1 , lies on the a 2 b2 x2 y 2 – =1. a 2 b2 bl along x-axis is is hyperbola. Thus, we proved that the equation of hyperbola with origin (0,0) and transverse axis © no N C tt E o R be T re pu $Note A hyperbola in which a = b is called an equilateral hyperbola. Discussion From the equation of the hyperbola we have obtained, it follows that, we x2 y2 = + 1 have for every point (x, y) on the hyperbola, 2 ≥ 1. a b2 i.e, x ≥ 1, i.e., x ≤ – a or x ≥ a. Therefore, no portion of the curve lies between the a lines x = + a and x = – a, (i.e. no real intercept on the conjugate axis). Similarly, we can derive the equation of the hyperbola in Fig 11.31 (b) as y2 x2 − =1 a2 b2 These two equations are known as the standard equations of hyperbolas. Note The standard equations of hyperbolas have transverse and conjugate $ axes as the coordinate axes and the centre at the origin. However, there are hyperbolas with any two perpendicular lines as transverse and conjugate axes, but the study of such cases will be dealt in higher classes. From the standard equations of hyperbolas (Fig11.29), we have the following observations: 1. Hyperbola is symmetric with respect to both the axes, since if (x, y) is a point on the hyperbola, then (– x, y), (x, – y) and (– x, – y) are also points on the hyperbola. 260 MATHEMATICS 2. The foci are always on the transverse axis. It is the positive term whose has transverse axis along x-axis of length 6, while x2 y 2 – =1 9 16 y 2 x2 – =1 25 16 has transverse axis along y-axis of length 10. 11.6.3 Latus rectum he denominator gives the transverse axis. For example, is Definition 9 Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola. 2b 2 . a Example 14 Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas: © no N C tt E o R be T re pu bl As in ellipse, it is easy to show that the length of the latus rectum in hyperbola is (i) x2 y 2 – = 1 , (ii) y2 – 16x2 = 16 9 16 Solution (i) Comparing the equation x2 y 2 – = 1 with the standard equation 9 16 x2 y 2 – =1 a2 b2 Here, a = 3, b = 4 and c = a 2 + b 2 = 9 + 16 = 5 Therefore, the coordinates of the foci are (± 5, 0) and that of vertices are (± 3, 0).Also, c 5 2b 2 32 = = = The eccentricity e = . The latus rectum a 3 a 3 (ii) Dividing the equation by 16 on both sides, we have Comparing the equation with the standard equation a = 4, b = 1 and c = a 2 + b 2 = 16 + 1 = 17 . y 2 x2 – =1 16 1 y 2 x2 – = 1 , we find that a 2 b2 CONIC SECTIONS Therefore, the coordinates of the foci are (0, ± (0, ± 4). Also, 261 17 ) and that of the vertices are (0, ± he 2b2 1 17 c = . . The latus rectum = The eccentricity e = = a 4 a 2 Example 15 Find the equation of the hyperbola with foci (0, ± 3) and vertices 11 ). 2 y 2 x2 =1 – a 2 b2 11 ), a = 2 11 2 © no N C tt E o R be T re pu Since vertices are (0, ± bl is Solution Since the foci is on y-axis, the equation of the hyperbola is of the form Also, since foci are (0, ± 3); c = 3 and b2 = c2 – a2 = 25 . 4 Therefore, the equation of the hyperbola is x2 y2 – = 1, i.e., 100 y2 – 44 x2 = 275. ⎛ 11 ⎞ ⎛ 25 ⎞ ⎜ ⎟ ⎜ ⎟ ⎝4⎠ ⎝ 4 ⎠ Example 16 Find the equation of the hyperbola where foci are (0, ±12) and the length of the latus rectum is 36. Solution Since foci are (0, ± 12), it follows that c = 12. 2b 2 = 36 or b2 = 18a Length of the latus rectum = a c2 = a2 + b2; gives 144 = a2 + 18a a2 + 18a – 144 = 0, i.e., a = – 24, 6. So Since a cannot be negative, we take a = 6 and so b2 = 108. Therefore Therefore, the equation of the required hyperbola is x2 y2 = 1 , i.e., 3y2 – x2 = 108 – 36 108 262 MATHEMATICS EXERCISE 11.4 In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. x2 y 2 y2 x2 – =1 – =1 2. 3. 9y2 – 4x2 = 36 16 9 9 27 4. 16x2 – 9y2 = 576 5. 5y2 – 9x2 = 36 6. 49y2 – 16x2 = 784. In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions. 7. Vertices (± 2, 0), foci (± 3, 0) 8. Vertices (0, ± 5), foci (0, ± 8) 9. Vertices (0, ± 3), foci (0, ± 5) 10. Foci (± 5, 0), the transverse axis is of length 8. 11. Foci (0, ±13), the conjugate axis is of length 24. bl is he 1. © no N C tt E o R be T re pu 12. Foci (± 3 5 , 0), the latus rectum is of length 8. 13. Foci (± 4, 0), the latus rectum is of length 12 14. vertices (± 7,0), e = 4 . 3 15. Foci (0, ± 10 ), passing through (2,3) Miscellaneous Examples Example 17 The focus of a parabolic mirror as shown in Fig 11.33 is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB (Fig 11.33). Solution Since the distance from the focus to the vertex is 5 cm. We have, a = 5. If the origin is taken at the vertex and the axis of the mirror lies along the positive x-axis, the equation of the parabolic section is y2 = 4 (5) x = 20 x Note that x = 45. Thus y2 = 900 Therefore y = ± 30 Hence AB = 2y = 2 × 30 = 60 cm. Fig 11.33 Example 18 A beam is supported at its ends by supports which are 12 metres apart. Since the load is concentrated at its centre, there Let the coordinate axis be chosen as shown in Fig 11. A point P(x. Coordinates of B are (x.. a = 12 ⎝ 100 ⎠ ⎝ 100 ⎠ Let AB be the deflection of the beam which is Therefore i. respectively.34.35 and P (x.CONIC SECTIONS 263 is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. we have (6)2 = 4a ⎜ ⎟ . ). y) is taken on the rod in such a way that AP = 6 cm. y) the point on it such that AP = 6 cm. From P draw PQ and PR perpendiculars on y-axis and x-axis. i. © no N C tt E o R be T re pu The equation of the parabola takes the form x2 = 4ay. x 2 = 4 × 300 × x= 24 1 2 m. Fig 11.34 bl is he Solution Let the vertex be at the lowest point and the axis vertical. ⎟ . Since it passes through ⎛ 3 ⎞ ⎛ 3 ⎞ 36 ×100 = 300 m ⎜ 6.e. 100 100 2 = 24 100 = 2 6 metres Example 19 A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. Solution Let AB be the rod making an angle θ with OX as shown in Fig 11. Show that the locus of P is an ellipse. How far from the centre is the deflection 1 cm? Fig 11.35 .e. Since AB = 15 cm. we have PB = 9 cm. where one vertex is at the vertex of the parabola. 8. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable. A rod of length 12 cm moves with its ends always touching the coordinate axes. Find the length of a supporting wire attached to the roadway 18 m from the middle. 2. Find the length of the side of the triangle. 4. bl or is ⎛x⎞ ⎛ y⎞ ⎜ ⎟ + ⎜ ⎟ =1 ⎝9⎠ ⎝6⎠ © no N C tt E o R be T re pu Miscellaneous Exercise on Chapter 11 1. Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum. the longest wire being 30 m and the shortest being 6 m. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The arch is 10 m high and 5 m wide at the base. An equilateral triangle is inscribed in the parabola y2 = 4 ax. 6. which is 3 cm from the end in contact with the x-axis. It is 8 m wide and 2 m high at the centre. 5.5 m from one end. find the focus. Find the equation of the posts traced by the man. A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m.MATHEMATICS From Δ PBQ. If a parabolic reflector is 20 cm in diameter and 5 cm deep. How wide is it 2 m from the vertex of the parabola? 3. sin θ = y 6 Since cos2 θ + sin2 θ = 1 2 he 264 2 x2 y2 + =1 81 36 Thus the locus of P is an ellipse. . cos θ = x 9 From Δ PRA. Determine the equation of the locus of a point P on the rod. An arch is in the form of a semi-ellipse. Find the height of the arch at a point 1. An arch is in the form of a parabola with its axis vertical. 7. k) and the radius r is (x – h)2 + (y – k)2 = r2. a 2 b2 a The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse. bl The equation of the parabola with focus at (a. The equation of a circle with centre (h. 0) a > 0 and directrix x = – a is © no N C tt E o R be T re pu y2 = 4ax. Length of the latus rectum of the parabola y2 = 4ax is 4a. Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola. Length of the latus rectum of the ellipse x2 y2 2b 2 + = 1 is . The equation of an ellipse with foci on the x-axis is x2 y 2 + =1. through the focus and whose end points lie on the parabola. An ellipse is the set of all points in a plane. in the plane. A hyperbola is the set of all points in a plane. the sum of whose distances from two fixed points in the plane is a constant. a 2 b2 Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse. he A circle is the set of all points in a plane that are equidistant from a fixed point is A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.CONIC SECTIONS 265 Summary In this Chapter the following concepts and generalisations are studied. the difference of whose distances from two fixed points in the plane is a constant. x2 y 2 The equation of a hyperbola with foci on the x-axis is : a 2 − b 2 = 1 . etc. He also gave the intercept form of the linear equation. Geometry as initially studied by the ancient Indians and Greeks. Descartes came to be regarded as the unique inventor of the analytical geometry. who made essentially no use of the process of algebra.. But the fundamental principle and method of analytical geometry were already discovered by Pierre de Fermat (1601-1665). The synthetic approach to the subject of geometry as given by Euclid and in Sulbasutras. Apollonius wrote a book called ‘The Conic’ which was all about conic sections with many important discoveries that have remained unsurpassed for eighteen centuries. Newton used method of undetermined coefficients to find equations of curves. Unfortunately. entitled Ad Locus Planos et So LIDOS Isagoge (Introduction to Plane and Solid Loci) was published only posthumously in 1679. Isaac Barrow avoided using cartesian method. Fermats treatise on the subject.C. a 2 b2 a he Length of the latus rectum of the hyperbola : The eccentricity of a hyperbola is the ratio of the distances from the centre of bl Historical Note is the hyperbola to one of the foci and to one of the vertices of the hyperbola. So. Euclid wrote his treatise on geometry around 300 B. x2 y 2 2b 2 − = 1 is : . Leibnitz used the terms ‘abscissa’. © no N C tt E o R be T re pu Geometry is one of the most ancient branches of mathematics. He used several types of coordinates including polar and bipolar. Clairaut (1729) was the first to give the distance formula although in clumsy form.. The Greek geometers investigated the properties of many curves that have theoretical and practical importance. Modern analytic geometry is called ‘Cartesian’ after the name of Rene Descartes (1596-1650) whose relevant ‘La Geometrie’ was published in 1637.C. Cramer (1750) . In the 200 B.266 MATHEMATICS Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola. ‘ordinate’ and ‘coordinate’. He was the first who organised the geometric figures based on certain axioms suggested by physical considerations. was continued for some 1300 years. L’ Hospital (about 1700) wrote an important textbook on analytical geometry. Monge (1781) gave the modern ‘point-slope’ form of equation of a line as y – y′ = a (x – x′) and the condition of perpendicularity of two lines as aa′ + 1 = 0. gave mE + m′E′ = 0 as the curve passing through the points of intersection of two loci E = 0 and E′ = 0. β) on y = ax + b as 1 + a2 .) studied conic sections for their own beauty. — — . The Greeks particularly Archimedes (287–212 B.) and Apollonius (200 B. of ⎝ ⎠ course. It is.C. surprising that one has to wait for more than 150 years after the invention of analytical geometry before finding such essential basic formula.C. Lacroix (1765–1843) was a prolific textbook writer. In 1818. S. have been linked to the conic sections. but his contributions to analytical geometry are found scattered. C. Lame. a civil engineer. © no N C tt E o R be T re pu ⎛ a′ – a ⎞ His formula for finding angle between two lines was tan θ = ⎜ 1 + aa′ ⎟ . These curves are important tools for present day exploration of outer space and also for research into behaviour of atomic particles.F.CONIC SECTIONS 267 is β′ – β (x – α) α′ – α (β – a – b ) bl y –β= he made formal use of the two axes and gave the equation of a circle as ( y – a)2 + (b – x)2 = r He gave the best exposition of the analytical geometry of his time. both in Mathematics and Science. He gave the ‘two-point’ form of equation of a line as and the length of the perpendicular from (α. Many important discoveries. 2005. namely the floor of the room and two adjacent walls of the room. we need two intersecting mutually perpendicular lines in the plane. Therefore. we do not have to deal with points lying in a plane only. if we were to locate the position of the Leonhard Euler (1707-1783) lowest tip of an electric bulb hanging from the ceiling of a room or the position of the central tip of the ceiling fan in a room. BELL © no N C tt E o R be T re pu You may recall that to locate the position of a point in a plane. The three numbers representing the three distances are called the coordinates of the point with reference to the three coordinate planes. “A Hand Book for designing Mathematics Laboratory in Schools”. consider the position of a ball thrown in space at different points of time or the position of an aeroplane as it flies from one place to another at different times during its flight. . So. In actual life. In this Chapter. Similarly. we will not only require the perpendicular distances of the point to be located from two perpendicular walls of the room but also the height of the point from the floor of the room. For example.T. NCERT.* * For various activities in three dimensional geometry one may refer to the Book. These lines are called the coordinate axes and the two numbers are called the coordinates of the point with respect to the axes. we need not only two but three numbers representing the perpendicular distances of the point from three mutually perpendicular planes.1 Introduction bl Mathematics is both the queen and the hand-maiden of all sciences – E.Chapter 12 is he INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 12. we shall study the basic concepts of geometry in three dimensional space. a point in space has three coordinates. . X′OY′Z′ and XOY′Z′. Then x. In Fig 12. as to how. z) lies in the octant XOYZ and so all x. called. z).3 Coordinates of a Point in Space Having chosen a fixed coordinate system in the space. These three planes intersect along the lines X′OX.. given a triplet of three numbers (x. Given a point P in space.1 the XOY plane as the plane of the paper and the line Z′OZ as perpendicular to the plane XOY. The distances measured from XY-plane upwards in the direction of OZ are taken as positive and those measured downwards in the direction of OZ′ are taken as negative. We may note that these lines are mutually perpendicular to each other. we associate with it three coordinates (x.y and z are called the x.INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 269 12. in front of the YZ-plane along OX as positive and to the back of it along OX′ as negative. If P was in any other octant. y and z-axes. the distance measured to the right of ZX-plane along OY are taken as positive. If the plane of the paper is considered as horizontal.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space © no N C tt E o R be T re pu bl is he Consider three planes intersecting at a point O such that these three planes are mutually perpendicular to each other (Fig 12. are known as the three coordinate planes. consisting of coordinate axes.. respectively the XY-plane. to the left of ZX-plane and along OY′ as negative. respectively. z are positive. The point O is called the origin of the coordinate system. meeting it at L. y. X′OYZ. of the point P in the space. The planes XOY.2 perpendicular PM on the XY-plane with M as the foot of this perpendicular (Fig 12. XOYZ′. YOZ and ZOX. We take Fig 12. The three coordinate planes divide the space into eight parts known as octants. the signs of x. y and z would change . VIII . II. 12. Then. XOY′Z. y. we locate a point in the space. y.y. called the x.2.1). then the line Z′OZ will be vertical. we drop a Fig 12. Y′OY and Z′OZ. from the point M. Let OL be x. we now explain. we draw a perpendicular ML to the x-axis. respectively.. we may note that the point P (x. These lines constitute the rectangular coordinate system. These octants could be named as XOYZ.z) and conversely. given a point in the space. y and z coordinates. respectively. and denoted by I. YZ-plane and the ZX-plane. how. X′OY′Z. Similarly. LM be y and MP be z. III.2). X′OYZ′. coordinate planes and the origin. Thus. z). y. Having reached the point M. given x. given any triplet (x. Let OA = x. z) of real numbers. z) of real numbers. z). to each point P in the space there corresponds an ordered triplet (x. we draw a perpendicular MP to the XY-plane and locate on it the point P corresponding to z. The coordinates of any point on the x-axis will be as (x. the point P will have the coordinates x. then locate the point M in the XY-plane such that (x. y) are the coordinates of the point M in the XY-plane.0) and the coordinates of any point in the YZ-plane will be as (0. z). OB = y and OC = z. Through the points A. respectively (Fig 12.0. y. Conversely. Then. ZX and XY planes. we draw three planes parallel to the coordinate planes. we locate the three points A. z) is any point in the space. y-axis and z-axis in the points A. Conversely.3). y. respectively. The point P so obtained has then the coordinates (x. y and z and we write P (x.0. corresponding to the ordered triplet (x. respectively. The point of interesection of these three planes. y. meeting the x-axis. we would first fix the point L on the x-axis corresponding to x. Thus. We observe that if P (x. Alternatively. there is a one to one correspondence between the points in space and ordered triplet (x. Remark The sign of the coordinates of a point determine the octant in which the point lies. then x.1 s ant Oct n a t e s rdi Coo I II III IV V VI VII VIII x + – – + + – – + y + + – – + + – – z + + + + – – – – . through the point P in the space. Note that LM is perpendicular to the x-axis or is parallel to the y-axis. BDPE and CEPF is obviously the point P. y. The following table shows the signs of the coordinates in eight octants. y. z). y and z. B and C. z). ZX-plane and XY-plane.0). B and C we draw planes parallel to Fig 12. y and z are perpendicular distances from YZ.270 MATHEMATICS © no N C tt E o R be T re pu bl is he accordingly. B and C on the three coordinate axes. $ Note The coordinates of the origin O are (0. y. Table 12. y. ADPF. namely.3 the YZ-plane. –7). (– 4.4 Distance between Two Points We have studied about the distance between two points in two-dimensional coordinate system. the coordinates of F are (2. the point (–3. 12. –2. z2) be two points referred to a system of rectangular axes OX.5). (1) Also. Let us now extend this study to three-dimensional system.2) and (–3. (4. Let P(x1.0. What are its y-coordinate and z-coordinates? 2. –5). 2. 5). in triangle PAQ. 2. . OY and OZ. 1. find the coordinates of F. Through the points P and Q draw planes parallel to the coordinate planes so as to form a rectangular parallelopiped with one diagonal PQ (Fig 12. (4. –1. z1) and Q ( x2.1. y1.4). (ii) The coordinates of points in the XY-plane are of the form _______.. 4.1.. the distance measured along OY is zero. Solution For the point F. (4. A point is in the XZ-plane. PQ2 = PA2 + AQ2 Fig 12. © no N C tt E o R be T re pu (1. (–3. – 2) lies in octant VI. 3).1. Therefore.1 A point is on the x -axis. (iii) Coordinate planes divide the space into ______ octants. since ∠PAQ is a right angle. Name the octants in which the following points lie: bl is 1. What can you say about its y-coordinate? 3.3. if P is (2. – 4.5).– 2) lie. 3). 2. EXERCISE 12.4 . triangle ANQ is right angle triangle with ∠ANQ a right angle. (– 4. Example 2 Find the octant in which the points (–3. 2) lies in second octant and the point (–3. –5). –5). Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as_______. it follows that. –2.1. he Solution From the Table 12. 2. Now.INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 271 Example 1 In Fig 12. 6) (2.4. y2. is In particular. – 41. 5). Q and R are collinear. we have . then OQ = which gives the distance between the origin O and any point Q (x2. PQ + QR = PR.272 MATHEMATICS Therefore From AQ2 = AN2 + NQ2 (1) and (2). z2). 20. bl Example 3 Find the distance between the points P(1. 5). i. Solution We know that points are said to be collinear if they lie on a line. y2. z1) and (x2. y1. 4) and Q (– 4.e. 30) and C (25. 4) and Q (– 4. we have AB2 = (10 – 3)2 + (20 – 6)2 + (30 – 9)2 = 49 + 196 + 441 = 686 BC2 = (25 – 10)2 + (– 41 – 20)2 + (5 – 30)2 = 225 + 3721 + 625 = 4571 . point P is origin O. 2 2 2 x2 + y 2 + z 2 . –3. 6.. Example 5 Are the points A (3. Hence. Solution The distance PQ between the points P (1. P.–3. (2) PQ2 = PA2 + AN2 + NQ2 Now PA = y2 – y1. Now. B (10.. 3. if x1 = y1 = z1 = 0. and PQ = (1 + 2) 2 + (2 − 3) 2 + (3 − 5) 2 = 9 + 1 + 4 = 14 QR = (7 − 1) 2 + (0 − 2) 2 + (−1− 3) 2 = 36 + 4 + 16 = 56 = 2 14 PR = (7 + 2) 2 + (0 − 3) 2 + (−1− 5) 2 = 81 + 9 + 36 = 126 = 3 14 Thus. 0. 3) and R (7. the vertices of a right angled triangle? Solution By the distance formula.. 2) is (−4 − 1) 2 + (1 + 3) 2 + (2 − 4) 2 © no N C tt E o R be T re pu PQ = = 25 + 16 + 4 = 45 = 3 5 units Example 4 Show that the points P (–2. Q (1. –1) are collinear. 1. 9). 2). AN = x2 – x1 and NQ = z2 – z1 2 PQ = (x2 – x1)2 + (y2 – y1) + (z2 – z1)2 Therefore PQ = ( x2 − x1 ) 2 +( y 2 − y1 ) 2 +( z 2 −z1 ) 2 he Hence 2 This gives us the distance between two points (x1. 1. z2). 2. y2. Draw PL. Let the point R (x. z1) and Q (x2. 2. 3). 3. z2). – 6) are the vertices of an isosceles triangle. 5) and (–1. y1. z) divide PQ in the given ratio m : n internally. 3. y. 6) and (– 4.e. 8) and (2. QM and RN perpendicular to .2 bl (x – 3)2 + (y – 4)2 + (z – 5)2 + (x + 1)2 + (y – 3)2 + (z + 7)2 = 2k2 1. 0. 6. 2. 4) are the vertices of a parallelogram. 10). –1). 3) and (–2. 5. 1. respectively.. (1. –1) (iv) (2.INTRODUCTION TO THREE DIMENSIONAL GEOMETRY CA 2 273 = (3 – 25)2 + (6 + 41)2 + (9 – 5)2 = 484 + 2209 + 16 = 2709 We find that CA2 + AB2 ≠ BC2. 3. 9. –3. y. (–1. –3. the sum of whose distances from A (4. 5). we have learnt how to find the coordinates of a point dividing a line segment in a given ratio internally. 3. 2x2 + 2y2 + 2z2 – 4x – 14y + 4z = 2k2 – 109. we have i. (1. 0. 4) (ii) (–3. –1. Now. Find the equation of the set of points which are equidistant from the points (1. (ii) (0. z). y2. –10). (4. 2. 3. 4. we extend this to three dimensional geometry as follows: Let the two given points be P(x1. –7). –1) are collinear. the triangle ABC is not a right angled triangle. 2. 1) (iii) (–1. –2. 7. 1). he Example 6 Find the equation of set of points P such that PA2 + PB2 = 2k2. Find the equation of the set of points P. 6) are the vertices of a right angled triangle. Show that the points (–2. 7. Hence. –7. – 4) and (1. Here PA2 = (x – 3)2 + (y – 4)2 + ( z – 5)2 is PB2 = (x + 1)2 + (y – 3)2 + (z + 7)2 By the given condition PA2 + PB2 = 2k2. 2. 3. (iii) (–1. 0) is equal to 10. 0. 2) and (2.5 Section Formula In two dimensional geometry. – 6) and (4. (1. where A and B are the points (3. 9. Find the distance between the following pairs of points: (i) (2. © no N C tt E o R be T re pu EXERCISE 12. 12. 5) and (4. 3) and (3. 7. 5). Solution Let the coordinates of point P be (x. 4. 6. 3) and (7. 0) and B (– 4. 4. Verify the following: (i) (0. m+n m + n ⎟⎠ ⎝ m+n If the point R divides PQ externally in the ratio m : n.y = 1 and z = 1 . by drawing perpendiculars to the XZ and YZ-planes. Also note that quadrilaterals LNRS and NMTR are parallelograms. as shown in Fig 12.5 bl m PR SP SL − PL NR – PL z − z1 = = = = = n QR QT QM − TM QM – NR z 2 − z is 274 This implies Similarly. The triangles PSR and QTR are similar. Therefore. . . mz 2 +nz1 m+n © no N C tt E o R be T re pu z= Fig 12. Line ST will intersect the line LP externally at the point S and the line MQ at T. then m : n = 1 : 1 so that x = x1 + x2 y + y2 z + z2 . 2 2 2 These are the coordinates of the mid point of the segment joining P (x1. then its coordinates are obtained by replacing n by – n so that coordinates of point R will be ⎛ mx2 − nx1 my2 − ny1 mz 2 − nz1 ⎞ . we get y = my2 + ny1 mx2 + nx1 and x = m+n m+n Hence. y1. the coordinates of the point R which divides the line segment joining two points P (x1. y2. z2) internally in the ratio m : n are ⎛ mx2 + nx1 my2 + ny1 mz2 + nz1 ⎞ ⎟ ⎜⎜ . y1. The points L.5. M and N will lie on a line which is the intersection of the plane containing PL. z1) and Q (x2. ⎜ ⎟ m−n m−n ⎠ ⎝ m−n Case 1 Coordinates of the mid-point: In case R is the mid-point of PQ. Obviously PL || RN || QM and feet of these perpendiculars lie in a XY-plane. Through the point R draw a line ST parallel to the line LM. RN and QM with the XY-plane. z2). z1) and Q (x2. y2. .MATHEMATICS he the XY-plane. z) be the point which divides line segment joining A(1. y= = 2+3 5 2+3 5 . Let the point P divides AB in the ratio k : 1. 4. 10). 0. we get k = − 3 k +1 2 . k 1 k 1 k 10 1 Let us examine whether for some value of k. . 3) and B (3. 10).INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 275 Case 2 The coordinates of the point R which divides PQ in the ratio k : 1 are obtained by taking k = m which are as given below: n he ⎛ k x 2 + x1 ky2 + y1 kz2 + z1 ⎞ ⎟ ⎜⎜ . –5) externally in the ratio 2 : 3. the point P coincides with point C. y. –2. B (2. . – 2. On putting 2k − 4 =14 . 6. Then Thus. y. 6) and (14. 1+k 1+k ⎟⎠ ⎝ 1+k Generally. ⎟ ⎝5 5 5 ⎠ (ii) Let P (x. –14. (2. . is Example 7 Find the coordinates of the point which divides the line segment joining the points (1. 4. 4. Then coordinates of the point P are 2k 4 4k 6 6k . –5) internally in the ratio 2 : 3. –2. z= 2(−5) + 3(3) − 1 = 2+3 5 © no N C tt E o R be T re pu x= bl Solution (i) Let P (x. 6) and C(14. 6. 3) and (3. and (ii) externally. Solution Let A (– 4. 0. the required point is (–3. 2(3) +3(1) 9 2(4) + 3(−2) 2 = . 4. – 2) be the given points. Example 8 Using section formula. Therefore ⎛ 9 2 −1⎞ ⎜ . 19). this result is used in solving problems involving a general point on the line passing through two given points. z) be the point which divides segment joining A (1. the required point is x = 2(3) + (−3)(1) 2(4) + (−3)(−2) 2(−5) + (−3)(3) = − 3. 4. –2) are collinear. prove that the three points (– 4. 3) and B (3. z = = 19 2 + (−3) 2 + (−3) 2 + (−3) Therefore. y = = − 14 . –5) in the ratio 2 : 3 (i) internally. C be (x1. 0. ⎜ ⎟ 2 2 ⎠ ⎝ 2 Let G be the centroid of the triangle. it divides the median AD in the ratio 2 : 1. 10. 10) and B (6. 8. . y1. C are collinear. y3. 10.Hence A. . C (14. Therefore. the coordinates of G are ⎛ ⎛ x2 + x3 ⎞ ⎞ ⎛ y2 + y3 ⎞ ⎛ z2 + z3 ⎞ ⎜ 2 ⎜ 2 ⎟ + x1 2 ⎜ 2 ⎟ + y1 2 ⎜ 2 ⎟ + z1 ⎟ ⎠ ⎠ ⎠ ⎜ ⎝ ⎟ . . Let D be the mid-point of BC. Let the coordinates of the vertices A. y1. z1). y3. then k +1 2 3 4( − ) + 6 2 =0 3 − +1 2 3 6( − ) + 10 6k + 10 2 = = −2 3 k +1 − +1 2 he 276 is Therefore. ⎜ ⎟ 3 3 3 ⎝ ⎠ Example 10 Find the ratio in which the line segment joining the points (4. respectively. B. B. Solution Let YZ-plane divides the line segment joining A (4. ⎜ ⎟ ⎝ k +1 k +1 k +1 ⎠ . z2) and (x3. z3). Hence coordinates of D are ⎛ x2 + x3 y2 + y3 z 2 + z3 ⎞ . Hence. y2. bl Example 9 Find the coordinates of the centroid of the triangle whose vertices are (x1. z) in the ratio k : 1. 8. (x2. z2) and (x3. – 8) is divided by the YZ-plane. – 8) at P (x. z1).When and MATHEMATICS 4k + 6 3 = k = − . Then the coordinates of P are ⎛ 4 + 6k 8 + 10k 10 − 8k ⎞ . ⎝ . © no N C tt E o R be T re pu Solution Let ABC be the triangle. –2) is a point which divides AB externally in the ratio 3 : 2 and is same as P. z3). ⎝ 2 +1 2 +1 2 +1 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ or ⎛ x1 + x2 + x3 y1 + y2 + y3 z1 + z2 + z3 ⎞ . (x2. y. y2. 10) and (6. 4. 2. 4 + 16 + 16 = 6 AB = (−1 − 1) 2 +(−2 − 2) 2 +(−1 − 3) 2 = BC = (2 + 1) 2 +(3 + 2) 2 +(2 + 1) 2 = CD = ( 4 − 2) 2 +(7 − 3) 2 +(6 − 2) 2 = 4 + 16 + 16 = 6 DA = (1 − 4) 2 +(2 − 7) 2 +(3 − 6) 2 = 9 + 25 + 9 = 43 9 + 25 + 9 = 43 Since AB = CD and BC = AD. i. ABCD is a parallelogram. –16. . 4. –10) are collinear. Q (5. Miscellaneous Examples Example 11 Show that the points A (1. 6) in the ratio (i) 2 : 3 internally. (ii) 2 : 3 externally. it is required to prove that ABCD is not a rectangle. 8). 6) are the vertices of a parallelogram ABCD. 5) and (1. 2. 2. – 6) and R (9. YZ-plane divides AB externally in the ratio 2 : 3. 2. Solution To show ABCD is a parallelogram we need to show opposite side are equal Note that.INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Since P lies on the YZ-plane. 3. we show that diagonals AC and BD are unequal. its x-coordinate is zero. 4 + 6k =0 k +1 2 3 Therefore. –2. Now.3 Find the coordinates of the point which divides the line segment joining the points (– 2. Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2. 7) and (3. show that the points A (2. –3. –1). – 6) and Q (10. B (–1. Using section formula. ⎝ 3 ⎠ Find the coordinates of the points which trisect the line segment joining the points P (4. 1) and © no N C tt E o R be T re pu bl is 1. 4).e. k =− he or 277 EXERCISE 12. B (–1. 3. For this. C (2. but it is not a rectangle. – 4. ⎛ 1 ⎞ C ⎜ 0. 2) and D (4. 8.2 ⎟ are collinear. 5.. – 4). 2. Given that P (3. –5. 7. 4. 3). Find the ratio in which Q divides PR. 3. We have . 6). 278 MATHEMATICS AC = (2 − 1) 2 +(3 − 2) 2 +(2 − 3) 2 = 1 + 1 + 1= 3 BD = ( 4 + 1) 2 +(7 + 2) 2 +(6 + 1) 2 = 25 + 81 + 49 = 155 . Since AC ≠ BD, ABCD is not a rectangle. he Note We can also show that ABCD is a parallelogram, using the property that $ diagonals AC and BD bisect each other. Solution If P (x, y, z) be any point such that PA = PB. ( x + 2) 2 + ( y − 1) 2 + ( z − 4) 2 ( x − 3) 2 + ( y − 4) 2 + ( z + 5) 2 = ( x + 2) 2 + ( y − 1) 2 + ( z − 4) 2 © no N C tt E o R be T re pu or ( x − 3) 2 + ( y − 4) 2 + ( z + 5) 2 = bl Now is Example 12 Find the equation of the set of the points P such that its distances from the points A (3, 4, –5) and B (– 2, 1, 4) are equal. or 10 x + 6y – 18z – 29 = 0. Example 13 The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C. Solution Let the coordinates of C be (x, y, z) and the coordinates of the centroid G be (1, 1, 1). Then x + 3−1 y − 5+ 7 z+7−6 = 1, i.e., x = 1; = 1, i.e., y = 1; = 1, i.e., z = 2. 3 3 3 Hence, coordinates of C are (1, 1, 2). Miscellaneous Exercise on Chapter 12 1. Three vertices of a parallelogram ABCD are A(3, – 1, 2), B (1, 2, – 4) and C (– 1, 1, 2). Find the coordinates of the fourth vertex. 2. Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0,4, 0) and (6, 0, 0). 3. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (– 4, 3b, –10) and R(8, 14, 2c), then find the values of a, b and c. 4. Find the coordinates of a point on y-axis which are at a distance of 5 2 from the point P (3, –2, 5). INTRODUCTION TO THREE DIMENSIONAL GEOMETRY 279 5. A point R with x-coordinate 4 lies on the line segment joining the points P(2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R. [Hint Suppose R divides PQ in the ratio k : 1. The coordinates of the point R are given he ⎛ 8k + 2 − 3 10k + 4 ⎞ by ⎜ , , ⎟ ]. ⎝ k +1 k +1 k +1 ⎠ 6.If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set of points P such that PA2 + PB2 = k2, where k is a constant. is Summary In three dimensions, the coordinate axes of a rectangular Cartesian coordinate bl system are three mutually perpendicular lines. The axes are called the x, y and z-axes. The three planes determined by the pair of axes are the coordinate planes, called XY, YZ and ZX-planes. © no N C tt E o R be T re pu The three coordinate planes divide the space into eight parts known as octants. The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y and z are the distances from the YZ, ZX and XY-planes. (i) Any point on x-axis is of the form (x, 0, 0) (ii) Any point on y-axis is of the form (0, y, 0) (iii) Any point on z-axis is of the form (0, 0, z). Distance between two points P(x1, y1, z1) and Q (x2, y2, z2) is given by PQ = ( x2 − x1 )2 + ( y2 − y1 )2 + ( z2 − z1 )2 The coordinates of the point R which divides the line segment joining two points P (x1 y1 z1) and Q (x2, y2, z2) internally and externally in the ratio m : n are given by ⎛ mx2 + nx1 my2 + ny1 mz2 + nz1 ⎞ ⎛ mx2 – nx1 my2 – ny1 mz2 – nz1 ⎞ , , , , ⎜ ⎟ and ⎜ ⎟ m+n m+n ⎠ m–n m–n ⎠ , ⎝ m+n ⎝ m–n respectively. The coordinates of the mid-point of the line segment joining two points ⎛ x1 + x2 y1 + y2 z1 + z2 ⎞ , , P(x1, y1, z1) and Q(x2, y2, z2) are ⎜ ⎟. 2 2 ⎠ ⎝ 2 280 MATHEMATICS The coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1) he ⎛ x1 + x2 + x3 y1 + y2 + y3 z1 + z2 + x3 ⎞ , , (x2, y2, z2) and (x3, y3, z3), are ⎜ ⎟. 3 3 3 ⎝ ⎠ Historical Note © no N C tt E o R be T re pu bl is Rene’ Descartes (1596–1650), the father of analytical geometry, essentially dealt with plane geometry only in 1637. The same is true of his co-inventor Pierre Fermat (1601-1665) and La Hire (1640-1718). Although suggestions for the three dimensional coordinate geometry can be found in their works but no details. Descartes had the idea of coordinates in three dimensions but did not develop it. J.Bernoulli (1667-1748) in a letter of 1715 to Leibnitz introduced the three coordinate planes which we use today. It was Antoinne Parent (1666-1716), who gave a systematic development of analytical solid geometry for the first time in a paper presented to the French Academy in 1700. L.Euler (1707-1783) took up systematically the three dimensional coordinate geometry, in Chapter 5 of the appendix to the second volume of his “Introduction to Geometry” in 1748. It was not until the middle of the nineteenth century that geometry was extended to more than three dimensions, the well-known application of which is in the Space-Time Continuum of Einstein’s Theory of Relativity. — — Chapter 13 he LIMITS AND DERIVATIVES bl 13.1 Introduction is With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature – WHITEHEAD © no N C tt E o R be T re pu This chapter is an introduction to Calculus. Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change. First, we give an intuitive idea of derivative (without actually defining it). Then we give a naive definition of limit and study some algebra of limits. Then we come back to a definition of derivative and study some algebra of derivatives. We also obtain derivatives of certain standard functions. 13.2 Intuitive Idea of Derivatives Sir Issac Newton (1642-1727) Physical experiments have confirmed that the body dropped from a tall cliff covers a distance of 4.9t2 metres in t seconds, i.e., distance s in metres covered by the body as a function of time t in seconds is given by s = 4.9t2. The adjoining Table 13.1 gives the distance travelled in metres at various intervals of time in seconds of a body dropped from a tall cliff. The objective is to find the veloctiy of the body at time t = 2 seconds from this data. One way to approach this problem is to find the average velocity for various intervals of time ending at t = 2 seconds and hope that these throw some light on the velocity at t = 2 seconds. Average velocity between t = t1 and t = t2 equals distance travelled between t = t1 and t = t2 seconds divided by (t2 – t1). Hence the average velocity in the first two seconds MATHEMATICS Table 13.1 Distance travelled between t2 = 2 and t1 = 0 = Time interval (t2 − t1 ) = t 0 1 1.5 1.8 1.9 1.95 2 2.05 2.1 2.2 2.5 3 4 (19.6 − 0 ) m = 9.8 m / s . ( 2 − 0) s Similarly, the average velocity between t = 1 and t = 2 is bl © no N C tt E o R be T re pu Likewise we compute the average velocitiy between t = t1 and t = 2 for various t1. The following Table 13.2 gives the average velocity (v), t = t1 seconds and t = 2 seconds. is (19.6 – 4.9 ) m = 14.7 m/s ( 2 − 1) s s 0 4.9 11.025 15.876 17.689 18.63225 19.6 20.59225 21.609 23.716 30.625 44.1 78.4 he 282 Table 13.2 t1 0 1 1.5 1.8 1.9 1.95 1.99 v 9.8 14.7 17.15 18.62 19.11 19.355 19.551 From Table 13.2, we observe that the average velocity is gradually increasing. As we make the time intervals ending at t = 2 smaller, we see that we get a better idea of the velocity at t = 2. Hoping that nothing really dramatic happens between 1.99 seconds and 2 seconds, we conclude that the average velocity at t = 2 seconds is just above 19.551m/s. This conclusion is somewhat strengthened by the following set of computation. Compute the average velocities for various time intervals starting at t = 2 seconds. As before the average velocity v between t = 2 seconds and t = t2 seconds is = Distance travelled between 2 seconds and t2 seconds t2 − 2 = Distance travelled in t2 seconds − Distance travelled in 2 seconds t2 − 2 LIMITS AND DERIVATIVES = 283 Distance travelled in t2 seconds − 19.6 t2 − 2 Table 13.3 4 v 29.4 3 2.5 2.2 2.1 2.05 2.01 24.5 22.05 20.58 20.09 19.845 19.649 is t2 he The following Table 13.3 gives the average velocity v in metres per second between t = 2 seconds and t2 seconds. © no N C tt E o R be T re pu bl Here again we note that if we take smaller time intervals starting at t = 2, we get better idea of the velocity at t = 2. In the first set of computations, what we have done is to find average velocities in increasing time intervals ending at t = 2 and then hope that nothing dramatic happens just before t = 2. In the second set of computations, we have found the average velocities decreasing in time intervals ending at t = 2 and then hope that nothing dramatic happens just after t = 2. Purely on the physical grounds, both these sequences of average velocities must approach a common limit. We can safely conclude that the velocity of the body at t = 2 is between 19.551m/s and 19.649 m/s. Technically, we say that the instantaneous velocity at t = 2 is between 19.551 m/s and 19.649 m/s. As is well-known, velocity is the rate of change of displacement. Hence what we have accomplished is the following. From the given data of distance covered at various time instants we have estimated the rate of change of the distance at a given instant of time. We say that the derivative of the distance function s = 4.9t2 at t = 2 is between 19.551 and 19.649. An alternate way of viewing this limiting process is shown in Fig 13.1. This is a plot of distance s of the body from the top of the cliff versus the time t elapsed. In the limit as the sequence of time intervals h1, h2, ..., approaches zero, the sequence of average velocities approaches the same limit as does the Fig 13.1 sequence of ratios 284 MATHEMATICS C1B1 C2 B2 C3 B3 , , , ... AC1 AC2 AC3 he where C1B1 = s1 – s0 is the distance travelled by the body in the time interval h1 = AC1, etc. From the Fig 13.1 it is safe to conclude that this latter sequence approaches the slope of the tangent to the curve at point A. In other words, the instantaneous velocity v(t) of a body at time t = 2 is equal to the slope of the tangent of the curve s = 4.9t2 at t = 2. 13.3 Limits © no N C tt E o R be T re pu lim f ( x ) = 0 bl is The above discussion clearly points towards the fact that we need to understand limiting process in greater clarity. We study a few illustrative examples to gain some familiarity with the concept of limits. Consider the function f(x) = x2. Observe that as x takes values very close to 0, the value of f(x) also moves towards 0 (See Fig 2.10 Chapter 2). We say x →0 (to be read as limit of f (x) as x tends to zero equals zero). The limit of f (x) as x tends to zero is to be thought of as the value f (x) should assume at x = 0. In general as x → a, f (x) → l, then l is called limit of the function f (x) which is f ( x) = l . symbolically written as lim x→a Consider the following function g(x) = |x|, x ≠ 0. Observe that g(0) is not defined. Computing the value of g(x) for values of x very near to 0, we see that the value of g(x) moves towards 0. So, lim g(x) = 0. This is intuitively x→0 clear from the graph of y = |x| for x ≠ 0. (See Fig 2.13, Chapter 2). Consider the following function. x2 − 4 , x≠2. x−2 Compute the value of h(x) for values of x very near to 2 (but not at 2). Convince yourself that all these values are near to 4. This is somewhat strengthened by considering the graph of the function y = h(x) given here (Fig 13.2). h( x) = Fig 13.2 LIMITS AND DERIVATIVES 285 he In all these illustrations the value which the function should assume at a given point x = a did not really depend on how is x tending to a. Note that there are essentially two ways x could approach a number a either from left or from right, i.e., all the values of x near a could be less than a or could be greater than a. This naturally leads to two limits – the right hand limit and the left hand limit. Right hand limit of a function f(x) is that value of f(x) which is dictated by the values of f(x) when x tends to a from the right. Similarly, the left hand limit. To illustrate this, consider the function bl Graph of this function is shown in the Fig 13.3. It is clear that the value of f at 0 dictated by values of f(x) with x ≤ 0 equals 1, i.e., the left hand limit of f (x) at 0 is is ⎧1, x ≤ 0 f ( x) = ⎨ ⎩ 2, x > 0 lim f ( x) =1 . x→0 © no N C tt E o R be T re pu Similarly, the value of f at 0 dictated by values of f (x) with x > 0 equals 2, i.e., the right hand limit of f (x) at 0 is lim f ( x) = 2 . x → 0+ Fig 13.3 In this case the right and left hand limits are different, and hence we say that the limit of f (x) as x tends to zero does not exist (even though the function is defined at 0). Summary We say xlim f(x) is the expected value of f at x = a given the values of f near →a – x to the left of a. This value is called the left hand limit of f at a. f ( x ) is the expected value of f at x = a given the values of We say xlim →a+ f near x to the right of a. This value is called the right hand limit of f(x) at a. If the right and left hand limits coincide, we call that common value as the limit of f(x) at x = a and denote it by lim f(x). x →a Illustration 1 Consider the function f(x) = x + 10. We want to find the limit of this function at x = 5. Let us compute the value of the function f(x) for x very near to 5. Some of the points near and to the left of 5 are 4.9, 4.95, 4.99, 4.995. . ., etc. Values of the function at these points are tabulated below. Similarly, the real number 5.001, 286 MATHEMATICS 5.01, 5.1 are also points near and to the right of 5. Values of the function at these points are also given in the Table 13.4. Table 13.4 4.9 4.95 4.99 4.995 5.001 5.01 5.1 f(x) 14.9 14.95 14.99 14.995 15.001 15.01 15.1 he x bl lim f ( x ) = 15 . x →5 – is From the Table 13.4, we deduce that value of f(x) at x = 5 should be greater than 14.995 and less than 15.001 assuming nothing dramatic happens between x = 4.995 and 5.001. It is reasonable to assume that the value of the f(x) at x = 5 as dictated by the numbers to the left of 5 is 15, i.e., Similarly, when x approaches 5 from the right, f(x) should be taking value 15, i.e., © no N C tt E o R be T re pu lim f ( x ) = 15 . x →5+ Hence, it is likely that the left hand limit of f(x) and the right hand limit of f(x) are both equal to 15. Thus, lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 15 . x → 5− x →5 x →5 This conclusion about the limit being equal to 15 is somewhat strengthened by seeing the graph of this function which is given in Fig 2.16, Chapter 2. In this figure, we note that as x approaches 5 from either right or left, the graph of the function f(x) = x +10 approaches the point (5, 15). We observe that the value of the function at x = 5 also happens to be equal to 15. Illustration 2 Consider the function f(x) = x3. Let us try to find the limit of this function at x = 1. Proceeding as in the previous case, we tabulate the value of f(x) at x near 1. This is given in the Table 13.5. Table 13.5 x 0.9 0.99 f(x) 0.729 0.970299 0.999 0.997002999 1.001 1.01 1.1 1.003003001 1.030301 1.331 From this table, we deduce that value of f(x) at x = 1 should be greater than 0.997002999 and less than 1.003003001 assuming nothing dramatic happens between Here again we note that the value of the function at x = 2 coincides with the limit at x = 2. again.3 As before we observe that as x approaches 2 from either left or right. We observe.6 x 1.e.4 strengthens this fact.4 .999 2. i.003 6. Let us try to find the limit of this function at x = 2. the value of f(x) seem to approach 6.11. the graph of the function f(x) = x3 approaches the point (1..e. that the value of the function at x = 1 also happens to be equal to 1.999 and 1. We record this as lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 6 x → 2− x→2 x →2 Its graph shown in Fig 13. It is reasonable to assume that the value of the f(x) at x = 1 as dictated by the numbers to the left of 1 is 1. x →1− Similarly. Let us try to find its limit at x = 2. when x approaches 1 from the right..01 2.1 f(x) 5. Chapter 2.001.97 5. Thus.95 1. The following Table 13. x →1− x →1 x →1 is Hence.9 1. it is likely that the left hand limit of f(x) and the right hand limit of f(x) are both equal to 1. we note that as x approaches 1 from either right or left. lim f ( x ) = 1 . f(x) should be taking value 1.997 6. © no N C tt E o R be T re pu bl This conclusion about the limit being equal to 1 is somewhat strengthened by seeing the graph of this function which is given in Fig 2.001 2.99 1. he lim f ( x ) = 1 . Illustration 4 Consider the constant function f(x) = 3.6 is now self-explanatory. This function being the constant function takes the same Fig 13.03 6. In this figure. 1). Table 13. Illustration 3 Consider the function f(x) = 3x.85 5.7 5.LIMITS AND DERIVATIVES 287 x = 0. i. x →1+ lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 1 . 7.e.71 1. in this case) everywhere. fact. again we observe that the lim f (x) = f (1) x→1 Now.. it is easily observed that lim x→a tabulate the values of f(x) near x = 1 in Table 13.5. From this also it is clear that the required limit is 3.2 = 2 = lim ⎡⎣ x ( x + 1) ⎤⎦ = lim ⎡⎣ x 2 + x ⎤⎦ .99 0. Also lim x.997001 1. We want to find lim x →1 From this it is reasonable to deduce that lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 2 . i. lim x = 1 and lim x + 1 = 2 x →1 x →1 x →1 Then lim x 2 + lim x = 1 + 1 = 2 = lim ⎡⎣ x 2 + x ⎤⎦ . convince yourself of the following three facts: Fig 13. Here.999 f(x) 1.9. Chapter 2.0301 © no N C tt E o R be T re pu x is f ( x ) .31 2. x →1 x →1 x →1 x →1 x →1 x →1 x →1 .2 2. 3) and is shown in Fig 2.01 1. lim ( x + 1) = 1. Table 13.7 0. 2).5 lim x 2 = 1.9701 1. its value at points close to 2 is 3. it is clear that as x approaches 1. x →1− x →1 x →1 From the graph of f(x) = x 2 + x shown in the Fig 13. In he f ( x ) = 3 for any real number a. the graph approaches (1.9 2. We Illustration 5 Consider the function f(x) = x2 + x.288 MATHEMATICS value (3.64 bl 0.1 1. Hence lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 3 x→ 2 x→2 x→2 Graph of f(x) = 3 is anyway the line parallel to x-axis passing through (0. In this case too.0950 From the Table 13. x→0 Here we tabulate the (approximate) value of f(x) near 0 (Table 13.1 – 0. we tabulate the (approximate) value of f(x) near this. we observe that lim sin x = 1.1 2 f(x) 0.8 x→ π 2 © no N C tt E o R be T re pu Table 13.9950 0. Table 13.9999 0.98995 0. π (Table 13.9950 Illustration 7 Consider the function f(x) = x + cos x. we may deduce that lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 1 x →0− x →0 x →0 In this case too. can you convince yourself that lim [ x + cos x ] = lim x + lim cos x is indeed true? x →0 x →0 x →0 .1 2 π − 0.01 0.9). x→ 2 where the angle is measured in radians. this is supported by the graph of f(x) = sin x which is given in the Fig 3.1 f(x) 0.001 0. We want to find the lim f (x). From 2 he Here.9989995 1.01 2 π + 0.8 bl (Chapter 3). We are interested in limπ sin x .0009995 1.8).9.9850 0. x→0 Now.01 2 π + 0.LIMITS AND DERIVATIVES 289 Illustration 6 Consider the function f(x) = sin x. is lim− f ( x ) = lim+ f ( x ) = lim f ( x ) = 1 x→ Further.001 0.9 x – 0.01 – 0. we observe that lim f (x) = f (0) = 1. x π − 0.00995 1.9999 0. we may deduce that π 2 x→ π 2 x→ π 2 . x→0 x2 Here.01 – 2.001 0.01 – 0. we deduce that the value of the function is decreasing to –2 and hence.01 bl x is he Illustration 8 Consider the function f ( x ) = lim f ( x ) = + ∞ x →0 We also remark that we will not come across such limits in this course.1 From the first three entries of the Table 13.10 1 0.001 2.01 2. we say 0. Table 13.1 – 2. lim f ( x ) = −2 x →0− . it does not make sense to talk of x approaching 0 from the left.1 f(x) – 2. x > 0 ⎩ As usual we make a table of x near 0 with f(x).001 0. x < 0 ⎪ f ( x) = ⎨ 0 . when we tabulate the values of f(x).11 x – 0. observe that the domain of the function is given to be all positive real numbers. What we mean here is that the value of f(x) may be made larger than any given number.001 2.01 0. Observe that for negative values of x we need to evaluate x – 2 and for positive values. x = 0 ⎪ x + 2.1 – 0.10 given below. Table 13. f(x) becomes larger and larger.1 f(x) 1 100 10–n 10000 102n © no N C tt E o R be T re pu Mathematically. where Illustration 9 We want to find lim x →0 ⎧ x − 2.290 MATHEMATICS 1 for x > 0 . we see that as x tends to 0. Below we tabulate the values of the function for positive x close to 0 (in this table n denotes any positive integer). Hence. f ( x ) . We want to know lim f (x). we need to evaluate x + 2. From the Table 13.11. x →1− Similarly. we Fig 13. From the values of f(x) for x less than 1.001 1. we remark that the value of the function at x = 0 is well defined and is. Table 13. but the limit of the function at x = 0 is not even defined.99 2.e. Illustration 10 As a final illustration..99 0. we find lim x →1 he Since the left and right hand limits at 0 do not coincide. we say that the limit of the function at 0 does not exist. lim f ( x ) = 3 ..LIMITS AND DERIVATIVES 291 From the last three entires of the table we deduce that the value of the function is increasing from 2 and hence lim f ( x ) = 2 x →0+ x ≠1 x =1 © no N C tt E o R be T re pu ⎧x + 2 f ( x) = ⎨ ⎩ 0 is where Fig 13.01 3. i. the value of f(x) should be 3 as dictated by values of f(x) at x greater than 1. it seems that the function should take value 3 at x = 1.12 x 0. indeed. equal to 0.e. x →1− x →1 x →1 Graph of function given in Fig 13. Here. Here. Graph of this function is given in the Fig13.7 strengthens our deduction about the limit.6 bl f ( x) .7 . i.999 3. x →1+ But then the left and right hand limits coincide and hence lim f ( x ) = lim+ f ( x ) = lim f ( x ) = 3 .6.001 3.1 f(x) 2.9 2. lim f ( x ) = 3 .999 1.1 As usual we tabulate the values of f(x) for x near 1.9 0.01 1. x →a x →a x →a (iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero). x →a x →a © no N C tt E o R be T re pu x →a (iii) Limit of product of two functions is product of the limits of the functions. lim x →a $ Note f ( x) g ( x) = lim f ( x ) x →a lim g ( x ) x→a In particular as a special case of (iii). lim [f(x) – g(x)] = lim f(x) – lim g(x). g(x)] = lim f(x). lim [f(x) + g (x)] = lim f(x) + lim g(x). subtraction.lim f ( x ) . We know that lim x = a. below we formalise these as a theorem without proof. multiplication and division as long as the limits and functions under consideration are well defined. In fact. Hence x →a . x →a x →a is Then (i) Limit of sum of two functions is sum of the limits of the functions. i. lim g(x). .e. .3. + anxn.e.. i. we have lim ⎡⎣( λ.3. i. 13. This is not a coincidence. lim [f(x) . f ) ( x ) ⎤⎦ = λ. we illustrate how to exploit this theorem to evaluate limits of special types of functions.. he 13.e.e.. for some real number λ . where ais are real numbers such that an ≠ 0 for some natural number n. x →a x →a bl x →a (ii) Limit of difference of two functions is difference of the limits of the functions.1 Algebra of limits In the above illustrations. when g is the constant function such that g(x) = λ . Theorem 1 Let f and g be two functions such that both lim f (x) and lim g(x) exist..2 Limits of polynomials and rational functions A function f is said to be a polynomial function if f(x) is zero function or if f(x) = a0 + a1x + a2x2 +. i. x →a x →a In the next two subsections.292 MATHEMATICS note that in general. we have observed that the limiting process respects addition. at a given point the value of the function and its limit may be different (even when both are defined). + an a n = f (a) (Make sure that you understand the justification for each step in the above!) g ( x) A function f is said to be a rational function... Thinking lim f ( x ) x→ a ⎡ a0 + a1 x + a2 x 2 + ... there are two scenarios – (i) when g(a) ≠ 0 and (ii) when g(a) = 0.. if f(x) = h x .. if h(a) = 0.x ) = lim x. where g(x) and h(x) ( ) are polynomials such that h(x) ≠ 0. + an x n ⎤ = lim ⎦ x →a ⎣ is of each of a0 ... Now.. a2 x 2 . In the latter case we can write g(x) = (x – a)k g1 (x). a = a 2 x →a x →a x →a x →a An easy exercise in induction on n tells us that lim x n = a n x →a he Now.. let f ( x ) = a0 + a1 x + a2 x 2 + .... an x n as a function. a1 x. we have bl a0 + lim a1 x + lim a2 x 2 + . + an lim x n = a0 + a1 lim x →a x →a x →a = a0 + a1a + a2 a 2 + . In the former case we say that the limit does not exist. Then lim f ( x ) = lim x →a x →a g ( x) h( x) = lim g ( x ) x→a lim h ( x ) = x→a g (a) h(a) However. + lim an x n = lim x →a x →a x →a x →a © no N C tt E o R be T re pu x + a2 lim x 2 + . h(x) = (x – a)l h1 (x) as h (a) = 0.LIMITS AND DERIVATIVES 293 lim x 2 = lim ( x.lim x = a. + an x n be a polynomial function.. we have lim f ( x ) = x→a lim g ( x ) x→a lim h ( x ) x→a lim ( x − a ) g1 ( x ) k = x→a lim ( x − a ) h1 ( x ) l x→a . where k is the maximum of powers of (x – a) in g(x) Similarly. if k > l. + ( −1)10 x →−1 ⎣ = 1 − 1 + 1. We have (iii) lim ⎡1 + x + x 2 + . 0 . we try to 0 rewrite the function cancelling the factors which are causing the limit to be of the form 0 . + x10 ⎤⎦ = 1 + ( −1) + ( −1)2 + .294 MATHEMATICS lim ( x − a ) = x→a ( k −l ) g1 ( x ) lim h1 ( x ) = x →a 0. the limit is not defined... 2 2 ⎢ x →1 ⎣ x − x x − 3x + 2 x ⎥⎦ Solution All the functions under consideration are rational functions... (iii) ⎡ x ( x + 1) ⎤⎦ (ii) lim x →3 ⎣ he ⎡ x3 − x 2 + 1⎤ Example 1 Find the limits: (i) lim ⎦ x →1 ⎣ lim ⎡1 + x + x 2 + . + x10 ⎤⎦ . + 1 = 1 . Hence the limits are the values of the function at the prescribed points.... If this is of the form 0 . Example 2 Find the limits: ⎡ x2 + 1 ⎤ ⎢ ⎥ (i) lim x →1 x + 100 ⎣ ⎦ ⎡ ⎤ x2 − 4 lim ⎥ (iii) x →2 ⎢ 3 2 ⎣ x − 4x + 4x ⎦ ⎡ x3 − 4 x 2 + 4 x ⎤ ⎢ ⎥ (ii) lim x →2 x2 − 4 ⎣ ⎦ ⎡ x3 − 2 x 2 ⎤ lim ⎥ (iv) x →2 ⎢ 2 ⎣ x − 5x + 6 ⎦ 1 ⎡ x−2 ⎤ − 3 (v) lim . we first evaluate these functions at the prescribed points. Hence.g1 ( a ) h1 ( a ) =0 If k < l.. x →−1 ⎣ © no N C tt E o R be T re pu ⎡ x ( x + 1) ⎤⎦ = 3 ( 3 + 1) = 3 ( 4 ) = 12 (ii) lim x →3 ⎣ bl (i) lim [x3 – x2 + 1] = 13 – 12 + 1 = 1 x→1 is Solution The required limits are all limits of some polynomial functions. ( x − 3) 2 − 3 −1 2 = lim x →2 . 0 ( x + 2 )( x − 2 ) x2 − 4 lim 2 = x→2 x ( x − 2) x3 − 4 x 2 + 4 x ( x + 2) = 2 + 2 = 4 = lim x →2 x ( x − 2 ) 2 ( 2 − 2) 0 which is not defined. 0 x2 ( x − 2) x3 − 2 x 2 lim 2 = lim x → 2 ( x − 2 )( x − 3) x →2 x − 5 x + 6 ( 2) 4 x2 = = = −4. we get it of the form Hence lim x →2 0 . it is of the form 0 . 4 © no N C tt E o R be T re pu = x ( x − 2) bl = lim x →2 (iii) Evaluating the function at 2. we get it of the form Hence 0 . (iv) Evaluating the function at 2.LIMITS AND DERIVATIVES 295 x2 + 1 12 + 1 2 = = x →1 x + 100 1 + 100 101 (i) We have lim Evaluating the function at 2. 0 he (ii) x ( x − 2) x3 − 4 x 2 + 4 x lim Hence = lim 2 x → 2 ( x + 2 )( x − 2 ) x →2 x −4 2 is as x ≠ 2 ( x + 2) 2 ( 2 − 2) 2+2 = 0 =0. 296 MATHEMATICS (v) First. xn − an = na n −1 . ( ) ⎤ ⎥ ⎥ ⎦ ⎡ x−2 ⎤ 1 = ⎢ x x −1 − x x −1 x − 2 ⎥ ) ( )( ) ⎦⎥ ⎣⎢ ( bl x2 − 4x + 3 = x x −1 x − 2 ( )( ) is ⎡ x2 − 4x + 4 − 1 ⎤ = ⎢ x x −1 x − 2 ⎥ )( ) ⎥⎦ ⎢⎣ ( 0 . Evaluation of an important limit which will be used in the sequel is given as a theorem below. 0 © no N C tt E o R be T re pu Evaluating the function at 1. x→a x − a lim Remark The expression in the above theorem for the limit is true even if n is any rational number and a is positive. we rewrite the function as a rational function. . we get it of the form Hence he ⎡ 1 1 ⎡ x−2 ⎤ ⎢ x−2 − − = 2 ⎢ x 2 − x x3 − 3x 2 + 2 x ⎥ ⎢ x ( x − 1) x x − 3 x + 2 ⎣ ⎦ ⎣ ⎡ x2 − 2 ⎤ x2 − 4x + 3 1 lim lim ⎢ 2 − 3 ⎥= x →1 x − x x − 3 x 2 + 2 x ⎦ x →1 x ( x − 1)( x − 2 ) ⎣ ( x − 3)( x − 1) = lim x →1 x ( x − 1)( x − 2 ) x−3 1− 3 = lim = x →1 x ( x − 2 ) 1(1 − 2 ) = 2. We remark that we could cancel the term (x – 1) in the above evaluation because x ≠1. Theorem 2 For any positive integer n. .. lim he = an – l + a an–2 +. . + x an–2 + an–1) x →a x − a x →a Thus. .. Then lim x →0 y −1 1+ x −1 = lim y →1 y – 1 x 1 1 y 2 − 12 = lim y →1 y −1 = 1 1 1 2 −1 (1) (by the remark above) = 2 2 . so that y → 1 as x → 0.. we see that xn – an = (x–a) (xn–1 + xn–2 a + xn–3 a2 + .LIMITS AND DERIVATIVES 297 Proof Dividing (xn – an) by (x – a). + x an–2 + an–1) xn − an = lim (xn–1 + xn–2 a + xn–3 a2 + . + an–2 (a) +an–l = an–1 + an – 1 +..+an–1 + an–1 (n terms) Example 3 Evaluate: x15 − 1 x →1 x10 − 1 Solution (i) We have (i) lim 1+ x −1 x bl (ii) lim is n −1 = na © no N C tt E o R be T re pu x →0 lim x →1 15 10 x15 − 1 lim ⎡ x − 1 ÷ x − 1 ⎤ ⎢ ⎥ = x →1 x −1 ⎦ x10 − 1 ⎣ x −1 ⎡ x15 − 1 ⎤ ⎡ x10 − 1 ⎤ lim lim ÷ ⎥ ⎢ ⎥ = x →1 ⎢ ⎣ x − 1 ⎦ x →1 ⎣ x − 1 ⎦ = 15 (1)14 ÷ 10(1)9 (by the theorem above) = 15 ÷ 10 = 3 2 (ii) Put y = 1 + x.. 9 Given below is a beautiful geometric proof of the following important inequality relating trigonometric functions. This is illustrated in Fig 13. cos x < sin x <1 x for 0 < x < π 2 (*) . For some real number a.8 Theorem 4 (Sandwich Theorem) Let f. For some a. Theorem 3 Let f and g be two real valued functions with the same domain such that he f (x) ≤ g( x) for all x in the domain of definition. x →a x →a x →a Fig 13. x→a x→a x→a Fig 13. g and h be real functions such that f (x) ≤ g( x) ≤ h(x) for all x in the common domain of definition.298 MATHEMATICS 13. then lim g(x) = l. if both lim f(x) and x →a © no N C tt E o R be T re pu bl is lim g(x) exist.4 Limits of Trigonometric Functions The following facts (stated as theorems) about functions in general come in handy in calculating limits of some trigonometric functions. if lim f(x) = l = lim h(x).8. then lim f(x) ≤ lim g(x).9. This is illustrated in Fig 13. Taking reciprocals throughout. x < OA. Then Area of ΔOAC < Area of sector OAC < Area of Δ OAB. we have sin x cos x sin x <1 x which complete the proof. Since length OA is positive. tan x. Further. Also tan x = and OA OA hence AB = OA.CD < . cos x < Theorem 5 The following are two important limits.e. Proof (i) The inequality in (*) says that the function . x →0 x (ii) lim sin x is sandwiched between the x function cos x and the constant function which takes value 1. CD < x . x →0 x (i) lim 1 − cos x =0. it is sufficient π ..10 is the angle AOC is x radians and 0 < x < he to prove the inequality for 0 < x < bl i. Hence.π. Thus OA sin x < OA. CD AB (since OC = OA) and hence CD = OA sin x.e. 2 In the Fig 13.10.AB .. Fig 13. join AC.LIMITS AND DERIVATIVES 299 Proof We know that sin (– x) = – sin x and cos( – x) = cos x. tan x. OA < AB. sin x =1.(OA)2 < OA. 2 2π 2 i. we have sin x < x < tan x. we have 2 x 1 < . O is the centre of the unit circle such that π . From Δ OCD. © no N C tt E o R be T re pu sin x = Since 0 < x < 1< π . Line segments B A and 2 CD are perpendiculars to OA. 1 x 1 OA. sinx is positive and thus by dividing throughout by sin x. we see that the proof of (i) of the theorem is x→0 complete by sandwich theorem. 4x → 0 and 2x → 0) x → 0 . since lim cos x = 1. ⎝2⎠ © no N C tt E o R be T re pu ⎛x⎞ sin ⎜ ⎟ 2 ⎛x⎞ = lim ⎝ ⎠ .lim ⎥÷⎢ ⎥ x →0 ⎢ ⎣ 4x ⎦ ⎣ 2x ⎦ ⎡ sin 4 x ⎤ ⎡ sin 2 x ⎤ ÷ lim ⎢ = 2.1.1 = 2 (as x → 0. . This 2 . Then is ⎛x⎞ ⎛x⎞ 2sin 2 ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ = lim ⎝ 2 ⎠ .0 = 0 x →0 x →0 x ⎝2⎠ 2 Observe that we have implicitly used the fact that x → 0 is equivalent to may be justified by putting y = Example 4 Evaluate: Solution (i) lim x →0 x . 4lim ⎥ ⎥ x →0 ⎢ 2 x → 0 ⎣ 4x ⎦ ⎣ 2x ⎦ = 2.sin ⎛ x ⎞ ⎜2⎟ x →0 x x ⎝ ⎠ 2 bl 1 − cos x lim = lim x →0 x →0 x he ⎛x⎞ To prove (ii).300 MATHEMATICS Further.2⎥ x → 0 sin 2 x ⎣ 4 x sin 2 x ⎦ ⎡ sin 4 x ⎤ ⎡ sin 2 x ⎤ = 2. we recall the trigonometric identity 1 – cos x = 2 sin2 ⎜ ⎟ . 2 (i) lim sin 4 x x →0 sin 2 x (ii) lim x →0 tan x x ⎡ sin 4 x 2 x ⎤ sin 4 x = lim ⎢ .lim sin ⎜ ⎟ = 1. lim x →2 3x 2 − x − 10 x2 − 4 22 ⎞ ⎛ x− ⎟ 2. i. If both are 0. see if we can write f(x) = f1 (x) f2(x) so that f1 (a) = 0 and f2 (a) ≠ 0. where q(x) ≠ 0. lim ax + b cx + 1 x → −1 x →3 x →0 1 10. Similarly. lim z →1 z 3 −1 1 z 6 −1 1 1 + 12. lim x →0 sin ax bx 14.1 = 1 x → 0 x → 0 x → 0 cos x x cos x x x A general rule that needs to be kept in mind while evaluating limits is the following..a+b+c ≠0 x →1 cx 2 + bx + a 11. lim ⎜ x→π ⎝ 7 ⎠ πr 2 3. lim r →1 ( x + 1)5 − 1 5. lim 7. where g1(a) = 0 and g2(a) ≠ 0. lim 13. lim x→ 0 x 8. given that the limit lim x→a g ( x ) g ( x) q ( x ) . he f ( x) exists and we want to evaluate this.1 Evaluate the following limits in Exercises 1 to 22. lim x10 + x5 + 1 x −1 6. lim x 2 x →−2 x + 2 ax 2 + bx + c . f ( x) = p (a) © no N C tt E o R be T re pu lim p ( x) = bl f ( x) is the value of f (a) and g(a).LIMITS AND DERIVATIVES (ii) We have lim x →0 301 sin x sin x 1 tan x . lim x 4 − 81 2 x2 − 5x − 3 9. x+3 1. First we check Say. EXERCISE 13. lim x →3 4x + 3 x →4 x − 2 4. lim x →0 sin ax .e. b ≠ 0 sin bx . a. we write g(x) = g1 (x) g2(x). then we see if we can get the factor which is causing the terms to vanish. lim = lim = lim = 1. Cancel out the common factors from f(x) and g(x) (if possible) and write Then x →a g ( x) q(a) . lim x→π π ( π − x ) 16. Find lim x →0 ⎪ 0. ⎪ b − ax . where f ( x ) = ⎨ 23. b. lim ax + x cos x b sin x x sec x 19. x = 0 ⎩ f ( x ) . where f ( x ) = ⎨ | x | 26. a + b ≠ 0 . lim x →0 x →0 lim tan 2 x π x− 2 bl 22. ⎪ 28. x≠0 f ( x ) . lim x →0 cos 2 x − 1 x →0 cos x − 1 cos x π−x 17. Suppose f ( x ) = ⎨4. ⎩ x <1 x =1 x >1 and if lim f (x) = f (1) what are possible values of a and b? x→1 . where f ( x ) = ⎨ x 25. x ≤ 0 f ( x ) and lim f ( x ) . Find lim x →0 x →1 ⎩3 ( x + 1) . x ≤ 1 = f x lim f x ( ) ⎨ 2 24. where f ( x ) = | x | −5 27. ax + sin bx (cosec x − cot x) 21. x = 0 ⎧ x ⎪ . x > 1 ⎧| x | ⎪ . lim sin ax + bx a.MATHEMATICS sin ( π − x ) 15. x > 0 ⎧⎪ x 2 − 1. Find x →1 ( ) . x≠0 f ( x ) . lim x →0 20. lim 18. Find lim x →5 ⎧ a + bx . π x→ 2 is x →0 he 302 © no N C tt E o R be T re pu ⎧ 2 x + 3. Evaluate lim x →0 ⎪⎩ 0. where ⎪⎩ − x − 1. . In these. that by knowing the position of a body at various time intervals it is possible to find the rate at which the position of the body is changing. compute lim f (x). x →1 © no N C tt E o R be T re pu x →1 f ( x) − 2 bl For what value (s) of a does lim f (x) exists? x →a is ⎧ x + 1.. For instance. Financial institutions need to predict the changes in the value of a particular stock knowing its present value. ⎧ mx 2 + n . .. x < 0 ⎪ x=0 . people maintaining a reservoir need to know when will a reservoir overflow knowing the depth of the water at several instances of time.LIMITS AND DERIVATIVES 303 29. x<0 ⎪ f ( x) If f ( x ) = ⎨ nx + m . Let a1. . ⎪ x − 1. a2. If f ( x ) = ⎨ 0.. ( x − an ) ..2. x→a →a he 1 31. It is of very general interest to know a certain parameter at various instants of time and try to finding the rate at which it is changing. There are several real life situations where such a process needs to be carried out. If the function f(x) satisfies lim x2 − 1 = π . x > 0 ⎩ 32. f ( x ) exist? and lim x →1 13. .. Rocket Scientists need to compute the precise velocity with which the satellite needs to be shot out from the rocket knowing the height of the rocket at various times. 0 ≤ x ≤ 1 . evaluate lim f ( x ) . 30. What is xlim f (x) ? For some a ≠ a1. The heart of the matter is derivative of a function at a given point in its domain of definition. a2. and many such cases it is desirable to know how a particular parameter is changing with respect to some other parameter. ..5 Derivatives We have seen in the Section 13. For what integers m and n does both lim x →0 ⎪ 3 x >1 ⎩ nx + m . an. an be fixed real numbers and define a function f ( x ) = ( x − a1 ) ( x − a2 ) . Derivative of f (x) at a is denoted by f′(a). The derivative of the function 3x at x = 2 is 3. Also prove that f ′ (0) + 3f ′ ( –1) = 0. The derivative of f at a is defined by lim f (a + h) − f (a) h →0 h provided this limit exists. h → 0 h h h →0 © no N C tt E o R be T re pu h →0 h →0 bl = lim h = lim is Solution We have he Observe that f′ (a) quantifies the change in f(x) at a with respect to x. We have f ' ( −1) = lim h →0 f ( −1 + h ) − f ( −1) h ⎡ 2 ( −1 + h )2 + 3 ( −1 + h ) − 5⎤ − ⎡ 2 ( −1)2 + 3 ( −1) − 5⎤ ⎦ ⎣ ⎦ = lim ⎣ h →0 h = lim h →0 and f ' ( 0 ) = lim h →0 2h 2 − h = lim ( 2h − 1) = 2 ( 0 ) − 1 = −1 h →0 h f (0 + h ) − f (0) h ⎡ 2 ( 0 + h )2 + 3 ( 0 + h ) − 5⎤ − ⎡ 2 ( 0 )2 + 3 ( 0 ) − 5⎤ ⎦ ⎣ ⎦ = lim ⎣ h →0 h . Example 6 Find the derivative of the function f(x) = 2x2 + 3x – 5 at x = –1. f′ (2) = lim f ( 2 + h ) − f ( 2) h →0 3( 2 + h ) − 3( 2) h 6 + 3h − 6 3h = lim = lim 3 = 3 . Solution We first find the derivatives of f(x) at x = –1 and at x = 0.304 MATHEMATICS Definition 1 Suppose f is a real valued function and a is a point in its domain of definition. Example 5 Find the derivative at x = 2 of the function f(x) = 3x. The Fig 13. Solution Since the derivative measures the change in function. limits are subjected to. f(a)) and Q = (a + h. f ( 0 + h ) − f ( 0) f ′(0) = lim h →0 h sin ( 0 + h ) − sin ( 0 ) h →0 sin h =1 h →0 h = lim © no N C tt E o R be T re pu h bl = lim is Solution Let f(x) = sin x. h →0 h h = lim 3−3 =0.11 .LIMITS AND DERIVATIVES 305 2h 2 + 3h = lim ( 2h + 3) = 2 ( 0 ) + 3 = 3 h →0 h →0 h = lim Clearly f ' ( 0 ) + 3 f ' ( −1) = 0 he Remark At this stage note that evaluating derivative at a point involves effective use of various rules. intuitively it is clear that the derivative of the constant function must be zero at every point.11 is now self explanatory. Let y = f(x) be a function and let P = (a. h h →0 h →0 We now present a geometric interpretation of derivative of a function at a point. This is indeed. Then Example 8 Find the derivative of f(x) = 3 at x = 0 and at x = 3. supported by the following computation. f ' ( 0 ) = lim f (0 + h) − f (0) h →0 Similarly f ' ( 3) = lim h→0 h f ( 3 + h ) − f ( 3) h = lim 3−3 0 = lim = 0 . Example 7 Find the derivative of sin x at x = 0. Fig 13. f(a + h) be two points close to each other on the graph of this function. The following illustrates this. This is referred to as derivative of f(x) dx dx or y with respect to x. Definition 2 Suppose f is a real valued function.306 MATHEMATICS We know that f ′ ( a ) = lim f (a + h) − f (a) h →0 f (a + h) − f (a) he h From the triangle PQR. the function defined by lim f ( x + h) − f ( x) h →0 h wherever the limit exists is defined to be the derivative of f at x and is denoted by f′(x). © no N C tt E o R be T re pu bl For a given function f we can find the derivative at every point. Hence = lim is lim f ′ ( a ) = tan ψ . the point Q tends to P and we have QR h→0 Q → P PR h This is equivalent to the fact that the chord PQ tends to the tangent at P of the curve y = f(x). Thus the limit turns out to be equal to the slope of the tangent. it is denoted by ( . It is also denoted by D (f (x) ). derivative of f at x = a is also denoted by d f ( x) dx a or df dx a ⎛ df ⎞ or even ⎜ ⎟ . Further. Formally. Thus f ' ( x ) = lim f ( x + h) − f ( x) h →0 h Clearly the domain of definition of f′ (x) is wherever the above limit exists. as h tends to 0. it defines a new function called the derivative of f . ⎝ dx ⎠ x = a Example 9 Find the derivative of f(x) = 10x. Solution Since f′ ( x) = lim h →0 f ( x + h) − f ( x) h . In the limiting process. This definition of derivative is also called the first principle of derivative. There are different notations for derivative of a function. If the derivative exists at every point. Sometimes f′ (x) is denoted by d dy f ( x ) ) or if y = f(x). we define derivative of a function as follows. it is clear that the ratio whose limit we are taking is precisely equal to tan(QPR) which is the slope of the chord PQ. LIMITS AND DERIVATIVES = lim 307 10 ( x + h ) − 10 ( x ) h →0 h 10h (10 ) = 10 . f ′(x) = lim f ( x + h) − f ( x) h →0 = lim h →0 h a−a 0 = lim = 0 as h ≠ 0 h →0 h h Example 12 Find the derivative of f(x) = Solution We have f ′(x) = lim 1 x f ( x + h) − f ( x) h →0 h 1 1 – = lim ( x + h) x h →0 h 1 ⎡ x − ( x + h) ⎤ ⎢ ⎥ = lim h →0 h ⎢⎣ x ( x + h ) ⎦⎥ −1 1 ⎡ −h ⎤ ⎢ ⎥ = lim = lim h → 0 h →0 h x ( x + h ) x ( x + h) ⎢⎣ ⎥⎦ = − 1 x2 . = lim h→ 0 h →0 h he = lim Example 10 Find the derivative of f(x) = x2. f ( x + h) − f ( x) h →0 ( x + h )2 − ( x )2 ( h + 2x ) = 2x = lim h →0 h bl = lim h →0 h is Solution We have. f ′(x) = lim © no N C tt E o R be T re pu Example 11 Find the derivative of the constant function f (x) = a for a fixed real number a. Solution We have. Then d d d ⎡⎣ f ( x) + g ( x) ⎤⎦ = f ( x) + g( x) . g ( x) + f ( x) . g ( x ) ⎤⎦ = f ( x) . d d d ⎡⎣ f ( x ) . Theorem 5 Let f and g be two functions such that their derivatives are defined in a common domain.308 MATHEMATICS 13. g ( x) − f ( x) g ( x) d ⎛ f ( x) ⎞ dx dx ⎜ ⎟= dx ⎝ g ( x) ⎠ ( g ( x) )2 The proofs of these follow essentially from the analogous theorem for limits.5. We will not prove these here. the quotient rule is . dx dx dx (iii) Derivative of product of two functions is given by the following product rule. dx dx dx Derivative of difference of two functions is difference of the derivatives of the functions. is (i) © no N C tt E o R be T re pu d d d ⎡⎣ f ( x ) − g ( x ) ⎤⎦ = f ( x) − g ( x) . we expect the rules for derivatives to follow closely that of limits. Then ( uv )′ = u′v + uv′ This is referred to a Leibnitz rule for differentiating product of functions or the product rule. As in the case of limits this theorem tells us how to compute derivatives of special types of functions. The last two statements in the theorem may be restated in the following fashion which aids in recalling them easily: Let u = f ( x ) and v = g (x). bl (ii) he Derivative of sum of two functions is sum of the derivatives of the functions. We collect these in the following theorem.1 Algebra of derivative of functions Since the very definition of derivatives involve limits in a rather direct fashion. Similarly. g ( x) dx dx dx (iv) Derivative of quotient of two functions is given by the following quotient rule (whenever the denominator is non–zero). d d f ( x) . where u is the constant function taking value 10 everywhere and v(x) = x.x + x. We have f(x) = x2 = x . we have the following theorem. ( x ) = dx dx dx dx = 1. Here.x and hence df d d d ( x... Write f(x) = 10x = uv.... we have f ' ( x ) = lim h →0 f ( x + h) − f ( x) h = lim h→0 ( x + h )n − x n h . let us tackle derivatives of some standard functions. It is easy to see that the derivative of the function f(x) = x is the constant f ( x + h) − f ( x) h →0 h x+h−x h →0 h he function 1.x + x. . This is because f ′ ( x ) = lim = lim is =1 .. By (i) of the above theorem (ten terms) © no N C tt E o R be T re pu d d x +. We note that this limit may be evaluated using product rule too. + 1 (ten terms) = 10. Theorem 6 Derivative of f(x) = xn is nxn – 1 for any positive integer n. Thus by the product rule we have = f ′ ( x ) = (10 x )′ = ( uv )′ = u ′v + uv′ = 0.1 = 10 On similar lines the derivative of f(x) = x 2 may be evaluated.+ x (ten terms) dx dx = 1 + . Proof By definition of the derivative function..1 = 2 x . + x ) bl We use this and the above theorem to compute the derivative of f(x) = 10x = x + .. + x (ten terms). More generally. f(x) = 10x = uv we know that the derivative of u equals 0.LIMITS AND DERIVATIVES 309 ⎛ u ⎞′ u ′v − uv′ ⎜ ⎟ = v2 ⎝v⎠ Now. = lim1 h→0 df ( x) d = dx dx ( x + ..x + 10. Also derivative of v(x) = x equals 1.x ) = ( x ) . + h n −1 he n ( ) ( ) © no N C tt E o R be T re pu d n d x.. x n−1 (by product rule) dx dx = 1..5. Theorem 7 Let f(x) = an x n + an −1 x n −1 + . We have = ( ) ( ) d d ( x ) . Thus n–1 df ( x) ( x + h ) − xn = lim dx h →0 h ( h →0 ) h ( ) nx n −1 + . dx Proof of this theorem is just putting together part (i) of Theorem 5 and Theorem 6.. the derivative function is given by df ( x) = nan x n −1 + ( n − 1) an −1 x x − 2 + .. Solution A direct application of the above theorem tells that the derivative of the above function is 600 x 99 − 55 x 54 + 1 . i..310 MATHEMATICS Binomial theorem tells that (x + h)n = hence (x + h) – x = h(nx n n–1 n +.. ( n − 1) x n − 2 (by induction hypothesis) = x n −1 + ( n − 1) x n −1 = nx n −1 . + h ( n ) C0 x n + ( C )x n 1 n −1 h + . which has been proved earlier. + 2a2 x + a1 . Example 13 Compute the derivative of 6x100 – x55 + x. .. + a1 x + a0 be a polynomial function. + h n −1 = nx n −1 ... 13... n can be any real number (but we will not prove it here)..x n −1 ( ) + x.. Remark The above theorem is true for all powers of x. = lim h →0 is = lim h nx n −1 + .e.2 Derivative of polynomials and trigonometric functions We start with the following theorem which tells us the derivative of a polynomial function.. x n−1 + x. Then. + ( n ) C n h n and ). we may also prove this by induction on n and the product rule as follows.x n −1 x = dx dx bl Alternatively. The result is true for n = 1. where ai s are all real numbers and an ≠ 0.. Solution A direct application of the above Theorem 6 tells that the derivative of the above function is 1 + 2x + 3x2 + . . . At x = 1 the value of this function equals ( 50 )( 51) 2 = 1275..lim . + x50 at x = 1. Therefore is Example 15 Find the derivative of f(x) = © no N C tt E o R be T re pu Example 16 Compute the derivative of sin x. Then f ( x + h) − f ( x) sin ( x + h ) − sin ( x ) df ( x) = lim = lim h →0 h →0 dx h h ⎛ 2x + h ⎞ ⎛ h ⎞ 2cos ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ (using formula for sin A – sin B) = lim h →0 h h sin h⎞ ⎛ 2 = cos x. We use the quotient rule with u = x + 1 and v = x. .1 = cos x lim cos ⎜ x + ⎟ . + 50x49..LIMITS AND DERIVATIVES 311 Example 14 Find the derivative of f(x) = 1 + x + x2 + x3 +. + 50 = x +1 x Solution Clearly this function is defined everywhere except at x = 0. Hence u′ = 1 and v′ = 1. Then f ( x + h) − f ( x) tan ( x + h ) − tan ( x ) df ( x) = lim = lim h →0 h →0 dx h h 1 ⎡ sin ( x + h ) sin x ⎤ lim = h→0 h ⎢ cos x + h − cos x ⎥ ( ) ⎢⎣ ⎥⎦ . Solution Let f(x) = tan x. + 50(1)49 = 1 + 2 + 3 + . = h →0 h → 0 h 2 ⎝ ⎠ 2 Example 17 Compute the derivative of tan x. he 1 + 2(1) + 3(1)2 + .. . . bl df ( x) d ⎛ x + 1 ⎞ d ⎛ u ⎞ u ′v − uv′ 1( x ) − ( x + 1)1 1 = ⎜ = =− 2 ⎟= ⎜ ⎟ = 2 2 dx dx ⎝ x ⎠ dx ⎝ v ⎠ v x x Solution Let f(x) = sin x. 312 MATHEMATICS ⎡ sin ( x + h ) cos x − cos ( x + h ) sin x ⎤ ⎢ ⎥ = lim h →0 h cos ( x + h ) cos x ⎢⎣ ⎥⎦ 1 = sec 2 x . (i) x 3 − 27 1 x2 5. EXERCISE 13. Find the derivative of x at x = 1. . For the function (iii) f ( x) = (ii) ( x − 1)( x − 2 ) (iv) x +1 x −1 x100 x99 x2 + +. Find the derivative of 99x at x = l00.2 1.+ + x +1 . 100 99 2 . cos 2 x bl = 1. 3. 2.lim h h→0 cos ( x + h ) cos x is = lim h →0 he sin ( x + h − x ) = lim (using formula for sin (A + B)) h →0 h cos ( x + h ) cos x Example 18 Compute the derivative of f(x) = sin2 x. . 4. Find the derivative of the following functions from first principle. Find the derivative of x2 – 2 at x = 10. © no N C tt E o R be T re pu Solution We use the Leibnitz product rule to evaluate this. df ( x) d = ( sin x sin x ) dx dx = ( sin x )′ sin x + sin x ( sin x )′ = ( cos x ) sin x + sin x ( cos x ) = 2sin x cos x = sin 2 x . sin h 1 . 7. where f is given by (i) f (x) = 2x + 3 x−2 (ii) f (x) = x + 1 x Solution (i) Note that function is not defined at x = 2. we have f ′ ( x ) = lim h →0 f ( x + h) − f ( x) h 2( x + h) + 3 2x + 3 − x−2 = lim x + h − 2 h →0 h . Find the derivative of cos x from first principle.LIMITS AND DERIVATIVES 313 Prove that f ′ (1) = 100 f ′ ( 0 ) . . Find the derivative of ( ax 2 +b ) 2 (iii) xn − an for some constant a. For some constants a and b. + a n −1 x + a n for some fixed real number a. find the derivative of (ii) 8. Find the derivative of 3 4 (ii) x −3 ( 5 + 3 x ) (v) x −4 3 − 4 x −5 3 ) ( x − 1) + 3x − 1 ( 5 −9 (iv) x 3 − 6 x ) © no N C tt E o R be T re pu (iii) (5x bl (i) 2 x − ( ) (vi) x−a x−b he ( x − a) ( x − b) is (i) 2 x2 − x + 1 3x − 1 10. 11. Find the derivative of the following functions: (i) sin x cos x (iv) cosec x (ii) sec x (iii) 5sec x + 4 cos x (v) 3cot x + 5cosec x (vi) 5sin x − 6 cos x + 7 (vii) 2 tan x − 7 sec x Miscellaneous Examples Example 19 Find the derivative of f from the first principle. But. Find the derivative of x n + ax n −1 + a 2 x n − 2 + . x−a 9. . 6. Example 20 Find the derivative of f(x) from the first principle.MATHEMATICS = lim h →0 ( 2 x + 2h + 3)( x − 2 ) − ( 2 x + 3)( x + h − 2 ) h ( x − 2 )( x + h − 2 ) = lim h →0 ( 2 x + 3)( x − 2 ) + 2h ( x − 2 ) − ( 2 x + 3)( x − 2 ) − h ( 2 x + 3) h ( x − 2 )( x + h − 2 ) –7 7 =− = lim h →0 ( x − 2 ) ( x + h − 2 ) ( x − 2 )2 is 1 ⎞ ⎛ 1⎞ ⎛ x+h+ −⎜x+ ⎟ ⎜ ⎟ f ( x + h) − f ( x) x+h⎠ ⎝ x⎠ = lim ⎝ = lim h →0 h →0 h h © no N C tt E o R be T re pu f ′( x) bl (ii) Again. where f(x) is (ii) x sin x (i) sin x + cos x Solution (i) we have f ' ( x ) = = lim h →0 = lim h →0 f ( x + h) − f ( x) h sin ( x + h ) + cos ( x + h ) − sin x − cos x h sin x cos h + cos x sin h + cos x cos h − sin x sin h − sin x − cos x h . note that the function f ′ is not defined at x = 0. The function is not defined at x = 0. note that the function f ′ is also not defined at x = 2. But. we have he 314 = lim h →0 1⎡ 1 1⎤ h+ − ⎥ ⎢ h⎣ x + h x⎦ ⎞⎤ 1⎡ x − x −h⎤ 1⎡ ⎛ 1 lim h lim h 1 + = − ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ = h →0 h x ( x + h ) ⎦⎥ h→0 h ⎣⎢ ⎜⎝ x ( x + h ) ⎠⎟ ⎦⎥ ⎣⎢ ⎡ ⎤ 1 1 ⎢1 − ⎥ =1− 2 = lim h →0 x ⎣⎢ x ( x + h ) ⎦⎥ Again. We use the quotient rule on this function sin x dg d d ⎛ cos x ⎞ = (cot x) = ⎜ dx dx dx ⎝ sin x ⎟⎠ .LIMITS AND DERIVATIVES sin h ( cos x − sin x ) + sin x ( cos h − 1) + cos x ( cos h − 1) = lim h→0 h h →0 ( cos h − 1) sin h ( cos h − 1) sin x ( cos x − sin x ) + lim + lim cos x h → 0 h →0 h h h = cos x − sin x h→0 = lim ( x + h ) sin ( x + h ) − x sin x h →0 h h ( x + h )( sin x cos h + sin h cos x ) − x sin x h →0 = lim = lim h is f ( x + h) − f ( x) x sin x ( cos h − 1) + x cos x sin h + h ( sin x cos h + sin h cos x ) h →0 h © no N C tt E o R be T re pu x sin x ( cos h − 1) bl f ' ( x ) = lim = lim he = lim (ii) 315 h →0 h + lim h→0 x cos x sin h + lim ( sin x cos h + sin h cos x ) h →0 h = x cos x + sin x Example 21 Compute derivative of (i) f(x) = sin 2x (ii) g(x) = cot x Solution (i) Recall the trigonometric formula sin 2x = 2 sin x cos x. g(x) = cot x = wherever it is defined. Thus df ( x) d d ( 2sin x cos x ) = 2 ( sin x cos x ) = dx dx dx = 2 ⎡( sin x )′ cos x + sin x ( cos x )′ ⎤ ⎣⎢ ⎦⎥ = 2 ⎡⎣( cos x ) cos x + sin x ( − sin x ) ⎤⎦ ( = 2 cos 2 x − sin 2 x (ii) By definition. ) cos x . Here. we use the fact tan x that the derivative of tan x is sec2 x which we saw in Example 17 and also that the derivative of the constant function is 0. Alternatively. h′( x) = ( x5 − cos x)′ sin x − ( x 5 − cos x) (sin x)′ (sin x) 2 .MATHEMATICS = (cos x)′ (sin x) − (cos x) (sin x)′ (sin x) 2 = (− sin x)(sin x) − (cos x)(cos x) (sin x)2 = − he 316 sin 2 x + cos 2 x = − cosec 2 x 2 sin x bl © no N C tt E o R be T re pu d d ⎛ 1 ⎞ dg (cot x) = = dx dx ⎜⎝ tan x ⎟⎠ dx is 1 . We use the quotient rule on this function wherever sin x it is defined. this may be computed by noting that cot x = = (1)′ (tan x) − (1)(tan x)′ (tan x) 2 (0) (tan x) − (sec x) 2 = (tan x) 2 = − sec 2 x = − cosec 2 x tan 2 x Example 22 Find the derivative of (i) x5 − cos x sin x Solution (i) Let h( x) = (ii) x + cos x tan x x5 − cos x . px 2 + qx + r ax + b 10.LIMITS AND DERIVATIVES = 317 (5 x 4 + sin x)sin x − ( x5 − cos x)cos x sin 2 x h′( x) = x + cos x wherever it is defined. cos x 1 + sin x . (x + a) ⎛r ⎞ 3. q. d. 8. r and s are fixed non-zero constants and m and n are integers): (i) − x (ii) (− x) −1 (iii) sin (x + 1) (iv) cos (x – 2. 2 1 ax + bx + c 2 a b − 2 + cos x 4 x x 11. ax + b px + qx + r 2 1+ 9. cx + d 1 x 6. b. 1 1− x 7. (ax + b) n (cx + d ) m 14. c. sin (x + a) 15. Find the derivative of the following functions from first principle: π ) 8 Find the derivative of the following functions (it is to be understood that a. (ax + b) n 13. ( ax + b )( cx + d ) ax + b 5. (px + q) ⎜ + s ⎟ ⎝x ⎠ 4. cosec x cot x 16. 4 x − 2 12. tan x ( x + cos x)′ tan x − ( x + cos x)(tan x)′ (tan x)2 bl (1 − sin x) tan x − ( x + cos x)sec 2 x = (tan x) 2 is (ii) We use quotient rule on the function he − x 5 cos x + 5 x 4 sin x + 1 = (sin x) 2 © no N C tt E o R be T re pu Miscellaneous Exercise on Chapter 13 1. p. 3x + 7cos x 28. Similarly the right hand limit.318 MATHEMATICS 17. For a function f and a real number a. sec x − 1 sec x + 1 19. x sin n x © no N C tt E o R be T re pu Summary he 2 is ) + 1 cos x bl 2 The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. one may be defined and not the other one). ( x + cos x ) ( x − tan x ) 4 x + 5sin x 26. lim f(x) and f (a) may not be same (In x →a fact. a + b sin x c + d cos x 21. (x 24. ( ax + sin x 25. sin x ( x + sec x ) ( x − tan x ) 30. sin n x 20. ) ( p + q cos x ) ⎛π ⎞ x 2 cos ⎜ ⎟ ⎝4⎠ 27. x 1 + tan x 29. sin( x + a) cos x 22. g ( x) ] = lim f ( x ). if they coincide. Limit of a function at a point is the common value of the left and right hand limits.lim g ( x ) x→a x→a x→a f ( x) ⎡ f ( x ) ⎤ lim x→a lim ⎢ = ⎥ x→a g ( x) g ( x) ⎣ ⎦ lim x →a Following are some of the standard limits xn − an = na n −1 x →a x − a lim . x 4 (5sin x − 3cos x) 23. sin x + cos x sin x − cos x 18. For functions f and g the following holds: lim [ f ( x) ± g ( x) ] = lim f ( x) ± lim g ( x) x →a x→a x →a lim [ f ( x). Leibnitz (1646 – 1717). The rigorous concept is mainly attributed to the great . After the advent of calculus many mathematicians contributed for further development of calculus. Issac Newton (1642 – 1727) and G. ⎜ ⎟ = v2 ⎝v⎠ Following are some of the standard derivatives.W.LIMITS AND DERIVATIVES lim x →0 319 sin x =1 x 1 − cos x =0 x →0 x The derivative of a function f at a is defined by f ( a + h) − f ( a ) h Derivative of a function f at any point x is defined by he lim f ′(a ) = lim (u ± v )′ = u ′ ± v′ © no N C tt E o R be T re pu (uv )′ = u ′v + uv′ bl df ( x) f ( x + h) − f ( x ) = lim h → 0 dx h For functions u and v the following holds: f ′( x) = is h →0 ⎛ u ⎞′ u ′v − uv′ provided all are defined. d n ( x ) = nx n −1 dx d (sin x) = cos x dx d (cos x) = − sin x dx Historical Note In the history of mathematics two names are prominent to share the credit for inventing calculus. Both of them independently invented calculus around the seventeenth century. Cauchy gave the foundation of calculus as we have now generally accepted in our textbooks. J.L. Cauchy gave examples such as the limit of α Δ y f ( x + i ) − f ( x) . — — . the “function derive’e. Starting with definition of a limit. © no N C tt E o R be T re pu Today not only the mathematics but many other subjects such as Physics. He wrote Δ x = i he sin α i → 0. Griffin. pioneered the teaching of calculus to first year students. Cauchy used D’ Alembert’s limit concept to define the derivative of a function.L. F.L. bl is Before 1900. and called the limit for for α = 0. John Perry and others in England started propagating the view that essential ideas and methods of calculus were simple and could be taught even in schools. Economics and Biological Sciences are enjoying the fruits of calculus. But just in 1900. Chemistry. So calculus became beyond the reach of youngsters. y′ for f ′ (x)”. A.Lagrange and Karl Weierstrass. Cauchy. it was thought that calculus is quite difficult to teach.320 MATHEMATICS mathematicians. This was regarded as one of the most daring act in those days. . it is as great a folly to make use of another. Let us start with two sentences: In 2003.1864) 14. The main asset that made humans “superior” to other species was the ability to reason. How well this ability can be used depends on each person’s power of reasoning. – ARTHENBOT 14. it is a sign that our knowledge of them is very small and confused and where a mathematical reasoning can be had. In mathematical language. All of us know that human beings evolved from the lower species over many millennia. George Boole (1815 .2 Statements The basic unit involved in mathematical reasoning is a mathematical statement. An elephant weighs more than a human being.Chapter 14 he MATHEMATICAL REASONING is There are few things which we know which are not capable of © no N C tt E o R be T re pu bl mathematical reasoning and when these can not. we shall discuss some fundamentals of deductive reasoning. the president of India was a woman. we shall discuss about some basic ideas of Mathematical Reasoning. we shall discuss the process of reasoning especially in the context of mathematics. In this Chapter. How to develop this power? Here. there are two kinds of reasoning – inductive and deductive. as to grope for a thing in the dark when you have a candle stick standing by you. We have already discussed the inductive reasoning in the context of mathematical induction.1 Introduction In this Chapter. That means this sentence is ambiguous. the second an order and the third a question. unless we know what x and y are. because the first one is an exclamation. they are statements. Sentences involving variable time such as “today”. In mathematics such sentences are called statements. Of these sentences. the sentence Tomorrow is Friday .322 MATHEMATICS © no N C tt E o R be T re pu bl is he When we read these sentences. Some examples are: Two plus two equals four. On the other hand. we are not in a position to determine whether it is true or false. There is no confusion regarding these. A sentence is called a mathematically acceptable statement if it is either true or false but not both. Can you think of an example of a sentence which is vague or ambiguous? Consider the sentence: The sum of x and y is greater than 0 Here. Therefore. it is false where x = 1. Such a sentence is not acceptable as a statement in mathematics. None of these is considered as a statement in mathematical language. y = –3 and true when x = 1 and y = 0. Now. For example. Regarding this sentence we cannot say whether it is always true or false . Therefore. For example. we come across many such sentences. All prime numbers are odd numbers. it is a “mathematically acceptable” statement. There is no ambiguity regarding these sentences. “tomorrow” or “yesterday” are not statements. we immediately decide that the first sentence is false and the second is correct. The sum of two positive numbers is positive. Whenever we mention a statement here. Where are you going? Are they statements? No. consider the sentence: Women are more intelligent than men. Some people may think it is true while others may disagree. the first two are true and the third one is false. While studying mathematics. But the sentence: For any natural numbers x and y. consider the following sentences : How beautiful! Open the door. This is because it is not known what time is referred here. the sum of x and y is greater than 0 is a statement. this sentence is not a statement. Therefore. therefore.. The sentence is correct (true) on a Thursday but not on other days. (iii) It is a scientifically established fact that sun is a star and. (ii) This sentence is also false since there are sets which are not finite. So.. This is also written as p: Fire is always hot. (iv) This sentence is subjective in the sense that for those who like mathematics. (iii) The sun is a star. For example. Kashmir is far from here. we denote the statement “Fire is always hot” by p. Give reasons for your answer. The same argument holds for sentences with pronouns unless a particular person is referred to and for variable places such as “here”. (iv) Mathematics is fun. (ii) Every set is a finite set. There are 40 days in a month. While dealing with statements. Example 1 Check whether the following sentences are statements.MATHEMATICAL REASONING 323 is not a statement. “there” etc. (v) There is no rain without clouds. this sentence is always true. is are not statements.. This means that this sentence is not always true. (vi) How far is Chennai from here? Solution (i) This sentence is false because 8 is greater than 6. what makes a sentence a statement is the fact that the sentence is either true or false but not both.. it may be fun but for others it may not be. (i) 8 is less than 6. the sentences he She is a mathematics graduate. q. Hence it is a statement. Here is another sentence © no N C tt E o R be T re pu bl Would you call this a statement? Note that the period mentioned in the sentence above is a “variable time” that is any of 12 months. this sentence is a statement. Hence it is not a statement. . For example. Hence it is a statement. we usually denote them by small letters p. Hence it is a statement. r. But we know that the sentence is always false (irrespective of the month) since the maximum number of days in a month can never exceed 31. “George Boole” discussed these methods in his book “The laws of Thought” in 1854. Hence it is a statement. An English mathematician. 14. (x) All real numbers are complex numbers. 2.3 New Statements from Old We now look into method for producing new statements from those that we already have. Here. Hence it is not a statement. (iii) The sum of 5 and 7 is greater than 10.1 1. Give three examples of sentences which are not statements. © no N C tt E o R be T re pu (v) The sides of a quadrilateral have equal length. Let us consider the statement: p: New Delhi is a city The negation of this statement is .324 MATHEMATICS (v) It is a scientifically established natural phenomenon that cloud is formed before it rains. (ix) Today is a windy day. he The above examples show that whenever we say that a sentence is a statement we should always say why it is so. This “why” of it is more important than the answer. (vi) Answer this question. This technique is to ask not only what it means to say that a given statement is true but also what it would mean to say that the given statement is not true. we look at an important technique that we may use in order to deepen our understanding of mathematical statements. (viii) The sum of all interior angles of a triangle is 180°. 14. bl (ii) Mathematics is difficult. As a first step in our study of statements. is (i) There are 35 days in a month. EXERCISE 14.1 Negation of a statement The denial of a statement is called the negation of the statement. (vi) This is a question which also contains the word “Here”. Therefore. (vii) The product of (–1) and 8 is 8. (iv) The square of a number is an even number. Give reasons for the answers. we shall discuss two techniques. Which of the following sentences are statements? Give reasons for your answer. this sentence is always true.3. This can simply be expressed as New Delhi is not a city. Definition 1 If p is a statement. (ii) 7 is rational. (ii) The negation of the statement in (ii) may also be written as It is not the case that 7 is rational. “It is not the $ case” or “It is false that” are also used. Solution (i) This statement says that in a rectangle. and read as ‘not p’. both the diagonals have the same length. is Note While forming the negation of a statement. Let us consider the statement p: Everyone in Germany speaks German. This means that if you take any rectangle. (i) Both the diagonals of a rectangle have the same length. This does not mean that no person in Germany speaks German. We shall consider more examples. It says merely that at least one person in Germany does not speak German. we may improve our understanding of it. phrases like. by looking at the negation of a statement.MATHEMATICAL REASONING 325 he It is not the case that New Delhi is a city This can also be written as It is false that New Delhi is a city. then both the diagonals have the same length. Example 2 Write the negation of the following statements. © no N C tt E o R be T re pu bl Here is an example to illustrate how. The negation of this statement is It is false that both the diagonals in a rectangle have the same length This means the statement There is atleast one rectangle whose both diagonals do not have the same length. This can also be rewritten as 7 is not rational. The denial of this sentence tells us that not everyone in Germany speaks German. then the negation of p is also a statement and is denoted by ∼ p. . (ii) There does not exist a quadrilateral which has all its sides equal. This can be written as The sum of 3 and 4 is not equal to 9. etc. (ii) The negation of the statement is It is not the case that there does not exist a quadrilateral which has all its sides equal. This is a false statement. We know that this statement is false. (iv) The negation is It is false that the sum of 3 and 4 is 9.3. This statement is true because we know that square is a quadrilateral such that its four sides are equal.2 Compound statements Many mathematical statements are obtained by combining one or more statements using some connecting words like “and”. © no N C tt E o R be T re pu bl is Solution (i) 14. This can be rewritten as There exists a natural number which is not greater than 0. (iii) The negation of the statement is It is false that every natural number is greater than 0. (iv) The sum of 3 and 4 is 9. he (iii) Every natural number is greater than 0. The negation of the statement is It is false that Australia is a continent. (i) Australia is a continent. Consider the following statement p: There is something wrong with the bulb or with the wiring.326 MATHEMATICS Example 3 Write the negation of the following statements and check whether the resulting statements are true. “or”. This statement tells us that there is something wrong with the bulb or there is . This statement is true. This also means the following: There exists a quadrilateral which has all its sides equal. This can also be rewritten as Australia is not a continent. This is a compound statement.MATHEMATICAL REASONING 327 bl is he something wrong with the wiring. Let us consider some examples. The connecting word is ‘and’. q: 7 is a prime number. q: All real numbers are complex. Example 4 Find the component statements of the following compound statements. (ii) The component statements are p: It is raining. q: The grass is green. These two statements can be combined with “and” r: 7 is both odd and prime number. each statement is called a component statement. r: There is something wrong with the wiring. (iii) The component statements are p: All rational numbers are real. q: It is cold. Solution Let us consider one by one (i) The component statements are p: The sky is blue. (iv)The component statements are . (iv) 0 is a positive number or a negative number. (ii) It is raining and it is cold. The connecting word is ‘and’. (i) The sky is blue and the grass is green. (iii) All rational numbers are real and all real numbers are complex. In this case. The connecting word is ‘and’. connected by “or” Now. That means the given statement is actually made up of two smaller statements: q: There is something wrong with the bulb. This leads us to the following definition: © no N C tt E o R be T re pu Definition 2 A Compound Statement is a statement which is made up of two or more statements. suppose two statements are given as below: p: 7 is an odd number. (iii) The component statements are p: A person who has taken Mathematics can go for MCA. Solution (i) The component statements are p: A square is a quadrilateral. Here the connecting word is ‘or’. Here the connecting word is ‘and’. We know that both these statements are true. Here the connecting word is ‘and’. (ii) The component statements are p: All prime numbers are odd numbers. © no N C tt E o R be T re pu (vi) 24 is a multiple of 2. bl (v) is (iii) A person who has taken Mathematics or Computer Science can go for MCA. 2 is a rational number or an irrational number. (ii) All prime numbers are either even or odd. (iv) Chandigarh is the capital of Haryana and UP. q: All prime numbers are even numbers. q: A person who has taken computer science can go for MCA.328 MATHEMATICS p: 0 is a positive number. The first statement is true but the second is false. (v) The component statements are . q: Chandigarh is the capital of UP. q: A square has all its sides equal. The connecting word is ‘or’. 4 and 8. (iv) The component statements are p: Chandigarh is the capital of Haryana. q: 0 is a negative number. Both these statements are true. he Example 5 Find the component statements of the following and check whether they are true or not. (i) A square is a quadrilateral and its four sides equal. Both these statements are false and the connecting word is ‘or’. (vi) The component statements are p: 24 is a multiple of 2. 3. Thus. (i) Number 3 is prime or it is odd. (v) Every natural number is an integer. © no N C tt E o R be T re pu EXERCISE 14. q: 24 is a multiple of 4. etc.4 Special Words/Phrases Some of the connecting words which are found in compound statements like “And”. “or”. 2. he The first statement is false and second is true. Find the component statements of the following compound statements and check whether they are true or false. (iv) The number 2 is greater than 7. Here the connecting word is ‘or’. Are the following pairs of statements negations of each other: (i) The number x is not a rational number. we observe that compound statements are actually made-up of two or more statements connected by the words like “and”. Write the negation of the following statements: (i) Chennai is the capital of Tamil Nadu. . The number x is not an irrational number.MATHEMATICAL REASONING p: q: 329 2 is a rational number. 11 and 5. (ii) The number x is a rational number. The number x is an irrational number. 1.2 bl is All the three statements are true. 2 is an irrational number. (iii) 100 is divisible by 3. r: 24 is a multiple of 8. (ii) 2 is not a complex number (iii) All triangles are not equilateral triangle. Here the connecting words are ‘and’. (ii) All integers are positive or negative. 14. These words have special meaning in mathematics. We shall discuss this mattter in the following section. Example 6 Write the component statements of the following compound statements and check whether the compound statement is true or false.4. p: 42 is divisible by 5. Let us look at another statement. (ii) 0 is less than every positive integer and every negative integer. This statement has following component statements q: 42 is divisible by 5. When we use these compound statements. Here. 14.1 The word “And” Let us look at a compound statement with “And”. We discuss this below. q: A line extends indefinitely in both directions. it is necessary to understand the role of these words. The component statement with ‘And’ is false if any of its component statements is false (this includes the case that some of its component statements are false or all of its component statements are false). (i) A line is straight and extends indefinitely in both directions. . 6 and 7. etc. are often used in Mathematical Statements. Solution (i) The component statements are p: A line is straight. he p: A point occupies a position and its location can be determined. Here. we know that the first is false while the other two are true. is bl r: Its location can be determined. The statement can be broken into two component statements as q: A point occupies a position.330 MATHEMATICS “Or”. © no N C tt E o R be T re pu r: 42 is divisible by 6. These are called connectives. s: 42 is divisible by 7. 2. The compound statement with ‘And’ is true if all its component statements are true. We have the following rules regarding the connective “And” 1. we observe that both statements are true. (iii) All living things have two legs and two eyes. We know that this is a true statement. q: 0 is less than every negative integer. 14.Sc. A student who has taken biology or chemistry can apply for M. This leads us to an important note. microbiology programme. then they are not parallel. The second statement is false. In order to understand statements with “Or” we first notice that the word “Or” is used in two ways in English language. Here is another statement. Alternatively. Therefore. This means that a person who does not want ice cream can have a pepsi along with Thali or one does not want pepsi can have an ice cream along with Thali. Now. © no N C tt E o R be T re pu Note Do not think that a statement with “And” is always a compound statement $ as shown in the above example. the compound statement is true. This sentence cannot be considered as a compound statement with “And”. who do not want a pepsi can have an ice cream. therefore. p: An ice cream or pepsi is available with a Thali in a restaurant.4. It is important to note the difference between these two ways because we require this when we check whether the statement is true or not. That is. Therefore. Let us first look at the following statement. if the two lines are not parallel. q: All living things have two eyes. we are using inclusive “Or”. (ii) The component statements are p: 0 is less than every positive integer.MATHEMATICAL REASONING 331 bl is he Both these statements are true. the compound statement is false. the compound statement is false. What does this mean? This means that if two lines in a plane intersect. then they intersect at a point. consider the following statement. Both these statements are false. That is this statement is true in both the situations. Here the word “And” refers to two things – alcohol and water.2 The word “Or” Let us look at the following statement. Therefore. . Here we mean that the students who have taken both biology and chemistry can apply for the microbiology programme. A person cannot have both ice cream and pepsi. as well as the students who have taken only one of these subjects. p: A mixture of alcohol and water can be separated by chemical methods. the word “And” is not used as a connective. p: Two lines in a plane either intersect at one point or they are parallel. In this case. This is called an exclusive “Or”. (iii) The two component statements are p: All living things have two legs. (iii) Two lines intersect at a point or are parallel. (iii) Here “Or” is exclusive because it is not possible for two lines to intersect and parallel together. A compound statement with an ‘Or’ is true when one component statement is true or both the component statements are true. © no N C tt E o R be T re pu (iv) Here also “Or” is exclusive because a student cannot take both French and Sanskrit. Its component statements are q: 125 is a multiple of 7. Consider another statement. bl is Solution (i) Here “Or” is inclusive since a person can have both a passport and a voter registration card to enter a country. Give reasons for your answer. Rule for the compound statement with ‘Or’ 1. (ii) Here also “Or” is inclusive since school is closed on holiday as well as on Sunday. r: Two lines are parallel. he (ii) The school is closed if it is a holiday or a Sunday. p: 125 is a multiple of 7 or 8. 2. (i) To enter a country. the compound statement p is false. (iv) Students can take French or Sanskrit as their third language. Example 7 For each of the following statements. . Therefore. A compound statement with an ‘Or’ is false when both the component statements are false. consider the following statement. r: 125 is a multiple of 8. For example. you need a passport or a voter registration card. the compound statement p is true. Both q and r are false. Therefore. determine whether an inclusive “Or” or exclusive “Or” is used. p: Two lines intersect at a point or they are parallel The component statements are q: Two lines intersect at a point. when q is true r is false and when r is true q is false.332 MATHEMATICS Let us look at an example. Then. we get that the statement is true. (iii) A rectangle is a quadrilateral or a 5-sided polygon. p: Mumbai is the capital of Kolkata or Karnataka. the compound statement is false. q: To get into a public library children need a letter from the school authorities. consider the following statement: p: The school is closed. the compound statement is true. © no N C tt E o R be T re pu Example 8 Identify the type of “Or” used in the following statements and check whether the statements are true or false: (i) 2 is a rational number or an irrational number. Therefore. Both q and r are true. The component statements are q: School is closed if there is a holiday. (ii) The component statements are p: To get into a public library children need an identity card. it is inclusive “Or” the compound statement is also true when children have both the card and the letter. When we look at the component statements. if there is a holiday or Sunday. Therefore. Consider another statement. Therefore. Children can enter the library if they have either of the two. therefore. Solution (i) The component statements are p: 2 is a rational number. Here.MATHEMATICAL REASONING 333 bl is he Again. r: School is closed if there is a Sunday. the compound statement is true. q: 2 is an irrational number. Let us consider some examples. The component statements are q: Mumbai is the capital of Kolkata. an identity card or the letter. r: Mumbai is the capital of Karnataka. we know that the first statement is false and the second is true and “Or” is exclusive. (iii) Here “Or” is exclusive. (ii) To enter into a public library children need an identity card from the school or a letter from the school authorities. . Both these statements are false. as well as when they have both. (iv) x = 2 and x = 3 are the roots of the equation 3x2 – x – 10 = 0.4. (ii) Square of an integer is positive or negative. Thus. A word closely connected with “there exists” is “for every” (or for all). compare the following two sentences: 1. in order for a piece of mathematical writing to make sense. For every positive number x there exists a positive number y such that y < x. we have y < x. This means that there is atleast one rectangle whose all sides are equal. Thus. We should also observe that it is important to know precisely where in the sentence a given connecting word is introduced. There exists a positive number y such that for every positive number x.3 1. © no N C tt E o R be T re pu bl In general. EXERCISE 14. especially when we have to check the validity of different statements. we have seen that many mathematical statements contain some special words and it is important to know the meaning attached to them. For each of the following compound statements first identify the connecting words and then break it into component statements. . p is an irrational number. (i) All rational numbers are real and all real numbers are not complex. The words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers. This means that if S denotes the set of all prime numbers. As a matter of fact.334 MATHEMATICS p: For every prime number p. For example. a mathematical statement that says “for every” can be interpreted as saying that all the members of the given set S where the property applies must satisfy that property. Although these statements may look similar. Another phrase which appears in mathematical statements is “there exists”. they do not say the same thing. p: There exists a rectangle whose all sides are equal. 2. then for all the members p of p is an irrational number. all of the symbols must be carefully introduced and each symbol must be introduced precisely at the right place – not too early and not too late. “There exists” and “For all”. is the set S. he 14. (iii) The sand heats up quickly in the Sun and does not cool down fast at night. (1) is true and (2) is false. Consider a statement. consider the statement. For example.3 Quantifiers Quantifiers are phrases like. Then the sentence “if p then q” says that in the event if p is true. (i) x + y = y + x is true for every real numbers x and y. Let p and q denote the statements p : a number is a multiple of 9. For example. . you should have a ration card or a passport. For example. 3.5 Implications In this Section. he (iii) There exists a capital for every state in India. Check whether the following pair of statements are negation of each other. r: If you are born in some country. When we look at this statement. To put it in other words” not happening of p has no effect on happening of q. x is less than x + 1. (iii) All integers are positive or negative. (i) Sun rises or Moon sets. 4. then it is a multiple of 3. if you are not born in the country. To apply for a driving licence.MATHEMATICAL REASONING 335 2. we shall discuss the implications of “if-then”. State whether the “Or” used in the following statements is “exclusive “or” inclusive. then you cannot say anything about q. Give reasons for your answer. q: a number is a multiple of 3. q : you are citizen of that country. r: If a number is a multiple of 9. Another point to be noted for the statement “if p then q” is that the statement does not imply that p happens. then you are a citizen of that country. “only if” and “if and only if ”. The statements with “if-then” are very common in mathematics. bl is (ii) There exists real numbers x and y for which x + y = y + x. Give reasons for your answer. There are several ways of understanding “if p then q” statements. One of the most important facts about the sentence “if p then q” is that it does not say any thing (or places no demand) on q when p is false. (i) There exists a number which is equal to its square. We shall illustrate these ways in the context of the following statement. consider the statement. we observe that it corresponds to two statements p and q given by p : you are born in some country. then q must be true. Identify the quantifier in the following statements and write the negation of the statements. (ii) For every real number x. © no N C tt E o R be T re pu (ii) 14. This says that a number is a multiple of 9 only if it is a multiple of 3. it is not divisible by 9.5. p only if q. 3. If the physical environment changes. (iii) If a triangle is not isosceles. 4. then it is not equilateral. Then the contrapositive of this statement is If the biological environment does not change. let us consider more examples. Next. (ii) If you are born in India. The symbol ⇒ stands for implies. then it is divisible by 3. it is isosceles. 2. This says that knowing that a number as a multiple of 9 is sufficient to conclude that it is a multiple of 3. bl This says that if a number is not a multiple of 3. then the biological environment changes. then you were not born in India. then ∼p”. Example 9 Write the contrapositive of the following statement: (i) If a number is divisible by 9. we shall consider another term called converse. then q is “if ∼q. then p. then it is not a multiple of 9. Solution The contrapositive of the these statements are (i) If a number is not divisible by 3. 14. then you are a citizen of India. For example. This says that when a number is a multiple of 9. then q” is if q. then the physical environment does not change. 5. p is a sufficient condition for q. The above examples show the contrapositive of the statement if p. (ii) If you are not a citizen of India. To understand this. if p then q is the same as the following: is he 1.336 MATHEMATICS Then. Note that both these statements convey the same meaning. (iii) If a triangle is equilateral. let us consider the following “if-then” statement. This says that a number is a multiple of 9 implies that it is a multiple of 3. ∼q implies ∼p. p implies q is denoted by p ⇒ q.1 Contrapositive and converse Contrapositive and converse are certain © no N C tt E o R be T re pu other statements which can be formed from a given statement with “if-then”. q is a necessary condition for p. . it is necessarily a multiple of 3. The converse of a given statement “if p. q : Triangle ABC is Isosceles. first identify the corresponding component statements. Example 11 For each of the following compound statements. you get an A grade in the class. then a – b is always a positive integer. (i) p if and only if q (ii) q if and only if p . (ii) If you do all the exercises in the book. (i) If a number n is even. the converse of the statement p: If a number is divisible by 10. ‘If and only if’. (ii) The component statements are given by p : a and b are integers. (iii) If two integers a and b are such that a – b is always a positive integer. then you have done all the exercises of the book. since the product of two integers is an integer and therefore a rational number. Then check whether the statements are true or not.MATHEMATICAL REASONING 337 For example. Solution (i) The component statements are given by p : Triangle ABC is equilateral. then it is divisible by 10. then it is isosceles. © no N C tt E o R be T re pu bl is Solution The converse of these statements are (i) If a number n2 is even. (i) If a triangle ABC is equilateral. then n2 is even. the compound statement is true. Let us consider some more examples. then n is even. represented by the symbol ‘⇔‘ means the following equivalent forms for the given statements p and q. it is divisible by 5 is q: If a number is divisible by 5. q : ab is a rational number. then a > b. Since an equilateral triangle is isosceles. then ab is a rational number. we infer that the given compound statement is true. he Example 10 Write the converse of the following statements. (ii) If a and b are integers. (iii) If two integers a and b are such that a > b. (ii) If you get an A grade in the class. then all its four sides are equal. q: If a number is divisible by 3. Write each of the following statements in the form “if-then” (i) You get a job implies that your credentials are good. © no N C tt E o R be T re pu (ii) A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Write the contrapositive and converse of the following statements. . Rewrite the following statement with “if-then” in five different ways conveying the same meaning. then x is odd. (ii) p: If the sum of digits of a number is divisible by 3. then the sum of its digits is divisible by 3.338 MATHEMATICS (iii) p is necessary and sufficient condition for q and vice-versa (iv) p ⇔ q Consider an example. Combine these two statements using “if and only if ”. (i) If x is a prime number. then the number is divisible by 3. then they do not intersect in the same plane. (ii) The Bannana trees will bloom if it stays warm for a month. (v) x is an even number implies that x is divisible by 4. (i) p: If a rectangle is a square. If a natural number is odd. (iv) You cannot comprehend geometry if you do not know how to reason deductively. (iii) Something is cold implies that it has low temperature. then the rectangle is a square. then its square is also odd. 2. (iv) To get an A+ in the class. it is necessary that you do all the exercises of the book. 3. (ii) If the two lines are parallel.4 1. EXERCISE 14. (iii) A quadrilateral is a parallelogram if its diagonals bisect each other. bl Solution (i) A rectangle is a square if and only if all its four sides are equal. q: If all the four sides of a rectangle are equal. is he Example 12 Given below are two pairs of statements. (ii) If you have winter clothes. show that q must be true. then in order to show that the statement “p and q” is true. Identify the statements given below as contrapositive or converse of each other. Rule 3 Statements with “If-then” . and which of the quantifiers “for every”. “or”. is bl 14. Step-2 Show that the statement q is true. “if-then”. (b) If a quadrilateral is a parallelogram. (i) If the diagonals of a quadrilateral do not bisect each other. then you live in Delhi. one must answer all the following questions. then in order to show that the statement “p or q” is true.MATHEMATICAL REASONING 339 4. Rule 1 If p and q are mathematical statements. then you do not live in Delhi. one must consider the following. Given statements in (a) and (b). Here. we shall discuss some techniques to find when a statement is valid. and which of the implications “if and only”. appear in the given statement. we will discuss when a statement is true. then it is a parallelogram. show that p must be true. Rule 2 Statements with “Or” If p and q are mathematical statements . If you live in Delhi. “there exists”. (i) If you do not have winter clothes. Case 2 By assuming that q is false. To answer this question. What does the statement mean? What would it mean to say that this statement is true and when this statement is not true? The answer to these questions depend upon which of the special words and phrases “and”. the following steps are followed. then the quadrilateral is not a parallelogram. We shall list some general rules for checking whether a statement is true or not.6 Validating Statements he (a) © no N C tt E o R be T re pu In this Section. Case 1 By assuming that p is false. then its diagonals bisect each other. Step-1 Show that the statement p is true. (ii) If the diagonals of a quadrilateral bisect each other. then you have winter clothes. 340 MATHEMATICS In order to prove the statement “if p then q” we need to show that any one of the following case is true. Thus we have shown that ∼q ⇒ ∼p $Note The above example illustrates that to prove p ⇒ q. for some integer m. prove that p must be false. then xy is odd. it is enough to show ∼q ⇒ ∼p which is the contrapositive of the statement p ⇒ q. p is true means that x and y are odd integers. Case 1 By assuming that p is true. then q is true and (ii) If q is true. y ∈ Z are such that x and y are odd. That is assume that if p is true. Then x = 2m + 1. we apply Case 1 of Rule 3. Example 14 Check whether the following statement is true or false by proving its contrapositive. y ∈ Ζ such that xy is odd. for some integer n. we need to show. Thus xy = (2m + 1) (2n + 1) = 2(2mn + m + n) + 1 This shows that xy is odd. © no N C tt E o R be T re pu q : xy is odd To check the validity of the given statement. This gives the statement ∼q : Product xy is even. (i) If p is true. This shows that p is not true. Therefore. then we will proceed as follows. If x. then both x and y are odd. This is possible only if either x or y is even. Suppose we want to check this by using Case 2 of Rule 3. then q is true. then p is true Now we consider some examples. Solution Let p : x. We assume that q is not true. prove that q must be true. Solution Let us name the statements as below .(Direct method) he Case 2 By assuming that q is false. y ∈ Z such that x and y are odd bl Example 13 Check whether the following statement is true or not. the given statement is true. If x. This implies that we need to consider the negation of the statement q.(Contrapositive method) is Rule 4 Statements with “if and only if ” In order to prove the statement “p if and only if q”. y = 2n + 1. where a and b have no common factors. bl Now what happens when we combine an implication and its converse? Next.. We can express this fact as If a tumbler is half empty. We have to check whether the statement p ⇒ q is true or not. then it is half empty. we have shown that ∼q ⇒ ∼p and hence the given statement is true. ∼q ⇒ ∼p he Now ∼q : It is false that both x and y are odd. Thus. ∼p is true.e. then the second happens and also if the second happens. q : A tumbler is half full. by checking its contrapositive statement i. That is we assume that such that 7 is rational. This means that there exists positive integers a and b a 7 = . Therefore. we conclude that p is true. we assume that p is not true i. This shows that xy is even.MATHEMATICAL REASONING 341 p : xy is odd. This implies that x (or y) is even.1 By Contradiction Here to check whether a statement p is true. If a tumbler is half full. p: 7 is irrational Solution In this method. 14. xy = 2ny for some integer n. We combine these two statements and get the following: A tumbler is half empty if and only if it is half full. Then. we assume that the given statement is false. Let us consider the following statements. That is ∼p is true. then the first happens. Squaring the equation. we shall discuss this.e. q : both x and y are odd. we arrive at some result which contradicts our assumption. © no N C tt E o R be T re pu p : A tumbler is half empty. is Therefore. Example 15 Verify by the method of contradiction.6. We know that if the first statement happens. that is. we discuss another method. Now. Then x = 2n for some integer n. then it is half full. b . For this purpose we need to show that if p then ∼q. $ However. Then a2 = 49c2 and a2 = 7b2 Hence. counter examples are used to disprove the statement. This shows that the he we get 7 = is assumption 7 is rational is wrong. we conclude that the given statement is false. (iii) method of contrapositive 2. But we have already shown that 7 divides a. If n is an odd integer. So n = 9 is a counter example. EXERCISE 14. then n is prime. bl Example 16 By giving a counter example. Therefore. show that the following statements are not true.342 MATHEMATICS a2 ⇒ a2 = 7b2 ⇒ 7 divides a. Hence. 4. Note In mathematics. Show that the following statement is true by the method of contrapositive. © no N C tt E o R be T re pu Solution The given statement is in the form “if p then q” we have to show that this is false. 3. we shall discuss a method by which we may show that a statement is false. we have discussed some techniques for checking whether a statement is true or not. (i) p: If all the angles of a triangle are equal. then x is also even. show that the following statement is false. 9 is one such number. Show that the statement p: “If x is a real number such that x3 + 4x = 0. . Next. The name itself suggests that this is an example to counter the given statement. Thus. This implies that 7 is a common factor of both of a and b which contradicts our earlier assumption that a and b have no common factors. generating examples in favour of a statement do not provide validity of the statement. a2 = b2 implies that a = b” is not true by giving a counter-example.5 1. there exists an integer c such b2 that a = 7c. p: If x is an integer and x2 is even. By giving a counter example. In the above. Show that the statement “For any real numbers a and b. (ii) method of contradiction. (ii) q: The equation x2 – 1 = 0 does not have a root lying between 0 and 2. To show this we look for an odd integer n which is not a prime number. the statement 7 is irrational is true. then x is 0” is true by (i) direct method. 7b2 = 49c2 ⇒ b2 = 7c2 ⇒ 7 divides b. The method involves giving an example of a situation where the statement is not valid. Such an example is called a counter example. then the triangle is an obtuse angled triangle. Miscellaneous Examples bl is Example 17 Check whether “Or” used in the following compound statement is exclusive or inclusive? Write the component statements of the compound statements and use them to check whether the compound statement is true or not. (i) p: Each radius of a circle is a chord of the circle. (ii) Negation of q is “it is false that q”. x2 > x. 343 t : 11 is a rational number. The component statements of the given statement are p : you are wet when it rains. Solution (i) The negation of p is “It is false that p is” which means that the condition x2 > x does not hold for all real numbers. This statement can be rewritten as ∼q: For all real numbers x. (iv) s: If x and y are integers such that x > y. the compound statement is true. q : You are wet when you are in a river. (iii) r: All birds have wings.MATHEMATICAL REASONING Which of the following statements are true and which are false? In each case give a valid reason for saying so. t: you are wet when it rains or you are in a river. Here both the component statements are true and therefore. Thus ∼q is the statement. Example 18 Write the negation of the following statements: (i) p: For every real number x. x2 ≠ 2 (iii) The negation of the statement is ∼r: There exists a bird which have no wings. (iii) r: Circle is a particular case of an ellipse. (ii) q: There exists a rational number x such that x2 = 2. ∼q: There does not exist a rational number x such that x2 = 2. This can be expressed as ∼p: There exists a real number x such that x2 < x. © no N C tt E o R be T re pu Solution “Or” used in the given statement is inclusive because it is possible that it rains and you are in the river. then –x < – y. Justify your answer. (v) he 5. (iv) s: All students study mathematics at the elementary level. . (ii) q: The centre of a circle bisects each chord of the circle. Case 1 If p. we have to check whether “if p then q” and “if q then p” is true. We check this by contrapositive method. That is driving over 80 km per hour is sufficient to get a fine. Also check whether the statement is true. then q also indicates that q is necessary for p. he Solution The necessary and sufficient condition that the integer n be odd is n2 must be odd. then n is odd. Therefore. Solution Let p and q denote the statements: p : you drive over 80 km per hour. Then n = 2k + 1 when k is an integer. We have to check whether this statement is true. The implication if p. Case 2 If q. n2 is one more than an even number and hence is odd. Let us assume that n is odd. q : you will get a fine. Here the sufficient condition is “driving over 80 km per hour”: Similarly. then p If q. then q If p. then q indicates that p is sufficient for q. if p. then you will get a fine. Example 19 Using the words “necessary and sufficient” rewrite the statement “The integer n is odd if and only if n2 is odd”. Thus n2 = (2k + 1)2 = 4k2 + 4k + 1 Therefore. t: If you drive over 80 km per hour. Example 20 For the given statements identify the necessary and sufficient conditions. then n2 is odd. To check the validity of “p if and only if q”.344 MATHEMATICS (iv) The negation of the given statement is ∼s: There exists a student who does not study mathematics at the elementary level. That is . We have to check whether this statement is true. The contrapositive of the given statement is: If n is an even integer. © no N C tt E o R be T re pu bl is q : n2 is odd. then n2 is an even integer n is even implies that n = 2k for some k. n2 is even. Let p and q denote the statements p : the integer n is odd. then q is the statement: If the integer n is odd. Then n2 = 4k2. then p is the statement If n is an integer and n2 is odd. . 2. then it is an obtuse angled triangle. (iii) r: If it is hot outside. then your mind is free and if your mind is free. (ii) q: There is traffic jam whenever it rains. then you feel thirsty. 3. then it is a rectangle and if a quadrilateral is a rectangle. then you watch television. Given below are two statements p : 25 is a multiple of 5. you will necessarily get a fine. In both cases check the validity of the compound statement. conveying the same meaning.MATHEMATICAL REASONING 345 When you drive over 80 km per hour. (iv) s: There exists a number x such that 0 < x < 1. (ii) q: I go to a beach whenever it is a sunny day. (iii) r: You can access the website only if you pay a subsciption fee. then q” (i) p: It is necessary to have a password to log on to the server. 7. the number x – 1 is also positive. then it is equiangular. Write each of the statements in the form “if p. p: If a triangle is equiangular. © no N C tt E o R be T re pu bl is he 1. Write the following statement in five different ways. it is necessary and sufficient that you do all the homework regularly. Rewrite each of the following statements in the form “p if and only if q” (i) p: If you watch television. Miscellaneous Exercise on Chapter 14 Write the negation of the following statements: (i) p: For every positive real number x. (i) p: The sum of an irrational number and a rational number is irrational (by contradiction method). 6. then n2 > 9 (by contradiction method). (ii) q: If n is a real number with n > 3. Check the validity of the statements given below by the method given against it. q : 25 is a multiple of 8. (ii) q: All cats scratch. (iii) r: For every real number x. 4. (iii) r: If a quadrilateral is equiangular. Write the compound statements connecting these two statements with “And” and “Or”. either x > 1 or x < 1. 5. Here the necessary condition is “getting a fine”. State the converse and contrapositive of each of the following statements: (i) p: A positive integer is prime only if it has no divisors other than 1 and itself. (ii) q: For you to get an A grade. p ⇒ q together with its converse. The smaller statements are called component statements of the compound statement. “Or”. A sentence with if p. .). – Compound statements and their related component statements: A statement is a compound statement if it is made up of two or more smaller statements. – The role of “And”. The following methods are used to check the validity of statements: (i) direct method (ii) contrapositive method (iii) method of contradiction (iv) using a counter example. then the negation of p is denoted by ∼p. “There exists” and “For every” in compound statements. Explained the terms: © no N C tt E o R be T re pu bl is he – Negation of a statement p: If p denote a statement. in the seventeenth century. who founded the modern subject of symbolic logic. gives p if and only if q. The converse of a statement p ⇒ q is the statement q ⇒ p. Historical Note The first treatise on logic was written by Aristotle (384 B.C. W.-322 B. – The meaning of implications “If ”. His idea was realised in the nineteenth century by the English mathematician George Boole (1815–1864) and Augustus De Morgan (1806–1871) . Leibnitz (1646 – 1716) conceived the idea of using symbols in logic to mechanise the process of deductive reasoning. German mathematician G.C. then q can be written in the following ways. “only if ”. “ if and only if ”. Later. – p implies q (denoted by p ⇒ q) – p is a sufficient condition for q – q is a necessary condition for p – p only if q – ∼q implies ∼p – The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p . It was a collection of rules for deductive reasoning which would serve as a basis for the study of every branch of knowledge.346 MATHEMATICS Summary A mathematically acceptable statement is a sentence which is either true or false. 80. i. 0. 53.Chapter 15 he STATISTICS 15. We can make decisions about the data by analysing and interpreting it. 42. 5. 52 Clearly. 50. Consider now the runs scored by two batsmen in their last ten matches as follows: Batsman A : 30. We have also studied the methods of finding a representative value for the given data.L. we have studied methods of representing data graphically and in tabular form. 53. 58. . 64. 117. the mean and median of the data are Batsman A Batsman B Mean 53 53 Median 53 53 Recall that. 46. 30.BOWLEY & A. 57. in order to make better interpretation from the data. Recall mean (arithmetic mean). median and mode are three measures of central tendency. 60.. But. 48. BODDINGTON © no N C tt E o R be T re pu We know that statistics deals with data collected for specific purposes. 91. we should also have an idea how the data are scattered or how much they are bunched around a measure of central tendency. This value is called the measure of central tendency. we calculate the mean of a data (denoted by x ) by dividing the sum of the observations by the number of observations. 71 Batsman B : 53.L.1 Introduction bl is “Statistics may be rightly called the science of averages and their estimates. A measure of central Karl Pearson tendency gives us a rough idea where data points are (1857-1936) centred. In earlier classes.” – A.e. This representation reveals certain salient features or characteristics of the data. We find the following diagrams: For batsman A For batsman B Fig 15.e. because the variability in the scores of batsman A is from 0 (minimum) to 117 (maximum).348 MATHEMATICS x= 1 n n ∑ xi i =1 Also. 53. Like ‘measures of central tendency’ we want to have a single number to describe variability. Thus. then the median is ⎜ ⎟ ⎝ 2 ⎠ observation. Can we say that the performance of two players is same? Clearly No. the measures of central tendency are not sufficient to give complete information about a given data.. In this Chapter. Whereas. we shall learn some of the important measures of dispersion and their methods of calculation for ungrouped and grouped data. . ⎝2 ⎠ We find that the mean and median of the runs scored by both the batsmen A and B are same i. This single number is called a ‘measure of dispersion’.1 Fig 15. while those corresponding to batsman A are scattered or more spread out. the range of the runs scored by batsman B is from 46 to 60. th he ⎛ n +1⎞ If the number of observations is odd. Variability is another factor which is required to be studied under statistics. the median is obtained by first arranging the data in ascending or descending order and applying the following rule. th and is ⎛n⎞ If the number of observations is even. then median is the mean of ⎜ ⎟ ⎝2⎠ th © no N C tt E o R be T re pu bl ⎛n ⎞ ⎜ + 1⎟ observations. Let us now plot the above scores as dots on a number line.2 We can see that the dots corresponding to batsman B are close to each other and are clustering around the measure of central tendency (mean and median). Range = 60 – 46 = 14. (ii) Quartile deviation. The important measures of dispersion. we must obtain the sum of the deviations. in the example of runs scored by two batsmen A and B. In case of batsman A. But.3 Range © no N C tt E o R be T re pu bl is Recall that. we find the deviations about a. Therefore. finding the mean of deviations about mean is not of any use for us. In this Chapter. An absolute measure of dispersion is the mean of these deviations. Thus. the scores are scattered or dispersed in case of A while for B these are close to each other. . Range of A > Range of B. 15.4 Mean Deviation Recall that the deviation of an observation x from a fixed value ‘a’ is the difference x – a. This difference is called the ‘Range’ of the data.STATISTICS 349 15. some of the deviations will be negative and some positive. we know that a measure of central tendency lies between the maximum and the minimum values of the set of observations. To obtain a single number for this. Range of a series = Maximum value – Minimum value. To find the mean. Thus. we find the difference of maximum and minimum values of each series. such measure must depend upon the difference (or deviation) of the values from the central tendency. we need some other measure of variability. Clearly. In order to find the dispersion of values of x from a central value ‘a’ . Clearly. Range = 117 – 0 = 117 and for batsman B. Also Sum of deviations 0 Mean of deviations = Number of observations = n = 0 Thus. The range of data gives us a rough idea of variability or scatter but does not tell about the dispersion of the data from a measure of central tendency. used there. (iii) Mean deviation. we had some idea of variability in the scores on the basis of minimum and maximum runs in each series. Let us discuss them in detail.2 Measures of Dispersion he The dispersion or scatter in a data is measured on the basis of the observations and the types of the measure of central tendency. Therefore. we shall study all of these measures of dispersion except the quartile deviation. as far as the measure of dispersion is concerned. (iv) Standard deviation. which depend upon the deviations of the observations from a central tendency are mean deviation and standard deviation. For this purpose. Moreover. 15. There are following measures of dispersion: (i) Range. the sum of the deviations from mean ( x ) is zero. the sum of deviations may vanish. xn − a Step 4 Find the mean of the absolute values of the deviations.. mean deviation from mean and median are commonly used in statistical studies. However. This mean is called the ‘mean deviation’. Step 2 Find the deviation of each xi from a... © no N C tt E o R be T re pu bl Remark Mean deviation may be obtained from any measure of central tendency. .. where x = Mean ∑ xi − M . i. This mean is the mean deviation about a.D. i.1 Mean deviation for ungrouped data Let n observations be x1.e. i..350 MATHEMATICS Sum of absolute values of deviations from 'a' .. x2 – a. Let it be ‘a’.e.. Number of observations is M. . if it is there.4. .. xn. in finding a suitable measure of dispersion. Thus. (a). to find the measure of dispersion from a fixed number ‘a’ we may take the mean of the absolute values of the deviations from the central value. drop the minus sign (–). we require the distance of each value from a central tendency or a fixed number ‘a’. where M = Median i =1 . x3 – a. The mean deviation from ‘a’ is denoted as M. x2.. that the absolute value of the difference of two numbers gives the distance between the numbers when represented on a number line. . i. x3. xn– a Step 3 Find the absolute values of the deviations. The following steps are involved in the calculation of mean deviation about mean or median: Step 1 Calculate the measure of central tendency about which we are to find the mean deviation.D.D.D.. Recall.e. Therefore. n M..(a) = he Remember that.D..(a ) = Thus and ∑ xi − a i =1 n M. .. x2 − a . ( x ) = M. x3 − a . x1 − a . Let us now learn how to calculate mean deviation about mean and mean deviation about median for various types of data 15. x1 – a. (M) = 1 n 1 n n ∑ xi − x i =1 n . Thus mean deviation about a central value ‘a’ is the mean of the absolute values of the deviations of the observations from ‘a’.e. 6. xi– x are 6 – 9.e. 10. 2.e. 4. 17. i... 5 Solution We have to first find the mean ( x ) of the given data x = 1 20 200 xi = ∑ = 10 20 i =1 20 The respective absolute values of the deviations from mean.Let us now illustrate the steps of the above method in following examples. 5.STATISTICS 351 Note In this Chapter. 5. we can carry on calculation. 10. xi − x are 2.e. i. 7. 2. 9. 3 Step 4 The required mean deviation about the mean is 8 M. 3. 4 – 9. –1. 1. 3. 5 . 12 – 9. $ step-wise without referring to steps. 8. 10. 17.75 8 8 Note Instead of carrying out the steps every time. 7. 1. 13. 12. 3. 19. 1. 8. i. 10 – 9. 3. 12 – 9. 4. 16. 8 – 9. 1. 9. 7. 13 – 9. 3 © no N C tt E o R be T re pu Step 3 The absolute values of the deviations. –2. 18. or –3. 7 – 9. –5. ( x ) = = ∑ xi − x i =1 8 3 + 2 + 1 + 3 + 4 + 5 + 1 + 3 22 = = 2. 6. 4. 7. 7. we shall use the symbol M to denote median unless stated $ otherwise. 20. 8.. 0. 4. Example 2 Find the mean deviation about the mean for the following data : 12. 15. 1. 1. 9. 12 bl Step 2 The deviations of the respective observations from the mean x . 2. 17. 7. 11.D. 3. Solution We proceed step-wise and get the following: Step 1 Mean of the given data is 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 72 = = 9 8 8 is x= he Example 1 Find the mean deviation about the mean for the following data: 6. xi − x are 3. 8. 7. 4.e. xn occurring with frequencies f1. x2. Let us discuss the method of finding mean deviation for both types of the data.. .. 10.4.. 10. 12.. 18. 9. This data can be represented in the tabular form as given below. 9. 0. and is called discrete frequency distribution: x : x1 x2 x3 . 6.352 MATHEMATICS 20 Therefore ∑ xi − x and M. ( x ) = = 124 i =1 he 124 = 6. 3. Arranging the data into ascending order. 2.D. xi − M are © no N C tt E o R be T re pu 6.. 1.2 20 Example 3 Find the mean deviation about the median for the following data: 3. 5. 21. 7.. 3.. 10.. 18. f2 . 12.. . 3. 21 The absolute values of the respective deviations from the median. fn (i) Mean deviation about mean First of all we find the mean x of the given data by using the formula . ( M ) = 1 11 1 xi − M = × 58 = 5. 9. 12 11 ∑ xi − M = 58 Therefore and i =1 M.2 Mean deviation for grouped data We know that data can be grouped into two ways : (a) Discrete frequency distribution.27 ∑ 11 i =1 11 15. (b) Continuous frequency distribution. fn respectively. 5. 7. 4. 19. 19. we have 3. 5.. Now th or 6th observation = 9 bl ⎛ 11 + 1 ⎞ Median = ⎜ ⎟ 2 ⎠ ⎝ is Solution Here the number of observations is 11 which is odd.. (a) Discrete frequency distribution Let the given data consist of n distinct values x1.D. 4. i. xn f : f1 f2 f3 . .D. ( x ) = ∑f i =1 i xi − x n ∑f i =1 1 n = ∑ f i xi − x N i =1 i (ii) Mean deviation about median To find mean deviation about median. After finding median. we find the median of the given discrete frequency distribution.. Thus n M. we obtain the mean of the absolute values of the deviations from median. .STATISTICS 353 n x= ∑ xi fi i =1 n ∑ fi = 1 N n ∑ xi fi . i.(M) = 1 N n ∑f i =1 i xi − M Example 4 Find mean deviation about the mean for the following data : xi 2 5 6 8 10 12 fi 2 8 10 7 8 5 Solution Let us make a Table 15. is where bl Then. the observation whose cumulative frequency is equal to or just greater than M. we find the deviations of observations xi from the mean x and take their absolute values.Thus. we identify N .. it is the required median. where 2 N is the sum of frequencies. © no N C tt E o R be T re pu After this.e. For this the observations are arranged in ascending order. After this the cumulative frequencies are obtained. n.. therefore. which is the required mean deviation about the mean. i =1 he i =1 n ∑x f i =1 i i denotes the sum of the products of observations xi with their respective frequencies fi and N = n ∑f i =1 i is the sum of the frequencies. xi − x for all i =1.D.. find the mean of the absolute values of the deviations. This value of the observation lies in the middle of the data. Then. 2.1 of the given data and append other columns after calculations. i =1 xi − x = 92 i © no N C tt E o R be T re pu i =1 22.2 xi 3 6 9 12 13 15 21 22 fi 3 4 5 2 4 5 4 3 c.2). we get (Table 15. Table 15. .f. 3 7 12 14 18 23 27 30 Now.5 15 8 7 56 0.5 20 6 10 60 1.5 40 300 6 ∑ i =1 92 6 ∑f f i xi = 300 .5 11 5 8 40 2. N=30 which is even. ( x ) = i =1 fi xi = 1 N 1 × 300 = 7.5 bl 6 N = ∑ f i = 40 .5 10 8 80 2. D.1 fi f ix i xi − x f i xi − x 2 4 5.3 40 xi − x = Example 5 Find the mean deviation about the median for the following data: xi 3 6 9 12 13 15 21 22 fi 3 4 5 2 4 5 4 3 Solution The given observations are already in ascending order.5 40 6 ∑ fi i =1 1 × 92 = 2.5 20 12 5 60 4.5 3.354 MATHEMATICS Table 15. Adding a row corresponding to cumulative frequencies to the given data. is 2 he xi 1 N 6 ∑ Therefore x= and M. D. absolute values of the deviations from median. Here also.3 f i xi − M 8 7 4 1 0 2 3 4 5 2 4 5 4 3 30 28 20 2 0 10 32 27 We have ∑f Therefore M. Median M = 15th observation + 16 th observation 2 13 + 13 = 2 = 13 Table 15. For example. . for which the corresponding observation is 13. i. Table 15. Both of these observations lie in the cumulative frequency 18.97. Therefore. 30 (b) Continuous frequency distribution A continuous frequency distribution is a series in which the data are classified into different class-intervals without gaps alongwith their respective frequencies. (M) = 8 ∑f i =1 i i =1 = 1 N 8 ∑ fi i =1 i 9 xi − M = 149 © no N C tt E o R be T re pu = 30 and 8 is fi 10 bl xi − M he Now..STATISTICS 355 Median is the mean of the 15th and 16th observations. Let us take the following example.3. we had made the assumption that the frequency in each class is centred at its mid-point. marks obtained by 100 students are presented in a continuous frequency distribution as follows : Marks obtained 0-10 10-20 20-30 30-40 Number of Students 12 18 27 20 40-50 17 50-60 6 (i) Mean deviation about mean While calculating the mean of a continuous frequency distribution. we write the mid-point of each given class and proceed further as for a discrete frequency distribution to find the mean deviation. xi − M are shown in xi − M 1 × 149 = 4.e. N = i =1 7 ∑ fi i =1 400 xi − x = 400 1 7 1800 fi xi = = 45 ∑ N i =1 40 1 7 1 fi xi − x = × 400 = 10 ∑ N i =1 40 Shortcut method for calculating mean deviation about mean We can avoid the tedious calculations of computing x by following step-deviation method.D. ∑ fi xi = 1800. Recall that in this method. Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Number of students 2 3 8 14 8 3 2 Table 15. Then deviations of the observations (or mid-points of classes) are taken from the .356 MATHEMATICS Example 6 Find the mean deviation about the mean for the following data. we take an assumed mean which is in the middle or just close to it in the data.4 from the given data : 2 15 30 60 20-30 3 25 75 20 60 30-40 8 35 280 10 80 © no N C tt E o R be T re pu bl 10-20 40-50 14 45 630 0 0 50-60 8 55 440 10 80 60-70 3 65 195 20 60 70-80 2 75 150 30 60 40 7 ∑ 1800 Here Therefore x= and M. ( x ) = i =1 7 fi = 40.4 Number of students Mid-points fi xi f ix i xi − x 30 fi xi − x is Marks obtained he Solution We make the following Table 15. Let.3 © no N C tt E o R be T re pu bl If there is a common factor of all the deviations.. as shown in Fig 15. become simpler. Then.4 Fig 15. we divide them by this common factor to further simplify the deviations. etc. This is nothing but the shifting of origin from zero to the assumed mean on the number line. the h mean x by step-deviation method is given by n ∑ fi di i x = a + =1 ×h N Let us take the data of Example 6 and find the mean deviation by using stepdeviation method.4 The deviations and step-deviations reduce the size of the observations. the new variable be denoted by d i = xi − a .STATISTICS 357 he assumed mean. These are known as step-deviations. multiplication. The process of taking step-deviations is the change of scale on the number line as shown in Fig 15. so that the computations viz. . where ‘a’ is the assumed mean and h is the common factor.3 is Fig 15. the median of continuous frequency distribution is obtained by first identifying the class in which median lies (median class) and then applying the formula . (x ) = 1 N 0 × 10 = 45 40 7 ∑ fi i =1 xi − x = 400 = 10 40 Note The step deviation method is applied to compute x . The data is first arranged in ascending order.358 MATHEMATICS Take the assumed mean a = 45 and h = 10. Let us recall the process of finding median for a continuous frequency distribution.D. (ii) Mean deviation about median The process of finding the mean deviation about median for a continuous frequency distribution is similar as we did for mean deviation about the mean.5 Number of Mid-points di = students xi − 45 10 fi di xi − x xi 10-20 2 15 –3 –6 30 20-30 3 25 –2 –6 20 30-40 8 35 –1 –8 10 40-50 14 45 0 0 50-60 8 55 1 8 60-70 3 65 2 70-80 2 75 3 Therefore 80 10 80 6 20 60 6 30 60 bl 0 0 400 7 ∑ fi di x = a + i =1 ×h N = 45 + and 60 0 © no N C tt E o R be T re pu 40 60 is fi f i xi − x he Marks obtained M .5. and form the following Table 15. Rest of the procedure $ is same. Table 15. The only difference lies in the replacement of the mean by median while taking deviations. Then. .e. the width of the median class and C the cumulative frequency of the class just preceding the median class. 5 23 138 13 15 13 91 15 28 25 3 45 30-40 16 44 35 7 112 40-50 4 48 45 17 68 50-60 2 50 55 27 54 Class Frequency Mid-points fi Cumulative frequency (c.) 0-10 6 6 10-20 7 20-30 50 xi 508 .D.6 xi − Med. l.f. After finding the median.STATISTICS 359 N −C 2 Median = l + ×h f where median class is the class interval whose cumulative frequency is just greater N . N is the sum of frequencies. f i xi − Med. respectively the lower 2 limit . (M) = bl 1 n ∑ fi xi − M N i =1 The process is illustrated in the following example: Then is of the deviations of mid-point xi of each class from the median i. © no N C tt E o R be T re pu Example 7 Calculate the mean deviation about median for the following data : Class 0-10 Frequency 6 10-20 20-30 30-40 40-50 50-60 7 15 16 4 2 Solution Form the following Table 15. f. h and C are. the absolute values he than or equal to M. xi − M are obtained. the frequency.6 from the given data : Table 15. 49 Find the mean deviation about the mean for the data in Exercises 5 and 6. 63.16 50 © no N C tt E o R be T re pu M.360 MATHEMATICS th The class interval containing N or 25th item is 20-30. C = 13. 5. 3. We know that he N −C ×h Median = l + 2 f 25 − 13 × 10 = 20 + 8 = 28 15 Thus. 38.1 Find the mean deviation about the mean for the data in Exercises 1 and 2. 9. f = 15. 4. 51. 12. 18. x i fi 7 4 6 3 5 10 30 50 70 90 4 24 28 16 8 Find the mean deviation about the median for the data in Exercises 7 and 8. is Here l = 20. 46. 42. 12. 16. 45. 20–30 is the median 2 class. Therefore. 1. x i 5 7 9 10 12 15 fi 8 6 2 2 2 6 15 21 27 30 35 3 5 6 7 8 8. 55. 14. h = 10 and N = 50 EXERCISE 15. 16. 11. 60. 13. 42. 44 Find the mean deviation about the median for the data in Exercises 3 and 4. x i fi . 48. 17. x i 5 10 15 20 25 fi 6.D. 10. 8. 13. 70. 13. 46. 7. 7. 17 2. 36. (M) = bl Therefore. 17 4. 40. 53. 72. 54. 46. 11. Mean deviation about median is given by Median = 20 + 1 N 6 ∑ fi i =1 xi − M = 1 × 508 = 10. 10. 3 Limitations of mean deviation In a series. is to take squares of all the deviations. Calculate the mean deviation about median age for the age distribution of 100 persons given below: Age 16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55 Number 5 6 12 14 26 12 16 9 [Hint Convert the given data into continuous frequency distribution by subtracting 0. in many cases. Also mean deviation is calculated on the basis of absolute values of the deviations and therefore. Another way to overcome this difficulty which arose due to the signs of deviations. the mean deviation about median calculated for such series can not be fully relied. the absolute values of the deviations were taken. mean deviation may give unsatisfactory results.5 Variance and Standard Deviation Recall that while calculating mean deviation about mean or median.4. 9. This implies that we must have some other measure of dispersion. 15. where the degree of variability is very high. cannot be subjected to further algebraic treatment. Find the mean deviation about median for the following data : Marks 0-10 10-20 20-30 30-40 40-50 50-60 Number of 6 8 14 16 4 2 Girls 12. Standard deviation is such a measure of dispersion.STATISTICS 361 © no N C tt E o R be T re pu bl is he Find the mean deviation about the mean for the data in Exercises 9 and 10.5 to the upper limit of each class interval] 15.5 from the lower limit and adding 0. Obviously all these squares of deviations are . the mean deviation about the mean is not very scientific. Height 95-105 105-115 115-125 125-135 135-145 145-155 in cms Number of 9 13 26 30 12 10 boys 11.Thus. Thus. Therefore. The sum of the deviations from the mean (minus signs ignored) is more than the sum of the deviations from median. otherwise the deviations may cancel among themselves. The absolute values were taken to give meaning to the mean deviation. Income 0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 per day Number 4 8 9 10 7 5 4 3 of persons 10. the median is not a representative central tendency. . there is a lower degree of dispersion. 24. 30. 32.. Then (x1 x)2 (x2 x)2 . if this sum is large. Here n = 15) 6 .. This implies that there is no n If ∑ (x − x) 2 i i =1 he dispersion at all as all observations are equal to the mean x . 16. 37.+ (44–30)2 +(45–30)2 = (–15)2 +(–14)2 + ... Let x1.. x2.. 35.+ 142 + 152 = 2 [152 + 142 + . 36. . 45. 42. 35.+ (–1)2 + 02 + 12 + 22 + 32 + . n (xn x)2 (xi x)2 . 25. xn be n observations and x be their mean..... 25. 22. x2. 21. 20.. then each ( xi − x ) has to be zero. is small . x3. 31. this indicates that the observations x1. The sum of squares of deviations from x for this set is 6 ∑ (x − x) i =1 i 2 = (5–30)2 + (15–30)2 + (25–30)2 + (35–30)2 + (45–30)2 +(55–30)2 = 625 + 225 + 25 + 25 + 225 + 625 = 1750 Let us now take another set B of 31 observations 15. 45. 44. the sum of squares of deviations of observations for set B from the mean y is given by 31 ∑(y i =1 i − y ) 2 = (15–30)2 +(16–30)2 + (17–30)2 + . 26....... On the contrary. 55.. i 1 If this sum is zero.. 39. 18. The mean of the observations is x = 30. 23. x3. Can we thus say that the sum is close to the mean x and therefore. 33. 19. 41.xn are n ∑ (x − x) i =1 bl from the mean x .. 40. 17. Now. 27. 29. 34. 15. 43. there is a higher degree of dispersion of the observations i 2 is a reasonable indicator © no N C tt E o R be T re pu of the degree of dispersion or scatter? Let us take the set A of six observations 5. 38.362 MATHEMATICS non-negative. 28. The mean of these observations is y = 30 Note that both the sets A and B of observations have a mean of 30. + 12] = 2× 15 × (15 + 1) (30 + 1) = 5 × 16 × 31 = 2480 6 (Because sum of squares of first n natural numbers = n (n + 1) (2n + 1) . To overcome this difficulty we take the mean of the squares of 1 n the deviations. we take ∑ ( xi − x ) 2 . we have bl is For the set A. it is × 2480 = 80.e. even though the observations in set A are more scattered from the mean (the range of deviations being from –25 to 25) than in the set B (where the range of deviations is from –15 to 15). Fig 15.STATISTICS n If ∑ (x − x) i =1 i 2 363 is simply our measure of dispersion or scatter about mean.. we can take 1 ( xi − x ) 2 as a quantity which leads to a proper measure ∑ n of dispersion. i. Therefore. In case of the set A.e. x2.6 Thus.67 and in case of the set B. Mean = Thus. we he will tend to say that the set A of six observations has a lesser dispersion about the mean than the set B of 31 observations. we can say that the sum of squares of deviations from the mean is not a proper measure of dispersion. 6 31 This indicates that the scatter or dispersion is more in set A than the scatter or dispersion in set B.. i.5 © no N C tt E o R be T re pu For the set B. the variance of n observations x1.. This is also clear from the following diagrams.. we have n i =1 1 1 × 1750 = 291.. mean of the squares of the deviations from mean is called the variance and is denoted by σ 2 (read as sigma square). xn is given by . This number. we have Fig 15.. which confirms with the geometrical representation of the two sets. 16. Therefore. standard deviation of ungrouped data.1 Standard Deviation In the calculation of variance. 18. 10. The number of observations is n = 10 Table 15. © no N C tt E o R be T re pu Example 8 Find the Variance of the following data: 6..7. we find that the units of individual observations xi and the unit of their mean x are different from that of variance. 20. (1) bl Let us take the following example to illustrate the calculation of variance and hence. 22. 24 Solution From the given data we can form the following Table 15.5. the standard deviation.7 xi di = xi − 14 2 Deviations from mean (x i– x ) (x i– x ) 6 –4 –9 81 8 –3 –7 49 10 –2 –5 25 12 –1 –3 9 14 0 –1 1 16 1 1 1 18 2 3 9 20 3 5 25 22 4 7 49 24 5 5 9 81 330 . 14. The mean is calculated by step-deviation method taking 14 as assumed mean. is given by is 1 n ( xi − x ) 2 ∑ n i =1 σ= .364 MATHEMATICS σ2= 1 n ( xi − x ) 2 ∑ n i =1 he 15. For this reason. since variance involves the sum of squares of (xi– x ). usually denoted by σ .. 12. the proper measure of dispersion about the mean of a set of observations is expressed as positive square-root of the variance and is called standard deviation. 8. 5.. .. Example 9 Find the variance and standard deviation for the following data: xi 4 8 11 17 20 24 32 fi 3 5 9 5 4 3 1 Solution Presenting the data in tabular form (Table 15.8 xi fi fi xi xi – x ( xi − x ) 2 4 8 11 17 20 24 32 3 5 9 5 4 3 1 30 12 40 99 85 80 72 32 420 –10 –6 –3 3 6 10 18 100 36 9 9 36 100 324 f i ( xi − x ) 2 300 180 81 45 144 300 324 1374 . x3 .. .STATISTICS 365 n ∑d Mean x = assumed mean + 1 Variance ( σ ) = n 2 and n i ×h = 14 + 5 × 2 = 15 10 10 ∑ ( xi − x )2 = 101 × 330 = 33 he Therefore i =1 i =1 f1.8).2 Standard deviation of a discrete frequency distribution Let the given discrete frequency distribution be x: x1.. we get Table 15. . . . xn 1 n ∑ fi (xi − x )2 N i =1 . x2. (2) © no N C tt E o R be T re pu In this case standard deviation (σ ) = n where N = ∑ fi . f3 .74 15. i =1 Let us take up following example. fn bl f: is Thus Standard deviation ( σ ) = 33 = 5. . f2. 7 7 i =1 i =1 ∑ fi xi = 420. ∑ fi ( xi − x ) 2 = 1374 7 1 × 420 = 14 30 1 variance (σ ) = N 2 = 7 ∑ fi (xi − he N = x )2 i =1 is Hence i =1 x= 1 × 1374 = 45. i =1 Another formula for standard deviation We know that 1 Variance (σ ) = N 2 n ∑ i =1 f i ( xi − x ) 2 = 1 ⎡ = N ⎢ ⎣ ∑ fi xi 2 + 1 ⎡ = N ⎢ ⎣ ∑ fi xi 2 + n i =1 n i =1 n 1 N n ∑ fi ( xi 2 + i =1 ∑ x 2 fi − i =1 n ∑ 2x i =1 x 2 − 2 x xi ) ⎤ fi xi ⎥ ⎦ n n ⎤ x 2 ∑ f i − 2 x ∑ xi fi ⎥ i =1 i =1 ⎦ .366 MATHEMATICS N = 30.8 = 6. the standard deviation will be obtained by the formula σ = 1 N n ∑ fi ( xi − i =1 x)2 .77 and © no N C tt E o R be T re pu 15. Then. the standard deviation is calculated by the technique adopted in the case of a discrete frequency distribution. If there is a frequency distribution of n classes each class defined by its mid-point xi with frequency fi. n where x is the mean of the distribution and N = ∑ fi .8 30 bl Therefore ∑ fi xi Standard deviation (σ ) = 45.3 Standard deviation of a continuous frequency distribution The given continuous frequency distribution can be represented as a discrete frequency distribution by replacing each class by its mid-point.5. STATISTICS N n ∑ i =1 2 f i xi + x 2 − 2 x 2 = ⎛ n ⎜ ∑ fi xi 1 n 2 2 = ∑ fi xi − ⎜⎜ i =1N σ N i −1 ⎜ ⎝ Thus. Table 15... N ∑ ∑ i i ⎢∑ i i ⎥ ⎢ N i =1 N ⎣ i =1 i =1 ⎦ ⎦ ⎣ 2 bl = 367 2 .9. (3) © no N C tt E o R be T re pu Example 10 Calculate the mean. we construct the following Table 15. variance and standard deviation for the following distribution : Class Frequency 30-40 40-50 50-60 60-70 70-80 80-90 90-100 3 7 12 15 8 3 2 Solution From the given data.9 Class Frequency Mid-point (f i ) (x i) f ix i (xi– x ) 2 f i(x i– x ) 2 30-40 3 35 105 729 2187 40-50 7 45 315 289 2023 50-60 12 55 660 49 588 60-70 15 65 975 9 135 70-80 8 75 600 169 1352 80-90 3 85 255 529 1587 90-100 2 95 190 1089 2178 50 3100 10050 . standard deviation (σ ) = 1 N 1 N n ∑ fi xi 2 − x 2 i =1 he or 1 2 ⎞ 2 ⎟ n ⎡ n ⎤ ⎟ = 1 ⎢ N ∑ fi xi 2 − ⎜⎛ ∑ f i xi ⎟⎞ ⎥ ⎟ N 2 ⎢ i =1 ⎝ i =1 ⎠ ⎥⎦ ⎣ ⎟ ⎠ ⎛ n ⎞ N ∑ fi xi − ⎜ ∑ f i xi ⎟ ⎝ i =1 ⎠ i =1 n is = n ⎤ 1 n ⎤ ⎡ 1⎡ n 2 Here or x f x xi fi = Nx ⎥ = f x x x x + − N 2 . we have σ = 1 N N ∑ fi xi − ( ∑ f i xi ) = 1 48 48 × 9652 − (614)2 = 1 463296 − 376996 48 2 2 .10: Table 15.10 xi fi f ix i x i2 f ix i2 3 7 21 9 63 8 10 80 64 640 13 15 195 169 2535 18 10 180 324 3240 23 6 138 529 3174 48 614 9652 Now.18 is 368 Example 11 Find the standard deviation for the following data : 3 8 13 18 23 fi 7 10 15 10 6 bl xi © no N C tt E o R be T re pu Solution Let us form the following Table 15.Mean x = 1 N 7 ∑fx i =1 7 1 2 Variance σ = N ( ) = and 3100 = 62 50 = i i ∑ fi (xi − x )2 i =1 he Thus MATHEMATICS 1 × 10050 = 201 50 Standard deviation (σ ) = 201 = 14. by formula (3). n ∑fx i x = We know that i i =1 N . (1) © no N C tt E o R be T re pu i. bl width of class-intervals).77 = 6.12 48 Therefore.e..STATISTICS 369 1 × 293.5. yi = xi − A or xi = A + hyi h . σ x = ∑ fi ( xi − x ) N i =1 1 = N n ∑ fi (A + hyi − A − h y )2 i =1 (Using (1) and (3)) . Standard deviation ( σ ) = 6.12 = 1 times (h being the h is Let the assumed mean be ‘A’ and the scale be reduced to he 15. we get n x ∑ f ( A + hy ) i = i i =1 N = n n n ⎞ ⎛ n ⎞ 1⎛ ⎜ ∑ fi A + ∑ h fi yi ⎟ = N ⎜ A ∑ fi + h ∑ fi yi ⎟ N ⎝ i =1 ⎠ i =1 i =1 ⎝ i =1 ⎠ 1 n = A. By using step-deviation method... it is possible to simplify the procedure. Let the step-deviations or the new values be yi. (2) Replacing xi from (1) in (2). Shortcut method to find variance and standard deviation Sometimes the values of xi in a discrete distribution or the mid points xi of different classes in a continuous distribution are large and so the calculation of mean and variance becomes tedious and time consuming. (3) n 2 2 Variance of the variable x..4.N +h N Thus Now ∑ fi yi i =1 N n ⎛ ⎞ because ⎜ ∑ fi = N ⎟ i =1 ⎝ ⎠ x=A+h y 1 ... σ x = hσ y From (3) and (4). Variance and Standard Deviation for the following distribution. we have ⎛ n ⎞ N ∑ fi yi − ⎜ ∑ fi yi ⎟ i =1 ⎝ i =1 ⎠ n 2 2 . (4) is or Let us solve Example 11 by the short-cut method and using formula (5) © no N C tt E o R be T re pu Examples 12 Calculate mean..11 from the given data : Table 15.MATHEMATICS = 1 N ∑ fi h2 (yi − y )2 = h2 N ∑ fi (yi − y )2 = h n i =1 n i =1 2 × variance of the variable yi he 370 2 2 σ x2 = h σ y i.11 Class Frequency fi Mid-point yi= xi − 65 10 y i2 fi y i f i y i2 xi 30-40 3 35 –3 9 –9 27 40-50 7 45 –2 4 – 14 28 50-60 12 55 –1 1 – 12 12 60-70 15 65 0 0 0 0 70-80 8 75 1 1 8 8 80-90 3 85 2 4 6 12 9 0-100 2 95 3 9 6 18 – 15 105 N=50 . (5) bl h σx = N .e. Classes 30-40 40-50 50-60 60-70 70-80 80-90 90-100 Frequency 3 7 12 15 8 3 2 Solution Let the assumed mean A = 65... Here h = 10 We obtain the following Table 15.. 18 © no N C tt E o R be T re pu EXERCISE 15. 6.STATISTICS = 50 ⎡ 2 ⎢ N ∑ f i yi − ( ∑ fi yi ⎣ ) 2⎤ ⎥ ⎦ (10 )2 ⎡50 × 105 − (–15) 2 ⎤ ⎢ ⎥⎦ 2 (50) ⎣ 1 [5250 − 225] = 201 25 is = h2 N2 50 and standard deviation (σ ) = 201 = 14. 7. First 10 multiples of 3 4. 13. xi 6 10 14 18 24 28 30 fi 2 4 7 12 8 4 3 xi 92 93 97 98 102 104 109 fi 3 2 3 2 6 3 3 6. 8. 12 2. Classes Frequencies 0-30 30-60 60-90 90-120 120-150 150-180 180-210 2 3 5 10 3 5 2 . First n natural numbers 3. 4.2 bl σ2 = Variance ∑ fi yi × h = 65 − 15 × 10 = 62 he x= A+ Therefore 371 Find the mean and variance for each of the data in Exercies 1 to 5. 10. 12. Find the mean and standard deviation using short-cut method. 1. 5. xi 60 61 62 63 64 65 66 67 68 fi 2 1 12 29 25 12 10 4 5 Find the mean and variance for the following frequency distributions in Exercises 7 and 8. 7. V. 48.5-40. 36.V. where σ and x are the standard deviation and mean of the data. of children 70-75 75-80 80-85 85-90 90-95 95-100 100-105 105-110 110-115 6 6 3 is Height in cms 10. variance and standard deviation using short-cut method 3 4 7 7 15 9 he No.5 .5 and then proceed.372 8. The measure of variability which is independent of units is called coefficient of variation (denoted as C. x ≠ 0 .5. we have studied about some types of measures of dispersion.6 Analysis of Frequency Distributions In earlier sections. For comparing the variability or dispersion of two series. of circles 33-36 37-40 15 17 41-44 21 45-48 49-52 bl Diameters 22 25 © no N C tt E o R be T re pu Calculate the standard deviation and mean diameter of the circles.V. we calculate the coefficient of variance for each series. Find the mean.5-44. Whenever we want to compare the variability of two series with same mean.48. The series having lesser C. MATHEMATICS Classes Frequencies 0-10 10-20 20-30 30-40 40-50 5 8 15 16 6 9. [ Hint First make the data continuous by making the classes as 32. The mean deviation and the standard deviation have the same units in which the data are given.5 .5-36. which are measured in different units.V.52. = σ x × 100 . . is said to be more variable than the other. is said to be more consistent than the other.5. 44.) The coefficient of variation is defined as C. The diameters of circles (in mm) drawn in a design are given below: No.5. 40. we do not merely calculate the measures of dispersion but we require such measures which are independent of the units.5. The series having greater C.] 15. the series with greater standard deviation (or variance) is called more variable or dispersed than the other. A or B is there greater variability in individual wages? Solution The variance of the distribution of wages in plant A ( σ 12 ) = 81 Therefore. Let us now take following examples: Example 13 Two plants A and B of a factory show following results about the number of workers and the wages paid to them. 1 C.6. the series with lesser value of standard deviation (or variance) is said to be more consistent than the other. Also.. can be compared on the basis of values C. (2nd distribution) = x × 100 2 he σ Then is σ Given x1 = x2 = x (say) C.STATISTICS 373 15.Vs.1 Comparison of two frequency distributions with same mean Let x1 and σ1 be the mean and standard deviation of the first distribution. of workers 5000 6000 Average monthly wages Rs 2500 Rs 2500 Variance of distribution of wages 81 100 In which plant. and x2 and σ2 be the mean and standard deviation of the second distribution. (2) x It is clear from (1) and (2) that the two C. Thus.V. (1st distribution) = x × 100 1 and 2 C.V.V... standard deviation of the distribution of wages in plant A ( σ 1 ) = 9 .V. we say that for two series with equal means. (2nd distribution) = © no N C tt E o R be T re pu and of σ 1 and σ 2 only. A B No.. (1) bl Therefore × 100 . (1st distribution) = σ1 x σ2 × 100 . and their standard deviations are 21 and 16. 70 = σ2 x2 ×100 16 16 × 100 or x2 = × 100 = 22.e. (2nd distribution) = 70. What are their arithmetic means. Given Variance of height = 127.85 70 x2 Example 15 The following values are calculated in respect of heights and weights of the students of a section of Class XI : Height Weight Mean Variance 162.V.1361 kg 127. standard deviation of the distribution of wages in plant B ( σ 2 ) = 10 Since the average monthly wages in both the plants is same.69 cm2 Can we say that the weights show greater variation than the heights? Solution To compare the variability. respectively. Thus.3 cm 2 .V.69cm2 Therefore Standard deviation of height = Also Variance of weight = 23. respectively. we have to calculate their coefficients of variation. the plant B has greater variability in the individual wages.2500. Rs.V.69cm = 11. (1st distribution) = 60.. σ 2 = 16 σ1 x1 × 100 © no N C tt E o R be T re pu C.36 kg 23. (2nd distribution) = i. the variance of the distribution of wages in plant B ( σ 2 2 ) = 100 he Therefore. C. (1st distribution) = bl Let x1 and x2 be the means of 1st and 2nd distribution. therefore. the plant with greater standard deviation will have more variability. σ 1 = 21 Solution Given is Example 14 Coefficient of variation of two distributions are 60 and 70.V.374 MATHEMATICS Also.1361 kg2 127.e. Then 21 21 ×100 or x1 = × 100 = 35 x1 60 Therefore 60 = and C. C. i.6 cm 52. we can say that weights show more variability than heights. of wage earners Mean of monthly wages Variance of the distribution Firm A Firm B 586 648 Rs 5253 Rs 5253 100 121 of wages (i) Which firm A or B pays larger amount as monthly wages? (ii) Which firm. A or B? © no N C tt E o R be T re pu 1.81 52.V.V. the coefficient of variations (C.81 kg Now.V.95 162. bl Clearly C.) are given by and (C.STATISTICS 375 Therefore Standard deviation of weight = 23.V. 10-20 20-30 30-40 35 54 52 53 56 58 52 50 51 49 Y 108 107 105 105 106 107 104 103 104 101 An analysis of monthly wages paid to workers in two firms A and B. gives the following results: No.18 is (C. shows greater variability in individual wages? . Marks 2. belonging to the same industry.1361 kg = 4. A or B.6 4.36 × 100 = 9.V. find out which is more stable in value: X 3.) in weights = Mean 11. in heights Therefore.3 × 100 he = Standard Deviation × 100 = 6.3 From the data given below state which group is more variable. in weights is greater than the C. 40-50 50-60 60-70 70-80 Group A 9 17 32 33 40 10 9 Group B 10 20 30 25 43 15 7 From the prices of shares X and Y below.) in heights = EXERCISE 15. e. xi = 1 yi 2 1 20 1 20 1 20 y = x .. Find which team may be considered more consistent? The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below: © no N C tt E o R be T re pu Miscellaneous Examples Example 16 The variance of 20 observations is 5. then yi = 2xi i.e. x20 and x be their mean. 5 = 2 i =1 1 20 ∑ (xi − x )2 20 i =1 20 or ∑ (xi − x )2 = 100 i =1 . We know that ( ) Variance σ 2 1 = n 20 ∑ (xi − x ) .8 .. we get . No. (1) If each observation is multiplied by 2. .. i.. mean number of goals scored per match was 2 with a standard deviation 1. the length or weight? 50 is 5. y=2x or x= 1 y 2 Substituting the values of xi and x in (1).. x2. of matches 1 9 7 5 3 50 ∑ xi = 212 . If each observation is multiplied by 2.e.25 goals.6 i =1 bl Which is more varying. i =1 50 ∑ yi = 261 . Given that variance = 5 and n = 20. 2 2 ∑ i 20 ∑ i = 20 ∑ xi n i =1 i =1 i =1 Therefore y= i. and the new resulting observations are yi . Solution Let the observations be x1. i =1 50 ∑ xi2 = 902. i =1 ∑ yi2 = 1457. of goals scored 0 1 2 3 4 No.376 MATHEMATICS The following is the record of goals scored by team A in a football session: 4.. find the new variance of the resulting observations. he For the team B.. i.6 )2 + x 2 + y 2 − 2 × 4. we have 2xy = 72 . (5) So. x2. we get x2 + y2 – 2xy = 97 – 72 i.. Now © no N C tt E o R be T re pu or Therefore 1+ 2 + 6 + x + y 5 22 = 9 + x + y x + y = 13 Mean x = 4. we have ..56 + x2 + y2 –8..24. i. 2 and 6. (1) 1 5 2 variance = 8. y.4 (x + y ) + 2 × ( 4.xn is increased by ‘a’. (x – y)2 = 25 or x–y= ±5 . where a is a negative or positive number. 2..STATISTICS 20 2 1 ⎞ ⎛1 ∑ ⎜⎝ 2 yi − 2 y ⎟⎠ = 100 . the series is 1.76 + 2. 6. from (1) and (5).8 × 13 + 38. (4) Subtracting (4) from (2).. we get x = 9. is Example17 The mean of 5 observations is 4.4 = bl Solution Let the other two observations be x and y.e...56 + 5.. find the other two observations..e.. If three of the observations are 1. the variance of the resulting observations becomes k2 times the original variance.e. . Example 18 If each of the observation x1. y = 9 when x – y = – 5 Thus. show that the variance remains unchanged.24 = . y = 4 when x – y = 5 or x = 4. 8. i =1 Thus the variance of new observations = 20 ∑(y i =1 i 377 − y ) 2 = 400 1 × 400 = 20 = 22 × 5 20 he $ Note The reader may note that if each observation is multiplied by a constant k.72 .4 )2 + ( 2.4 and their variance is 8. Therefore. x.4 )2 + (1.4 )2 ⎤⎦ ⎣ 5 or 41.20 = 11.. the remaining observations are 4 and 9. Also ... (2) Therefore x2 + y2 = 97 But from (1)..24 = ∑ ( xi −x ) n i =1 1⎡ ( 3. (3) x2 + y2 + 2xy = 169 From (2) and (3). 378 MATHEMATICS Solution Let x be the mean of x1.1 We know that i.xn . $ Note We may note that adding (or subtracting) a positive number to (or from) each observation of a group does not affect the variance. Then 1 n n ∑ yi = n ∑ (xi + a) i =1 i =1 n ⎤ 1⎡ n 1 x a⎥ = + ∑ ∑ i ⎢ = n n i =1 ⎦ ⎣ i =1 n ∑x + i =1 na =x+a n . the variance of the new observations is same as that of the original observations. x= 1 n ∑ xi n i =1 40 = 1 100 ∑ xi or 100 i =1 100 ∑ xi = 4000 i =1 . the variance of the new observations i is 1 n bl y= 1 1 n 2 σ 2 = n ∑ (yi − y ) = n i =1 2 1 = n n ∑ (x + a − i =1 i .. Example 19 The mean and standard deviation of 100 observations were calculated as 40 and 5.1. respectively by a student who took by mistake 50 instead of 40 for one observation. Then the variance is given by 1 n n ∑ (xi − x ) 2 i =1 If ‘a is added to each observation.. y = x+a Thus..e. the new observations will be yi = xi + a Let the mean of the new observations be y . What are the correct mean and standard deviation? Solution Given that number of observations (n) = 100 Incorrect mean ( x ) = 40. (2) © no N C tt E o R be T re pu i.. x2. (1) he σ 12 = x − a) 2 [Using (1) and (2)] n ∑ (xi − x )2 = σ 12 i =1 Thus. Incorrect standard deviation (σ) = 5. ....e. 01 − 1592.e.9) 2 1617. 1 n 2 is Correct mean = bl Hence or 100 1 26.1 = n ∑ x − (x ) 2 i 2 i =1 × Incorrect ∑ xi − (40) 2 2 © no N C tt E o R be T re pu i.01 = 25 =5 2 he = 4000 – 50 + 40 = 3990 .01 = 100 i =1 n × Incorrect ∑x 2 i – 1600 i =1 n Incorrect ∑ xi = 100 (26.01 + 1600) = 162601 Therefore 2 i =1 n Correct ∑ xi = Incorrect Now 2 i =1 n ∑x i =1 2 i – (50)2 + (40)2 = 162601 – 2500 + 1600 = 161701 Therefore Correct standard deviation = = = Correct ∑x i 2 − (Correct mean) n 161701 100 − (39.9 100 100 1 n 2 1 ⎛ n ⎞ xi − 2 ⎜ ∑ xi ⎟ ∑ n i =1 n ⎝ i =1 ⎠ Standard deviation σ = Also 1 n = 5.e.STATISTICS i. Thus 379 Incorrect sum of observations = 4000 the correct sum of observations = Incorrect sum – 50 + 40 correct sum 3990 = = 39. Subject 7. If five of the observations are 2. respectively.9 Standard deviation 12 15 20 which of the three subjects shows the highest variability in marks and which shows the lowest? The mean and standard deviation of a group of 100 observations were found to be 20 and 3.xn. axn are a x and a2 σ2.. 21 and 18. respectively. Range = Maximum Value – Minimum Value Mean deviation for ungrouped data M. Summary Measures of dispersion Range. On rechecking. Later on it was found that three observations were incorrect. The mean and variance of 7 observations are 8 and 16. 10. Quartile deviation. The mean and standard deviation of marks obtained by 50 students of a class in three subjects. respectively. . he 3. is 2.25. ax3. find the remaining two observations. it was found that an observation 8 was incorrect. respectively. M. (ii) If it is replaced by 12.380 MATHEMATICS Miscellaneous Exercise On Chapter 15 4. (a ≠ 0). respectively.. 12.. which were recorded as 21. (x ) = ∑ xi – x n . Find the mean and standard deviation if the incorrect observations are omitted. Prove that the mean and variance of the observations ax1. x2. If six of the observations are 6. Given that x is the mean and σ2 is the variance of n observations x1. mean deviation. The mean and standard deviation of six observations are 8 and 4. 10. standard deviation are measures of dispersion. (M) = ∑ xi – M n . If each observation is multiplied by 3. Mathematics Physics Chemistry Mean 42 32 40. 7. The mean and standard deviation of 20 observations are found to be 10 and 2.D. Mathematics. The mean and variance of eight observations are 9 and 9. . 4. 14.. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. ax2.. 12 and 13. 12. © no N C tt E o R be T re pu 6.. 5. find the new mean and new standard deviation of the resulting observations.D. respectively.. Find the remaining two observations. Physics and Chemistry are given below: bl 1. variance. particularly. about 2000 years ago. (M) = 1 1 2 (xi – x ) 2 . The system of collecting data related to births and deaths is mentioned in Kautilya’s Arthshastra (around 300 B.) = σ × 100. where N = ∑ f i N N Variance and standard deviation for ungrouped data M. the series with lesser standard deviation is more consistent or less scattered.). x ≠ 0.D.C. Historical Note ‘Statistics’ is derived from the Latin word ‘status’ which means a political state. perhaps the first census was held in Egypt.) A detailed account of administrative surveys conducted during Akbar’s regime is given in Ain-I-Akbari written by Abul Fazl. σ 2 ⎢ ⎥⎦ N ⎣ where yi = = h N N ∑ f i yi2 − ( ∑ fi yi ) . In the year 3050 B. M.. σ= ∑ N N Shortcut method to find variance and standard deviation. σ= ( xi – x ) ∑ ∑ n n Variance and standard deviation of a discrete frequency distribution he σ2 = 1 1 2 2 fi ( xi − x ) . during the regime of Chandra Gupta Maurya (324-300 B. x For series with equal means.STATISTICS 381 Mean deviation for grouped data ∑ fi xi – x ∑ fi xi – M . This suggests that statistics is as old as human civilisation.C.D.V.C. © no N C tt E o R be T re pu σ2 = σ2 = 2 h2 ⎡ N∑ f i yi2 − ( ∑ fi yi ) ⎤ . . 2 xi − A h Coefficient of variation (C. ( x ) = . σ = fi ( xi − x ) ∑ ∑ N N Variance and standard deviation of a continuous frequency distribution bl is σ2 = 2 1 1 2 N ∑ fi xi2 − ( ∑ fi xi ) fi ( xi − x ) . In India also. we had an efficient system of collecting administrative statistics. Francis Galton (1822-1921). pioneered the use of statistical methods.382 MATHEMATICS © no N C tt E o R be T re pu bl is he Captain John Graunt of London (1620-1674) is known as father of vital statistics due to his studies on statistics of births and deaths. Education. Fisher (1890-1962). an Englishman. Jacob Bernoulli (1654-1705) stated the Law of Large numbers in his book “Ars Conjectandi’. Karl Pearson (1857-1936) contributed a lot to the development of statistical studies with his discovery of Chi square test and foundation of statistical laboratory in England (1911). etc. The theoretical development of statistics came during the mid seventeenth century and continued after that with the introduction of theory of games and chance (i. published in 1713. probability). in the field of Biometry. known as the Father of modern statistics. applied it to various diversified fields such as Genetics.. — — . Biometry. Agriculture. Sir Ronald A.e. e. Recall that the outcomes are called equally likely when we have no reason to believe that one is more likely to occur than the other. in 1933. – JOHN ARBUTHNOT © no N C tt E o R be T re pu In earlier classes. This is logically not a correct definition. to the total Kolmogorov number of equally likely outcomes. .N. He . we used equally likely or equally probable outcomes. The outcomes in favour of the event of ‘getting an even number’ are 2. when you have a candle in your hand. we studied about the concept of probability as a measure of uncertainty of various phenomenon. This theory of probability (1903-1987) is known as classical theory of probability. we learnt to find the probability on the basis of observations and collected data.Chapter 16 he PROBABILITY Where a mathematical reasoning can be had. In Class IX. these theories can not be applied to the activities/experiments which have infinite number of outcomes. For instance. to define probability. we assume that all outcomes have equal chance (probability) to occur. as to grope for a thing in the dark..3. a Russian mathematician.. In classical theory we assume all the outcomes to be equally likely. In other words. Kolmogorov. This is called statistical approach of probability. We have obtained the probability of getting an even number in throwing a die as 3 1 i. three in number).4. it is as great a folly to 16. Both the theories have some serious difficulties. In general. we find the ratio of the number of outcomes favourable to the event.6 (i.2. Thus.e.5 and 6 (six in number).1 Introduction bl is make use of any other. to obtain the probability of an event.4. another theory of probability was developed by A. Here the 6 2 total possible outcomes are 1. Thus. An experiment is called random experiment if it satisfies the following two conditions: (i) It has more than one possible outcome. or 6. Find a sample space. we shall refer the random experiment by experiment only unless stated otherwise. we can definitely say that the sum of measure of angles is 180°. without knowing the three angles.2. To understand this approach we must know about few basic terms viz. each outcome of the random experiment is also called sample point. we will study about this approach called axiomatic approach of probability. in what follows next. 4. Let us now consider some examples. we perform many activities which have a fixed result no matter any number of times they are repeated. sample space. For example. Example 1 Two coins (a one rupee coin and a two rupee coin) are tossed once. 16. Since either coin can turn up Head (H) or Tail(T). We also perform many experimental activities. 4. In other words. but we are not sure which one of these results will actually be obtained. Thus. Consider the experiment of rolling a die. Sample space is denoted by the symbol S. 5. the possible outcomes may be . (ii) It is not possible to predict the outcome in advance. in his book ‘Foundation of Probability’ published in 1933. 2. he 16. In this Chapter. Such experiments are called random experiments. when a coin is tossed it may turn up a head or a tail. where the result may not be same. For example given any triangle. 3. Solution Clearly the coins are distinguishable in the sense that we can speak of the first coin and the second coin.384 MATHEMATICS laid down some axioms to interpret probability. 5. Check whether the experiment of tossing a die is random or not? In this chapter. when they are repeated under identical conditions. Let us learn about these all. The set of outcomes {1. if we are interested in the number of dots on the upper face of the die. the set of all possible outcomes of a random experiment is called the sample space associated with the experiment. The outcomes of this experiment are 1. events.2 Random Experiments © no N C tt E o R be T re pu bl is In our day to day life. etc. 6} is called the sample space of the experiment. 3. random experiment.1 Outcomes and sample space A possible result of a random experiment is called its outcome. 2. Each element of the sample space is called a sample point. (4.T) = HT Tail on first coin and Head on the other = (T.H) = HH Head on first coin and Tail on the other = (H. Similarly. (6. (6. (6.3).1).4).5). (4.5). Therefore. (3.2). So. (5. the second draw may be H or R. a 2 rupee coin and a 5 rupee coin in his pocket. (2.1). © no N C tt E o R be T re pu bl Solution Suppose 1 appears on blue die and 2 on the red die.3). (4.6) (3. (5. . Also. In general each outcome can be denoted by the ordered pair (x. (3.5). For the $ sake of simplicity the commas are omitted from the ordered pairs.5). (1. Solution (i) Let Q denote a 1 rupee coin. the second draw may be H or Q. (4.1).4). (2. (1. The number of elements of this sample space is 6 × 6 = 36 and the sample space is given below: {(1.2).6)} Example 3 In each of the following experiments specify appropriate sample space (i) (ii) A boy has a 1 rupee coin. The first coin he takes out of his pocket may be any one of the three coins Q.4).5). the outcomes may be HQ or HR.3). the second draw may be Q or R. H or R.T) = TT Thus. Lastly. if ‘3’ appears on blue die and ‘5’ on red. (3.3).5). this sample space is given by S = {(x.3). A person is noting down the number of accidents along a busy highway during a year.2). TH. (1. (2. TT} he Note The outcomes of this experiment are ordered pairs of H and T.4). the outcomes may be RH or RQ.5). (4. corresponding to R.2). corresponding to H. Similarly.H) = TH Tail on both coins = (T. find the number of elements of this sample space. (5. (5. the outcome is denoted by the ordered pair (3. (1.2). He takes out two coins out of his pocket. (3. (6.6) (5. (2.1). (2. is Example 2 Find the sample space associated with the experiment of rolling a pair of dice (one is blue and the other red) once.6). (6. H denotes a 2 rupee coin and R denotes a 5 rupee coin.4). y): x is the number on the blue die and y is the number on the red die}. (6. (2. Corresponding to Q. So the result of two draws may be QH or QR.6). y).2). (4.3). (1.1). one after the other. the sample space is S = {HH.2).PROBABILITY 385 Heads on both coins = (H. We denote this outcome by an ordered pair (1. HT. (5. Therefore. where x is the number appeared on the blue die and y is the number appeared on the red die.4).6).1). (3. TTH. If it shows head. Ti means tail on the coin and the number i on the die. RQ} (ii) The number of accidents along a busy highway during the year of observation can be either 0 (for no accident ) or 1 or 2. TH. A coin is tossed and a die is thrown. 5. HB2.} he Example 4 A coin is tossed. A coin is tossed four times. A die is thrown two times. TTTH.2. A coin is tossed and then a die is rolled only in case a head is shown on the coin. HWi means head on the coin and ball Wi is drawn. HW1. Specify the sample space for the experiment in which a room is selected and then a person. 7. Hence. 2.1 In each of the following Exercises 1 to 7.1.. W3. the sample space is S={QH.. W2. HW2. Then a sample space of the experiment is S = { HB1. describe the sample space for the indicated experiment. 8.. its colour and the number on its uppermost face is noted. Describe the sample space of this experiment. Describe the sample space. HQ. QR. (i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births? . a sample space associated with this experiment is S= {0.} EXERCISE 16.. 2 boys and 2 girls are in Room X. HW4. HW3. Describe the sample space. one of white colour and one of blue colour are placed in a bag. or the 2nd toss. T4. Solution In the experiment head may come up on the first toss. B3 and the white balls by W1. T3. is Solution Let us denote blue balls by B1. T5. we draw a ball from a bag consisting of 3 blue and 4 white balls. T2. 6. 3. or some other positive integer. © no N C tt E o R be T re pu Example 5 Consider the experiment in which a coin is tossed repeatedly until a head comes up. One die is selected at random and rolled. T1. if it shows tail we throw a die. bl Here HBi means head on the coin and ball Bi is drawn. One die of red colour. B2. HB3. Thus.. or the 3rd toss and so on till head is obtained. and 1 boy and 3 girls in Room Y. TTTTH. A coin is tossed three times. T6}. An experiment consists of recording boy–girl composition of families with 2 children. W4. 4. RH. HR.. Similarly. 1. the desired sample space is S= {H.386 MATHEMATICS Thus. without replacement. A die is thrown repeatedly untill a six comes up. What is the sample space if we are interested in the number of girls in the family? A box contains 1 red and 3 identical white balls. An experiment consists of tossing a coin and then throwing it second time if a head occurs.3 Event We have studied about random experiment and sample space associated with an experiment. Find the sample space. one after the other. Two balls are drawn at random in succession without replacement. then a die is rolled once. 12. A coin is tossed. An associated sample space is S = {HH. TH. the die is thrown again. If the out come is a head. A coin is tossed. 15. We find that HT and TH are the only elements of S corresponding to the occurrence of this happening (event). The slips are put in a box and mixed thoroughly. 2. Write the sample space for this experiment. a die is thrown. HT. Now suppose that we are interested in those outcomes which correspond to the occurrence of exactly one head. 16. . What is the sample space for this experiment? he (ii) 14. is 10. These two elements form the set E = { HT. The sample space serves as an universal set for all questions concerned with the experiment. Write the sample space for this experiment. Describe the sample space for the experiment. TT}. A person draws two slips from the box. If it shows a tail. Each bulb is tested and classified as defective (D) or non – defective(N). An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. the coin is tossed twice. © no N C tt E o R be T re pu 13. 16.PROBABILITY 387 11. bl 9. 3 and 4 are written separatly on four slips of paper. If the number on the die is odd. Consider the experiment of tossing a coin two times. TH} We know that the set E is a subset of the sample space S . we find the following correspondence between events and subsets of S. Write the sample space of this experiment. Find the sample space for this experiment. If the die shows up an even number. Similarly. If it shows head. If a tail occurs on the first toss. What is the sample space for the experiment? The numbers 1. we throw a die. Suppose 3 bulbs are selected at random from a lot. we draw a ball from a box which contains 2 red and 3 black balls. . HT.. TT} D = { HT. 3. Definition Any subset E of a sample space S is called an event. the whole sample space is called the sure event. 6} Let E be the event “ the number appears on the die is a multiple of 7”. Can you write the subset associated with the event E? Clearly no outcome satisfies the condition given in the event. Now let us take up another event F “the number turns up is odd or even”. TH. 5. The associated sample space is S = {1.e. In the light of this we define an event as follows. we say that the empty set only correspond to the event E. If actually ‘1’ had appeared on the die then we say that event E has occurred. 4. we say that event E has occurred Thus. In other words we can say that it is impossible to have a multiple of 7 on the upper face of the die. Impossible and Sure Events The empty set φ and the sample space S describe events. © no N C tt E o R be T re pu bl 16. To understand these let us consider the experiment of rolling a die. we say that the event E has not occurred. 1.MATHEMATICS Description of events Number of tails is exactly 2 Number of tails is atleast one Number of heads is atmost one Second toss is not head Number of tails is atmost two Number of tails is more than two Corresponding subset of ‘S’ A = {TT} B = {HT. As a matter of fact if outcomes are 2 or 3. i. i. In fact φ is called an impossible event and S. Thus. TH. no element of the sample space ensures the occurrence of the event E. Clearly . If the outcome ω is such that ω ∉ E. 16.3. the event E of a sample space S is said to have occurred if the outcome ω of the experiment is such that ω ∈ E. TT} S = {HH. TH.3. Let E denotes the event “ a number less than 4 appears”.2 Types of events Events can be classified into various types on the basis of the elements they have.1 Occurrence of an event Consider the experiment of throwing a die. 2. TT} C = {HT. the event E = φ is an impossible event. TT} φ he 388 is The above discussion suggests that a subset of sample space is associated with an event and an event is associated with a subset of sample space.e. Thus. we have studied about different ways of combining two or more sets. complement of a set etc. TTH} Each of the above subsets contain more than one sample point. C be events associated with an experiment whose sample space is S. HTH. it is called a simple (or elementary) event. THH} be the event ‘only one tail appears’ Clearly for the outcome HTT. But we may say that the event ‘not A’ has occurred. HHT. 5. Thus.e. THT. union. it is called a Compound event. the event F = S is a sure event. THT. © no N C tt E o R be T re pu bl is he 2. i. In a sample space containing n distinct elements. It is also called the event ‘not A’. .TTH} F={HTT. THH. Complementary Event For every event A. TH. 3. HTT.3 Algebra of events In the Chapter on Sets. Simple Event If an event E has only one sample point of a sample space. 3. 4. For example. E2={HT}. the event A has not occurred. difference. An associated sample space is S = {HHH. 6. HTH. all outcomes of the experiment ensure the occurrence of the event F. For example. Thus. with every outcome which is not in A. Compound Event If an event has more than one sample point. THH.THT. take the experiment ‘of tossing three coins’. HHH} G= {TTT. intersection. a sample space is S={HH.3. TTH. TTH. HHT. there corresponds another event A ′ called the complementary event to A. TTT} Let A={HTH. For example in the experiment of tossing two coins. E3= { TH} and E4={TT}. there are exactly n simple events. HT. TT} There are four simple events corresponding to this sample space. Like-wise we can combine two or more events by using the analogous set notations. The subsets of S associated with these events are E={HTT. Let A. 2.THT. B. viz. hence they are all compound events. HTT. we say that ‘not A’ occurs. are all compound events.PROBABILITY 389 F = {1. These are E1= {HH}.} = S. 1. 16.. in the experiment of “tossing a coin thrice” the events E: ‘Exactly one head appeared’ F: ‘Atleast one head appeared’ G: ‘Atmost one head appeared’ etc. HHT. 5} (ii) ‘A and B’ = A ∩ B = {3. Let A be the event ‘getting a prime number’. and B = {(5. HTT.6)} so A ∩ B = {(6. 3. (6. A ∩ B = {ω : ω ∈ A and ω ∈ B} For example. (6. If A and B are two events.6)}.4. 5} and B = {1. THT.5). When the sets A and B are two events associated with a sample space. TTT} A′ = {ω : ω ∈ S and ω ∉A} = S – A.6). 4.5). (6.5} (iii) ‘A but not B’ = A – B = {2} (iv) ‘not A’ = A′ = {1. Solution Here S = {1. or Event ‘A or B’ = A ∪ B = {ω : ω ∈ A or ω ∈ B} is Therefore he 2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A ∪ B contains all those elements which are either in A or in B or in both. then the set A ∩ B denotes the event ‘A and B’. B be the event ‘getting an odd number’. the set A–B may denote the event ‘A but not B’. Write the sets representing the events (i) Aor B (ii) A and B (iii) A but not B (iv) ‘not A’. Thus. then ‘A ∪ B’ is the event ‘either A or B or both’.e.5). which belong to both ‘A and B’. TTH. (6.3). 3.1). 3.6} .390 MATHEMATICS Thus the complementary event ‘not A’ to the event A is A′ = {HHH. A = {2. (6.6)} may represent the event ‘the score on the first throw is six and the sum of the scores is atleast 11’.6)} Note that the set A ∩ B = {(6. The Event ‘A and B’ We know that intersection of two sets A ∩ B is the set of those elements which are common to both A and B.2). (6.. (6.4). 2.We know that A – B = A ∩ B´ Example 6 Consider the experiment of rolling a die. Therefore. © no N C tt E o R be T re pu bl 3. 5. The Event ‘A but not B’ We know that A–B is the set of all those elements which are in A but not in B. 3. 6}. This event ‘A ∪ B’ is also called ‘A or B’. in the experiment of ‘throwing a die twice’ Let A be the event ‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then A = {(6. (6. 4. 2. i.5). (6. 5} Obviously (i) ‘A or B’ = A ∪ B = {1. . B and C are called exhaustive events. Such events A.. En are n events of a sample space S and if n E1 ∪ E 2 ∪ E 3 ∪ . events Ei and Ej are pairwise disjoint and n ∪ E i = S . . 3}. 2. i.5 Exhaustive events Consider the experiment of throwing a die. We observe that A ∪ B ∪ C = {1.. 2. consider the events A ‘an odd number appears’ and event B ‘a number less than 4 appears’ Obviously A = {1..PROBABILITY 391 16... 2. 3} © no N C tt E o R be T re pu Now 3 ∈ A as well as 3 ∈ B Therefore. In general. Remark Simple events of a sample space are always mutually exclusive.4 Mutually exclusive events In the experiment of rolling a die. 4. In other words. 5} and B = {2. 5. 4. 3} ∪ {3. B = {3. 3.e. if Ei ∩ Ej = φ for i ≠ j i. B: ‘a number greater than 2 but less than 5 appears’ and C: ‘a number greater than 4 appears’. i.3. if E1. if they can not occur simultaneously.. We have S = {1. 6}. E2. .. 2. Here he A = {1. E2. En are said to be exhaustive if atleast one of them necessarily occurs whenever the experiment is performed.. 4. 3. 3. .4} and C = {5.. 6}. there is no outcome which ensures the occurrence of events A and B simultaneously. 16. 6} bl is Clearly A ∩ B = φ. In this case the sets A and B are disjoint.In other words. 5. . E2. 3.. En are called mutually exclusive and exhaustive i =1 events. then events E1. A ‘an odd number appears’ and B ‘an even number appears’ Clearly the event A excludes the event B and vice versa. two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event. En are called exhaustive events..e. Let us define the following events A: ‘a number less than 4 appears’. 5} and B = {1. 2.3. events E1. In general. a sample space is S = {1. E2. Consider events. A and B are disjoint sets. ... 6}.. Further.e. Then A = {1... 4} ∪ {5. Again in the experiment of rolling a die. ∪ E n = ∪ E i = S i =1 then E1. A and B are not mutually exclusive events. 6} = S.. (5.. 1). 1). i. (B. B: ‘the sum is a multiple of 3’. (1. HTT. HHT. consider the following events. (A.e. 1). C). (4. A. THT. HTH. TTH. Also C ∩ D = φ and so C and D are mutually exclusive events. A. (A. 2). they are mutually exclusive. Also. (4. TTT} and A = {TTT}. D: ‘the sum is greater than 11’. Similarly A ∩ C ≠ φ. 5. (4. (4. (3. 3). (6. (2. (B. D). 2. (6. (3. A: ‘No head appears’. HTH. B: ‘Exactly one head appears’ and C: ‘Atleast two heads appear’. (5. (5. (2. 1). THT. 3). (4. (1. Hence. HHH} Now A ∪ B ∪ C = {TTT. 6)} B = {(1. the events are pair-wise disjoint. y = 1. 5). (2. 4. 4). 6}. (2. A ∩ B = φ. 4). THH. C = {HHT. C: ‘the sum is less than 4’. A ∩ D ≠ φ. 3. Do they form a set of mutually exclusive and exhaustive events? Solution The sample space of the experiment is S = {HHH. THT. 2)} and D = {(6. (6. Example 8 A coin is tossed three times. (2. 3). TTH}. 4). he Example 7 Two dice are thrown and the sum of the numbers which come up on the dice is noted. B = {HTT. B ∩ C ≠ φ and B ∩ D ≠ φ. (6. B and C are exhaustive events. (6. HTT. 2). 4). 6)} ≠ φ Therefore. THH. 6). 5). the pairs. 1). 6)} C = {(1. HHT. Which pairs of these events are mutually exclusive? © no N C tt E o R be T re pu bl is Solution There are 36 elements in the sample space S = {(x. (1. 6). Let us consider the following events associated with this experiment A: ‘the sum is even’. 6)} We find that A ∩ B = {(1. 3). TTH. HTH. 2). 2). B and C form a set of mutually exclusive and exhaustive events. HHH} = S Therefore. 6). (6. (3. 4). 5). D) are not mutually exclusive events. 5). 2). 1). (2. 1). 3). . (5. (1. 5). y): x. THH. Then A = {(1. (3. Thus. 5). A ∩ C = φ and B ∩ C = φ Therefore.392 MATHEMATICS We now consider some examples. (4. 2). (5. (3. (3. C). (2. 4). (5. 3). A and B are not mutually exclusive events. 1). A die is thrown. Refer to question 6 above. C denote the event” three tails show and D denote the event ‘a head shows on the first coin”.2 1. A – C. D ∩ E. A ∩ B. B denote the event “two heads and one tail show”. state true or false: (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually exclusive and exhaustive (iii) A = B′ . The events A. (iii) Two events. E ∩ F′. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive? 2. which are not mutually exclusive. Describe the following events: A: the sum is greater than 8. Three coins are tossed once. B: getting an odd number on the first die. Describe the following events: (ii) (iv) (vi) B: a number greater than 7 D: a number less than 4 F: a number not less than 3 he (i) A: a number less than 7 (iii) C: a multiple of 3 (v) E: an even number greater than 4 Also find A ∪ B. 6. B and C are as follows: A: getting an even number on the first die. Let A denote the event ‘three heads show”. Which pairs of these events are mutually exclusive? 4. B: 2 occurs on either die C: the sum is at least 7 and a multiple of 3. Describe the events (i) A ′ (ii) not B (iii) A or B (iv) A and B (v) A but not C (vi) B or C (vii) B and C (viii) A ∩ B′ ∩ C′ 7. (iv) Two events which are mutually exclusive but not exhaustive. An experiment involves rolling a pair of dice and recording the numbers that come up. (v) Three events which are mutually exclusive but not exhaustive. C: getting the sum of the numbers on the dice ≤ 5. B ∪ C. E ∩ F. Describe (i) Two events which are mutually exclusive. F′ © no N C tt E o R be T re pu bl is 3. Two dice are thrown. Which events are (i) mutually exclusive? (ii) simple? (iii) Compound? 5. (ii) Three events which are mutually exclusive and exhaustive. D – E. Three coins are tossed. A die is rolled.PROBABILITY 393 EXERCISE 16. e. (vi) A′. . i. It follows from (iii) that P(φ) = 0. i i i For example. we take F = φ and note that E and φ are disjoint events.. In earlier classes. 16. sample space and events associated with these experiments. ωn .. ω2.. Axiomatic approach is another way of describing probability of an event. P(A) = ∑ P(ωi ).e. in ‘a coin tossing’ experiment we can assign the number 1 2 to each of the outcomes H and T. Therefore..e. C are mutually exclusive and exhaustive. The probability P is a real valued function whose domain is the power set of S and range is the interval [0. then P(E ∪ F) = P(E) + P(F). S = {ω1.. we have studied some methods of assigning probability to an event associated with an experiment having known the number of total outcomes.. In our day to day life we use many words about the chances of occurrence of events. P(H) = 1 2 and P(T) = 1 2 (1) Clearly this assignment satisfies both the conditions i.1] satisfying the following axioms (i) For any event E. we have considered random experiments. Let S be the sample space of a random experiment.e. P (φ) = 0. i.. ωi ∈ A.4 Axiomatic Approach to Probability © no N C tt E o R be T re pu bl is he In earlier sections. In this approach some axioms or rules are depicted to assign probabilities.. from axiom (iii). B′.394 MATHEMATICS (iv) A and C are mutually exclusive (v) A and B′ are mutually exclusive. we write P(ω ) for P({ω })... each number is neither less than zero nor greater than 1 and . ωn} It follows from the axiomatic definition of probability that (i) 0 ≤ P (ωi) ≤ 1 for each ωi ∈ S (ii) P (ω1) + P (ω2) + . P (E) ≥ 0 (ii) P (S) = 1 (iii) If E and F are mutually exclusive events. To prove this. Note It may be noted that the singleton {ω } is called elementary event and $ for notational convenience. ω2 . Probability theory attempts to quantify these chances of occurrence or non occurrence of events. + P (ωn) = 1 (iii) For any event A. we get P (E ∪ φ) = P (E) + P (φ) or P(E) = P(E) + P (φ) i. Let S be a sample space containing outcomes ω1 .. . ω2. ω6}. we can assign the numbers p and (1 – p) to both the outcomes such that 0 ≤ p ≤ 1 and P(H) + P(T) = p + (1 – p) = 1 This assignment. in this case we can say that probability of H = 1 1 ..2 0. in this case.6 − 1 4 − 1 3 Solution (a) Condition (i): Each of the number p(ωi) is positive and less than one. satisfies both conditions of the axiomatic approach of probability. (2) 4 4 Does this assignment satisfy the conditions of axiomatic approach? If we take P(H) = 1 is 3 .. we can say that there are many ways (rather infinite) to assign probabilities to outcomes of an experiment. probability of H = (a) 1 6 1 6 1 6 1 6 1 6 1 6 (b) 1 0 0 0 0 0 (c) 1 8 2 3 1 3 1 3 (d) 1 12 1 12 1 6 1 6 1 6 3 2 (e) 0... Hence. In fact.PROBABILITY P(H) + P(T) = 395 1 1 + =1 2 2 Therefore.3 0. Condition (ii): Sum of probabilities = 1 1 1 1 1 1 + + + + + =1 6 6 6 6 6 6 .1 0.. and probability of T = 2 2 he 1 3 and P(T) = .4 0. We now consider some examples.Which of the following assignments of probabilities to each outcome are valid? Outcomes ω 1 ω2 ω3 ω4 ω5 ω6 and probability of T = © no N C tt E o R be T re pu bl Yes. too. Example 9 Let a sample space be S = {ω1.5 0. 4 4 We find that both the assignments (1) and (2) are valid for probability of H and T. GGB} and B = {BBG.5 + 0. GGB. the assignment is valid (b) Condition (i): Each of the number p(ωi) is either 0 or 1. the assignment is not valid 2 Since. A sample space associated with this experiment is S = {BBB. 2 or 3 defective pens as result of this examination.3 + 0. 1.1 + 0. HT.4 + 0. BGB. BBB} P(A) = ∑ P(ωi ).6 = 2. the assignment is not valid.1.396 MATHEMATICS (d) 3 > 1.1 Probability of an event Let S be a sample space associated with the experiment ‘examining three consecutive pens produced by a machine and classified as Good (non-defective) and bad (defective)’. BGB. GBB. GBB. We may get 0.∀ ωi ∈ B 1 1 1 1 4 1 + + + = = 8 8 8 8 8 2 Let us consider another experiment of ‘tossing a coin “twice” = P(BBG) + P(BGB) + P(GBB) + P(BBB) = The sample space of this experiment is S = {HH. TT} Let the following probabilities be assigned to the outcomes .4. Let the probabilities assigned to the outcomes be as follows Sample point: BBB BBG BGB GBB BGG GBG GGB GGG Probability: 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 Let event A: there is exactly one defective pen and event B: there are atleast two defective pens. BGG. Hence Now A = {BGG. the assignment is not valid © no N C tt E o R be T re pu bl 16. TH.∀ωi ∈ A = P(BGG) + P(GBG) + P(GGB) = and 1 1 1 3 + + = 8 8 8 8 P(B) = ∑ P(ωi ). Since p(ω6) = is (e) he Therefore.2 + 0. sum of probabilities = 0. GBG. Condition (ii) Sum of the probabilities = 1 + 0 + 0 + 0 + 0 + 0 = 1 Therefore. GGG}. GBG. BBG. where B stands for a defective or bad pen and G for a non – defective or good pen. the assignment is valid (c) Condition (i) Two of the probabilities p(ω5) and p(ω6) are negative. P(HT) = .e. TT} Now P(E) = Σ P(wi)..e.e. then it follows that P(E) = m n = Number of outcomes favourable to E Total possible outcomes 16.. P (A ∪ B) Let A = {HHT.PROBABILITY P(HH) = 397 1 1 2 9 .3 Probability of the event ‘A or B’ Let us now find the probability of event ‘A or B’. HHH} be two events associated with ‘tossing of a coin thrice’ Clearly A ∪ B = {HHT. If each out come is equally likely.e.4. p + p + . Here E = {HH... ω2.4... P(TH) = .2 Probabilities of equally likely outcomes Let a sample space of an experiment be S = {ω1. for all ωi ∈ S where 0 ≤ p ≤ 1 n Since ∑ P(ωi ) =1 i. P(ωi) = p. THH.e. P(TT) = 4 7 7 28 1 9 4 + = 4 28 7 For the event F: ‘exactly two heads’. we have F = {HH} he Clearly this assignment satisfies the conditions of axiomatic approach. i.. p = 1 n Let S be a sample space and E be an event.. + p (n times) = 1 i=1 or np = 1 i. HTH. Now. Let all the outcomes are equally likely to occur. HHH} Now P (A ∪ B) = P(HHT) + P(HTH) + P(THH) + P(HHH) . THH. such that n(S) = n and n(E) = m. THH} and B = {HTH. i. ωn}.. the chance of occurrence of each simple event must be same. HTH. for all wi ∈ E P(F) = P(HH) = 1 4 bl and is = P(HH) + P(TT) = © no N C tt E o R be T re pu 16.. i. let us find the probability of the event E: ‘Both the tosses yield the same result’. if A and B are any two events associated with a random experiment. P(A ∪ B) = [ ∑ P(ωi )∀ ωi ∈ (A–B)] + [ ∑ P(ωi )∀ ωi ∈ A ∩ B] + [ ∑ P(ωi )∀ ωi ∈ B – A ] (because A–B. [using (1)] .. Thus to get the probability P(A ∪ B) we have to subtract the probabilities of the sample points in A ∩ B from P(A) + P(B) P(A ∪ B) = P(A) + P(B) − ∑ P(ωi ) . A ∩ B and B – A are mutually exclusive) . then Also P(A) = P(HHT) + P(HTH) + P(THH) = 3 8 and P(B) = P(HTH) + P(THH) + P(HHH) = 3 8 Therefore P(A) + P(B) = is It is clear that 3 3 6 + = 8 8 8 he 1 1 1 1 4 1 P ( A ∪ B) = + + + = = 8 8 8 8 8 2 P(A ∪ B) ≠ P(A) + P(B) © no N C tt E o R be T re pu bl The points HTH and THH are common to both A and B . ∀ωi ∈ A ∩ B i. ∀ωi ∈ A ∪ B . Since we have A ∪ B = (A–B) ∪ (A ∩ B) ∪ (B–A) .e. = P( A) + P(B) − P(A ∩ B) Thus we observe that. the elements of A ∩B are included twice. In the computation of P(A) + P(B) the probabilities of points HTH and THH.398 MATHEMATICS If all the outcomes are equally likely. we have P ( A ∪ B ) = ∑ p ( ωi ) . i. P(A ∪ B) = P(A) + P(B) − P(A ∩ B) In general.. then by the definition of probability of an event. (1) Also P(A) + P(B) = [ ∑ p (ωi ) ∀ ωi ∈ A ] + [ ∑ p(ωi ) ∀ ωi ∈ B] = [ ∑ P(ωi )∀ωi ∈ (A–B) ∪ (A ∩ B) ] + [ ∑ P(ωi )∀ωi ∈ (B – A) ∪ (A ∩ B) ] = [ ∑ P(ωi )∀ωi ∈ (A – B) ] + [ ∑ P(ωi )∀ωi ∈ (A ∩ B) ] + [ ∑ P(ωi )∀ωi ∈ (B–A) ] + [ ∑ P(ωi )∀ωi ∈ (A ∩ B)] = P(A ∪ B) + [ ∑ P(ωi )∀ ωi ∈ A ∩ B] = P(A ∪ B) + P(A ∩ B) .e.. . Clearly the sample space is S = {1. which is Axiom (iii) of probability.10 are considered to be equally likely... Alternatively. . (3) Subtracting (3) from (2) gives P (A ∪ B) – P(B) = P(A) – P (A ∩ B) or P(A ∪ B) = P(A) + P (B) – P (A ∩ B) The above result can further be verified by observing the Venn Diagram (Fig 16.4 Probability of event ‘not A’ Consider the event A = {2. we have P( A ∪ B) = P(A ) + P( B) . 16.1 If A and B are disjoint sets. and B = (A ∩ B) ∪ (B – A). ... Using Axiom (iii) of probability.10} If all the outcomes 1.4. 3.. 2. where A ∩ B and B – A are mutually exclusive.PROBABILITY 399 © no N C tt E o R be T re pu bl is he Hence P(A ∪ B) = P (A) +P(B) – P(A ∩ B) . 6. then the probability of each outcome is 1 10 . (2) and P(B) = P ( A ∩ B) + P (B – A) .. 8} associated with the experiment of drawing a card from a deck of ten cards numbered from 1 to 10. 4. it can also be proved as follows: A ∪ B = A ∪ (B – A)..e. where A and B – A are mutually exclusive. i. for mutually exclusive events A and B. 2.. then A ∩ B = φ Therefore P(A ∩ B) = P (φ) = 0 Thus.. they are mutually exclusive events. we get P (A ∪B) = P (A) + P (B – A) .1) Fig 16.. e. P(A′) = © no N C tt E o R be T re pu bl is Thus.400 Now MATHEMATICS P(A) = P(2) + P(4) + P(6) + P(8) 1 1 1 1 4 2 + + + = = 10 10 10 10 10 5 Also event ‘not A’ = A ′ = {1. 5. a spade) (iv) not a diamond (v) not a black card.. 10} Now P(A′) = P(1) + P(3) + P(5) + P(7) + P(9) + P(10) = he = 6 3 = 10 5 3 2 = 1 − = 1 − P(A) 5 5 Also. 7. Solution When a card is drawn from a well shuffled deck of 52 cards. We know that P(B′) = 1 – P(B) = 1 − 4 1 12 =1 − = 52 13 13 . If each outcome is equally likely. (i) Let A be the event 'the card drawn is a diamond' Clearly the number of elements in set A is 13.. we know that A′ and A are mutually exclusive and exhaustive events i. or P( A ′ ) = P(not A) = 1 – P(A) We now consider some examples and exercises having equally likely outcomes unless stated otherwise. A ∩ A′ = φ and A ∪ A′ = S or P(A ∪ A′) = P(S) Now P(A) + P(A′) = 1. a club or. the number of possible outcomes is 52.e. Probability of a diamond card = 1 4 (ii) We assume that the event ‘Card drawn is an ace’ is B Therefore ‘Card drawn is not an ace’ should be B′. 9. Example 10 One card is drawn from a well shuffled deck of 52 cards. by using axioms (ii) and (iii). 3. calculate the probability that the card will be (i) a diamond (ii) not an ace (iii) a black card (i.e. P(A) = 13 1 = 52 4 i. Therefore. e. Solution There are 9 discs in all so the total number of possible outcomes is 9..PROBABILITY 401 (iii) Let C denote the event ‘card drawn is black card’ Therefore. A disc is drawn at random from the bag. P(B) = . n (B) = 2 Hence P(A) = 2 9 (iii) The number of blue discs = 3. B. (iv) not blue. C be defined as A: ‘the disc drawn is red’ B: ‘the disc drawn is yellow’ C: ‘the disc drawn is blue’. so the event ‘card drawn is not a diamond’ may be denoted as A' or ‘not A’ (v) 1 1 = 2 2 © no N C tt E o R be T re pu We know that P(not C) = 1 – P(C) = 1 − bl is 1 3 = 4 4 The event ‘card drawn is not a black card’ may be denoted as C′ or ‘not C’. n (A) = 4 4 9 (ii) The number of yellow discs = 2..e. Now P(not A) = 1 – P(A) = 1 − Therefore. 2 he i.e. Probability of a black card = 1 . i. Calculate the probability that it will be (i) red. number of elements in the set C = 26 26 1 = 52 2 Thus. (v) either red or blue. i. Let the events A. P(C) = (iv) We assumed in (i) above that A is the event ‘card drawn is a diamond’. n(C) = 3 Therefore.e. (i) The number of red discs = 4. The discs are similar in shape and size. Probability of not a black card = 1 2 Example 11 A bag contains 9 discs of which 4 are red. i. 3 are blue and 2 are yellow.. (iii) blue. (ii) yellow. (b) Atleast one of them will not qualify the examination and (c) Only one of them will qualify the examination.05 and that Ashima will qualify the examination is 0. A and C are mutually exclusive events.10 and P(E ∩ F) = 0.10.02. P(C) = 1 2 = 3 3 (v) The event ‘either red or blue’ may be described by the set ‘A or C’ Since. Solution Let E and F denote the events that Anil and Ashima will qualify the examination.05 + 0. respectively.02. Then (a) The event ‘both Anil and Ashima will not qualify the examination’ may be expressed as E´ ∩ F´. Given that P(E) = 0.10 – 0.402 MATHEMATICS 3 1 = 9 3 (iv) Clearly the event ‘not blue’ is ‘not C’.02 = 0.13 Therefore P(E´ ∩ F´) = P(E ∪ F)´ = 1 – P(E ∪ F) = 1 – 0. Ashima will not qualify the examination.02 = 0. The probability that Anil will qualify the examination is 0. we have P(A or C) = P (A ∪ C) = P(A) + P(C) = 4 1 7 + = 9 3 9 he P(not C) = 1 − is Therefore © no N C tt E o R be T re pu bl Example 12 Two students Anil and Ashima appeared in an examination.13 = 0.. P(F) = 0. E´ is ‘not E’.e. The probability that both will qualify the examination is 0.e. i.87 (b) P (atleast one of them will not qualify) = 1 – P(both of them will qualify) = 1 – 0. Find the probability that (a) Both Anil and Ashima will not qualify the examination. We know that P(not C) = 1 – P(C) Therefore. i. Since.. Anil will not qualify the examination and F´ is ‘not F’.05. Also E´ ∩ F´ = (E ∪ F)´ (by Demorgan's Law) Now or P(E ∪ F) = P(E) + P(F) – P(E ∩ F) P(E ∪ F) = 0.98 . © no N C tt E o R be T re pu (a) Out of two women.02 = 0. Together they can be selected in 2 C1 × 2 C1 ways. ω4 .05 – 0. two can be selected in 2 C 2 = 1 way.10 – 0. Therefore (c) P ( One man ) = 2 C1 × 2 C1 2 × 2 2 = = 4 2×3 3 C2 Two men can be selected in 2 C 2 way. where E ∩ F´ and E´ ∩ F are mutually exclusive. two can be selected in 4 C 2 ways. Which of the following can not be valid assignment of probabilities for outcomes of sample Space S = {ω1 . ω6 . Therefore. ω2 . No men in the committee of two means there will be two women in the committee. P(only one of them will qualify) = P(E ∩ F´ or E´ ∩ F) he = P(E ∩ F´) + P(E´ ∩ F) = P (E) – P(E ∩ F) + P(F) – P (E ∩ F) = 0. ω3 . and Ashima will not qualify) or (Anil will not qualify and Ashima will qualify) i.. E ∩ F´ or E´ ∩ F. What is the probability that the committee will have (a) no man? (b) one man? (c) two men? bl Solution The total number of persons = 2 + 2 = 4.3 1. ω7 } . Therefore (b) P ( no man ) = 2 4 C2 1 × 2 × 1 1 = = 4×3 6 C2 One man in the committee means that there is one woman. One man out of 2 can be selected in 2 C1 ways and one woman out of 2 can be selected in 2 C1 ways.02 + 0.11 is Example 13 A committee of two persons is selected from two men and two women. Out of these four person. Hence P ( Two men ) = 2 4 C2 1 1 = 4 = C2 C2 6 EXERCISE 16. ω5 .e.PROBABILITY 403 (c) The event only one of them will qualify the examination is same as the event either (Anil will qualify. 05 0.01 0. There are four men and six women on the city council.03 0. what is the probability of the event ‘not A’. Find the probability that letter is (i) a vowel (ii) a consonant 9. and a person win Re 1 for each head and lose Rs 1. (v) A number less than 6 will appear.1 1 7 0.3 0.2 he 404 1 2 3 4 5 6 15 14 14 14 14 14 14 14 2.1 0. A letter is chosen at random from the word ‘ASSASSINATION’. how likely is it that it is a woman? 7.1 – 0. what is the probability that atleast one tail occurs? 3.6 1 7 0.2 1 7 0. © no N C tt E o R be T re pu bl is (e) 4.3 1 7 0. Three coins are tossed once.3 1 7 0.6 0. 8. 11 10. (iv) A number more than 6 will appear. If . A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. 5.01 0. From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.1 1 7 0.2 0. A fair coin is tossed four times. (c) Calculate the probability that the card is (i) an ace (ii) black card. Find the probability of getting (i) 3 heads (ii) 2 heads (iii) atleast 2 heads (iv) atmost 2 heads (v) no head (vi) 3 tails (vii) exactly two tails (viii) no tail (ix) atmost two tails 2 is the probability of an event. A card is selected from a pack of 52 cards. find the probability of following events: (i) A prime number will appear. find the probability that the sum of numbers that turn up is (i) 3 (ii) 12 6. If one council member is selected for a committee at random. A coin is tossed twice.4 0.7 0.MATHEMATICS Assignment (a) (b) (c) (d) ω1 ω2 ω3 ω4 ω5 ω6 ω7 0. (ii) A number greater than or equal to 3 will appear.5 – 0.50 for each tail that turns up.2 0.4 1 7 0. (a) How many points are there in the sample space? (b) Calculate the probability that the card is an ace of spades. (iii) A number less than or equal to one will appear. A die is thrown. 7. Events E and F are such that P(not E or not F) = 0.25.4. In an entrance test that is graded on the basis of two examinations.8 13.] 12. find the probability that he will be studying Mathematics or Biology. Given P(A) = is 1 3 bl (i) he 11.5 and the probability of passing neither is 0. In a lottery..35 .8 and the probability of passing the second examination is 0. he wins the prize. 15.5 0. P(B) = 0. What is the probability of passing both? The probability that a student will pass the final examination in both English and Hindi is 0.35 . and if these six numbers match with the six numbers already fixed by the lottery committee. P(A ∩ B) = 0.48 and P(A and B) = 0. P(B) = 0. if A and B are mutually exclusive 5 5 © no N C tt E o R be T re pu 14. 19. a person choses six different natural numbers at random from 1 to 20..6 (ii) P(A) = 0.PROBABILITY 405 1 5 1 15 ..5. 0.42.. Find P(A or B). A and B are events such that P(A) = 0.. (ii) 0. Determine (i) P(not A).1.7. 0. Check whether the following probabilities P(A) and P(B) are consistently defined (i) P(A) = 0.16. State whether E and F are mutually exclusive. the probability of a randomly chosen student passing the first examination is 0. 1 1 1 .5.75. 18. what is the probability of passing the Hindi examination? .95. What is the probability of winning the prize in the game? [Hint order of the numbers is not important. The probability of passing atleast one of them is 0.. 20. If the probability of passing the English examination is 0. 17. If a student is selected at random from the class. Fill in the blanks in following table: P(A ∪ B) P(A) P(B) P(A ∩ B) events.25 0. P(B) = 0. P(A ∪ B) = 0. (ii) P(not E and not F).7 3 1 and P(B) = . If E and F are events such that P(E) = 16.6 (iii) 0. 10% of the class study both Mathematics and Biology. (ii) P(not B) and (iii) P(A or B) In Class XI of a school 40% of the students study Mathematics and 30% study Biology. find 4 2 8 (i) P(E or F). P(F) = and P(E and F) = . CDAB. What is the probability that she visits © no N C tt E o R be T re pu Solution The number of arrangements (orders) in which Veena can visit four cities A. CADB. CDBA DABC. BCDA CABD. B. BDAC. find the probability that The student opted for NCC or NSS. The student has opted NSS but not NCC. (iv) and (v). or D is 4! i.406 MATHEMATICS 21. ADBC} n ( F) 4 1 Therefore. 32 opted for NSS and 24 opted for both NCC and NSS. DBAC. DCAB.Therefore. In a class of 60 students. ABDC. n (S) = 24. P ( F ) = n S = 24 = 6 ( ) Students are advised to find the probability in case of (iii). DCAB. CABD. CDAB. E = {ABCD. 30 opted for NCC..e. BCAD. BDCA. ADBC. B. BADC. DABC. ACDB. he (i) (ii) (iii) Miscellaneous Examples A before B? A first and B last? A just before B? (ii) (iv) A before B and B before C? A either first or second? bl (i) (iii) (v) is Example 14 On her vacations Veena visits four cities (A. CADB. A sample space for the experiment is S = {ABCD. ACDB. Since the number of elements in the sample space of the experiment is 24 all of these outcomes are considered to be equally likely. DCBA} (i) Let the event ‘she visits A before B’ be denoted by E Therefore. DACB. If one of these students is selected at random. ADCB} Thus (ii) P(E) = n(E) n ( S) = 12 1 = 24 2 Let the event ‘Veena visits A before B and B before C’ be denoted by F. The student has opted neither NCC nor NSS. DACB ACBD. DABC. C. Here F = {ABCD. ABDC. DBCA. . CBDA. ACBD. ADBC. CBAD. 24. ABDC. C and D) in a random order. ADCB BACD. ...PROBABILITY 407 Example 15 Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards. (2) Also A ∩ E = A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) [using distribution property of intersection of sets over the union]. (1) Now P (E ) = P ( B ∪ C) = P ( B) + P ( C ) − P ( B ∩ C ) .. B. prove that P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) +P ( C ) − P ( A ∩ B ) − P ( A ∩ C ) – P ( B ∩ C) + P ( A ∩ B ∩ C) Solution Consider E = B ∪ C so that P (A ∪ B ∪ C ) = P (A ∪ E ) = P(A) + P(E) − P(A ∩ E) . C7 (i) Number of hands with 4 Kings = 4 C4 × from the rest 48 cards) 48 C3 (other 3 cards must be chosen 4 Hence P (a hand will have 4 Kings) = C 4 × 48 C3 1 = 52 7735 C7 he 52 is Solution Total number of possible hands = 4 Therefore P (3 Kings) = C3 × 48 C 4 9 = 52 1547 C7 © no N C tt E o R be T re pu (iii) P(atleast 3 King) = P(3 Kings or 4 Kings) = P(3 Kings) + P(4 Kings) bl (ii) Number of hands with 3 Kings and 4 non-King cards = 4 C3 × 48 C4 = 9 1 46 + = 1547 7735 7735 Example 16 If A. it contains (i) all Kings (ii) 3 Kings (iii) atleast 3 Kings. Thus P ( A ∩ E ) = P ( A ∩ B ) + P ( A ∩ C ) – P ⎡⎣( A ∩ B) ∩ ( A ∩ C ) ⎤⎦ . C are three events associated with a random experiment. There will be 3! arrangements for A. respectively. So P (A. Therefore. 4 cards are drawn from a well – shuffled deck of 52 cards. 60 A. 5 marbles are drawn from the box. ABC. B and C are first three to finish (in any order) (Assume that all finishing orders are equally likely) is (a) he – P ( A ∩ B) − P ( A ∩ C ) + P ( A ∩ B ∩ C ) 5! = 5 × 4 × 3 = 60 sample points. B and C finish first. we will have 5 P3 .. B and C finish first. (b) What is the probability that A. 60 A. A box contains 10 red marbles.e. each with ( 5 − 3)! © no N C tt E o R be T re pu three places. bl Solution If we consider the sample space consisting of all finishing orders in the first 1 . respectively.408 MATHEMATICS = P ( A ∩ B ) + P ( A ∩ C ) – P [ A ∩ B ∩ C] . (3) Using (2) and (3) in (1).e. B and C are the first three finishers. 20 blue marbles and 30 green marbles. second and third respectively) = 3! 6 1 = = 60 60 10 Miscellaneous Exercise on Chapter 16 1.. second and third. B. we get P [ A ∪ B ∪ C] = P ( A ) + P ( B ) + P ( C ) − P ( B ∩ C ) Example 17 In a relay race there are five teams A. second and third. i.. B and C. what is the probability that (i) all will be blue? (ii) atleast one will be green? 2. the sample points corresponding to this event will be 3! in number. What is the probability that A. What is the probability of obtaining 3 diamonds and one spade? . B and C finish first. There is only one finishing order for this. a probability of (a) (b) 1 . D and E. C.. B and C are first three to finish) = Thus P(A. i. A and B are two events such that P(A) = 0.000 tickets are sold and ten equal prizes are awarded. Out of 100 students. Sheetal F 46 4. What is the probability that the spokesperson will be either male or over 35 years? 9. Three letters are dictated to three persons and an envelope is addressed to each of them.. If you and your friend are among the 100 students. two sections of 40 and 60 are formed. what is the probability of forming a number divisible by 5 when. From the employees of a company. 5 persons are selected to represent them in the managing committee of the company. 5. Alis F 28 5. (i) the digits are repeated? (ii) the repetition of digits is not allowed? 10. What is the probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets. each labelled with ten digits i. determine (i) P(2) (ii) P(1 or 3) (iii) P(not 3) he 4. If die is rolled once. Find (i) P(A ∪ B) (ii) P(A´ ∩ B´) (iii) P(A ∩ B´) (iv) P(B ∩ A´) 8. Salim M 41 A person is selected at random from this group to act as a spokesperson. What is the probability of a person getting the right sequence to open the suitcase? .PROBABILITY 409 3. © no N C tt E o R be T re pu 7.54. The number lock of a suitcase has 4 wheels. Rohan M 33 3.000 are randomly formed from the digits 0. 1. No.e. If 4-digit numbers greater than 5. what is the probability that is (a) you both enter the same section? (b) you both enter the different sections? bl 6. A die has two faces each with number ‘1’. The lock opens with a sequence of four digits with no repeats. Particulars of five persons are as follows: S. 3. In a certain lottery 10. the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope. 1. Name Harish Sex M Age in years 30 2. 5. from 0 to 9. P(B) = 0.35. and 7. three faces each with number ‘2’ and one face with number ‘3’.69 and P(A ∩ B) = 0. we studied about the axiomatic approach of probability... If A and B are any two events. E2. where n(A) = number of elements in n(S) the set A.. The main features of this Chapter are as follows: © no N C tt E o R be T re pu bl is he Sample space: The set of all possible outcomes Sample points: Elements of sample space Event: A subset of the sample space Impossible event : The empty set Sure event: The whole sample space Complementary event or ‘not event’ : The set A′ or S – A Event A or B: The set A ∪ B Event A and B: The set A ∩ B Event A and not B: The set A – B Mutually exclusive event: A and B are mutually exclusive if A ∩ B = φ Exhaustive and mutually exclusive events: Events E1. P(A ∪ B) = P(A) + P(B) – P(A ∩ B) If A and B are mutually exclusive.∪ En = S and Ei ∩ Ej = φ V i ≠ j Probability: Number P (ωi) associated with sample point ω i such that (i) 0 ≤ P (ωi) ≤ 1 (iii) P(A) = (ii) ∑ P ( ω ) for all ω ∈ S = 1 i i ∑ P ( ω ) for all ω ∈ A... then P(A or B) = P(A) + P(B) If A is any event. n(S) = number of elements in the set S.410 MATHEMATICS Summary In this Chapter. En are mutually exclusive and exhaustive if E1 ∪ E2 ∪ .. then P(A or B) = P(A) + P(B) – P(A and B) equivalently. then P(not A) = 1 – P(A) . Equally likely outcomes: All outcomes with equal probability Probability of an event: For a finite sample space with equally likely outcomes Probability of an event P(A) = n(A) . The number P (ω ) is called probability i i i of the outcome ωi. A. It had its origin in the 16th century when an Italian physician and mathematician Jerome Cardan (1501–1576) wrote the first book on the subject “Book on Games of Chance” (Biber de Ludo Aleae). evolved out of practical consideration. Besides. a Frenchman Pierre Laplace (1749–1827).PROBABILITY 411 Historical Note © no N C tt E o R be T re pu bl is he Probability theory like many other branches of mathematics. A Markov (1856–1922) and A.L Chebyshev (1821–1897). the Russian P. introduces probability as a set function and is considered a classic. Pascal became interested in these problems and discussed with famous French Mathematician Pierre de Fermat (1601–1665). It was published in 1663 after his death. a gambler Chevalier de Metre approached the well known French Philosopher and Mathematician Blaise Pascal (1623–1662) for certain dice problem. Kolmogorov is credited with the axiomatic theory of probability. Pascal and Fermat. — — . Bernoulli (1654–1705). His book ‘Foundations of Probability’ published in 1933. Both Pascal and Fermat solved the problem independently. a Dutchman. N Kolmogorove (1903–1987). In 1654. outstanding contributions to probability theory were also made by Christian Huygenes (1629– 1665). De Moivre (1667–1754). J. called Binomial Series.1 Introduction 1 he Appendix ∞ ∑a © no N C tt E o R be T re pu a1 + a2 + a3 + . .1.. is called an infinte series associated with infinite sequence. an.. It gives us a particular type of infinite series. m ( m − 1) 1. by examples. = bl As discussed in the Chapter 9 on Sequences and Series. . a sequence a1. 2. 3 x 3 + .e. a2.. . i. . then the combinations n Cr do not make any sense. the Binomial Theorem. giving an infinite series in which the index is negative or a fraction and not a whole number.e.. we shall study about some special types of series which may be required in different problem situations... a1 + a2 + a3 + . We illustrate few applications. Theorem The formula (1 + x )m = 1 + mx + holds whenever x < 1 . A. + n C n xn Here. we state a more general form of the theorem in which the index is not necessarily a whole number.1. This series can also be expressed in abbreviated form using the sigma notation. n is non-negative integer. i.2 Binomial Theorem for any Index In Chapter 8.INFINITE SERIES is A.. . having infinite number of terms is called infinite sequence and its indicated sum. + an + . we discussed the Binomial Theorem in which the index was a positive integer. We know the formula (1 + x)n = n C0 + n C1 x + . k k =1 In this Chapter. . In this Section.. .2 x2 + m ( m − 1)( m − 2 ) 1 ... .. + an + . Observe that if we replace index n by negative integer or a fraction... . We now state (without proof). (1 – x) – 2 = 1 +2x + 3x2 + 4x3 + . 2.. (1 – x) – 1 = 1 + x + x2 + x3 + . . . . a The general term in the expansion of (a + b)m is m ( m − 1)( m − 2 ) .e.. .2 a m − 2b 2 + . ( m − r + 1) a m − r b r 1.r We give below certain particular cases of Binomial Theorem. .2. b < 1 or equivalently when | b | < | a |. . This is not possible m ( a + b) ⎡ ⎛ b ⎞⎤ b⎞ m⎛ = ⎢ a ⎜1 + ⎟ ⎥ = a ⎜1 + ⎟ ⎝ a⎠ ⎣ ⎝ a ⎠⎦ m bl Consider m is 2. (1 + x) – 2 = 1 –2 x + 3x2 – 4x3 + . he (1 − 2 )−2 = 1 + ( −2 )( −2 ) + 1= 1 + 4 + 12 + ..2 ⎢⎣ m m −1 = a + ma b + This expansion is valid when m ( m − 1) 1.. . when we assume x < 1 .3. when m is a negative integer or a fraction 2 ⎤ ⎛b⎞ ⎜ a ⎟ + . .. . Note carefully the condition | x | < 1.2 ( −2 )2 + . i. (1 + x) – 1 = 1 – x + x2 – x3 + . .⎥ ⎝ ⎠ ⎥⎦ © no N C tt E o R be T re pu ⎡ b m ( m − 1) m = a ⎢1 + m a + 1. For example.INFINITE SERIES 413 Remark 1. Note that there are infinite number of terms in the expansion of (1+ x)m.... 3. when | x | < 2. – 1< x < 1 is necessary when m is negative integer or a fraction.. . we obtain or ( −2 )( −3) 1. if we take x = – 2 and m = – 2. x⎞ ⎛ Example 1 Expand ⎜ 1 − ⎟ ⎝ 2⎠ − 1 2 . 4. these are left to students as exercises: 1... we state the formula to find the sum of infinite geometric series a + ar + ar2 + . and illustrate the same by examples.P. Here a = 1. where a is first term and r. we have discussed the formula to find the sum of finite series a + ar + ar2 + ..... r = 2 4 .. Particularly... 3 9 2 ... We have 3 n ⎛2⎞ 1− ⎜ ⎟ n ⎝ 3 ⎠ = 3 ⎡1 − ⎛ 2 ⎞ ⎤ Sn = ⎢ ⎜ ⎟ ⎥ 2 ⎢⎣ ⎝ 3 ⎠ ⎦⎥ 1− 3 n . 3.. the common ratio of G.3 Infinite Geometric Series is From Chapter 9. a sequence a 1.5..414 MATHEMATICS Solution We have − 1 2 ⎛ 1⎞ ⎛ 1 ⎞⎛ 3 ⎞ ⎜ − ⎟ −x ⎜ − ⎟⎜ − ⎟ x 2 2 ⎛ ⎞ ⎝ ⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎛ − ⎞ + . 2.P. In this section... Section 9. ⎝3⎠ . if we take a1 = a. .. ....P. ar.P. Earlier. ar2. .... a 3. + arn – 1 which is given by Sn = ( a 1− rn 1− r ). n–1. 4 32 he ⎛ x⎞ ⎜1 − ⎟ ⎝ 2⎠ A. . if bl ak +1 = r (constant) for k = 1.1. 1.2 ⎝ 2⎠ = 1+ x 3x 2 + + .. a 2. + arn – 1 + .. = 1+ ⎜ ⎟+ ⎜ ⎟ 1 ⎝ 2 ⎠ 1 . then the ak © no N C tt E o R be T re pu resulting sequence a. Let us consider the G. (1) ⎛2⎞ Let us study the behaviour of ⎜ ⎟ as n becomes larger and larger.... a n is called G. arn–1 is taken as the standard form of G.. ⎜ ⎟ → 0 . In other words.INFINITE SERIES n ⎛2⎞ ⎜ ⎟ ⎝3⎠ 415 1 5 10 20 0. we find that ⎝3⎠ © no N C tt E o R be T re pu the sum of infinitely many terms is given by S = 3. . ar2. if numerical value of common ratio r is less than 1.1316872428 0. for infinite geometric progression a. Sn → S= we have a 1− r For example (i) 1+ (ii) 1 − 1 1 1 1 + + + .00030072866 n n he ⎛2⎞ We observe that as n becomes larger and larger. as n → ∞. sum to infinity of infinite geometric series is denoted by S. Mathematically. Therefore. ⎜ ⎟ becomes ⎝3⎠ n bl ⎛2⎞ sufficiently small. = =2 1 2 2 2 23 1− 2 1 1 1 + 2 − 3 + .01734152992 0..... Thus.6667 0. then ( a 1− rn Sn = ) 1− r a ar n − = 1− r 1− r ar n → 0 . In this case. Consequently.. r n → 0 as n → ∞ since | r | <1 and then 1− r a as n → ∞ . ar. we say that as n becomes sufficiently large. Thus. = 2 2 2 1 1 2 = = 1 3 ⎛ 1⎞ 1− ⎜ − ⎟ 1+ 2 ⎝ 2⎠ ... 1− r Symbolically. ⎜ ⎟ becomes closer and closer to ⎝3⎠ n is ⎛2⎞ zero. 4 16 64 −5 1 and r = − . the great Swiss mathematician introduced the number e in his calculus text in 1748. 1 5 1+ 4 4 A. it is clear that its sum is also positive.. Consider the two sums 1+ and 1 1 1 1 + + + . Since every term of the series (1) is positive.. which gives 8 4! 23 2 4! 24 1 1 1 1 1 1 = and 4 = .. Also | r | <1 .. .. (3) Observe that 1 1 1 1 1 1 = and 2 = ... which gives < 4 . + + . (2) 1 1 1 1 + 3 + 4 + . −5 5 −5 . 2 2 2 2 2 ... the sum to infinity is 4 = 4 = −1 ....P.4 Exponential Series © no N C tt E o R be T re pu bl Leonhard Euler (1707 – 1783)..1. which gives < 2 4 3! 2 2 3! 6 1 1 1 1 1 1 < = and 3 = .416 MATHEMATICS Example 2 Find the sum to infinity of the G. 3! 4! 5! n! . The number e is useful in calculus as π in the study of the circle... + n−1 + . . . Consider the following infinite series of numbers 1 1 1 1 + + + + . 4 4 he Solution Here a = is −5 −5 Hence. ...... (1) 1! 2! 3! 4! The sum of the series given in (1) is denoted by the number e Let us estimate the value of the number e.. 16 5! 2 2 5! 120 . + n −1 + ..... + + .. by analogy.. ⎟ ⎬ 2 2 2 ⎠⎭ ⎩⎝ 1! 2! ⎠ ⎝ 2 . when n > 2 n! 2 We observe that each term in (2) is less than the corresponding term in (3)... Therefore e < 3 and also e > 2 and hence 2 < e < 3. (4) is Therefore he 1 1 < n −1 . Solution In the exponential series e = 1+ x replacing x by (2x + 3)... we can say that 1⎞ ⎛ 1 1 1 1 ⎛1 1 1 ⎞ ⎜ 3! + 4! + 5! + ........ + n! ⎟ < ⎜ 2 + 3 + 4 + . ⎝ 1! 2! ⎠ bl 1 ⎛ 1 1⎞ ⎛1 1 1 ⎞ ⎜1 + + ⎟ + ⎜ + + + ... + n−1 + ..INFINITE SERIES 417 Therefore. 1! 2! 3! . + + .. © no N C tt E o R be T re pu ⎧⎛ 1 1⎞ ⎛ 1 1 1 1 ⎞⎫ < ⎨⎜ 1 + + ⎟ + ⎜ 2 + 3 + 4 + .. + n−1 + . we get. (5) ⎧ ⎛ 1 1 1 1 1 ⎞⎫ = ⎨1 + ⎜ 1 + + 2 + 3 + 4 + . we get x x 2 x3 + + + .. n! 1! 2! 3! Example 3 Find the coefficient of x2 in the expansion of e2x+3 as a series in powers of x. ⎟ ⎬ 2 2 2 ⎠⎭ ⎩ ⎝ 2 2 = 1+ 1 1 1− 2 =1+ 2 = 3 Left hand side of (5) represents the series (1). ⎟ n! ⎝ 1! 2! ⎠ ⎝ 3! 4! 5! ⎠ . Remark The exponential series involving variable x can be expressed as ex = 1 + x x 2 x3 xn + + + ..... ⎟ 2 2 2 ⎝ ⎠ ⎝2 ⎠ Adding ⎛ 1 1⎞ ⎜1 + + ⎟ on both sides of (4).. Alternatively e2x+3 = e3 . n! 1! 2! 3! . ⎣ ⎦ n! n=2 he 418 ∞ = 2 3n –2 [using n! = n (n – 1) (n – 2)!] n = 2 ( n − 2 )! ∑ ⎡ 3 32 33 ⎤ 2 = ⎢1 + + + + ... = 2e 3 Thus. Therefore. the coefficient of x2 in the expansion of e2x+3 is 2! Example 4 Find the value of e2. This can be expanded by the n! = Binomial Theorem as C2 3n − 2 22 . + + ..⎥ ⎣ 1! 2! 3! ⎦ = 2e3 . Solution Using the formula of exponential series involving x.....MATHEMATICS e 2 x +3 = 1+ Here. e2x 3 ⎡ 2 x (2 x) 2 (2 x)3 ⎤ e3 ⎢1 + + + + .. the coefficient of x2 is is 1⎡ n n 2 n 3 + C1 3n −1 ( 2 x ) + n C2 3n − 2 ( 2 x ) + . Thus 2e is the coefficient of x2 in the expansion of e2x+3. the coefficient of x2 in the whole n! series is ∞ n ∑ n−2 © no N C tt E o R be T re pu ∑ ∞ n ( n − 1) 3 C 2 3n − 2 2 2 2 = n! n! n=2 bl n Here..... the general term is ( 2 x + 3) ( 2 x + 3) 2 1! ( 2 x + 3)n n! + 2! + . + ( 2 x ) ⎤ . (3+ 2 x) n . we have ex = 1 + x x 2 x3 xn + + + .⎥ = ⎣ 1! 2! 3! ⎦ 2 3 2 e . rounded off to one decimal place. 5 Logarithmic Series Another very important series is logarithmic series which is also in the form of infinite series... 2 3 4 . is 7. the value of e2.. Substituting x = 1 in the $ expansion of log (1+x). we get 2 2 2 23 2 4 25 2 6 + + + + + + .4 . we have bl ⎛ ⎞ 2 2 2 23 2 4 ⎞ 2 5 ⎛ 2 2 2 2 3 e2 < ⎜1 + + + + ⎟ + ⎜ 1 + + 2 + 3 + ..355 and 7.. A. Theorem If | x | < 1.. rounded off to one decimal place. we get e e 1 1 1 log e 2 = 1 – + – + . then x 2 x3 + − .. log e (1 + x ) = x − Note The expansion of log (1+x) is valid for x = 1.1.. 1! 2! 3! 4! 5! 6! = 1+ 2 + 2 + 4 2 4 4 + + + + . 3 3 15 45 he e2 = 1 + ≥ the sum of first seven terms ≥ 7. is On the other hand.4. ⎟ 6 ⎝ 1! 2! 3! 4! ⎠ 5! ⎝ 6 6 ⎠ © no N C tt E o R be T re pu ⎛ ⎞ 2 ⎞ 4 ⎛ 1 ⎛1⎞ 4⎜ 1 ⎟ 2 ⎟ = 7 + = 7. Therefore... = 7 + ⎜⎜ 1 + + ⎜ ⎟ + .. ⎟⎟ = 7 + ⎜ 1 15 ⎝ 3 ⎝ 3 ⎠ 15 5 ⎜⎜ 1 − ⎟⎟ ⎠ 3⎠ ⎝ Thus..4. e2 lies between 7. We state the following result without proof and illustrate its application with an example. 2 3 The series on the right hand side of the above is called the logarithmic series.355.INFINITE SERIES 419 Putting x = 2. .⎥ ⎥ ⎢ Solution Right hand side = ⎢ 2 3 2 3 ⎣ ⎦ ⎣ ⎦ ( (1 + px + qx ) 2 = log e 1 + ( α + β ) x + αβx 2 = Left hand side. 2 3 he ⎡ ⎤ ⎡ ⎤ α 2 x 2 α3 x3 β 2 x 2 β3 x 3 x . — — . bl = log e ) is = log e (1 + α x ) + log (1 + βx ) Here.... x α − + − + β − + − . prove that ( ) log e 1 + px + qx 2 = ( α + β ) x − α 2 + β 2 2 α 3 + β3 3 x + x − . We have also assumed that both | α x | < 1 and | β x | < 1.. we have used the facts α + β = p and αβ = q . We know this from the © no N C tt E o R be T re pu given roots of the quadratic equation.. β are the roots of the equation x 2 − px + q = 0 .420 MATHEMATICS Example 5 If α. 2. then we discuss the steps involved in the process of modelling. Suppose a surveyor wants to measure the height of a tower. Egyptians. both of which have facilitated the handling of lengthy and complicated problems. So. We shall first talk about what a mathematical model is. The process of translation of a real-life problem into a mathematical form can give a better representation and solution of certain problems. the other option is to find out the factors that are useful to find the height. A. Thus. From his knowledge of trigonometry. then the problem can be solved as given in Example 1. Here we shall familiaries you with the steps involved in this process through examples.1 Introduction Much of our progress in the last few centuries has made it necessary to apply mathematical methods to real-life problems arising from different fields – be it Science. his job is now simplified to find the angle of elevation to the top of the tower and the distance from the foot of the tower to the point where he is standing. he knows that if he has an angle of elevation and the distance of the foot of the tower to the point where he is standing. It is physically very difficult to measure the height using the measuring tape.Appendix 2 MATHEMATICAL MODELLING A. then he can calculate the height of the tower. The use of Mathematics in solving real-world problems has become widespread especially due to the increasing computational power of digital computers and computing methods. So.2. Both of which are easily measurable.2 Preliminaries Mathematical modelling is an essential tool for understanding the world. Management etc. The architects. if he measures the angle of elevation as 40° and the distance as 450m. Indians. artisans and craftsmen based many of their works of art on geometric prinicples. The process of translation is called Mathematical Modelling. Babylonians and Greeks indulged in understanding and predicting the natural phenomena through their knowledge of mathematics. Finance. In olden days the Chinese. . distance and angle of elevation. ∠AOB is the angle of elevation. We also know that the angle of elevation. Step 1 We first try to understand the real problem. Let h denote the height. The real problem is to find the height h of the tower using the known distance d and the angle of elevation θ. Then we used the trigonometric ratio for the “tangent” function and found the relation as h = d tan θ So. AB denotes the tower. So we look for a relation connecting these three quantities. in this step we formulated the problem mathematically. Let us now look at the different steps used in solving the problem. It is given that the horizontal distance of the foot of the tower from a particular point O on the ground is 450 m. In the problem a tower is given and its height is to be measured. . Step 2 The three quantities mentioned in the problem are height. distance and angle of elevation. is 40°. (1) Fig 1 Step 3 We use Equation (1) to solve h. Then d = 450m.422 MATHEMATICS Example 1 The angle of elevation of the top of a tower from a point O on the ground. This is obtained by expressing it geometrically in the following way (Fig 1).. h and d. That means we found an equation representing the real problem. and d = 450m. we have studied the real problem and found that the problem involves three parameters height.6m Step 4 Thus we got that the height of the tower approximately 378m. . we used some geometry and found that the problem can be represented geometrically as given in Fig 1. Solution We shall solve this in different steps. We have θ = 40°. OA gives the horizontal distance from the point O to foot of the tower. Let d denotes this distance. Then we get h = tan 40° × 450 = 450 × 0. is 40°. In step 1. Find the height of the tower. That means in this step we have studied the real-life problem and identified the parameters. denoted by θ.839 = 377. In the Step 2. Then we have tan θ = h or h = d tan θ d This is an equation connecting θ.. which is 450 m away from the foot of the tower. hygrometers to measure humidity. That is we found Solution of the problem. 4. In the last step. The instruments are used to measure these parameters which include thermometers to measure temperature. We call this as Interpreting the mathematical solution to the real situation In fact these are the steps mathematicians and others use to study various reallife situations. 1. assuming that the batsman is not there. anemometers to measure wind speed.MATHEMATICAL MODELLING 423 In Step 3. a physiologist in the first problem and work out a mathematical equivalent of the problem. air pressure. Any constriction in the blood vessel or any change in the characteristics of blood vessels can change the flow and cause damages ranging from minor discomfort to sudden death. Department of Agriculture wants to estimate the yield of rice in India from the standing crops. humidity. wind speed. We shall consider the question. Meteorology department makes weather predictions based on mathematical models. Based on some statistical techniques decisions are made on the average yield of rice. In cricket a third umpire takes decision of a LBW by looking at the trajectory of a ball. The problem is to find the relationship between blood flow and physiological characteristics of blood vessel. Proper flow of blood is essential to transmit oxygen and other nutrients to various parts of the body in humanbeings as well as in all other animals. Scientists identify areas of rice cultivation and find the average yield per acre by cutting and weighing crops from some representative fields. which are called the mathematical models. we interpreted the solution of the problem and stated that the height of the tower is approximately 378m. This equivalent consists of one or more equations or inequalities etc. for example. barometers to measure airpressure. This simulated model is used to take decision of LBW. Then . based on the known paths of balls before it hits the batsman’s leg. Mathematical equations are arrived at. Once data are received from many stations around the country and feed into computers for further analysis and interpretation.6m. 2. etc. “why is it necessary to use mathematics to solve different situations. How do mathematicians help in solving such problems? They sit with experts in the area. we solved the mathematical problem and got that h = 377.” Here are some of the examples where mathematics is used effectively to study various situations. 3. Some of the parameters which affect change in weather conditions are temperature. simulated. Example 2 (Bridge Problem) Konigsberg is a town on the Pregel River. In 1736 Euler proved that the walk was not possible to do. An interesting geometric model is illustrated in the following example. and 3 join to riverbank B. visiting all the vertices. Euler proved it could not be done because he worked out that. He used four dots (vertices) for the two river banks and the two islands.424 MATHEMATICS solve the model and interpret the solution in terms of the original problem. You can see that 3 bridges (arcs) join to riverbank. to have an odd vertex you would have to begin or end the trip at that vertex. He proved this by inventing a kind of diagram called a network. Since the bridge problem has 4 odd vertices. On Euler’s network this meant tracing over each arc only once. we shall discuss what a mathematical model is. A. heard about the problem. (Think about it). that is made up of vertices (dots where lines meet) and arcs (lines) (Fig3). there can only be two odd vertices if you are to trace over each arc only once. The seven lines (arcs) are the seven bridges. which in the 18th century was a German town. People tried to walk around the town in a way that only crossed each bridge once. and 3 join to island D. These have been marked A. Within the town are two river islands that are connected to the banks with seven bridges as shown in (Fig 2). A mathematical model is a representation which comprehends a situation. but it proved to be difficult problem. Remember that the problem was to travel around town crossing each bridge only once. 5 bridges (arcs) join to island C. D. Before we explain the process. so Fig 3 they are called odd vertices (An even vertex would have to have an even number of arcs joining to it). Since there can only be one beginning and one end. Leonhard Euler. but now is Russian. B and C. This means that all the vertices have an odd number of arcs. a Swiss Fig 2 mathematician in the service of the Russian empire Catherine the Great. it just not possible to do! . we consider the modelling done to study the motion of a simple pendulum. using each bridge only once? Here the situation will be as in Fig 4. Let us now understand the different processes involved in Mathematical Modelling. Understanding the problem This involves. it is possible for the Konigsbergians to go around the city using each bridge exactly once. Fig 4 However. All of us are familiar with the simple pendulum. Conversion of physical situation into mathematics with some suitable conditions is known as mathematical modelling.2. joining the land areas of river banks A and B (Fig 4). Mathematical modelling is nothing but a technique and the pedagogy taken from fine arts and not from the basic sciences. Four steps are involved in this process. 1. The invention of networks began a new theory called graph theory which is now used in many ways. After the addition of the new edge. much water has flown under the bridges in Konigsberg. A. D and C still have odd degree. So. This pendulum is simply a mass (known as bob) attached to one end of a string whose other end is fixed at a point. As an illustrative example. for example. an extra bridge was built in Konigsberg. Based on this.MATHEMATICAL MODELLING 425 After Euler proved his Theorem. So. We have studied that the motion of the simple pendulum is periodic. we shall define what mathematical modelling is and illustrate the different processes involved in this through examples. Formulation Consists of two main steps. Is it possible now for the Konigsbergians to go round the city. Identifying the relevant factors In this. what we need to find is the period of oscillation. we find out what are the factors/ . including planning and mapping railway networks (Fig 4). both the vertices A and B have become even degree vertices. we give a precise statement of the problem as Statement How do we find the period of oscillation of the simple pendulum? The next step is formulation. In 1875. understanding the process involved in the motion of simple pendulum. Definition Mathematical modelling is an attempt to study some part (or form) of the real-life problem in mathematical terms.3 What is Mathematical Modelling? Here. The period depends upon the length of the string and acceleration due to gravity. We measure the period of oscillation. Mathematical description This involves finding an equation. experiments were conducted in which the values of period T were measured for different values of l.. we have identified four parameters for studying the problem. .426 MATHEMATICS parameters involved in the problem. in the case of pendulum. In the case of simple pendulum. Now.. We make the observation that there is no appreciable change of the period with mass. the mass of the bob (m). This gives the equation g T = 2π l g . effective length (l ) of the pendulum which is the distance between the point of suspension to the centre of mass of the bob. This process of searching the essential parameters is necessary before we go to the next step. For example. our purpose is to find T. which is assumed to be constant at a place. we consider the length of string as effective length of the pendulum and acceleration due to gravity (g). (2) Equation (2) gives the mathematical formulation of the problem. Now. So. we perform the same experiment on equal mass of balls but take strings of different lengths and observe that there is clear dependence of the period on the length of the pendulum. Here. In the case of simple pendulums the solution involves applying the formula given in Equation (2). Usually we have to do some operation which involves solving an equation. calculation or applying a theorem etc. These values were plotted on a graph which resulted in a curve that resembled a parabola. It implies that the relation between T and l could be expressed T2 = kl It was found that k = . the factors are period of oscillation (T).. inequality or a geometric figure using the parameters already identified. For this we need to understand what are the parameters that affect the period which can be done by performing a simple experiment. 2.. Finding the solution The mathematical formulation rarely gives the answer directly. This indicates that the mass m is not an essential parameter for finding period whereas the length l is an essential parameter. We take two metal balls of two different masses and conduct experiment with each of them attached to two strings of equal lengths. (1) 4π 2 . 042 Now. or else.36 = 0.MATHEMATICAL MODELLING 427 The period of oscillation calculated for two different pendulums having different lengths is given in Table 1 Table 1 l 225 cm 275cm T 3. we conduct some experiments on the pendulum and find out period of oscillation. error = 3. . Once we accept the model. In other words. Many times model equations are obtained by assuming the situation in an idealised context. and mass m = 385 gm. In this case. In the case of simple pendulum. we compare the measured values in Table 2 with the calculated values given in Table 1. T = 3. improve it. we will reject it. The difference in the observed values and calculated values gives the error. T = 3. the essential characteristic of a real life problem. Table 2 Periods obtained experimentally for four different pendulums Mass (gms) Length (cms) Time (secs) 385 275 225 3.371 3. This process is called validation of the model. The process of describing the solution in the context of the real situation is called interpretation of the model. Interpretation/Validation A mathematical model is an attempt to study. for l = 275 cm. The model will be useful only if it explains all the facts that we would like it to explain. we have to interpret the model. we can interpret the solution in the following way: (a) The period is directly proportional to the square root of the length of the pendulum.04 sec 3. with the known facts about the real problem.36 sec The table shows that for l = 225 cm. For example.04 sec and for l = 275 cm.056 230 275 225 3. then test it again. Otherwise.011 which is small and the model is accepted.352 3.371 – 3.36 sec. we measure the effectiveness of the model by comparing the results obtained from the mathematical model. The results of the experiment are given in Table 2. in such situation we look for a better model and this process continues. We study the simple situation with this model expecting that we can obtain a better model of the situation. But we found that there is some error in the calculated result and measured result. This is because we have neglected the mass of the string and resistance of the medium. This leads us to an important observation.428 MATHEMATICS (b) It is inversely proportional to the square root of the acceleration due to gravity..02 Rs 10 Soyabean .09 . We just pick one or two main factors to be completely accurate that may influence the situation.. we summarise the main process involved in the modelling as (a) Formulation (b) Solution (c) Interpretation/Validation The next example shows how modelling can be done using the techniques of finding graphical solution of inequality. x. So. The real world is far too complex to understand and describe completely. Determine the daily minimum cost of the food mix. y are related by the equation z = 10x + 20y . Our validation and interpretation of this model shows that the mathematical model is in good agreement with the practical (or observed) values. The special food is a mixture of corn and soyabean with the following compositions Table 3 Material Nutrients present per Kg Protein Nutrients present per Kg Fibre Cost per Kg Corn . So the variables (factors) that are to be considered are x = the amount of corn y = the amount of soyabean z = the cost Step 2 The last column in Table 3 indicates that z. (1) The problem is to minimise z with the following constraints: .06 Rs 20 The dietary requirements of the special food stipulate atleast 30% protein and at most 5% fibre. Solution Step 1 Here the objective is to minimise the total daily cost of the food which is made up of corn and soyabean.60 . Now. Example 3 A farm house uses atleast 800 kg of special food daily. Then try to obtain a simplified model which gives some information about the situation. 02x + 0. y. x = 470. Step 3 This can be solved graphically. (2) (b) The food should have atleast 30% protein dietary requirement in the proportion as given in the first column of Table 3. The shaded region in Fig 5 gives the possible solution of the equations.03x – . x + y ≥ 800 ..30y ≤ 0 0.6 and y = 329.6 + 20 × 329.4) i. This gives 0.6y ≥ 0.01y ≥ 0 This gives the formulation of the model.3 (x + y) .21x – ...05 (x + y) . (3) (c) Similarly the food should have atmost 5% fibre in the proportion given in 2nd column of Table 3.e. From the graph it is clear that the minimum value is got at the Fig 5 point (470..6... Then the problem can be restated in the following mathematical form.06 y ≤ 0.4 = 11294 .329.. (4) We simplify the constraints given in (2). This gives the value of z as z = 10 × 470.MATHEMATICAL MODELLING 429 (a) The farm used atleast 800 kg food consisting of corn and soyabean i. This gives 0.4. Statement Minimise z subject to x + y ≥ 800 0..e.09x + 0. (3) and (4) by grouping all the coefficients of x. . . The birth rate is the same for all intervals. Likewise. let p (t) denote the population in that particular year.” In the next example. and a constant d. We find the population by Jan. (1) 2. Let B(t) denote the number of births in the one year between t and t + 1 and D(t) denote the number of deaths between t and t + 1.430 MATHEMATICS This is the mathematical solution. There is no migration into or out of the population. Step 4 The solution can be interpreted as saying that. t = 1 stands for the next year etc. protein and fibre is Rs 11294 and we obtain this minimum cost if we use 470.6 kg of corn and 329... Let t denote the time in years. B (t ) P (t ) is called the birth rate for the time interval t to t + 1. 2. This means that there is a constant b. Example 4 Suppose a population control unit wants to find out “how many people will be there in a certain country after 10 years” Step 1 Formulation We first observe that the population changes with time and it increases with birth and decreases with deaths. called the birth rate.4 kg of soyabean. For any time t. Then t takes values 0. Suppose we want to find the population in a particular year.. “The minimum cost of the special food with corn and soyabean having the required portion of nutrient contents. say t0 = 2006. 1. D (t) P (t) is called the death rate for the time interval t to t + 1. 1st. t = 0 stands for the present time. the death rate is the same for all intervals. i. Then we get the relation P (t + 1) = P (t) + B (t) – D (t) Now we make some assumptions and definitions 1. 2005.. Assumptions 1. How will we do that. the only source of population . We want to find the population at a particular time. 2. Add the number of births in that year and subtract the number of deaths in that year. b= B (t ) P (t ) and d= D (t ) P (t ) .. we shall discuss how modelling is used to study the population of a country at a particular time. for all t ≥ 0.e. called the death rate so that. 02 and d = 0. . this result is absurd. What will the population be in 10 years? Using the formula.000) = (1. the Malthusian parameter. t = 0.25 of a person.. Step 2 – Solution Suppose the current population is 250. As a result of assumptions 1 and 2. 2. in more high-flown language.. in honor of Robert Malthus who first brought this model to popular attention. In terms of r.. Here.000. (5) P(t) is an example of an exponential function.000. we get P(t) = (1 + b – d)t P (0) .155. .. since one can’t have 0. ...104622125) (250.155.MATHEMATICAL MODELLING 431 change is birth and death. P (t + 1) = P(t) + B(t) – D(t) = P(t) + bP(t) – dP(t) = (1 + b – d) P(t) .000 and the rates are b = 0. (4) for t = 0.25 Step 3 Interpretation and Validation Naturally. P(10) = (1.. we are not getting the exact answer because of the assumptions that we have made in our mathematical model. 1. The constant 1 + b – d is often abbreviated by r and called the growth rate or.000) = 276...01)10 (250. . where c and r are constants.000. So. (2) Setting t = 0 in (2) gives P(1) = (1 + b – d)P (0) . we deduce that.531 (approximately).. 1. Equation (4) becomes P(t) = P(0)r t .. (3) Setting t = 1 in Equation (2) gives P(2) = (1 + b – d) P (1) = (1 + b – d) (1 + b – d) P (0) (Using equation 3) = (1 + b – d)2 P(0) Continuing this way. for t ≥ 0. we calculate P(10). is an exponential function.01. we do some approximation and conclude that the population is 276. 2. The above examples show how modelling is done in variety of situations using different mathematical techniques.531. Any function of the form cr t.. Equation (5) gives the mathematical formulation of the problem. we try to identify the sources of the shortcomings.e..432 MATHEMATICS Since a mathematical model is a simplified representation of a real problem. when the obtained results are interpreted physically whether or not the model gives reasonable answers. Thus mathematical modelling can be a cycle of the modelling process as shown in the flowchart given below: START ↓ < ASSUMPTIONS/AXIOMS ↓ FORMULATION ↓ SOLUTION ↓ INTERPRETATION < ↓ VALIDATION ↓ NO < YES SATISFIED —v — < STOP . If a model is not accurate enough. new mathematical manipulation and hence a new evaluation. by its very nature. It may happen that we need a new formulation. Obviously. has built-in assumptions and approximations. the most important question is to decide whether our model is a good one or not i. M. –2. 71. 2 } (iv) D = { L. f. 4. 6. –1. T. { b }. 3 }. (iv). (viii). (vi). } (ii) B = {0. 5. O. R. { 1 }. 5} (v) E = {T. June. . 5. (i). 1. 80} (iv) D = {2. –10) (iii) [ 0.1 © no N C tt E o R be T re pu bl is he 1. 2. (vii). 4. 4. 6] (ii) (– 12. n∈N and 1 ≤ n ≤ 4 } (iv) { x : x is an even natural number} (v) { x : x = n2. . 3 } (vi) ⊂ (vi) True (iv) φ (– 4. (xi) φ. 3. E. { a }. (i) { x : x ∈ R. 3. (vii) and (viii) are sets. { a } (ii) φ. (i) A = {–3. O. 6 } (ii) B = {1.2 1. G. Y} (vi) F = {B.3 1. 2. n∈N and 1 ≤ n ≤ 4 } (ii) { x : x = 2n. c. d. (i) ↔ (c) (ii) ↔ (a) (iii) ↔ (d) (iv) ↔ (b) EXERCISE 1. { 2 }. 26. 3. R} 4. 2 }. 0. 2. 2. (i). B= D. 3. (v). 35. (i) { x : x = 3n. 62. (i) (i) (i) (i) (iii). – 3 < x < 0 } (ii) { x : x ∈ R. . 5. b } φ. 4 } (iii) C = { –2. 7 ) (iv) [ 3.ANSWERS EXERCISE 1. { 1. 3. 5} (iii) C = {17. 2. April. (i) (iii) 1 (i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄ (v) ⊄ ⊂ False (ii) True (iii) False (iv) True (v) False (v). g. 6 < x ≤ 12 } (iv) { x R : – 23 ≤ x < 5 } 9. (iv) Finite (ii) Infinite (ii) Yes (ii) No (ii) Infinite (iii) Finite (iv) Infinite Finite (iii) Infinite (iv) Finite No (iii) Yes (iv) No Yes 6. E = G (v) Finite (v) Infinite EXERCISE 1. A } (v) E = { February. h. 1. { 2. 4 ] 7. November } (vi) F = {b. j } 6. 3 }. { 1. E. 44. 0. Y. 2. (i) A = {1. 5. 6 ≤ x ≤ 12 } (iii) { x : x ∈ R. { 1. (i) (vii) (i) (i). { 3 }. N. { a. 3. (i) ∈ (ii) ∉ (iii) ∉ (vi) ∈ (v) ∈ (vi) ∉ 3. (iii) . n∈N and 1 ≤ n ≤ 5 } (iii) { x : x = 5n. 1. n∈N and 1 ≤ n ≤ 10 } 5. I. 4. 53. –1. September. 2. 3. (ix). (i) { a. 9. 9. 20 } (viii) { 4. 11 } (iv) {4. 21} (v) { 2. 14. 8. 2. c } (ii) {f. 4. 6. b. 9. c. 8 } (vii) { 3. 18. h } (iii) { b. 6. 9. 5 or a multiple of 3 } (iv) A ∪ B = {x : 1 < x < 10. 8. 5. 3. (i) X ∪ Y = {1. b. 3. 4. 2. 9} (ii) {1. 6. (ii) {3. 15. d. 10} (v) {1. 8. b. h } { b. 8. 6. 9 } (iii) {7. 2. 11 } 7. { 5. 16 } (vi) { 5.8 } (iii) {3.5 (i) (iv) 2. 7. 12. 5. 4. 3. 5. (i) {3. A ∪ B = { a. 9. 11. c } (i) { 1. 7. 3 } (ii) A ∩ B = { a } 6. 8. 9. 12.4 1. 2. 2. 10. 8. 6. 5. 6. 3. 5. Set of irrational numbers 12. d } (iii) T (iv) T EXERCISE 1. 9. 10} is (iv) {3. f. (i) F (ii) F (iii) { b . 9. 2. 10. o. 10. B he 2. 6. d . 8 } (vi) {1. 20 ) (vii) { 20 } (x) { 5. 3 } 4. 6. 9 } (v) { 1. 14} (xi) {2. 11 } (iii) { 3 } (iv) φ (iii) φ (iv) { 11 } (vii) φ © no N C tt E o R be T re pu (viii) { 7. 5. Yes. 16 } (ix) { 2. 6. 6 } (ii) {1. 9. 7. 9. 16. x ∈ N} (v) A ∪ B = {1. 3. 8. 16} (xii) {5. 18. 5. 7. 4. f. 9. (i) B (v) { 2 } (ix) {7. h} (ii) { a. e ) { x : x is an odd natural number } { x : x is an even natural number } { x : x ∈ N and x is not a multiple of 3 } { x : x is a positive composite number and x = 1 ] . 15 } (ii) C (iii) D (iv) φ (vi) { x : x is an odd prime number } 8. 9 } { 5. 7. 4. 6. 21} (v) φ bl 5. 5. e. 14. u } A ∪ B = {x : x = 1. (i) { 7. 3. 3. 5. 4. 2. g. 9 } { d. 4. 3. 11 } (ii) { 11. c. c. (i) (iv) 3. 2. 8. 12. 5 } (iii) (ii) A ∪ B = { a. 10. 9. 6. 10 } (i) X ∩ Y = { 1. 15. (i) (ii) (iii) (iv) 1. 20} 10. 15. 4. 7. i. 7. 7. 4.434 MATHEMATICS EXERCISE 1. e. 13 } (v) φ (vi) { 7. 7. 4. 21 } (iii) (iii) {3. 18. 6. 15 } (x) { 7. 4. 7. 4 } (vi) { 1. g } 11. 8. 3. 7). –1). 1). (–1. (m. z} and B = {1. 50 4. –1. A ⊂ C. (4. –1. 2). (1. 2 9 } 2 he (xi) { x : x ∈ N and x ≤ (x) { x : x ∈ N and x < 7 } is (v) (vi) (vii) (viii) 435 © no N C tt E o R be T re pu Miscellaneous Exercise on Chapter 1 1. False 12. 1. We may take A = { 1. 2)} H × G = {(5. (2. (2. The number of elements in A × B is 9. 8). n). n). 1)} 6. 35 8. 1). (–1. (i) U (ii) A (iii) φ (iv) φ EXERCISE 1. –1. (– 1. 3 }. 8)} 4. y} 8. (2. 1. 5 3. 7). (–1. 11 EXERCISE 2. 7. 9. A × B = {(1. 4)} A × B will have 24 = 16 subsets. y. B ⊂ C. A' is the set of all equilateral triangles. 1). – 1). (n.6 2. (ii) 30 16. (7. A × A = {(– 1. (7. C = { 2 . (i) False P × Q = {(m. 25. 42 5. 8). (1.ANSWERS { x : x is a positive integer which is not divisible by 3 or not divisible by 5} { x : x ∈ N and x is not a perfect square } { x : x ∈ N and x is not a perfect cube } { x : x ∈ N and x ≠ 3 } (ix) { x : x ∈ N and x ≠ 2 } 6. – 1). B = { 1. A = {a. (1. (8. 60 bl 1. (i) False (ii) False (iii) True (iv) False (v) False (vi) True 7. D ⊂ A. (5. 4). (4. m)} (ii) True (iii) True 5. 1. x = 2 and y = 1 2. B = {x. (8. (n. m). (1. 3). b}. (1. 4). G × H = {(7. 3 } 13. 5).1 1. (2. 4). 5).2} . D ⊂ B. 1)} A × A × A = {(–1. 30 6. 19 7. A ⊂ B. 325 14. 125 15. –1. 7). 1. –1). 3). (1. 3. 2 }. (i) 52. 1). A = {x. D ⊂ C 2. –1). –1). (1. (8. 6). Range = (– ∞. 0] (ii) Domain of function = {x : –3 ≤ x ≤ 3} Range of function = {x : 0 ≤ x ≤ 3} (i) f (0) = –5 (ii) f (7) = 9 (iii) f (–3) = –11 4. 4. 14. 5.} Range of R = {5. 0). R = {(1. 3). of relations from A into B = 26 9. 3. R = {(1. 6)} (ii) Domain of R = {1. (5. Domain of R = Z Range of R = Z EXERCISE 2. 1). 3. (3. 2. (1. 7). Range = {1} (ii) yes. –1). (0. (–1. (1. 8. 6.4). (6. Domain of R = {5. 2. 2). 7. –1).. 7}. –1). (0. 6} (iii) Range of R = {1. (2. (3. 3). (1. 8)} Domain of R = {1. 14} is 2. y) : y = x – 2 for x = 5. 1}. (2 4). 12} Co domain of R = {1. (1. 4. (3. 343)} 8. (1. 6). 5. (2. (3. (i) Domain = R. 2. Range of R = {3. 4. 4. 5. 1). 1) EXERCISE 2.3 1. (3. 6. 4. 2. Range = {1. 4. 11. 3. 2. 9). 2) 412 (iii) 5 (ii) Range = [2. 10. (2. 5} 5. 17}. Domain = {2. Domain of R = {0. 1. (1. 4). 0. (7. 6. 4). (4. 6). ∞) (ii) t (28) = t (–10) = 14 (iii) Range = R (iv) 100 . (1. (i) t (0) = 32 5. (i) Range = (– ∞. 6). Domain = (2.. No. (3.2). 7} (iii) No. 3} Range of R = {6. (1. remaining elements of A × A are (–1. 8} bl 3.2 he 1. 9. (4. (i) yes. (3. (6. . A = {–1. 2. 10} 7. (5. 4). 8.3). 6). 6)} (i) R = {(x. 9). 6. 8. (2. 8). 6. 6. 14}. 27). 125). 2. 0). 7. (i) R = {(1. (5. (2. 6). 6. 3. R = {(1. 7} (ii) R = {(5. 12. 6). 6} © no N C tt E o R be T re pu 4. 4). 12)} Domain of R = {1. 3. (7. 3. R = {(2. 4} Range of R = {3. 3).5)}. 9.. 4). 2.436 MATHEMATICS 10. tan x = − . cos x = − . tan x = 5 4 5 3 3 4. 2. 6. No 12. 12° 36′ 4π 3 (iii) 300° (iii) 26π 9 (iv) 210° (iv) © no N C tt E o R be T re pu (i) Range of f = {3.1 5π 36 (i) 39° 22′ 30″ 19π 72 (ii) –229° 5′ 29″ (ii) – 3.ANSWERS 437 Miscellaneous Exercise on Chapter 2 2. (i) Yes. 20π 3 (iii) 6. 13 } bl 10.1 3. ∞) Domain = R. cosec x = – . tan x = 3 . x≠ ⎜ ⎟x= 2x − 3 2 ⎝g⎠ 1. 7. 4. 11. cot x = − 13 12 13 5 12 . 7.2 1. 5. cos x = – . is ⎛ f ⎞ x +1 3 . (ii) No 11. Domain of function is set of real numbers except 6 and 2. tan x = − . sin x = − . cosec x = – . 5 : 4 7 25 EXERCISE 3. sin x = − 12 13 5 12 5 . sec x = − . a = 2. cosec x = . Domain = [1. EXERCISE 3. cosec x = – . Range = [0. sin x = − 3 2 1 . 12π 4. sec x = −2 . cos x = . (i) 2 15 (ii) 1 5 5. sec x = − . cot x = 2 3 3 5 4 5 3 4 2. (i) No (ii) No (iii) No (f – g) x = – x + 4 he 2. b = – 1 9. cot x = − 3 5 4 4 3 4 5 3 5 4 3. 5. ∞). Range = non-negative real numbers Range = Any positive real number x such that 0 ≤ x < 1 (f + g) x = 3x – 2 8. n∈Z 6 6 6 5. . sin x = 6. 5 13 12 13 12 . . sec x = − . cos x = − . nπ + . cosec x = . . 5 2 5 . 3 2 10.– . 6 3 . x= nπ or x = nπ. bl EXERCISE 3. cot x = − 13 5 13 12 5 1 2 7. n ∈ Z 3 3 3 2.– 2 3 3 10. .3 (i) 3 +1 2 2 (ii) 2 – 3 is 5. x = nπ + ( − 1)n 8.2 5 5 9. nπ + . 2 8. n ∈ Z 4 3 9. n ∈ Z 3 4.n ∈ Z 3 3 Miscellaneous Exercise on Chapter 3 8. π π x = (2n + 1) . 4 + 15 4 4 . 8 + 2 15 8 − 2 15 . nπ + (–1)n .n∈Z 2 2 8 nπ π .n∈Z 6 6 6 6. or + . . . π 4π π . 1 he 438 EXERCISE 3. n ∈ Z 3 3 3 © no N C tt E o R be T re pu 1. or n π ± . x= 7π π or (2n + 1) .MATHEMATICS 5. n ∈ Z 6 2 nπ nπ 3π . or 2nπ ± . 2nπ ± .4 3. x= 7. π 5π π . 3 9. 5π 11π 5π . 7π 11π 7π . − 13. 14 + 28 i 3. 17 5 +i 3 3 8. − 4. −3π −3π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ 4 4 ⎠ ⎝ π π⎞ ⎛ π π 7 . −22 107 −i 3 27 11. 9. 2. 3 4. i 14. −1 ± 7 i 4 3. 3. 3π 3π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ 4 4 ⎠ ⎝ 5. 7.2 −2π 3 5 3 −i 14 14 2 . ± 3 i 5. 0 5. 27 19 21i − 5 10 bl 9. 2 ⎜ cos + i sin ⎟ 8 . 1± 7 i 2 7. −1 ± 7 i 2 2 –1 ± 7i –2 2 ± 34 i 2 3 . 6. –7 2 i 2 EXERCISE 5.3 1. −3 ± 11 i 2 1 2. cos + i sin 6 6 2 2 ⎝ ⎠ 6. 2 – 7 i 6.1 2. –4 is 242 − 26i 10. −1 ± 7 i 2 2 8. 5π 6 –π –π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ 4 4 ⎠ ⎝ © no N C tt E o R be T re pu 1 . 2 2 1 i 2 10. −3 ± 3 3 i 2 4.ANSWERS 439 EXERCISE 5. 4 3 +i 25 25 12. i he 1. 3 (cos π + i sin π) EXERCISE 5. 2. 1 1. x ≥ –1.. 3. 1. 0. –6) 11. ∞) 16. – 2. 2 2 i 2 7 . (i) 3π 3π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ 4 4 ⎠ ⎝ 8. (ii) 4 4 ⎠ ⎝ 5. 1. (4.} (ii) (–2. 14. (i) {1.8). (6. 4] 2 . 19.} 2.. (10. x = 3. 1} (ii) (–∞. (–∞. (i) No Solution (ii) {.10). (–∞.12) 25. .3. 2) 4. 4} (ii) {. 2] 12. (–4. – 2. –1. 307 + 599 i 442 2 14 ± i 3 21 is 3. 7. −2 . 2 – 2 i 18. 3. –3) 9.. 1± 12. (–∞. (i) {–1.440 MATHEMATICS Miscellaneous Exercise on Chapter 5 (i) 6. –3] 18. 9. 0 20. – 4. (– ∞. (7. x ≥ – x > – 1.. (4.. Greater than or equal to 35 22. (– ∞.. – 1. ∞) 17. – 3} 3. 4 © no N C tt E o R be T re pu EXERCISE 6. 10. (5.4.9) 24. . 0. ∞) 5. (8. Greater than or equal to 8 but less than or equal to 22 . y = – 3 bl 1. (– ∞.7). 7 21. 0. x < 3.2. 1 5 2 ± i 27 27 1 3π 2 4 13. (–∞. 20. Greater than or equal to 82 23. (ii) 0 5 17. 2] 13. 2. (i) {.. 9 cm 26. 2 4 ± i 3 3 2 15. 6) 10. 2.. he 3π 3π ⎞ ⎛ 2 ⎜ cos + i sin ⎟ . ∞) 6. 8. 2] 15. (–∞. 120] 14. – 3. (–∞. 4. 3.2 2. . 5.ANSWERS 441 EXERCISE 6. 6. © no N C tt E o R be T re pu bl is he 1. 9. 10.3 1. . 2. © no N C tt E o R be T re pu bl is he 7. EXERCISE 6.442 MATHEMATICS 8. 4. is bl © no N C tt E o R be T re pu ANSWERS 443 . 6.he 3. 7. 8. 5. 12.444 MATHEMATICS 10. 14. . © no N C tt E o R be T re pu bl is he 9. 11. 13. 108 5.5 litres but less than 900 litres. 2. 2] ⎡ 11 ⎤ ⎢1. (i) 125. More than 562. 5. 9. 7) 9. (5. 3 ⎥ ⎣ ⎦ © no N C tt E o R be T re pu 6. 7. 11] 11. 13. 3] 2.8 EXERCISE 7. More than 320 litres but less than 1280 litres. 1] 4.ANSWERS 445 is he 15. ⎜ 3 ⎥⎦ ⎝ 3 bl 3. ∞) 10. (–5. 2] ⎛ – 80 – 10 ⎤ . [2. 14.6 ≤ MA ≤ 16. 336 . 5040 4. 20 3. (– 23. [– 7. (0. (ii) 60. [– 4. Miscellaneous Exercise on Chapter 6 1. Between 20°C and 25°C 12. 8 6.1 1. (–1. 5) 8. (ii) 4 (i) 360. 210 7. 40 6 . 200 © no N C tt E o R be T re pu 1. 2880 3. 104060401 11. x6 6 x 4 15 x 2 6. x5 5 x3 10 10 5 1 + + + 3+ 5 x+ 243 81 27 9 x 3x x 5. 35 4. 33810 (i) 1814400. 11040808032 10. 4. 1440 5. 45 5. 2000 9. 64 x –576 x + 2160 x – 4320 x3 + 4860 x2 – 2916 x + 729 2. 120. (iii) 25401600 bl 1. 2(x6 + 15x4 + 15x2 + 1).446 MATHEMATICS EXERCISE 7. (1. (iii) 1632 6. (ii) 2419200.1 1. 64 EXERCISE 7. (i) 5. 778320 3. 60 56 6.3 Miscellaneous Exercise on Chapter 7 1. 9509900499 20 15 x2 6 1 4 x x6 7. 40 8. (ii) 6 6. 3960 4. (i) 504. 198 10000 > 1000 8. (i) 40320. 151200 EXERCISE 8.4 2. (i) 30. 11. he EXERCISE 7. (ii) 588. 5. 420 10. (ii) 720. 1–10x + 40x2 – 80x3 + 80x4 – 32x5 32 40 20 5 3 x5 x x 5 x 8 32 x5 x3 6 5 4 3. 3600 4. 4536 3. 9 7. 8(a3b + ab3). (iii) 240 10. 120 9.2 1. 22 C 7 + 22 C 10 11. 884736 9. 9. (ii) 15120 2. 40320 is 504 2. 50400 7. 30.1) 12. (ii) 18 5. 4C 1 × 48 C 4 2. No 3. (i) 3. 48 8. 28 4. 907200 8. y r −105 9 35 12 x . . 625. 65. 729 10. 0. –125. 25. . 61236 x5y5 10. 15. 24. 8. . r = 3 9 7 is Miscellaneous Exercise on Chapter 8 1. −1. 2. – 101376 3. 1512 16 8 32 16 x 2 x3 x 4 + 2 − 3 + 4 − 4x + + + −5 x x 2 2 16 x x 10. 18564 8. 27x6 – 54ax5 + 117a2x4 – 116a3x3 + 117a4x2 – 54a5x + 27a6 9. m = 4 7. 4. 15625 6.. 16 and 32 8. x 8 48 12. 5. 323. 12. he 1. 11. 107. .2 4. EXERCISE 9. –3125. ( −1)r 12Cr . 35 2. 2. 396 6 7. 49 128 360 23 3 + 11 + 35 + 107 + 323 + . 1 1 1 5 7 4. . .ANSWERS 447 EXERCISE 8. 7. 3 9 21 75 . b = 5. a = 3. .x12−2r . 2a8 + 12a6 – 10a4 – 4a2 + 2 8. n = 6 2. n = 10 171 © no N C tt E o R be T re pu bl 3. −1 −1 −1 −1 ⎛ −1 ⎞ ⎛ −1 ⎞ ⎛ −1 ⎞ ⎛ −1 ⎞ . a = 5. 35. 2 3 4 5 6 1. . 3. 11. 3.x24−r . – 1+ ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + . n = 7. . and 6 6 2 6 6 5. .9510 6. 21 and 2 2 2 2 9.y r 6... 93 3.. – 1760 x9y3 ( −1)r 6Cr .1 1 2 3 4 5 . − . 8. 2 6 24 120 ⎝ 2 ⎠ ⎝ 6 ⎠ ⎝ 24 ⎠ ⎝ 120 ⎠ . 1. 3 3 3 81 9 19. –1 30. . 3072 bl EXERCISE 9.. 1⎡ 20 1 − ( 0.. 64. . 14 7.3 1. 2. –1. 9 and 27 16. 2059 −4 −8 −16 80 8 . ( n ( n + 1) ( n + 2 ) ( n + 3) 4 . 2 5 5 2 2 5 ( ) 16 16 n . 10n − 1 − n 18. 30 (2n) 2 32. 1. rR 21. – 2187 © no N C tt E o R be T re pu 2. 2. ± 1 ⎛ n ⎞ 3 + 1 ⎜ 3 2 − 1⎟ ⎜ ⎟ ⎝ ⎠ ) ( 9. 2..2 2.or 4. x2 –16x + 25 = 0 27. (a) 13th .2. 496 20. Rs 500 (1. (c) 9th ( 8.1) ⎤ ⎦ 6⎣ ( x3 1 − x 2 n 10. 17. –6. . 98450 4. 1− x ) 2 5 2 2 5 5 2 12. n= n ( n + 1) ( n + 2 ) 3 2. 9 8. 1002001 ⎡1 − ( −a )n ⎤ ⎣ ⎦ 1+ a 7.16. 2q 10. 4 6. Terms are . 3.. and 2 3 5 2 + 2 + 1 + 0 + (–1) + . 120. 20 and 23 17. EXERCISE 9. 7 2 11. ) 3 11 3 −1 2 14. he 1. 15. (b) 12th. 27 16.448 MATHEMATICS 13..4 1. 480. 1 is n ( 5n + 7 ) 2 13.1. ) 31. 5 or 20 6. 11. 0. . 3 5 8 14. 5. 4 179 321 14. Rs 245 18. 22 + 13. − 32 . or . r = or .− 8. 2 −1 7 7 15.. 12. 14. 0 9. 220 2n 4.1)10 EXERCISE 9. 1. 5 5 . –24 26. or − and – 1 2 2 14. 11 7. 3n (n + 1) (n + 3) 8. 135° 1 1 and 1. ⎜ . 32 © no N C tt E o R be T re pu 2. 449 4. ) ( 50 n 5n 2n 2 10 − 1 − 1 − 10− n − . 1210 10. is 3. 9. or 8. 104. – a) and − 3a. (ii) x2 − x1 7. (0. x = 1 ⎛ 15 ⎞ 4. 20. a). 11 5. a). 160.5 Crores 2 . 4 bl Miscellaneous Exercise on Chapter 9 ( 23. – a). Rs 16680 31. 8. 2840 2 ( n + 2) 12 he ( n ( n + 1) 3n2 + 5n + 1 6 ) ) n ( n + 1) ( 2n + 1) + 2 2n − 1 6 n ( 2n + 1) ( 2n − 1) 3 10. − 1 2 10.ANSWERS ) 6. 8729 8. 121 square unit. 1 and 2.000 EXERCISE 10. (i) ( n 2 n + 3n + 5 3 27. (i) y2 − y1 . 0 ⎟ ⎝2 ⎠ ( 3a. 6 12. (0. 5. Rs 17000. n ( n + 1) 12 ( 3n 2 n n +1 4. ± 3 6. 4 11.0 or (0. 3050 9. 8. + 23n + 34 ( 5. Rs 39100 32. Rs 5120 ) 25. 1680 ( n 2n 2 + 9n + 13 24 29. (0. or – 1 and –2. n ( n + 1) 7. – 3 11.0 ) 5. 1 . and 3. 16.1 1. 2 ( ) 2. Rs 43690 ) 30. 28. (ii) 81 9 3 27 21. 25 days ) 22. x – 2y + 10 = 0 3. 0. © no N C tt E o R be T re pu EXERCISE 10.2 4. x + y = 5 3x + y − 2 = 0 and 3x + y + 2 = 0 L= 16. x + 2y – 6 = 0. 5 units ( ) ( ) 3 + 2 x + 2 3 – 1 y = 8 3 + 1 or ( ) ( ) 3 − 2 x + 1 + 2 3 y = –1 + 8 3 . 5x – y + 20 = 0 8. 4 6 (iii) x y 3 (ii) 3 + −2 =1. 2. 0) 6. 120° (ii) x cos 90° + y sin 90° = 2. 1. (iii) x cos 315° + y sin 315° = 2 2 . 0 7 7 3 3 2. intercept with y-axis = − and no intercept with x-axis. (i) 1 1 5 5 y = − x + 0. 2 . 1340 litres. 2 2 . 315° 65 1 p+r units. is 12. 3x – 4y + 8 = 0 3 x + y = 10 11.450 MATHEMATICS EXERCISE 10. 2x + y – 6 = 0 10. −2.942 90 15. y + 7x = 21 9. − 2. 3 3 (i) x cos 120° + y sin 120° = 4. (ii) units. − . 2x – 9y + 85 = 0 bl 14. 4. y = mx ) 5. y = 0 and x = 0 17. 0. (i) 7. ( ) ( 6. (ii) y = −2 x + . . 90°. (i) x y + = 1.3 19. 12. he 1. 2 2 2 y = − . 5x + 3y + 2 = 0 7. 0) and (8. (1 + n)x + 3(1 + n)y = n +11 13.6. (iii) y = 0x + 0. 2kx + hy = 3kh.192 ( C − 20 ) + 124. 3. . 17 2 l 5. 4. 4 . 30° and 150° 22 9 10. 2x + y + 6 = 0 3 −1 y = 4 ( ) 3 –1 9. 3x – 4y + 18 = 0 8. x − 3y + 2 3 = 0 3 +1 x − 2. (– 2. ⎜ .1 1. 2x – 3y + 18 = 0 © no N C tt E o R be T re pu 6. (c) 6 or 1 5. m = 2. r = 451 65 6. 2 x − 3 y = 6. x2 + y2 + 2ax + 2by + 2b2 = 0 7. x2 + y2 – 2x – 2y = 0 5. 36x2 + 36y2 – 36x – 18y + 11 = 0 4. 3). r = 4 4 11. − ⎟ . − ⎟ ⎝ 25 25 ⎠ 13. x2 + y2 – 6x – 8y + 15 = 0 53 9. k2 square units 12.c = 2 2 2 Miscellaneous Exercise on Chapter 10 7π . x2 + y2 + 4x – 6y –3 = 0 3. 15. 23 5 units 18 1± 5 2 7 16. x2 + y2 – 7x + 5y – 14 = 0 12.1 6 he 17. 19. 3. y = 1. c(–5. (–1. ⎜ . 119x + 102y = 125 EXERCISE 11. (b) ± 2. r = 10. x2 + y2 – 4y = 0 2. x2 + y2 + 4x – 21 = 0 & x2 + y2 – 12x + 11 = 0 . The line is parallel to x .0 ⎟ ⎝5 ⎠ 24. x = 1. 8.axis or parallel to y-axis 17. 0) . ⎜ 0 . − 3 x + 2 y = 6 8 ⎞ ⎛ 32 ⎞ ⎛ 4. ⎟ 3⎠ ⎝ 3 ⎠ ⎝ x=− 5 22 7. 3x – y = 7. 4). c(2. c ( 1 1 . 18x + 12y + 11 = 0 ⎛ 13 ⎞ 22. (a) 3. x + 3y = 9 14. r = 6 8. c(4. – 5). bl sin ( – θ) –θ 2 sin 2 is 1. y – x = 1. – 4). 1 : 2 15. 5 11. ⎜ 0.ANSWERS ⎛ 68 49 ⎞ 14. 13x + 13y = 6 9. 18. 2x + y = 5 1 5 . 21. 0) axis . –4).axis.0). Major axis = 12. Minor axis = 8 . axis .3 1.x .x . e = Latus rectum = 16 3 2. 0). axis . x2 + y2 – ax – by = 0 14. directrix y = . axis .y . EXERCISE 11. length of the Latus rectum = 6 2 2 is 3. length of the Latus rectum = 12 3 3 ). F ( . length of the Latus rectum = 16 –9 9 ) . since the distance of the point to the centre of the circle is less than the radius of the circle. x2 + y2 – 4x – 4y = 5 15. length of the Latus rectum = 8 4. directrix x = 2. Major axis = 8. ± 5). 4 . directrix x = – 3. directrix x = – 7. y2 = 12x 12. 0). 5 . y2 = 24x 10. 0).y . 5 8 5 3. 0). Major axis = 10. directrix y = 4. he 1.y . F (0. x2 = – 12y 11. 0). F (0. F (0. axis . V ( ± 4.axis.452 MATHEMATICS 13.x . y2 = –8x 8. axis . Inside the circle. e = Latus rectum = 20 .axis. 2x2 = 25y EXERCISE 11. 6 9 2 7 .2 2. directrix y = – . length of the Latus rectum = 9 4 4 © no N C tt E o R be T re pu 6. V (± 6. 2y2 = 9x 9. F (3. F ( ± 7 . F (–2. F (0.axis.axis. ± 21 ). e = Latus rectum = 21 .axis. length of the Latus rectum = 10 2 bl 5 2 5. F (± 20 . Minor axis = 4 . Minor axis = 6 . V (0. Minor axis = 2 . Minor axis = 10 . x2 y 2 + =1 25 9 11. Minor axis = 4 . ± 4 2 ).0). Minor axis = 20 . 2 Latus rectum = 5 Latus rectum = 13 . 2 is 6. F (0. e = bl Latus rectum = 10 © no N C tt E o R be T re pu 7. V (0.± 20). Major axis =12 . F (0. 4 1 2 9. Major axis = 6 . Minor axis = 4 . ± 10 3 ). Major axis = 20. x2 y 2 + =1 36 20 13. V ( ± 3. 3 . e = Latus rectum = 2 2 . Major axis = 8 . x2 y2 + =1 169 144 16.± 6). 7 he 5. x2 y 2 + =1 16 7 18. e = Latus rectum = ( 4 3 ) 8. Minor axis = 12 . V (0. Major axis =14 . F ( ± 5 . ± 75 ). e = 72 7 3 . Major axis =40 . x2 y 2 + =1 1 5 15. e = 453 3 . V (0.0).± 15 . F 0. F (0. x2 y2 + =1 144 169 12. x2 y 2 + =1 9 4 14. e = Latus rectum = 15 . x2 y 2 + =1 25 9 5 . x2 y 2 + =1 64 100 17. 3 8 3 10. V (0. 0).± 10).ANSWERS 4. 0). ± 4). V (± 7. F (± 13 . y 2 x2 − =1 9 16 10. Latus rectum = 2 4 7. Latus rectum = 18 1. y 2 x2 − =1 25 39 9. e = he EXERCISE 11. Vertices (0. Foci (0 ± 6). 1. e = is 3. x2 y 2 + =1 25 9 .± ). Focus is at the mid-point of the given diameter. MATHEMATICS x2 y 2 + =1 10 40 2 2 20. Vertices (0. Vertices (± 6. x2 y2 − =1 4 12 14. Vertices (0. ± 3). x2 y2 − =1 25 20 13. 18 sq units 4. 2.4 6. 0). ± 13 ).11 m (approx.23 m (approx. x2 y2 − =1 16 9 11. Vertices (0. e = 2. ± 4). Latus rectum = 4 2 2. e = . Foci (0. ± 65 ). Foci (± 10.) 5. e = 13 . x + 4y = 52 or x2 y 2 + =1 52 13 9 5 .454 19. Latus rectum = 5 5 3 3 © no N C tt E o R be T re pu 5. 0). Foci (0. ± 2).56m (approx. Latus rectum = 3 3 bl 4.± 5 64 . y2 x2 − =1 25 144 12. x2 y 2 − =1 4 5 8. y 2 x2 − =1 5 5 Miscellaneous Exercise on Chapter 11 1. 0). Foci (0.) 7. x2 y 2 + =1 81 9 8. Foci (± 5. Vertices (± 4. 8 3a 6. e = 49 65 . 9. Latus rectum = 9 2 6 2 14 14 4 5 ).) 3. 0). x2 9 y 2 − =1 49 343 15. 2. plane (ii) (x. x2 + y 2 + z 2 − 2x − 7 y + 2z = k 2 – 109 2 EXERCISE 13. y and z . π 6.ANSWERS 455 EXERCISE 12. 2 14. 0) 5. 2 : 3 5.3 ) 2. (1. a b 11 4 11. 2) Miscellaneous Exercise on Chapter 12 1. 1 π 1 4 .coordinates are zero 3. b = − 2. b 13. II. a b 1 2 22 ⎞ ⎛ 2. 5 7. – 10. (8. 10.coordinate is zero (iii) Eight 43 (iii) 2 26 (iv) 2 5 5. 1 : 2 ⎝ 5 5 5 ⎠ 3. − 16. – 6. c=2 3 4. y . a = – 2. 19 2 8. 1 π 4. 2. (i) 2 5 (ii) 4. VI. y. 0) and (0. 7. VIII. (i) ⎜ . 0) 2. III.17 . IV. ⎜ π − ⎟ 7 ⎠ ⎝ 3. 7 16 . .2 EXERCISE 12.3 © no N C tt E o R be T re pu bl ⎛ −4 1 27 ⎞ 1. 34 . 6) 6. 6 5. – 4. V. – 2. I. 9x2 + 25y2 + 25z2 – 225 = 0 is 1.1 1. (4. (6. 1 15. 8) 3. ⎟ (ii) ( − 8. − 9. 108 7 12. (i) XY . VII 4.1 1. x – 2z = 0 he EXERCISE 12. (0. – 2). – 2. + a n −1 7. 2 Limit does not exist at x = 1 Limit does not exist at x = 0 0 28. 24. 1 4. For lim f (x) to exists. (i) 3x2 (ii) 2x – 3 (iii) −2 x3 (iv) −2 ( x − 1)2 6. ( 2 (ii) 4ax ax + b (i) (iii) (v) (vii) (ii) 20x3 – 15x2 + 6x – 4 (iii) –2 x(3x – 2) –12 36 – + 2 (vi) ( x + 1) (3x – 1)2 x5 x10 cos 2x 5sec x tan x – 4sin x – 3cosec2 x – 5 cosec x cot x 2sec2 x – 7sec x tan x −3 ( 5 + 2x ) x4 10..2 bl 32. a = 0. 3. (i) 2x − a − b 8.. 29. 4 21. (i) 2 (v) 11. 2 of m and n. b = 4 17. 27. – sin x (ii) sec x tan x (iv) – cosec x cot x (vi) 5cos x+ 6sin x (iv) 15x4 + 24 x5 . © no N C tt E o R be T re pu EXERCISE 13. Limit does not exist at x = 0 he 456 lim f (x) = 0 and lim f (x) = (a – a ) (a – a ) . a −b ) (iii) ( x − b )2 nx n − anx n −1 − x n + a n ( x − a )2 9 .. 1 23. nx n −1 + a(n − 1)x n − 2 + a 2 (n − 2)x n −3 + . lim f (x) exists for any integral value x→0 x→1 1. 20 2. 99 3. we need m = n. lim f (x) exists for all a ≠ 0. 6 26. 0 18.MATHEMATICS a +1 b 0 22.. 19. 20. x →a is 31. (a – a ) 1 2 x x →a x → a1 30. 25. na ( ax + b ) 2 n −1 ax 2 10. x 3 5 x cos x 3 x sin x 20 sin x 12cos x 23. 22.ANSWERS 457 Miscellaneous Exercise on Chapter 13 4. 1 x2 he 1. cx d 2. 2 −apx 2 − 2bpx + ar − bq ( px 2 ( x – 1) 6. + qx + r ) 2 2 apx 2 9. tan 2 x x cos x 26. x ≠ 0.1 7. 8. 1 1 sin x 19. 2 sin x p q cos x 2a x cos x x tan x 1 sin x 35 15 x cos x 28 cos x 28 x sin x 15sin x 3x 7 cos x 2 bx c 2 is qr x2 π⎞ ⎛ (iii) cos (x + 1) (iv) −sin ⎜ x − ⎟ 8⎠ ⎝ 2b sin x x3 bl 3. x 2 sin x sin x 2 x cos x 24. 2sec x tan x 18. 2 2ax b . n–1 n sin x cos x . 2c (ax+b) (cx + d) + a (cx + d)2 ps −2 ad bc 5. 4a x5 © no N C tt E o R be T re pu 11. (i) – 1 (ii) 14. x ( ax + b )n−1 ( cx + d )m−1 ⎣⎡ mc ( ax + b ) + na ( cx + d )⎦⎤ 15. 2bpx bq ar ax b 12. 2 sin x cos x bc cos x ad sin x bd c d cos x 2 sec x 1 2 cos a cos 2 x 21. 1 13. 20. cos (x+a) 16. – cosec3 x – cosec x cot2 x 17. q sin x ax 2 25. This is not a statement because from the context it is not clear which room is referred here and the term bold is not precisely defined. (ix) It is not clear from the context which day is referred and therefore. it is a statement. . (ii) She is an engineering student. it is not a statement.1 © no N C tt E o R be T re pu bl (i) This sentence is always false because the maximum number of days in a month is 31. Therefore. (v) This sentence is sometimes true and sometimes false. it is not a statement. For example. 2. it is a statement. (iv) This sentence is sometimes true and sometimes not true. is not a statement. For example the square of 2 is even number and the square of 3 is an odd number. squares and rhombus have equal length whereas rectangles and trapezium have unequal length. Therefore. 1 tanx 2 x tan x . (ii) This is not a statement because for some people mathematics can be easy and for some others it can be difficult. 1. 1 sec x tan x he 458 sin x n x cos x sin n 1 x is EXERCISE 14. (iii) “cos2θ is always greater than 1/2”. Therefore. it is not a statement. (iii) This sentence is always true because the sum is 12 and it is greater than 10. 29. it is a statement. it is a statement. Therefore. (viii) This sentence is always true and therefore.MATHEMATICS x cos 27. (vi) It is an order and therefore. we cannot say whether the sentence is true or not. This is also not a statement because who ‘she’ is. π 2 sin x x cos x 4 sin 2 x x sec x 1 sec 2 x 1 tanx x sec 2 x 28. 30. (x) This is a true statement because all real numbers can be written in the form a + i × 0. The three examples can be: (i) Everyone in this room is bold. Therefore. we know what θ is. (vii) This sentence is false as the product is (–8). Unless. (i) “There exists”.3 (i) “And”. The component statements are: 2 x = 2 is a root of the equation 3x – x – 10 = 0 2 x = 3 is a root of the equation 3x – x – 10 = 0 2. all integers are negative (False). All real numbers are not complex. 1. the given pairs are negations of each other. The negation is There exists a real number x such that x is not less than x + 1. (iv) “And”. (i) bl 1. is (ii) (iii) (iv) (v) 2. . number 3 is odd (True). Square of an integer is negative. All triangles are equilateral triangles. The sand does not cool down fast at night. The negation is There exists a state in India which does not have a capital. Number 3 is prime. Therefore. The component statements are: Square of an integer is positive. Every natural number is not an integer. the pairs are negations of each other.2 (i) Chennai is not the capital of Tamil Nadu.100 is divisible by 11 and 100 is divisible by 5 (False).ANSWERS 459 EXERCISES 14. (ii) “Or”. The negation of the first statement is “the number x is a rational number. (iii) “There exists”.” which is the same as the second statement” This is because when a number is not irrational. (iii) “And”. it is a rational. (ii) “For every”. The number 2 is not greater than 7. 2 is a complex number. the component statements are: The sand heats up quickily in the sun. EXERCISE 14. The negation of the first statement is “x is an irrational number” which is the same as the second statement. The component statements are: All rational numbers are real. All integers are positive. (ii) (i) (ii) (iii) he © no N C tt E o R be T re pu 3. The negation is There does not exist a number which is equal to its square. Therefore. 100 is divisible by 3. then it will bloom. then they are not parallel The converse is If two lines do not interesect in the same plane. A natural number is odd implies that its square is odd. This statement can be written as “If x is an even number. then your credentials are good. The converse is If a number x in odd. 4. If x is not divisible by 4. The converse is If you do not know how to reason deductively. For the square of a natural number to be odd. If x is divisible by 4. then they are parallel The contrapositive is If something is not at low temperature. then x is divisible by 4”. it is sufficient that the number is odd If the square of a natural number is not odd. he EXERCISE 14. The negation of the statement in (i) is “There exists real number x and y for which x + y ≠ y + x”. then the natural number is not odd. then it is cold The contrapositive is If you know how to reason deductively. If you get a job. If the banana tree stays warm for a month.460 MATHEMATICS No. (v) (i) © no N C tt E o R be T re pu 2. The contrapositive is If a number x is not odd. then it is a prime number. (i) Exclusive (ii) Inclusive (iii) Exclusive 3. then you can not comprehend geometry. then you can comprehend geometry. The converse is. The contrapositive is If two lines intersect in the same plane. then x is not a prime number. For a natural number to be odd it is necessary that its square is odd.4 (ii) (iii) (iv) (v) 3. is (i) (ii) (iii) (iv) bl 1. (i) (ii) . then x is an even number. A natural number is odd only if its square is odd. then it is not cold The converse is If something is at low temperature. instead of the statement given in (ii). then x is not an even number. The contrapositive is. therefore 11 is irrational. Miscellaneous Exercise on Chapter 14 (i) (ii) (iii) (iv) 2. then it is not hot outside. If you get A+ in the class. a (i) (ii) b (i) (ii) EXERCISE 14.ANSWERS If diagonals of a quadrilateral bisect each other. then it is not a sunny day. A chord which is not a dimaeter gives the counter example. The contrapositive is If I do not go to a beach. (iii) The converse is If you feel thirsty. By definition of the chord. then I go to a beach. The statement can be written as “If a positive integer is prime. (i) 1. then it is hot outside. This can be shown by giving a counter example. In the equation of an ellipse if we put a = b. There exists a real number x such that neither x > 1 nor x < 1. then it is a sunny day. Since 11 is a prime number. The converse of the statement is If a positive integer has no divisors other than 1 and itself. There exists a positive real number x such that x–1 is not positive. then you do all the exercises in the book.5 (i) False. The converse of the statement is If I go to beach. The contrapositive is If you do not feel thirsty. by the rule of inequality bl is 5. Contrapositive Converse Contrapositive Converse he (iii) (iv) 4. then it is a circle (Direct Method) (iv) True. then it has no divisors other than 1 and itself. . (ii) False. 461 © no N C tt E o R be T re pu (v) False. it should intersect the circle in two points. then it is a parallelogram. There exists a cat which does not scratch. The contrapositive of the statement is If positive integer has divisors other than 1 and itself then it is not prime. There does not exist a number x such that 0 < x < 1. (iii) True. (ii) The given statement can be written as “If it is a sunny day. then it is a prime. 9. 1.23 8. 5. (ii) B Miscellaneous Exercise on Chapter 15 1.33 4. 3. 9.4 EXERCISE 15. 2. 7 5. 6. 4. 24. 8. The compound statement with “Or” is 25 is a multiple of 5 or 8 This is true statement.52. (i) 10. 43.25 7. 8 2.34 12. B 4. 7. 6. 3.99 (ii) 10. 43. 107. (iii) If you can access the website. 132 EXERCISE 15.69 9. 2. (ii) You get an A grade if and only if you do all the homework regularly.09 6. Highest Chemistry and lowest Mathematics 7. (i) B.32 6. 5.5.35 © no N C tt E o R be T re pu 1.1.4 3. 3 9. Weight 3. 16. 1.25 n + 1 n2 − 1 . 2 12 5.2 1.28 EXERCISE 15. 20. 93. 74. 16 7. 105. (ii) If it rains. 27. 157.1 11. 1. then you have a password. A 2. 100.2. 29. 12 5.1 bl is he 3.5 4. (iii) A quadrilateral is equiangular if and only if it is a rectangle. 11. The compound statement with “And” is 25 is a multiple of 5 and 8 This is a false statement.55. 10.4 8. (i) You watch television if and only if your mind in free. 19. 10. 2276 10. 8 3.036 . 5. then there is traffic jam.3 1.92 10. Same as Q1 in Exercise 14. Y 5. 7. 64.98 6.462 MATHEMATICS (i) If you log on to the server. 4.27 3. then you pay a subscription fee. 2. .4. H5. H5. 2. NDN. H2. (6. 13.2). YB3.2. HTH. NNN} {T. . TTHH. .4). {HHH. (2. T2. (4. A = {(3. XG1. B1. TB1.2.2.1 he is © no N C tt E o R be T re pu 4.1). H6. H6} {6. THHT. B and C are mutually exclusive. 4. and “getting at least two tails” (ii) “Getting no heads”. HTTT. (1..4. 7. (6. H1.6). H1. H26. THT.3). H6. 3HT.6)..3)..6). C and D (ii) A and C (iii) B and D (i) “Getting at least two heads”. TTH.4). 1TH. 4T. H2. . T5.1). H65. NND. T2. D∩E = φ. 6. H41. E∩ F′ = φ. T1. TB2. “getting exactly one head” and “getting at least two heads” . WR. R6.6).2). A and C. or 3.6).4). (5. (1.1).1). T} {XB1.5}.. (1.3).. (1.5. T5.3). } bl 1. (2. (5. NDD.. B4. GB. DDN. R2.3). 5. 6} (ii) φ (iii) {3. TTHT. y) : x. 6T} {TR1. TB3. H42. HTT.6). 6}. H43. F′ = {1. 2). W6. H21..6).5). .2 1.6). TTTT} {H1. 5HT. B and C.1. 2} 3.5.2. A – C = {1.1). (2. HHTT. B6} (i) {BB.. (2. (6. D – E = {1. THH.6). W3.... . (6. 12. (4. (5. (5.4). 10. HTTH. 3.5. 15. y = 1. 6H. 1. (2. (2.2).. (4.6). (6. W1. (4. T4. (2. (3. R3. 5.3)} {1HH. TR2. YG3. H22. R4. (3. H66} {(1.6).4). EXERCISE 16.. 5TH. 2. T3. H23.6)} A and B. (2. 2. 3} (v) {6} (vi) {3. (2. 4H.3}. 3TH. (6. 4). (1. 5. 1). No. YG4. (i) A and B. T1.6).6). (5. H64. GG} (ii) {0. HHTH.6).6).2).6).2.. 5TT.5). 4. H4. DND.3). HTHT. (3.6).6)} B = {(1. A∩B = φ. 2T.5). T4. (6. TTT} {(x. (4. H24.2).2). W4. 2). (5. 6}.2). 6} (iv) {1. T3. THHH. (2.. (1. 5. (5. HT. 11. H5. BG. A∪B = {1.. (1. 2. H61. . 8.. H5.. B2.1. (2. (3. 4.1. W2. B5. XG2.5). 4). DNN. H63. 5HH. .2). 9. (5. HHT. 4. (4. (2. 1HT. 2. B∪C = {3. H4. H45. (2. THTH.5).6).. T6} {DDD. 14. B3. E∩F = {6}. XB2.ANSWERS 463 EXERCISE 16. (1. H2.6} {(1. WW} [HH. (2. (4. (6. (6. H44. . 6}. TTTH.. H4. H3.. T6} {H1. 1TT. THTT. (4. 3TT. 5. (i) {1. H3. (6. H46. (1.. H62. R5.1).6)} {HHHH. 3. YG5} {R1. HTHH. 2.6)} C ={(3. 3HH. (3. (2. 2} {RW. H3. 16. W5.2). HHHT. H25. 2H. (2.3.2). H3.6). (2.6).3). (3.2).5).464 MATHEMATICS (iii) “Getting at most two tails”.2).2).3).4). (3.5). (3.2). (2. (1. (5. 1). (4.2). (5.1). (4. (4. (6. (6.1).2). (4. (2. (1. (1.6).2). (5. (6. (5. (5. (6. (6. (2. (3. (5. (4.2).6).1).3). (1. (2. (4. (3. (1. A = {(2. (3. (4. (1.4).2).3). (2. (6.5). (4.1)} (i) A′ = {(1.4).6). (1.4).6).1). (4.5).5).4).5). (1. © no N C tt E o R be T re pu bl is he 6. (6.5). (3.4).6). (2.1). (5.4).4). (3. (3. (1.3). (5.2). (6.6).1).4).4).3). (1. (3.2). (i) 1 2 1 5 (ii) (iii) (iv) 0 (v) 2 3 6 6 5.5). (4. (3.2).6).1).2). (6.6). (6.3).3).5). (1. (6.5). (2.3). (3. (4. (6.2).3). (2. (5.2).6)} C = {(1.4). (4.5).2). (6.1).6). (6. (4. (2. (6. (6.4). (2. (6. (5. (1. (6. (4. (4.6).3 1. (2. (6.2).1). (1.5).2).2).1). and getting exactly three tails” $Note There may be other events also as answer to the above question.6)} (vii) B ∩ C = {(1. (2. (1.4).1).6)} 7.4). (5.2).5). (3.1).2).6).3). (6.3). (4.3).1).6).1). (5.4).1). (5. (6. (1. (2. (6. (1.2). (6.6)} B = {(1. (1. (2. (1. (1.3). (5. (4. (6. (4. (a) 52 (b) 1 1 1 (ii) (c) (i) 52 13 2 . (6. (6. (1.2).6)} (vi) B ∪ C = {(1. (1. (1. (4. (5.2). (5.3). (i) True (ii) True (iii) True (iv) False (v) False (vi) False EXERCISE 16.6)} = B (ii) B′ = {(2. (3.5).6)} = S (iv) A ∩ B = φ (v) A – C = {(2.1).1).3).3).6).3).2).3).1). (2.3). (2. (5.5).5). (2.3). (3. (6. (3.4).4).2).3). (3. (3.5). (5. (4. (3.4).4).2).4). (1.5). (3.4). (5.6).6). (3.4).1). (3. (5.3). (4. (i) 1 1 (ii) 12 12 6.3). (2. (2.2). (2.2). (6.1). (2.5). 3 5 (e) No 2. (3.6).6). (2.1). 3 4 4. (a) Yes (b) Yes (c) No (d) No 3.6).3). (4. (4. (1.1).4). 1). (4.5).1).5). (2.5).5).5).4).4).1). (3.1).1).4). (4. (6.2)} (viii) A ∩ B′ ∩ C′ = {(2.3). (1. (4. and “getting exactly two tails” (iv) “Getting exactly one head” and “getting exactly two heads” (v) “Getting exactly one tail”.5). (3. (5. (3. (3. (4. (2. (4. (5. (3. (6.3).5). 1). (4.6)} = A (iii) A∪B = {(1. (5.3). (5. “getting exactly two tails”.1).4). (4. (1.5).2).4).3). 4 5 © no N C tt E o R be T re pu 13. 999 1000 33 83 (ii) 3 8 10.65 2 15 Miscellaneous Exercise on Chapter 16 20 30 C3 .00 gain.50 loss.88 (ii) 0.74 19. (a) 17 33 6. (i) 19 30 11 30 (ii) (iii) 16. 0.00) = 7 15 (ii) 0. C2 10000 C2 2 3 7. P(Winning Rs 1.12 (iii) 0.00 loss. 1 38760 is 8. (i) 6 13 (ii) 7 13 11. (i) 2.50) = 9. P (Losing Rs 6. No 17. (ii) 1 − (b) 16 33 C5 60 C5 13 1.52 (iii) 0.5 (iii) 0. P (Losing Rs 3.58 (ii) 0.15 14. (a) 5. Rs 3.00) = 16 4 8 he P ( Winning Rs 4. (i) No. 0. (i) 1 1 3 . Rs 4. (i) 0. (i) bl 12. Rs 1. 13 C1 52 C4 C5 60 C5 3. Re 1.6 21. 4 16 1 3 1 7 1 1 3 1 7 (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) 8 8 2 8 8 8 8 8 8 9 11 10.ANSWERS 465 7. (i) 9990 (b) 1 5040 8.50 gain. 1. 4 5 (c) 9990 C10 C10 10000 . because P(A∩B) must be less than or equal to P(A) and P(B). (ii) Yes 15. P (Losing Re. (i) 1 1 5 (ii) (iii) 2 2 6 4.00 loss. 1 1 .19 (iv) 0.50) = . (i) 0. (i) 5 8 3 8 (ii) 18. 0.55 20. Rs 6.34 9.00) = .
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