CJC MATHEMATICS DEPARTMENT2014 JC2 H2 MATHEMATICS MYE REVISION 1 Image taken from https://www.pinterest.com/pin/47006389832156653/ Topic 1 Differentiation & its Applications Q1 The curve C has parametric equations 2 2 x t t = + , 2 (1 ) y t = ÷ , where t e . (a) Find the range of values of t for which C is decreasing. [3] (b) Find the equation of the normal to the curve at 2 t = . Determine if the normal meets C again. [4] Q2 The variables x and y are related by ( ) 3 ln y x xy = + . Find the value of d d y x when both x and y are equal to 1. [4] Q3 A circular cylinder is inscribed in a right circular cone of base radius 2c and vertical height c. One of the circular ends of the cylinder lies on the base of the cone and the other end is in contact with the inner surface of the cone. If the cylinder has base radius x, find expressions for the volume V and the total surface area S of the cylinder in terms of c and x. If c is fixed and x varies, find the maximum value of V. [8] Prove that S cannot exceed 8πc 2 . [2] Q4 In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo under construction to house deadly weapons. The circumference of the top of the silo lies just within the dome as shown in the diagram below. The power supply is such that the radius r of the dome generated is fixed at 0.5 km. Let x km and y km be the radius and height of the silo respectively. Show that the volume of the silo can be expressed as 3 1 4 y y t | | ÷ | \ . km 3 . Find the exact maximum volume of the silo which can be constructed. [5] A power surge causes the dome to be enlarged (still retaining its hemispherical shape) such that its volume is increasing at a constant rate of 0.75 km 3 s ÷1 . Find the exact rate of increase of surface area of the dome when 1 d 3 kms d 4 r t t ÷ = . [5] [The formula for the volume and surface area of a sphere are V = 3 4 3 r t and A = 4 2 r t .] Instructions Dear students, Work through all of the sums and we will discuss them in Week 7 Lecture & Tutorial Revision. Remember, the learning will only be RICH if you had done the thinking yourself. What we hope ever to do with ease, we must learn first to do with diligence ~ Samuel Johnson~ y x r CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 2 Topic 2 Integration & its Applications Q1 <2013/I/Q11(iii)-(iv)> A curve C has parametric equations x = 3t 2 , y = 2t 3 . A curve L has equation x = y 2 + 1. The diagram shows the parts of C and L for which 0 y > . The curves C and L touch at the point M. (iii) Show that 6 2 4 3 1 0 t t ÷ + = at M. Hence, or otherwise, find the exact coordinates of M. [3] (iv) Find the exact value of the area of the shaded region bounded by C and L for which 0 y > . [6] Q2 2012/I/Q2 Topic 3 Differential Equations Q1 CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 3 Q2 N99/2/14b Use the substitution y = u – 2x to find the general solution of the differential equation dy dx = – 8x + 4y + 1 4x + 2y + 1 . Q3 N2001/2/14(b) A rectangular tank has a horizontal base. Water is flowing into the tank at a constant rate and flows out at a rate which is proportional to the depth of the water in the tank. At time t seconds, the depth of water in the tank is x metres, If the depth is 0.5m, it remains at this constant value. Show that dx dt = –k(2x – 1) , where k is a positive constant. When t = 0, the depth of water in the tank is 0.75 m and is decreasing at a rate of 0.01 ms -1 . Find the time at which the depth of water is 0.55 m. Topic 3 Complex Numbers Q1 (a) Given that ( ) 3 1 i 3 i z + = ÷ , find the exact value of the modulus of z and show that the argument of z is 11 12 t . [4] By first expressing z in the form i x y + , where x and y are real numbers, find the exact real and imaginary parts of z and hence find the exact value of 11 tan 12 t . [4] (b) On a single Argand diagram, sketch the following loci given by (i) 4 3i 5 z ÷ ÷ = , (ii) i 7i z z + > ÷ . [2] Hence, or otherwise, find the greatest and least possible values of arg (z ÷ 6), giving your answers in radians correct to 3 decimal places. [4] Q2 2013 NJC/I/11(b) Solve the equation 4 1 3i z = ÷ + . [3] Hence deduce the roots of the equation ( ) 4 3 i i w + = ÷ . [2] [Express all answers in the form iθ e r , where 0 r > and π θ π. ÷ < s Give r and θ in exact form.] Q3 2013 ACJC/II/2 The polynomial ( ) 4 3 P 2 i 2 i z z a z z a = + + + , ae , has factor 2 i z ÷ . (i) Find the exact value of a. [2] (ii) Solve the equation ( ) P 0 z = , leaving your answers in the form , i re u where 0 and r t u t > ÷ < s . [4] (iii) One of the roots, 1 z , is such that ( ) 1 0 arg 2 z t < < . The locus of points representing z, where 1 arg( ) z z k ÷ = , passes through the origin. Find the exact value of k, and the cartesian equation of this locus. [3] CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 4 Topic 4 Sampling Methods Q1 A company has 6 departments and a total of 240 staff. Each department has a different number of staff. The management, concerned by a large turnover, decides to survey a sample of 30 staff to seek their opinions on the working conditions in the company. Two methods are suggested on how the sample can be selected. Method 1: Use Simple Random Sampling Method 2: Use Stratified Sampling (i) Describe how each method can be carried out. [4] (ii) Give one reason for choosing (a) Method 1 over Method 2, [1] (b) Method 2 over Method 1. [1] Q2 2008 NYJC Prelim/II/Q9a Nanyang Hotel employs 450 hotel staff consisting of 360 general workers, 30 administrative staff and 60 technicians. A random sample of 15 employees is to be chosen to represent the views of all the employees about the hotel. This sample is to be made so that there is as close a representation of all the staff as possible without bias to any particular individual or group. Explain how this can be done, justifying your working. Q3 (a) A school consisting of 600 students has 400 boys and 200 girls. The principal wants to find out what students think of the meals provided in the canteen. On one particular day she selects a sample of 12 pupils. Explain briefly whether a stratified sample might be preferable to a random sample in this context. [2] (b) In order to carry out a statistical test on the leaves of a tree, a student examines 150 leaves that are within reach. Comment on the validity of the sampling method chosen in this context. [1] Topic 5 Correlation & Linear Regression Q1 2013/Prelim/RI/II/9 The data below shows the average height, measured in feet, of cherry trees from age 1 year to age 11 years. Age ( x ) 1 2 3 4 5 6 7 8 9 10 11 Average height ( y ) 6 9.5 13 15 16.5 17.5 18.5 19 19.5 19.7 19.8 (i) Calculate r , the value of the product moment correlation coefficient betweeen x and y . State, with a reason, whether the value of r would be different if the average height is measured in meters instead. [2] (ii) Give a sketch of the scatter diagram for the data. [1] (iii) It is desired to predict the average height of cheery trees beyond age 11 years. Explain why neither a linear nor a quadratic model is likely to be appropriate [2] It is suggested that the average height y can be modelled by the formula ln y a b x = + . (iv) Find the equation of the regression line for the suggested model, and the product moment correlation coefficient between ln x and y . Estimate the average height of cherry trees age 5.5 years, and comment on the reliability of your answer. [4] Q2 2013/Prelim/TPJC/II/9 A particular hospital in Town A observed that the number of patients seeking medical treatment for respiratory problems has changed due to changes in air quality. The hospital decided to monitor the situation and the table below shows the results: Pollutants Index (x) 200 138 124 112 83 73 66 42 Number of patients (y) 140 134 115 87 k 48 33 10 The least square regression line of y on x has equation 14.894 0.89875 y x = ÷ + correct to 5significant figures. The line passes through ( x , y ) and the mean pollutants index, x , is 104.75. (i) Show that k = 67. [2] (ii) Draw a scatter diagram to illustrate the data. [2] (iii) Explain, giving your reasons, why a model of the form y = a ln x + b fits the data better. [1] (iv) Using the model in part (iii), calculate the equation of the least square regression line and the value of r. [3] (v)The hospital decided that they will need to purchase new equipment if the number of patients reaches 120. Explain, with reason, whether it is advisable for the hospital to purchase new equipment if the pollutant index is 150. [1] CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 5 Q3 2013/Prelim/RVHS/II/11 It is believed that the probability p of a randomly chosen person having presbyopia (vision disorder due to aging) is related to the person’s age x, in years. The table gives the observed values of p for six different values of x. x 25 28 30 35 40 45 p 0.00235 0.00648 0.01000 0.03500 0.10000 0.45600 (i) Give a sketch of the scatter diagram for the above data and comment whether x and p have a linear relationship. [2] (ii) State, with a reason, which of the following would be an appropriate model to represent the above data. (A) x b a p + = , (B) x b a p ÷ + = e , (C) bx a p + = ln where a and b are constants and b > 0. [2] (iii) For the appropriate model, calculate the values of a and b, and find the product moment correlation coefficient. [2] (iv) Find an estimate of the probability of a 20 year-old person having presbyopia. Comment on the reliability of the estimate. [3] Topic 6 PnC Q1 2007/PJC (a) Peter attempts to break a four-digit code, which is made up of numbers 1, 2,..,9. Given that the digits may be repeated, in how many ways can the code be formed if (i) the last digit of the code is a ‘3’, [1] (ii) the code is an odd-number greater than 6000, [3] (b) The number 1086624 can be expressed in prime factors as 5 2 3 1 2 3 7 11 × × × . Excluding 1 and 1086624, how many positive integers are factors of 1086624? [3] Q2 TJC/2012 2 men and 5 women go to a restaurant for a meal. They choose an outdoor round table with 7 seats. Find the number of ways the group can be seated if (i) the two men are not seated next to each other, [2] (ii) one of the women, Mary, is to be seated between the two men. [2] Before their orders arrive, they request to shift to a table in the 'non-smoking' section of the restaurant. They are then given a round table with 10 seats. Find the number of ways they can be seated if (iii) the empty seats are adjacent to each other, [2] (iv) none of the empty seats are adjacent to each other and there must be more than 1 person between any two empty seats. [2] Q3 HCI/ 2006 block test Consider the name of a French mathematician “POISSON”. (a) How many seven letter words can be formed using all the letters in “POISSON”? [2] (b) How many seven letter words can be formed such that no vowels are adjacent to each other? [3] (c) How many three letter words can be formed using the letters in “POISSON”? [4] Topic 7 Probability Q1 9233/N07/2/25 The numbers of men and women studying Chemistry, Physics and Biology at a college are given in the following table. Chemistry Physics Biology Men 12 16 32 Women 8 12 20 One of these students is chosen at random by a researcher. Events M , W , C and B are defined as follows. M : the student chosen is a man W : the student chosen is a woman C : the student chosen is studying Chemistry B : the student chosen is studying Biology Find (i) P( | ) W B , (ii) P( | ) B W , (iii) P( ) B W . [1, 1, 2] State, with a reason in each case, whether W and B are independent, and whether M and C are mutually exclusive. [4] CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 6 Q2 J86/2/6 A box contains 25 apples, of which 20 are red and 5 are green. Of the red apples, 3 contain maggots and of the green apples, 1 contains maggots. Two apples are chosen at random from the box. Find, in any order, (i) the probability that both apples contain maggots. [2] (ii) the probability that both apples are red and at least one contains maggots. [4] (iii) the probability that at least one apple contains maggots, given that both apples are red. [4] (iv) the probability that both apples are red given that at least one apple is red. [4] Q3 Hari, a car salesman, has found that he can make a sale to 65% of his male customers but to only 45% of his female customers. All of Hari’s sales are independent. On Tuesday morning Hari has two male customers and one female customer. Find the probability that Hari makes exactly two sales. [4] 60% of Hari’s customers are male. Find the probability that, on Wednesday morning, Hari makes a sale to his first customer. [3] Find the probability that, on Thursday morning, Hari makes his first sale to his fourth customer. [3] For a randomly chosen customer, find the probability that the customer is female given that Hari makes a sale to that customer. [4] Topic 8 Binomial Distributions Q1 2013 MJC/II/Q8(a) The random variable Y has a binomial distribution with mean 1.6 and ( ) P 0 0.1296 Y = = . Find ( ) P 2 Y > . [4] Q2 2013 HCI/II/Q6 In a carton of apples, a sample of 8 apples is taken and examined for spoilt apples. (i) State, in context, an assumption for the number of spoilt apples in the sample to be modelled by a binomial distribution. [1] The number of spoilt apples in a random sample of size 8 may be modelled by the distribution ( ) B 8, p . If at least 2 apples in a sample are found to be spoilt, the carton is rejected. It is known that the probability of a carton being rejected is 0.04 . (ii) Write down an equation satisfied by p and find the value of p . [3] 60 cartons of apples are loaded onto a lorry. Use a suitable approximation to find the probability that more than 56 cartons loaded onto the lorry are not rejected. [3] Q3 2013 RI/II/Q11 Records had shown that 18% of the eggs from a particular farm were classified as “small” while 8% of the eggs from the farm were classified as “large”. The rest were classified as “medium” eggs, of which % m were substandard, 20 m< . (i) Mrs Egg randomly selects 10 medium eggs from the farm. [2] Given that the probability that Mrs Egg found 2 substandard medium eggs is 0.3, write down an equation involving m and hence find m. A two-stage inspection scheme for accepting or rejecting a large batch of eggs from the farm is as follows: Stage 1: Randomly select and inspect a tray of 30 eggs. Accept the batch if the tray contains at most 5 “small” eggs. Proceed to stage 2 if the tray contains 6 or 7 “small” eggs. Reject the batch otherwise. Stage 2: Randomly select and inspect a further sample of 10 eggs. Accept the batch if the further sample contains at least 1 “large” egg. Reject the batch otherwise. CJC MATHEMATICS DEPARTMENT 2014 JC2 H2 MATHEMATICS MYE REVISION 7 (ii) Find the probability that the batch is eventually accepted. [3] Mr Egg randomly selects 60 eggs from the farm. (iii) By using a suitable approximation, find the probability that the total number of “small” eggs and “medium” eggs in his sample is more than 55. [3] A supermarket which opens from 7 am to 5pm daily sells eggs from the farm in trays of 30. The average hourly demand of eggs in the morning (7 am to 12 noon) is 19 trays while in the afternoon (12 noon to 5pm), the average hourly demand of eggs reduces to 9 trays. You may assume that both the demand of eggs in the morning and that in the afternoon follow Poisson distributions. (iv) State an assumption needed for the daily demand of eggs to follow a Poisson distribution. [1] (v) Calculate the least number of trays of eggs the supermarket must stock up each day to ensure that the probability of meeting the daily demand of eggs is more than 0.9. [4] Topic 9 Poisson Distributions Q1 2013 A level/P2/Q12(i),(ii),(iii) A company has two departments and each department records the number of employee absent through illness each day. Over a long period of time it is found that the average numbers absent of a day are 1.2 for Administration Department and 2.7 for the Manufacturing Department. (i) State, in context, two conditions that must be met for the numbers of absences to be well modelled by Poisson distributions. Explain why each of your two conditions may not be met. [3] For the remainder of the question assume that these conditions are met. You should also assume that absences in the two departments are independent of each other. (ii) Find the smallest number of days for which the probability that no employee is absent through illness in from Administration Department is less than 0.01. [2] Each employee absent of a day represents one “day of absence’. So, one employee absent for 3 days contributes 3 days of absence, and 5 employees absent on 1 day contribute 5 days of absence. (iii) Find the probability that, in a 5-day period, the total number of days of absence in the two departments is more than 20. [3] Q2 2007 AJC/MYE/Q12 In a certain town A, the number of road accidents occurs at an average rate of one every 2 days. The occurrence of any accident is independent of the occurrence of any other accident. Taking a week to consist of 7 days, find the probability that (i) there are more than 3 accidents in a randomly chosen week; [2] (ii) in a ten-week period, there are at least 3 but less than 7 weeks with more than 3 accidents per week. [3] In another town B, the number of road accidents occurs at an average rate of one every day. Given that there are a total of 5 accidents in towns A and B in a randomly chosen week, find the probability that at least 4 of them occur in town B. [3] Q3 2007 TPJC/MYE/Q9b Cars travelling from the town centre pass a check point at an average rate of 3.1 per 30-minute interval. (i) Give two reasons why a Poisson distribution may be an appropriate model. [2] (ii) Calculate the probability of recording at least 6 cars travelling from the town centre in a 60-minute interval. [3] Cars travelling out of the town centre pass the same check point at an average rate of 1 per 20 minute interval. (iii) Find the probability that there will be no more than 10 cars passing the check point between 8 am to 9 am on a particular day. [3]