37965451 Bansal Classes 12th Standard Maths DPPs

March 27, 2018 | Author: Niranjan Savarirajalu | Category: Trigonometric Functions, Equations, Triangle, Sine, Quadratic Equation
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| JgBANSAL CLASSESIpTargef CLASS: XII SIT J E E 2 0 0 7 MATHEMATICS Daily Practice Problems TIME: 50 Min. DPP. NO.-53 (ABCdT DATE: 11-12/12/2006 Q. 1 Q.2 Revision Dpp on Permutation & combination Select the correct alternative. (Only one is correct) Number of natural numbers between 100 and 1000 such that at leas t one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D)none The number of way s in which 100 persons may be seated at 2 round tables T, and T 2 , 5 0 persons being seated at each is : ( A ) f m M ! m l ! Q. 3 There are six periods in each working day of a school. Number of ways in which 5 subjects can be arranged if each subject is allotted at least one period and no period remains vacant is (A)210 (B)1800 (C)360 (D)120 The number of ways in whi ch 4 boys & 4 girls can stand in a circle so that each boy and each girl is one after the other is: (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 ! . 4 ! If letters ofthe word "PARKAR" are written down in all possible manner as they are in a dictiona ry, then the rank of the word "PARKAR" is: (A) 98 (B) 99 (C) 100 (D) 101 The num ber of different words of three letters which can be formed from the word "PROPO SAL", if a vowel is always in the middle are: (A) 53 (B) 52 (C) 63 (D) 32 Consid er 8 vertices of aregular octagon and its centre. If T denotes the number of tri angles and S denotes the number of straight lines that can be formed with these 9 points then T - S has the value equal to (A) 44 (B)48 (C) 52 (D)56 A polygon h as 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C) 20 (D) 25 Q. 4 Q.5 Q. 6 Q.7 Q. 8 Q. 9 The number of ways in which a mixed double tennis game can be arranged from amon gst 9 married couple if no husband & wife plays in the same game is; (A) 756 (B) 1512 (C) 3024 (D) 4536 4 normal distinguishable dice are rolled once. The numbe r of possible outcomes in which atleast one die shows up 2 is: (A) 216 (B) 648 ( C) 625 (D) 671 Il-l OQ X nr x . p is equal to : Q. 10 Q-l l f Q. 12 ( B ) f ^ ( Q ^ There are counters available in x different colours, The counters are all alike except for the colour. The total number of arrangements consisting of y counters , assuming sufficient number of counters of each colour, if no arrangement consi sts of all counters of the same colour is: (A) x y - x (B) x y - y (C) y x - x ( D)yx-y In a plane a set of 8 parallel lines intersects a set of n parallel lines , that goes in another direction, forming a total of 1260 parallelograms. The va lue of n is: (A) 6 (B) 8 (C) 10 (D) 12 Q. 13 Q. 14 A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. If internal arrangement inside the car does not matter the n the number of ways in which they can travel, is (A) 91 (B) 126 (C) 182 (D)3920 In a conference 10 speakers are present, If S5 wants to speak before S 2 & S 2 wants to speak after S 3 , then the number of ways all the 10 speakers can give their speeches with the above restriction if the remaining seven speakers have n o obj ection to speak at any number is (A) 10 Q. 15 C3 (B) 10 Pg (C) I0 P3 (D) i i l Q. 16 There are 8 different consonants and 6 different vowels. Number of different wor ds of 7 letters which can be formed, ifthey are to contain 4 consonants and 3 vo wels if the three vowels are to occupy even places is (A) 8 P 4 . 6 P 3 (B) 8 P 4 . 6 C 3 (C) s P 4 . 7 P 3 (D) 6 P 3 . 7 C 3 . 8 P 4 Number of ways in which 5 different books can be tied up in three bundles is (A) 5 (B) 10 (C) 25 (D) 50 Ho w many words can be made with the letters of the words "GENIUS" if each word nei ther begins with G nor ends in S is : (A) 24 (B) 240 (C) 480 (D) 504 Number of n umbers greater than 1000 which can be formed using only the digits 1,1,2,3,4,0 t aken four at a time is (A) 332 (B) 159 (C) 123 (D) 112 Select the correct altern ative. (.More than one are correct) Q.17 Q. 18 Q. 19 Q.20 Identify the correct statement(s). (A) Number of naughts standing at the end of 1125 is 30. (B) Atelegraph has 10 arms and each aim is capable of 9 distinct pos itions excluding the position of rest. The number of signals that can be transmi tted is 1010 - 1 . (C) In a table tennis tournament, every player plays with eve ry other player. If the number of games played is 5050 then the number of player s in the tournament is 100. (D) Number of numbers greater than 4 lacs which can be formed by using only the digits 0,2,2,4, 4 and 5 is 90. n+ Q.21 Q.22 '-Cg + «C4 > n+2 C 5 - n C 5 for all ' n ' greater than : (B) 9 (C) 10 (D) 11 (200^ (A) 8 The number of ways in which 200 different things can be divided into groups of 1 00 pairs is: (10fl (102^1 (103^1 (A) 2 ( 1 . 3 . s..199) (C) -,100 /·lnn\ i 200! 00 <>I t J r r J _ (D) 200! ->100 to n factors is equal to : (B) 2»Cn ( D ) 2 n · (1 - 3 - 5 B I T Q.23 2' (100)! The continued product, 2 . 6 . 1 0 . 1 4 (A) 2n P n (C) ( n + 1)(n + 2 ) ( n + 3) (n + n) 2n-l) Q.24 The Number of ways in which five different books to be distributed among 3 perso ns so that each person gets at least one book, is equal to the number of ways in which (A) 5 persons are allotted 3 different residential flats so that and each person is alloted at most one flat and no two persons are alloted the same flat . (B) number of parallelograms (some of which may be overlapping) formed by one set of 6 parallel lines and other set of 5 parallel lines that goes in other dir ection. (C) 5 different toys are to be distributed among 3 children, so that eac h child gets at least one toy. (D) 3 mathematics professors are assigned five di fferent lecturers to be delivered, so that each professor gets at least one lect urer. 4 Q.l (a) (b) J BANSAL CLASSES {Target BIT JEE 2007 DATE: 22-23/11/2006 MATHEMATICS Daily Practice Problems TIME: 75 Min. DPR NO.-S2 CLASS: XII (ABCD) This is the test paper ofClass-XI (PQRS & J) held on 19-11-2006. Take exactly 75 minutes. Consider the quadratic polynomial f (x) = x 2 - 4ax + 5 a 2 - 6a. Find the small est positive integral value of'a' for which f (x) is positive for every real x. Find the largest distance between the roots of the equation f (x) = 0. [2.5 + 2. 5] Q.2(a) Find the greatest value of c such that system of equations x 2 + y 2 = 25 x+y= c has a real solution. (b) The equations to a pair of opposite sides of a parallelogram are x 2 - 7x + 6 = 0 and y 2 - 1 4 y + 40 = 0 find the equations t o its diagonals. Q. 3 [2.5+2.5] Find the equation of the straight line with gradient 2 if it intercepts a chord of length 4^/5 on the circle x 2 + y 2 - 6x - 1 Oy + 9 = 0. [5] Q.4 The value ofthe expression, cos^ 2x + 3 cos 2x 7 7 wherever defined is independent of x. Without allotting c os x - s i n x a particular value of x, find the value of this constant. [5] Fin d the general solution of the equation sin 3 x(l + cot x) + cos 3 x(l + tan x) = cos 2x. Q. 5 [5] Q. 6 If the third and fourth terms of an arithmetic sequence are increased by 3 and 8 respectively, then the first four terms form a geometric sequence. Find (i) the sum of the first four terms ofA.P. (ii) second term of the G.P. [2.5+2.5] Q.7(a) Let x = Ð or x = - 15 satisfies the equation, log 8 (&x 2 +wx + / ) = 2 . I f k, w and/are relatively prime positive integers then find the value of k+w +f. (b) The quadratic equation x 2 + mx + n - 0 has roots which are twice those of x 2 + px + m = 0 and n m, n and p* 0. Find the value of ~ . [2.5+2.5] x y Lme Ð + Ð = 1 intersects the x and y axes at M and N respectively. If the coordinates of t he point P 6 8 lying inside the triangle OMN (where 'O' is origin) are (a, b) su ch that the areas of the triangle POM, PON and PMN are equal. Find (a) the coord inates of the point P and (b) the radius of the circle escribed opposite to the angle N. [2.5+2.5] Starting at the origin, a beam oflight hits a mirror (in the fomi of a line) at the point A(4,8) and is reflected at the point B(8,12). Compu te the slope of the mirror. [5] Q. 8 Q. 9 Q. 10 Q.ll Find the solution set of inequality, log x + 3 (x 2 - x) < 1. [5] If the first 3 consecutive terms of a geometrical progression are the roots of t he equation 2x 3 - 1 9 x 2 + 57x - 5 4 = 0 find the sum to infinite number of te rms of G.P. [5] Find the equation to the straight lines joining 1 lie o- "m to t he points of intersection of the straight line 2L + L = i and the circle 5(x 2 + y 2 + bx+ay) = 9ab. Also find the linear relation between a and b so that a b th ese straight lines may be at right angle. [3+2] Q. 12 Q. 13 L e t / ( x ) = | x - 2 | + | x - 4 | Ð | 2 x - 6 j . Find the sum of the largest and smallest values of f (x) if x e [2, 8], [5] x+1 x+2 If x+3 x+2 x+3 x+4 x+a x + b = 0 then all lines represented by ax + by + c = 0 pass through a fixed poin t. x+c [5] Q.14 Find the coordinates of that fixed point. Q. 15 If Sj, S 7 , S 3 ,... S ,.... are the sums of infinite geometric series whose fi rst terms are 1,2,3,... n,... and 1 1 1 1 whose common ratios are Ð, - , Ð,...., ,.. . respectively, then find the value of 2 J nr * O ** 1 T A 5 B 20 In any triangl e if tan Ð = 7 and tan Ð = Ð then find the value of tan C. 2 6 2 3/ 2(1-1 r=l - . [5] Q. 16 [5] Q.17 The radii r p r 2 , r 3 of escribed circles of a triangle ABC are in harmonic pr ogression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides. [5] Find the equation of a circle passing through the origin if t he line pair, xy - 3x + 2y - 6 = 0 is orthogonal to it. If this circle is orthog onal to the circle x 2 + y 2 - kx + 2ky - 8 = 0 then find the value of k. [5] Fi nd the locus of the centres of the circles which bisects the circumference of th e circles x 2 + y 2 - 4 and x 2 + y 2 Ð 2x + 6y + 1 = 0. [5] Find the equation of the circle whose radius is 3 and which touches the circle x 2 + y 2 - 4x Ð 6y - 12 =0 internally at the point ( - 1 , - 1 ) . [5] Find the equation of the line suc h that its distance fiom the lines 3x - 2y - 6 = 0 and 6x - 4 y - 3 = 0 is equal . [5] Find the range of the variable x satisfying the quadratic equation, x 2 + (2 cos (j))x - sin2c|> = 0 V <j) e R. ( n y^ (n sin x(3 + sin 2 x) If tan ~ + ~ ! = t a r r ~ + ~ then prove that s i n y = 5 . 2.) \ 4 J,) l + ^sin^x Q. 18 Q. 19 Q.20 Q.21 Q. 22 [5] Q.23 [5] 1 Q.l Q. 2 i BANSAL CLASSES Target IIT JEE 2007 DATE: 10-11/11/2006 MATHEMATICS Daily Practice Problems TIME: 60 Min. DPP. NO.-51 CLASS : XII (ABCD) Select the correct alternative. (Only one is correct) There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. 1 1 1 1 1 Find the sum of the infinite series 7 + 7T: + T r + 7 7 + 7 r + T 9 18 30 45 63 (A) } (B) i (C) | (D) f Number of degrees in the smallest positive angle x such that 8 sin x cos 5 x - 8 sin5x cos x = 1, is (A) 5° (B) 7.5° (C)10° (D) 15° Q. 3 There exist positive integers A, B and C with no common factors greater than 1, such that Alog 200 5 + B log 200 2 = C. The sumA + B + C equals (A) 5 ~ (B) 6 (C ) 7 (D) 8 A triangle with sides 5,12 and 13 has both inscribed and circumscribed circles. The distance between the centres of these circles is (A) 2 (B)| (C) V6 5 (D)^f y Q. 4 Q. 5 The graph of a certain cubic polynomial is as shown. If the polynomial can be wr itten in the form / ( x ) = x 3 + ax2 + bx + c, then (A) c = 0 (B) c < 0 (C) c > 0 (D) c = - 1 Q. 6 The sides of a triangle are 6 and 8 and the angle 0 between these sides varies s uch that 0° < 0 < 90°. The length of 3rd side x is (A) 2 < x < 14 (B) 0 < x < 10 (C) 2 < x < 10 (D)0<x<14 The sequence a t , a^ a 3 ,.... satisfies a{ = 19. first n - 1 terms. Then a2 is equal to (A) 179 (B) 99 = 99, and for all n > 3, a n is t he arithmetic mean of the (C) 79 (D)59 Q.7 Q.8 If b is the arithmetic mean between a and x; b is the geometric mean between 'a' and y; 'b' is the harmonic mean between a and z, (a, b, x,y,z> 0) then the valu e of xyz is (A) a 3 (B,b3 ( C ) ' t a 2b-a 2a-b Q.9 Given A(0,0), ABCD is a rhombus of side 5 units where the slope of AB is 2 and t he slope of AD is 112. The sum of abscissa and ordinate of the point C is (A) 4 V5 (B)5V5 (C)6V5 (D) 8V5 Q. 10 A circle of finite radius with points (-2, -2), (1,4) and (k, 2006) can exist fo r (A) no value of k (B) exactly one value of k (C) exactly two values of k (D) i nfinite values of k If a A ABC is formed by 3 staright lines u = 2x + y - 3 = 0; v = x - y = 0 and w = x - 2 = 0 then for k = - 1 the line u + kv = 0 passes thr ough its (A) incentre (B) centroid (C) orthocentre (D) circumcentre x2 + 1 0 x - 3 6 a b c Ð = Ð + ÐÐ~ + If a, b and c are numbers for which the equation - Ð Ð Ð+ + x(x - 3 ) x x-3 (x-3) then a + b + c equals (A) 2 (B) 3 (C)10 (D)8 Q. 11 Q. 12 is an identity, Q. 13 1 1 1 If a, b, c are in G.P. then ~ , Ð, b - a 2 b b - c (A) A. P. (B) G.P. are in (C)H.P. (D) none Q. 14 How many terms are there in the G.P. 5,20, 80, 20480. (A) 6 (B)5 (C) 7 1 (D)8 Q. 15 The sum of the first 14 terms of the sequence 1 1 j= + h t= + 1 + Vx 1-X 1 Ðv x 7 is ( A ) 14 ( B ) ^ f > (C) (l + V x ) ( l - x ) ( l - V x ) 10 (D)none Q. 16 If x, y > 0, logyx + logxy = Ð and xy = 144, then arithmetic mean of x and y is (A ) 24 (B) 36 (C)12V2 (D)13V3 Q. 17 A circle of radius R is circumscribed about a right triangle ABC. If r is the ra dius of incircle inscribed in triangle then the area of the triangle is (A)r(2r + R) (B)r(r + 2R) (C)R(r + 2R) (D)R(2r + R) The simplest form of 1 + 1 (A) a for a * 1 (C) - a for a * 0 and a * 1 Ð is Ð 1-a (B) a for a * 0 and a * 1 (D)lfora*l £ Q. 18 Select the correct alternatives. (More than one are correct) Q. 19 If the quadra tic equation ax2 + bx + c = 0 (a > 0) has sec29 and cosec 2 0 as its roots then which of the following must hold good? (A) b + c = 0 (B) b 2 - 4ac > 0 (C) c > 4 a (D) 4a + b > 0 Which of the following equations can have sec29 and cosec29 as its roots (9 e R)? (A) x 2 - 3x + 3 = 0 (B) x 2 - 6x + 6 = 0 (C) x 2 - 9x + 9 = 0 (D) x 2 - 2x + 2 = 0 The equation | x - 2 | 10x2_1 = | x - 2 | 3x has (A) 3 in tegral solutions (C) 1 prime solution Q. 22 (B) 4 real solutions (D) no irration al solution Q.20 Q.21 Which of the following statements hold good? (A) If Mis the maximum and m is the minimum value of y = 3 sin2x + 3 sin x · cos x + 7 cos2x then the mean of M and m is 5, 71 .71 (B) The value of cosecÐ sec Ð is a rational which is not integral. 18 ^ 18 (C) If x lies in the third quadrant, then the expression 1/4 s i n 4 independent ofx. x + sin 2 2x + 4 cos 2 4 2 is (D) There are exactly 2 values of 9 in [0, 2tt] which satisfy 4 cos 2 9 - 2 -Jl cos 9 - 1 = 0 . MATCH THE COLUMN INSTRUCTIONS: Column-I and column-II contains four entries each . Entries of column-I are to be matched with some entries of column-El. One or m ore than one entries of column-I may have the matching with the same entries of column-H and one entry of column-I may have one or more than one matching with e ntries of column-II. Column-I Column-II (A) Area of the triangle formed by the s traight lines (P) 1 x + 2y - 5 = 0, 2x + y - 7 = 0 and x - y + 1 = 0 in square u nits is equal to (Q) 3/4 (B) (C) Abscissa of the orthocentre of the triangle who se vertices are the points (-2, -1); (6, - 1) and (2, 5) Variable line 3x(A. + 1 ) + 4y(A. - 1) - 3 ( 1 - 1) = 0 for different values of A, are concurrent at the point (a, b). The sum (a + b) is The equation ax2 + 3xy - 2y2 - 5x + 5y + c = 0 represents two straight lines perpendicular to each other, then | a + c | equal s (R) (S) 2 3/2 Q.l (D) Column-I Q.2 (A) In a triangle ABC, AB = 2^3 , BC = 2-J6 , AC > 6, and area of t he triangle ABC is 3 V<5 . Z B equals (B) (C) (D) In a triangle ABC is b = S , c = 1 andA= 30° then angle B equals In a A ABC if (a + b + c)(b + c - a) = 3bc then Z A equals Area of a triangle ABC is 6 sq. units. If the radii of its excircles are 2,3 and 6 then largest angle of the triangle is Column-I The sequence a, b, 10, c,d is an arithmetic progression. The value o f a + b + c + d The sides of right triangle form a three term geometric sequence. The shortest side has lengt h 2. The length of the hypotenuse is of the form a + Vb where a e N and 7 b is a surd, then a + b equals (C) The sum of first three consecutive numbers of an in finite G .P. is 70, if the two extremes be multipled each by 4, and the mean by 5, the products are in A.P. The first term of the GP. is The diagonals of a para llelogram have a measure of 4 and 6 metres. They cut off forming an angle of 60°. If the perimeter of the parallelogram is 2[-Ja + Vb) where a, b e N then (a + b) equals 2 2 Column-II (P) (Q) (R) (S) 60° o 90 120 o 75° Q.3 (A) (B)' Column-II (P) 10 (Q) (R) (S) 20 26 40 (D) J g BANSAL CLASSES I B Target I1T JEE 2007 CLASS: XII (ABCD) Q. 1 DATE: 04-07/10/2006 Pa/7/ Practice Problems TIME: 40 Min.for each MATHEMATICS DPP. NO.-49, 50 -49 8 clay targets have been arranged in vertical column, 3 being in the first colum n, 2 in the second, and 3 in the third. In how many ways can they be shot (one a t a time) if no target below it has been shot. [4] Q.2 Evaluate: /x(sin 2 (sinx) + cos 2 (cosx))dx o Evaluate: jx(sin(cos 2 x)cos(sin 2 x ) ) d x [4] Q.3 [4] Q.4 J - . x dx * V xYQ111 YJ-fAQY . sin x + cos x / 0 <f 1 VI Prove that 2 ^ 1 3 n + 1 [6] Q.5 _ J _ 1 = 3n + 2 j 71 ^ [9] - S O Q. 1 If cos A, cos B and cos C are the roots of the cubic x 3 + ax 2 + bx + c = 0 where A, B, C are the angles of a triangle then find the value of a 2 - 2b - 2c. [4] Find all f u n c t i o n s , / : R - > R satisfying ( x / ( x ) - 2 F ( x ) ) ( F ( x ) - X 2 ) = 0 V x e R where f (x) = F'(x). [4] Q.2 0 '3 Q 3 Q.4 J j f ^ ¥ 2l3-xJ f * HI 0 0 J JÐ^ dx reduces to zero by a substitution x = 1 /t. Using this or ¹ ax" + b x + a o aAx For a > 0, b > 0 verify that °f fax otherwise evaluate: i 2 0 tan - 1 x x "\3 d [7] Q.5 1 v dx y [81 A JABANSAL CLASS ES l ^ P T a r g e t HT J E E CLASS: XII (ABCD) 2007 DATE: 29-30/9/2006 MATHEMATICS Daily Practice Problems DPP. NO.-47 This is the test paper-1 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60 minutes. P A R X - A Select the correct alternative. (Only one is correct) [24 x 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for each wrong answ er. Q. 1 The area of the region of the plane bounded above by the graph of x 2 + y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 is (A) jc/4 (B) ti2/4 (C) 7c/2 (D) it Q.27' Consider straight line ax + by = c where a , b , c e R+ and a , b, c are distinct. This line meets the coordinate axes at P and Q respectively . If area of AOPQ, 'O' being origin does not depend upon a, b and c, then (A) a ; b. c are in G.P. (B) a, c, b are in G.P. (C) a, b. c are in A.P. (D) a, c, b a re in A.P. If x and y are real numbers and x2 + y2 = 1, then the maximum value o f (x + y)2 is (A) 3 Q.4 (B) 2 (C) 3/2 (D) J 5 Q. y dx The value of the definite integral j n (a > 0) is q (1 + x )(1 + x ) (A) ti/4 (B) nil (C) tc (D) some function of a. a b e cos Ð cosÐcos Let a, b, c are non zero constant number then Lim Ð-Ð Ð equals r-»co sinÐsin r r . . . _ _ _ J (D) independent of a, band c . b . C ... a 2 + b 2 - c 2 (A) 2bc Q.6 ^ (B) c2 + a 2 - b 2 2bc ^xb2+c2-a2 (C) 2bc A curve y =/(x) such that/"(x) = 4x at each point (x, y) on it and crosses the x -axis at (-2, 0) at an angle of 450. The value of / (1), is (A) - 5 (B) - 15 (C) - f (D) y Q.7/ v sinx cosx tanx cotx = The minimum value of the function/(x) = 1 + / + 7 + ~T as 2 9 Vl-cos x vl-sin x vsec x - 1 Vcosec x - 1 x varies over all numbers in the l argest possible domain of / ( x ) is (A) 4 (B) - 2 (C) 0 (D) 2 A non zero polyno mial with real coefficients has the property that f (x) = / ' (x) · f"(x). The lea ding coefficient of / (x) is (A) 1/6 ' (B) 1/9 (C) 1/12 (D) 1/18 l_ Q.8 Q-9 r tan -1 (nx) ^ 2 X 2 Let Cn = J s i n - V ) then Lim n -C f l "n+l equais (A) 1 / Q. 10 (B) 0 (C) - 1 (D) 1/2 2 2 2 | = | z31 = 1 then z, + z 2 + z 3 , (D) equal to 1 Let Zj, z2, z3 be complex numbers suchthat zx + z2 + z3 = 0 and | zx \ - \ is (A ) greater than zero (B) equal to 3 (C) equal to zero Q.ll Number of rectangles with sides parallel to the coordinate axes whose vertices a re all of the form (a, b) with a and b integers such that 0 < a, b < n, is (n e N) (A) Q.12 ^.13 n 2 (n + l)2 (B) (n - l ) 2 n 2 (C) + (n + 1)2 (D) n2 Number of roots of the function/(x) ~ (A) 0 2 1 ^ 3 - 3x + sin x is (C)2 (D) more than 2 (B) 1 If p (x) = ax + bx + c leaves a remainder of 4 when divided by x, a remainder of 3 when divided by x + 1, and a remainder of 1 when divided by x - 1 then p(2) i s (A) 3 (B) 6 (C) - 3 (D) - 6 Let/(x) be a function that has a continuous deriva tive on [a, b],/(a) and/(b) have opposite signs, and / ' (x) * 0 for all numbers x between a and b, (a < x < b). Number of solutions does the equation / ( x ) = 0 have (a < x < b). (A) 1 (B) 0 (C) 2 (D) cannot be determined -3tt/2 j Sin(3x + 7t)dx V^l 5 Which of the following definite integral has a positive value? 2it/3 0 0 Jsin(3x + 7i)dx (g) Jsin(3x + 7t)dx ^ q Jsin(3x + Jt)dx 2tc/3 -3it/2 ^ Q. 16 Let set A consists of 5 elements and set B consists of 3 elements. Number of fun ctions that can be defined from A to B which are neither injective nor surjectiv e, is (A) 99 (B) 93 (C) 123 (D) none A circle with center A and radius 7 is tang ent to the sides of an angle of 60°. A larger circle with center B is tangent to t he sides of the angle and to the first circle. The radius of the larger circle i s (A) 30V3 (B) 21 (C) 20V3 (D) 30 The value of the scalar (p x q)-(r x s) can be expressed in the determinant form as q-r (A) p-r qs p-s 1 1/x 0 p-r (B) q-s y p 1/x p-s q-r vXl7 \J2[- 18 (C) q-r p-s p-r q-s p-r (D) q-r p-s q-s Q.19 a/x jf Lim x · In 0 x->00 1 _1 5, where a, p, y are finite real numbers then (C) a e R, p=l, yeR (C) 4 (D) a e R, p = 1, y = 5 (D) 1 (A) a = 2, p=l, yeR Q.20 Q.21 (A) 2 (B) a =2, p=2, y = 5 (B) 2 / 2 If / (x. y) = sin ( | x [ + | y |), then the area of the domain of / is A, B and C are distinct positive integers, less than or equal to 10. The arithmetic mean of A and B is 9. The geometric mean of A and C is 5 / 2 · The harmonic mean of B and C is ¹9 (A) 9Ð (B) (C) (D) 2-^r v v_/ 2 ~ ' 19 9 19 17 If x is real and 4y2 + 4x y + x + 6 = 0, then the complete set of values of x for which y is real, is (A) x < 2 or x > 3 ( B ) x < - 2 or x > 3 ( C ) - 3 < x < 2 ( D ) x < - 3 or x > 2 I alternatively toss a fair coin and throw a fair die until I, either toss a head or throw a 2. If I toss the coin first, the probability that I throw a 2 before I toss a head, is (A) 1/7 " (B) 7/12 (C) 5/12 (D) 5/7 Let A, B. C, D be (not ne cessarily square) real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix S = ABCD, consider the two statements. I S3 = S II s2 = s4 (A ) II is true but not I (B) I is true but not II (C) both I and II are true (D) b oth I and II are false. Q.22 Q.23 Q.24 J s B A N S A L CLASS ES V S Target NT JEE 2 0 0 7 CLASS: XII (ABCD) DATE: 02-03/10/2006 MATHEMATICS Daily Practice Problems DPP. NO.-48 This is the test paper-2 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60 minutes. Select the correct alternative. (More than one is/are correct) There i s NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q. 1 [ 3 x 6 = 18] The function/(x) is defined for x > 0 and has its inverse g (x) which is differe ntiable. I f / (x) satisfies g(x) J f (t) dt = X2 and g (0) = 0 then (A)/(x) is an odd linear polynomial (C)/(2) = 1 (B)/(x) is some quadratic polynomial (D)g(2 ) = 4 Q. 2 Consider a triangle ABC in xy plane with D, E and F as the middle points of the sides BC, CA and AB respectively. If the coordinates of the points D, E and F ar e (3/2, 3/2); (7/2,0) and (0, -1/2) then which of the following are correct? (A) circumcentre of the triangle ABC does not lie inside the triangle. (B) orthocen tre, centroid, circumcentre and incentre of triangle DEF are collinear but of tr iangle ABC are non collinear. (C) Equation of a line passes through the orthocen tre of triangle ABC and perpendicular to its plane is r = 2(i - j) + A.k 5V2 (D) distance between centroid and orthocentre of the triangle ABC is ÐÐ. Q. 3 X X If a continuous function/ ( x ) satisfies the relation, j t / ( x - t ) dt = j / ( t ) dt + s j n X-+ cos x - x - 1 ¹ for 0 0 . all real numbers x, then which of the following does not hold good? it (A)/(0) = 1 ( B ) / ' (0) = 0 (C)f" (0) = 2 (D) J / ( x ) d x = e * 0 MATCH THE COLUMN [ 3 x 8 = 24] There is NEGATIVE marking. 0.5 mark will be deducted for each wrong match within a question. Q.l ,.. (A) T. Column I Y VY Ðtl X-*co Column II (P) 0 Lim In x r dt Ð IS V X JJ3 /n tt In e (B) (C) ' ¹vx +1 ¹xz+l /~T7 2 -e Lim e is is where n e N \ (Q) 4n ¹ : 1 Lim (-1)" s i n f W n 2 + 0.5n + l l sin J tan -1 / (R) (D) The value of the integral j 0 VX + ly 9A A f tan" 1 l + 2x-2x dx is (S) non existent Q.2 Consider the matrices A= andB : a 0 b T 1 and let P be any orthogonal matrix and Q = PAP Column II (P) G.P. with com mon ratio a (Q) A. P. with common difference 2 (R) GP. with common ratio b (S) A . P. with common difference - 2. and R = P T Q K P also S = PBP T and T = P T S K P Column I (A) If we vary K fro m 1 to n then the first row first column elements at Rwill form (B) If we vary K from 1 to n then the 2 nd row 2nd column elements at Rwill form (C) If we vary K from 1 to n then the first row first column elements of T will form (D) If we vary K from 3 to n then the first row 2nd column elements of T will represent th e sum of Q.3 (A) (B) (C) (D) Column I Column II (P) (Q) (R) (S) 30° 45° 60° Given two vectors a and b such that | a | = | b | = |a + b | = 1 The angle betwe en the vectors 2a + b and a is In a scalene triangle ABC, if a c o s A = b c o s B then Z C equals In a triangle ABC, BC = 1 and AC = 2. The maximum possible va lue which the Z A can have is In a A ABC Z B = 75° and BC = 2AD where AD is the al titude from A, then Z C equals 90° SUBJECTIVE: Q.l 96V · 2 1 SupposeV= J x sin x Ð dx, find the value of 71 2 tc/2 [ 5 x 1 0 = 50] Q. 2 " , where m r is non zero integer and n and r are relatively prime natural numbe rs. Find the value of m + n + r. A circle C is tangent to the x and y axis in th e first quadrant at the points P and Q respectively. BC and AD are parallel tang ents to the circle with slope - 1 . If the points A and B are on the y-axis whil e C and D are on the x-axis and the area of thefigureABCD is 900 V2 sq. units th en find the radius of the circle. One of the roots of the equation 2000x6 + 100x5 + 1 Ox3 + x - 2 = 0 is of the fo rm m + Q.3 Q. 4 Let/(x) = ax2 - 4ax+b (a > 0) be defined in 1 < x < 5. Suppose the average of th e maximum value and the minimum value of the function is 14, and the difference between the maximum value and minimum value is 18. Find the value of a 2 + b2. 1 x Q.5 If the Lim x-*0 1 + ax Vl + x 1 + bx 1 2 3 exists and has the value equal to I, then find the value of Ð - y + Ð . JGBANSAL CLASS>ES Target I I T JEE 2 0 0 7 CLASS: XII (ABCD) Q.l Q. 2 DATE: 27-28/9/2006 MATHEMATICS Daily Practice Problems DPR NO.-46 This is the test paper of Class-XI (J-Batch) held on 24-09-2006. Take exactly 75 minutes. If tan a . tan P are the roots of x 2 - px + q = 0 and cot a,cot p are the roots of x 2 - rx + s = 0 then find the value of rs in terms ofp and q. [4] Let P(x) = ax2 + bx + 8 is a quadratic polynomial. If the minimum value of P(x) is 6 when x = 2 , find the values of a and b. 14] ( \_\ Q.3 LetP= f j 102" n=l .n-i then find log 001 (P). sec 8A - 1 sec 4A - 1 tan 8 A tan 2 A [4] Q. 4 Q.5 Q.6 Prove the identity [4] [4] [4] Find the general solution set of the equation loglan x (2 + 4 cos hi) - 2. Find the value of sin a + sin 3a + sin 5a + Ð cos a + cos 3a + cos 5a + + sinl7a n Ð - wh en a = Ð . + cosl7a 24 Q.7(a) Sum the following series to infinity 1 1-4-7 + 1 4-7-10 + 1 7-10-13 + (b) Sum the following series upton-terms. 1 -2-3-4 + 2-3-4-5 + 3-4-5-6 + Q.8 Q. 9 The equation cos 2 x - sin x + a = 0 has roots when x e (0, rc/2) find 'a'. Th e geometric mean ofA and C is 5 / 2 · Fi n d the harmonic mean of B and C. Q. 10 Q . 11 Q. 12 Express cos 5x in terms of cos x and hence find general solution ofth e equation cos 5x = 16 cos 5 x. [3 + 3] [6] {6] A, B and C are distinct positive integers, less than or equal to 10. The arithme tic mean ofA and B is 9. [6] If x is real and 4y 2 + 4xy + x + 6 = 0, then find the complete set of values of x for which y is real. [6] Find the sum of all the integral solutions of the inequality 21og3x-41ogx27<5. [6] Ð, show that 2 (i-f)HI) l + tan Ð [ i 2 j + f 1 Ð tanÐ1 I 2J ( y^ t » | ] l + tan Ð I 2> sin a + sin P + sin y - 1 cos a + cos p +cosy [7] j y i 4(a) In any A ABC prove that C C c 2 = (a - b) 2 cos 2 Ð + (a + b) 2 sin 2 Ð. (b) In any A ABC prove that a 3 cos(B - C) + b 3 cos(C - A) + c 3 cos(A - B) = 3 abc. [4 + 4] d Q. 1 Q.2 Q.3 l BANSAL CLASSES 5Targe* liT JEE 2007 DATE: 20-21/9/2006 MATHEMATICS Daily Practice Problems DPP. NO.-44 CLASS: XII (ABCD) This is the test paper of Class-XI (PQRS) held on 17-09-2006. Take exactly 75 mi nutes. Evaluate £ 8 n -.2 r · 5 s where 5 r s = r=l s=l Will the sum hold i f n - > oo? n n r O i f r ^ S 1 if r = s [4] |4J x x Find the general solution of the equation, 2 + tan x · cot Ð + cot x · tan Ð = 0. Gi ven that 3 sin x + 4 cos x = 5 where x e (0, n/2). Find the value of 2 sin x + c os x + 4 tan x. 14] Q.4 Find the integral solution of the inequality log 0 3 ( x - 1 ) ' <Ð: 1 ==· < 0. V2x~- x 2 +8 K {4] Q.5 (a) (b) Q. 6 Q. 7 Q.8 Q. 9 In A ABC, suppose AB = 5 cm, AC = 7 cm, Z ABC : Find the length of the side BC. Find the area of A ABC. [4] [4] [5] The sides of a triangle are n- \,n and n + 1 and the area is n-Jn · Determine n. W ith usual notions, prove that in a triangle ABC, r + r { + r 2 - r 3 = 4R cos C. Find the general solution of the equation, sin %x + cos nx = 0. Also find the su m of all solutions in [0,100], [5] Find all negative values of'a' which makes th e quadratic inequality sin2x + a cos x + a 2 > 1 + cos x true for every x e R So lveforx, s i°g 2 * 2 ^ l o g J x V s ) = ^ l o g ^ x 2 _ 5 i o g 2 * 1 , ¹ ™ In a tria ngle ABC if a 2 + b 2 = 101 c 2 then find the value of & cot C . cot A + cot B [ 5] [5] [5] 1 1 Q.10 Q. 11 Q.12 Solve the equation for x, 5 2 +52 +!0g5(smx) = 152 +l08l5(C0Sx) [5] Z~n . n=l 6 00 [5] Q. 14 Q . 15 Suppose that P(x) is a quadratic polynomial such that P(0) = cos 3 40°, P( 1) = (c os 40°)(sm240°) and P(2) t 0 . Find the value of P(3). [8] If /, m, n are 3 numbers in G.P. prove that the first term of an A.P. whose 7th, mth, nth terms are in H. P. is to the common difference as (m + 1) to 1. [8] BAN SAL CLASSES y g Target I IT JEE 2007 CLASS : XII (ABCD) Q. 1 DATE: 22-23/9/2006 MATHEMATICS Daily Practice Problems TIME: 55 to 60 Min. DPP. NO.-45 Let a, b, c, d, e, f e R such that ad + be + cf = ^ ( a 2 + b 2 + c 2 ) ( d 2 + e 2 + f 2 ) use vectors or otherwise to prove that, a+b+c Va2+b2+c2 d+e+f Vd2+e2 +f2 ' Q.2 Let the equation x 3 - 4x 2 + 5x - 1.9 = 0 has real roots r, s, t. Find the area of the triangle with sides r, s, and t. 50 2 Q. 3 Suppose x + ax + bx + c satisfies f (-2) = - 1 0 and takes the extreme value Ð whe re x = Ð . Find the value of a, b and c. J 2 Q- 4 f i-y Hv L e t I x l d X ^/nx +xy- and r / n x x + xy _ I 1Ðy dy x d dy where ~ = x y . Show that I · J = (x + d)(y + c) where c, d e R. Hence show that Ð (I J) = I + J Ð y dx dx Q.5 Let a;, i = 1, 2, 3, 4, be real numbers such that aj + % + % + a 4 = 0. Show tha t for arbitrary real numbers bi5 i = 1,2, 3 the equation a, + bjx + 3a 2 x 2 + b 2 x 3 + Sa^x4 + b 3 x 5 + 7a 4 x 6 = 0 has at least one real root which lies on the interval - 1 < x < 1. V3 Q.6 Evaluate: x 2Ðl l x Ðt = x J I x + x +3x" + X r dx + 1 Q. 7 Let x, y e R in the interval (0, 1) and x + y = 1. Find the minimum value of the expression x x + yy r | (1 - sin x)(2 - sin x) ^ y (1 + sin x)(2 + sin x) ^ i l l SBANSAL CLASS: XII (ABCD) C L A S SE S l U l a r g e t NT JEE 2 0 0 7 DATE: 08-12/9/2006 M A T H E M A T I C S Daily Practice Problems DPP. NO.-42, 43 TIME : 60 Min. DATE : 08-09/09/2006 O P P - 4 2 This is the test paper of Class-XIII (XYZ) held on 27-08-2006. Take exactly 60 minutes. S^'S-yV Select the correct alternative, (Only one is correct) There is NEGATIVE marking. 1 mark will be deducted for ea ch wrong answer. sin 2 (x 3 + x 2 + x - 3 ) Li rn ~ ~ ~ ~~ has the value equal t o M x->i 1 Ð cos(x Ð 4x + 3) (A) 18 (B) 9/2 (C) 9 (D) none [16 x 3 = 48] Q. I dt Q.2 / Let/(x)= r . . If g'(x) is the inverse of / ( x ) then g'(0) has the va lue equal to 4 2 * 3-v t +3t +13 (A) 1/11 (B) 11 (C)Vl3 (D) l / V n Q.3 The func tion/(x) has the property that for each real number x in its domain, 1/x is also in its domain and /(x) + /(l/x) = x. The largest set of real numbers that can b e in the domain of /(x), is (A){x|x*0) (B) { x | x > 0) (C) { x | x * - l a n d x * 0 a n d x * 1) Q.4 j 6/ Let w = Z 2 (D) {-1, 1} z +1 37 + 6 , and z = 1 + i. then | w | and amp w respectively are (B) , - 71/4 (C) 2, 3TC/4 ( D) ^ , 3n/4 (A) 2, - n /4 1 - cos a - tan 2 (a/2) k cos a Q.5 A If . j/ " ~= where k, w and p have no comm on factor other than 1, then the F ./! sin (a/2) w + pcosa 2 2 2 value of k + w + p is equal to (A) 3 (B)4 (C)5 (D)64 Q.6 In a birthday .party, each man shook h ands with eveiyone except his spouse, and no handshakes took place between women . If 13 married couples attended, how many handshakes were there among these 26 people? (A) 185 (B)234 (C)312 (D)325 If x and y are real numbers such that x2 + y2 = 8, the maximum possible value of x - y, is (A) 2 (B) (C) V2/2 (D) 4 Let w(x ) and v(x) are differentiable functions such that u(x) = 7. If U^x) ~ P and ' u( x) v(x) = q, then Q.7 Q.8/ p+q M p - q has the value equal to (A) 1 Q.9 Q. 10 (B)0 (C)7 (D)-7 The coefficient of x9 when (x + (2/Vx j)30 is expanded and simplified is (A) 30C |4 · 29 (B) 30C]6 · 214 (C) 30 C 9 -2 21 (D) 10C9 Let C be the circle described by ( x - a)2 + y2 = r2 where 0 < r < a. Let m be the slope of the line through the or igin that is tangent to C at a point in the first quadrant. Then r Va2-r2 r (A) m = r ^ 7 (B) m = Ð (C) m = (D) m = Va - r r a What can one say abo ut the local extrema of the function/(x) = x + (1/x)? (A) The local maximum off (x) is greater than the local minimum of/(x). (B) The local minimum off (x) grea ter than the local maximum off (x). (C) The function/(x) does not have any local extrema. (D)/(x) a r Q. 11 has one asymptote. Q.l 2 / r_2^ + arctan(5) equals tan arc tan I 3 v (A) - / 3 (B)-l (C)l (D)V3 / y/Q- ip A line passes through (2, 2) arid cuts a triangle of area 9 square units from th e first quadrant. The sum of all possible values for the slope of such a line, i s (A) - 2.5 (B) - 2 (C) - 1.5 (D) - 1 Which of the following statement is/are tr ue concerning the general cubic / ( x ) = ax3 + bx2 + cx + d (a * 0 & a, b, c, d e R) I The cubic always has at least one real root II The cubic always has exac tly one point of inflection (A) Only I (B) Only II (C) Both I and II are true If S = 1 + 3 + 5 + (A) S + 2550 2 2 2 2 2 2 2 ^gf. 14 (D) Neither 1 nor II is true Q. 15 Q. 16 + (99) then the value of the sum 2 + 4 + 6 + + (100)2 is (B)2S (C) 4S (D) S + 50 50 Through the focus of the parabola y2 = 2px (p > 0) a line is drawn which interse cts the curve at A(x,, y,) y\y 2 and B(x,, v.). The ratio x x equals l 2 (A) 2 ( B) - 1 (C) - 4 (D) some function of p Select the correct alternative. (Only one is correct) There is NEGATIVE marking. 1 mark will-be deducted for each wrong an swer, i · n-3n ^ i If 6 N) ^n(x-9)»+n-3D+1-3n = 3 ^ ^ ^ ° f X iS (A) [2,5)' ' (B) (1,5 ) (C) (-1,5) (D)(-co,oo) The area of the region(s) enclosed by the curves v = x2 and y = ^ | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1 Suppose that the domain of the function/(x) is set D and the range is the set R, where D and R are the subsets of real numbers. Consider the functions:/(2x),/(x + 2), 2/(x), /(x/2), / ( x ) / 2 - 2 . If m is the number of functions listed above that must have the same d omain as/and n is the number of functions that must have the same range as f (x) , then the ordered pair (m, n) is (A) (1,5) (B) (2, 3) (C)(3,2) (D) (3, 3) r x 2 + 2mx - 1 for x < 0 / : R -» R is defined as / ( x ) = - mx - 3 for x > 0 If / ( x ) is one-one then m must lies in the interval (A) ( oo, 0) Ð (B) ( oo, 0] Ð (C)(0,oo) (D) [0, co) B = { x | (m - 1)X2 + m x + 1 = 0, X e R } Let A = { x | x 2 + (M - l ) x - 2(m + 1 ) = 0 , X G R } ; [ 9 x 4 = 36 j '! 7 18 Q.l 9 Q.20 Q.21 . Number of values of m such that A u B has exactly 3 distinct elements, is (A) 4 (B) 5 (C) 6 (D) 7 ^Q.22 If the function/(x) = 4x2 - 4x - tarra has the minimum value equal to - 4 then the most general values of 'a' are given by (A) 2n7t + ti/3 (B) 2nn - rc/3 (C) im ± n/3 (D) 2nn/3 where n e I Direction for Q.23 to Q.25. ^/Q.23 sinx-xcosx x Consider the function defined on [0, i] -> R, / ( x ) = 5 * 0 anc® f (0) = 0 The function/(x) (A) has a removable discontinuity at x = 0 (B) has a non removable finite discontinuity at x=0 (C) has a non removable infinite discontinuity at x = 0 (D) is continuous at x = 0 1 ^jQ.24 J / ( x ) d x equals (A) 1 - sin (1) t (B) sin (1) - 1 (C) sin (1) (D)-sin(l) ^.25 1 L i m z j / ( x ) d x equals t->o t 7 1 0 (A) 1/3 (B) 1/6 (C) 1/12 (D) 1/24 DATE : 11-12/09/2006 i>B>S>-43 TIME : 60 Min. [ 7 x 4 = 28] Select the correct alternative. (More than one are correct) xe x Q.26 Let / (x) = L There is NO NEGATIVE marking. Marks will be awarded only if all the correct alte rnatives are selected. x<0 then the correct statement is x + x2 - xJ x > 0 ( A ) / is continuous and differentiate for all x. ( B ) / is continuous but not diff erentiate at x = 0. ( C ) / ' is continuous and differentiate for all x. ( D ) / ' is continuous but not differentiate at x = 0. x2-l Suppose/ is defined from R Ð [Ð1, 1] as / ( x ) = Ðz > where R is the set of real number. Then the x" + 1 statem ent which does not hold is ( A ) / is many one onto ( B ) / increases for x > 0 and decrease for x < 0 (C) minimum value is not attained even though f is bounde d (D) the area included by the curve y = f (x) and the line y = 1 is n sq. units . 2 Q.27 Q.28 The value of the definite integral r , (3 + cosx V J x ' n i 3 _ c o s x J > is v 0 ]dx J ^3-cosx J ) ( CV z e r o ' (D) V* 0 V3 + c o s x ; (A) n ] l n ( Jdx J V3 Ð cosx J (B) 0-29 r x 3 (l-x)sin(l/x 2 J if 0 < x < l f : [0. 1] -> R is defined as / ( x ) = j __ , then 0 if x = 0 (A)/ is continuous but not derivable in [0, 1 ] ( C ) / is bo unded in [0, 1 ] ( B ) / is differentiate in [0, 1 ] ( D ) / ' is bounded in [0, 1] Q.30 Let 2 sin x + 3 cos v = 3 and 3 sin y + 2 cos x = 4 then (A) x + y = (4n + 1)TE/ 2, n e l (B) x + y = (2n + l)rc/2, n E I (C) x and y can be the two non right an gles of a 3-4-5 triangle with x > y. (D) x and v can be the two non right angles of a 3-4-5 triangle with y > x. The equation cosec x + sec x = 2V2 has (A) no s olution in (0, n/4) (C)no solution in (n/2, 3n/4) (B) a solution in [tc/4 , n/2) (D) a solution in [37r/4, tc) 2 Q.31 Q.32 For the quadratic polynomial / ( x ) = 4x - 8kx + k, the statements which hold g ood are (A) there is only one integral k for which/(x) is non negative V x e R ( B) for k < 0 the number zero lies between the zeros of the polynomial. (C)/(x) = 0 has two distinct solutions in (0, 1) for k e (1/4, 4/7) (D) Minimum value of y V k e R is k(l + 12k) I^A. l^TI-S^ MATCH THE COLUMN [ 3 x 8 = 24] Q. i Column-I contain four functions and column-II contain their properties. Match ev ery entry of column-1 with one or more entries of column-II. Column-I Column-II (A) / ( x ) = sin"](§in x) + cos""1 (cos x) (P) range is [0,71] (B) g (x) = sin-'j -x | + 2 tair'j x | (Q) is increasing V x e (0, 1) (C) (D) ( 2x 1 h (x) = 2sirr> ! Ð j j , x 6 [0, 1] k (x) = cot(cor'x) (R) (S) period is 2% is decreasing V x e ( 0, 1) Q.2 (A) (B) Column-I Column-II +c a+b+c 3 Centre of the parallelopipeci whose 3 coterminous edges OA, OB and (P) OC have p osition vectors a, b and c respectively where O is the origin, is OABC is a tetr ahedron where O is the origin. Positions vectors of its angular points A, B and C are a, b and c respectively. Segments joining each vertex with the centroid of the opposite face are concurrent at a point P whose p. v.'s are Let ABC be a tr iangle the position vectors of its angular points are a, b and c respectively. I f\a-b\ = \b-c\=\c-a\then the p.v.of the orthocentre of the triangle is Let a, b, c be 3 mutually perpendicular vectors of the same magnitude. If an unbiown vecto r x satisfies the equation a x[fx -b)xaj+b x[(x-c)xbj+c x({x -a)xc) = G. Then x is given by Column-I (Q) (C) (R) a+b+c 2 (D) (S) Q.3 (A) Column-II (a - b)(b - c)(c - a)(a + b + c) then the solution 1 (x-b)2 (x-c)(x-a) 1 (x-c)2 =0, is (x-a)(x-b) (P) a +b+c If 1 a a~ 1 b 1 (x-a)2 of the equation (x-b)(x-c) (B) The value of the limit, f , a X + b X + cX (Q) L X ™ (^/(x + a)(x + b)(x + c) - x), iis (R) (S) a +b+c (C) (D) Lim x->0 equals Let a, b, c are distinct reals satisfying a3 + b3 + c3 = 3abc. If the quadratic equation (a + b - c)x2 + (b + c - a)x + (c + a - b) = 0 has equal roots then a r oot of the quadratic equation is SUBJECTIVE: Q.l [ 4 X 6 = 24] Let / ( x ) = (x + l)(x + 2)(x + 3)(x + 4) + 5 where x e [-6, 6], If the range o f the function is [a, b] where a, b e N then find the value of (a + b). tu/4 Q.2 Q.3 Let I o j (TCX - 4x ) /n(l + tan x)dx. If the value of 1 2 7i "7n 2 k where k e N, find k. Suppose/and g are two functions such that f g : R -> R, 2 / ( x ) ^/n^l + V l ^ ] and ( fiW g(x) = /n! x + \ / l T x 2 then find the value of x egW + g'(x) at x = 1. 120ti is equal to Ð-Ð, find the value of k. K Q.4 If the value of limit L,m -1 l + 7 ( k - l ) k ( k + lXk + 2) Z cos k(k + l) k=2 / JHBANSAL CLASSIES ^ T a r g e t 1ST JEE 2 0 0 7 CLASS: XII (ABCD) DATE: 04-07/9/2006 MATHEMATICS Daily Practice Problems DPR N0.-40, 41 DATE: 04-05/09/2006 TIME: 50 Min. Q. 1 Let/(x) = 1 - x - x 3 . Find all real val ues of x satisfying the inequality, 1 - / ( x ) - / 3 ( x ) > / ( 1 - 5x) g2x _ gX j Integrate: j Ð dx 3 (e x sin x + cos x)(e x cos x - sin x) The circle C : x 2 + y 2 + kx + (1 + k)y - (k + 1) = 0 passes through the same two points for ever y real number k. Find the coordinates of these two points. the minimum value of the radius of a circle C. i Comment upon the nature of roots of the quadratic eq uation x + 2 x = k + J| t + k | dt depending on the 0 value of k e R. 2 Q.2 Q.3 (i) (ii) Q. 4 Q.5 1/n a C¹ Given Lim = Ð where a and b are relatively prime, find the value of (a + b) . 2n f\ b n->oo \ ny 3n DFP-41 DATE: 06-07/09/2006 Q. 1 TIME: 50 Min. Let a, b, c be three sides of a triangle. Suppose a and b are the roots of the e quation x 2 - (c + 4)x + 4(c + 2) = 0 and the largest angle of the triangle is 9 degrees. Find 0. 7 1 Find the value of the definite integral j|V2sinx + 2 c o s x jdx. o 1 Let tan a · tan (3 = 7 ^ 5 . Find the value of (1003 - 1002 cos 2a)(10 03 - 1002 cos 2(3) 1+V5 Q.2 Q.3 0 * 4 Q.5 2 r / X2 + l Ð /. j( .l n + X Ð X +1 V x Ð n dx XJ and e 2 is 60°. The angle Two vectors Sj and e 2 with | e ( | = 2 and \ e 2 | = 1 and angle between between 2t e, + 7 e 2 and ej +1 e 2 belongs to the interval (90°, 180°). Find the ra nge of t. Q.6 Afimction fix) continuous on Rand periodic with period 2% satisfie s f (x) + sin x - / ( x + n) = sin 2 x. Find/(x) and evaluate f / ( x ) d x . CLASS: XII (ABCD) DATE; 30-31/8/2006 TIME: 60 Min. DPP. NO.-39 This is the test paper of Class~XI (J-Batch) held on 27-08-2006. Take exactly 60 minutes. Q. 1 Fi nd the set of values of'a' for which the quadratic polynomial (a + 4)x 2 - 2ax + 2a - 6 < 0 V x e R . x+1 x+5 Solve the inequality by using method of interval, ÐÐ- ^ · [3] I31 [3] [3] 4 | BAN SAL CLASSES glTarget SIT JEE 2007 MATHEMATICS^ Daily Practice Problems Q. 2 Q.3 Q.4 Q.5 Find the minimum vertical distance between the graphs of y = 2 + sin x and y = c os x. d (3 ^ cos x - c o s J x Solve: dx 4 whenx = 18°. If p, q are the roots of t he quadratic equation x + 2bx + c = 0, prove that 2 l o g [ j y - p + y f y - q } = log2 + log(y + b + j, 2 [4] [4] Q. 6 Q.7 Q. 8 Q.9 Q.10 Q.ll Q. 12 Q. 13 x 2 +14x + 9 Find the maximum and minimum value of y = Ð, VxeR. x +2x + 3 Suppose that a and b are positive real numbers such that log 2 7 a + log 9 b = 7 /2 and log 27 b + log 9 a=2/3. Find the value of the ab. [4] Given sin 2 y=sin x · sin z where x, y, z are in an A.P. Find all possible values of the common diffe rence of the A.R and evaluate the sum of all the common differences which lie in the interval (0,315). [4] tan 86 Prove that = (1 + sec29) (1 + sec40) (1 + sec8 6). [4] ·jl 371 571 In Find the exact value of tan 2 Ð: + tan 2 Ð + tan2Ð~ + tan 2 Ð . 16 16 16 16 89 i [4] 151 Evaluate Y ^ l + (tann°) 2 Find the value of k for which one root of the equation of x 2 - (k + 1 )x + k 2 + k-8=0 exceed 2 and other is smaller than 2. [5] Let an be the 0 th term of an arithmetic progression. Let Sn be the sum of the first n terms of the arithmetic progression with aj = 1 and a 3 = 3a g . Find the largest possible value of S n . [5] V Z. J ( C^ C A B Q. 14(a) IfA+B+C = n & sin A + Ð = k sin Ð, then find the value of tan Ð -t an Ð in terms of k. ( \ X +x (b) Solve the inequality, log. log 6 - <0. '0.5 x+4 Q . 15 [2 + 4] Given the product p of sines of the angles of a triangle & product q of their co sines, find the cubic equation, whose coefficients are functions o f p & q & who se roots are the tangents of the angles of the triangle. [6] If each pair of the equations x 2 +pjX + qj = 0 x2 + p 2 x + q2 = 0 x2+p3x-i-q3 = 0 has exactly one root in common then show that (p, + p 2 + p 3 ) 2 = 4(pjp 2 + p 2 p 3 + p3pj - q, - q 2 - q3). [6] Q. 16 4 Q. 1 Q.2 | BANSAL CLASSES j Target III JEE 2 0 0 7 DATE: 23-24/8/2006 MATHEMATICS Daily Practice Problems TIME: 60 Min. DPP. NO.-38 CLASS: XII (ABCD) r 2 1/2 Find the value of a and b where a < b, for which the integral j (24 - 2x - x ) dx } i a s the largest a value. Solve the differential eqaution: y' + sin x - cos x Ve -cosx y= e x - c o s x Ð Q.3 Integrate: J. -dx (x cos x - sin x)(x sin x + cos x) Q.4 In a A ABC, given sin A: sin B : sin C = 4 : 5 : 6 and cos A: cos B : cos C = x : y : z. Find the ordered pair that (x, y) that satisfies this extended proporti on. QQ.6 5 V sin 1 V x FCNdx X X Find the general solution of the equation, 2 + tan x · cot Ð + cot x · tan Ð = 0 Q.7 Let a , (3 be the distinct positive roots of the equation tan x = 2x then evalua te J(sinax-sin[3x)dx , o independent of a and {3. J | BANSAL CLASSES I g g T a r g e f HT JEE 2 0 0 7 MATHEMATICS Daily Practice Problems DPR NO.^37 CLASS: XII (ABCD) " DATE: 18-19/8/2006 TIME: 75 Min. This is the test paper of C lass-XI (PQRS) held on 13-07-2006. Take exactly 75 minutes. Q. 1 Q.2 Q.3 The sum of the first five terms of a geometric series is 189, the sum of the fir st six terms is 381, and the sum of the first seven terms is 765. What is the co mmon ratio in this series. [4] Form a quadratic equation with rational coefficie nts if one of its root is cot 2 l 8°. 1 ( a + 1)(p + 1 } [4] Let a and (3 be the roots ofthe quadratic equation ( x - 2 ) ( x - 3)+(x-3)(x + l ) + ( x + l)(x-2)=0.Find the value of + (a 1 _ 2)(p _2) + (a _m 1 _ 3) · W [4] Q.4 Q.5 Q. 6 Q.7 If a sin2x +Mies in the interval [-2,8] foreveryx < R then find the value of ( a - b ) . = For x > 0, what is the smallest possible value of the expression log(x 3 - 4x 2 + x + 26) - log(x + 2)? [4] The coefficients of the equation ax 2 + bx + c = 0 w here a * 0, satisfy the inequality (a + b + c)(4a - 2b + c) < 0. Prove that this equation has 2 distinct real solutions. [4] In an arithmetic progression, the third term is 15 and the eleventh term is 55. An infinite geometric progression can be formed beginning with the eighth term o f this A.P. and followed by the fourth and second term. Find the sum of this geo metric progression upto n terms. Also compute Srjo if it exists. [5] Find the so lution set of this equation log)sin X|(x2 - 8x + 23) > l o g ( s i n x j ( 8 ) i n x e [0,2n). Find the positive integers p, q, r, s satisfying tan Ð = ( j p - Jq) (yfr - s)Find the sum to n terms of the series. 1 + Q.8 Q.9 Q. 10 [5] [5] 2 Ð + - 3 + Ð 4 + Ð 5 + 2 4 8 16 32 Also find the sum if it exist if n -> oo. Q. 11 Q. 12 Q.13 (a) (b) ( c) Q. 14 2 [5] If sin x, sin 2x and cos x · sin 4x form an increasing geometric sequence, find th e numerial value of cos 2x. Also find the common ratio of geometric sequence. [5 ] Find all possible parameters 'a' for which, f (x) = (a 2 + a - 2)x 2 - (a + 5) x - 2 is non positive for every x e [0,1 ]. st nd rd 2 st nd [5 j The 1 , 2 and 3 terms of an arithmetic series are a, band a where 'a' is negativ e. The 1 , 2 and 3rd terms of a geometric series are a, a 2 and b find the val u e of a and b sum of infinite geometric series if it exists. If no then find the sum to n terms of the G P sum ofthe 40 term ofthe arithmetic series. [5] j) The n th term, a n of a sequence of numbers is given by the formula a n = a n _ } + 2n for n > 2 and aj = 1. Find an equation expressing an as a polynomial in n. Al so find the sum to n terms ofthe sequence. [8] 2x x Let/(x) denote the sum of th e infinite trigonometric series, / ( x ) = ^ sin Ð sin Ð . 3 n=J 3 Find/ ( x ) (inde pendent of n) also evaluate the sum ofthe solutions ofthe equation f (x) = 0 lyi ng in the interval (0,629). [8] 00 Q. 15 . k B A N S A L CLASSES I B Target I I T JE£ 2 0 0 7 CLASS: XII (ABCD) I > I* I " - 3 DATE: 16-17/08/2006 Q.l Q.2 Q.3 x 3 MATHEMATICS Daily Practice Problems DPP. NO.-35, 36 5 TIME: 45 Min. d If y = Jx 2 V^nt dt, find at x = e . l Find the equation of the normal to the curve y = (l +x) y + sin -1 (sin2 x) at x = 0. x Find the real number 'a' such that 6 + J a 2 f f(t)dt -jÐ = 2 v x · Q.4 Q.5 7 The tangent to y = ax + bx + - at (1,2) is parallel to the normal at the point (-2, 2) on the curve y = x 2 + 6x + 10. Find the value of a and b. Let f be a r eal valued function satisfying f(x) + f(x+4) = f(x + 2) + f(x + 6) then prove th at the function x+8 g(x) = | f(t) dt is a constant function. X Q. 6 A tangent drawn to the curve C l = y = x 2 + 4 x + 8 at its point P touches the curve C 2 = y = x 2 + 8x + 4 at its point Q. Find the coordinates of the point P and Q, on the curves C j and C 2 . 3«S DATE: 16-17/08/2006 TIME: 45 Min. 2 3 4 Q. 1 Given real numbers a and r, consi der the following 20 numbers: ar, ar , ar , ar , , ar20. If the sum of the 20 nu mbers is 2006 and the sum of the reciprocal of the 20 number is 1003, find the p roduct of the 20 numbers. Q.2 Let f(x) and g(x) are differentiable functions sat isfyingthe conditions; (i)f(0) = 2 ; g ( 0 ) = l (ii)f'(x) = g(x) & Find the fun ctions f(x) and g(x). 3 (iii)g'(x) = f(x). Q.3 Let f(x) = L 2x-3 (b3-b2+b-l) ÐY (b 2 + 3b + 2 j _ ,0<x<l ,1 < x < 3 Find all possible real values of b such that f(x) has the smallest value at x = 1. Q. 4 There is a function f defined and continuous for all real x, which satis fies an equation ofthe form J f(t) dt = j t f(t)dt + _ _ + _ + c , where C is a constant. Find an explicit formula for f(x) and o x 8 9 also the value of the co nstant. Q.5 Q. 6 r Given Jf(tx) dt = nf( x ) then find f(x) where x > 0. o Tange nt at a point P j [other than (0,0)] on the curve y = x 3 meets the curve again at P 2 . The tangent at P 2 meets the curve at P 3 & so on. Show that the abscis sae of P,, P 2 , P 3 , P n , form a GP. Also find the ratio ^(P^P,) area (P 2 P 3 P 4 ) Xf V 2 X16 X18 ft 4 Q.l Q.2 | BANSAL CLASSES |Target 8iT JEE 2007 DATE: 11-12/8/2006 MATHEMATICS Daily Practice Problems TIME: 60 Min. DPP. NO.-34 CLASS 7 XII (ABCD) Let F (x) = jV4 +1 2 dt and G (x) = JV4 +1 2 dt then compute the value of (FG)' (0) where dash -1 X denotes the derivative. 10 identical balls are to be distributed in 5 different boxes kept in a row and labelled A, B, C, D and E. Find the number of ways in which the balls can be dis tributed in the boxes if no two adjacent boxes remain empty. Q. 3 Iff (x) = 4x 2 + ax + (a - 3) is negative for atleast one negative x, find all p ossible values of a. Q.4 (a) (b) (c) Let/(x) = sin 6 x + cos 6 x + k(sin 4 x + cos 4 x) for some real number k. Deter mine all real numbers k for which/(x) is constant for all values of x. all real numbers k for which there exists a real number 'c' such that f (c) = 0. I f k = - 0 . 7 , determine all solutions to the equation/(x) = 0. 7 T Q.5 , Letx 0 = 2cosÐ a n d x n = ^ 2 + x ^ , n = 1 , 2 , 3 , n-*>o find Lim 2< n+1) -V2^ T n ~. Q.6 f Let/(x)= Ð Ð - then find the value of the sumy j 20C>6 / + ^ 1 1 f 2 1 +f U 0 0 6 j ^ ^2006 J f 3 ^ [2006J (2005^ 2006 J Q.7 V j ^ d * 8 + sin x x . Q.8 Va For a > 0, fmdthe minimum value ofthe integral J(a 3 + 4 x - a 5 x 2 ) e a x dx. 0 I BANSAL CLASSES Target liT JEE 2007 CLASS: XII (ABCD) O P P 1 MATHEMATICS Daily Practice Problems DPP. NO.-33 E K X H E W E DATE: 31/7/2006 to 5/08/2006 O F This is the test paper of Class-XIII (XYZ) held on 30-07-2006. Take exactly 2 Ho urs. N O T E : Leave Star ( *) marked problems. " P A R T ' - A . Select the cor rect alternative. (Only one is correct) Q.l Number of zeros of the cubic f (x) = x3 + 2x + k V k e R, is (A) 0 (B) 1 (C) 2 /x Q.2 The value of Lim dr, is x->°° dx y L(r + l ) ( r - l ) (A) 0 Q.3 (B) 1 (C) 1/2 -2 x 4 (D) non existent 5 - 1 equal to 86. The sum of 2x (D)9 (D)3 [26 x 3 = 78] t There are two numbers x making the value of the determinant these two numbers, i s (A)-4 (B)5 (C)-3 Q.4 A function / (x) takes a domain D onto a range R if for each y e R , there is so me x e D for which / (x) = y. Number of function that can be defined from the do main D = {1,2,3} onto the range R = {4, 5} is (A) 5 (B)6 (C)7 (D)8 Suppose/,/' a nd/" are continuous on [0, e] and that/' (e) = / ( e ) = / ( l ) = 1 and j e 1 Q.5 f/(x),¹ 2 X = Z, then 1 1 the value of f / " ( x ) / n x d x equals I 5 1 3 1 (B) j Q.6 (C) 1 1 (D) 1 - 1 A circle with centre C (1, 1) passes through the origin and intersect the x-axis at A and y-axis at B. The area of the part of the circle that lies in the first quadrant is (A) n + 2 (B) 2n - 1 (C) 2n - 2 (D) n + 1 The planes 2x - 3y + z = 4 and x + 2y - 5z = 11 intersect in a line L. Then a vector parallel to L, is (A ) 13i + l l j + 7 k (B) 1 3 i + l l j - 7 k (C) 1 3 i - l l j + 7 k (D) i + 2 j - 5 k &Q.8 A fair dice is thrown 3 times. The probability that the product of the thre e outcomes is a prime number, is (A) 1/24 (B) 1/36 (C) 1/32 (D) 1/8 Q.9 Period o f the function, / ( x ) = [x] + [2x] + [3xj + + [nx] n(n +1) n J. where n e N and [ J denotes the greatest integer function, is (A) 1 (B) n (C) 1/ n Q. 10 (D) non periodic 2i - i 1 Let Z be a complex number given by, Z = 3 i - 1 the statement which doe s not hold good, is (A) Z is purely real 10 1 1 (B) Z is purely imaginary (C) Z is not imaginary (D) Z is complex with sum of its real and imaginary part equals to 10 Let/(x, y) = xy2 if x and y satisfy x2 + y2 = 9 then the minimum value o f f (x, y) is (A) 0 (B) - 3-^3 (Q-6V3 (D)-3V6 Q. 11 Q. 12 Vl + 3 x - l - x Eim Ð Ð ^ has the value equal to x^o (1 + x) -l-101x (A)3 5050 (B)5 050 (C) 5051 (D) 4950 Q. 13 Number of positive solution which satisfy the equation log 2 x · log 4 x · log 6 x = log 7 x · log 4 x + log 2 x · log 6 x + log 4 x · loggX? (A) 0 Q.14 Q. 15 (B) 1 (C) 2 _1 3 (D) infinite Number of real solution of equation 16 sin"'x tan x cosec"'x = n is/are (A) 0 (B ) 1 (C) 2 (D) infinite Length of the perpendicular from the centre of the ellips e 27x2 + 9y2 = 243 on a tangent drawn to it which makes equal intercepts on the coordinates axes is (A) 3/2 f, (B) 3/V2 1Ðx 2n 2 (C) 3V2 2x 1-x2 (D) 6 Q.l 6 Let/(x) = cos"1 (A) 0 1+ x + tan (B) ti/4 where x e (-1, 0) then/simplifies to (C) n/2 (D) 7t Q. 17A person throws four standard six sided distinguishable dice. Number of way s in which he can throw if the product of the four number shown on the upper fac es is 144, is (A) 24 (B) 36 (C) 42 (D)48 Q.18 a Let A = p x b q y 4x c r and sup pose that det.(A) = 2 then the det.(B) equals, where B = 4y 4z z (B) det(B) = - 8 (C) det(B) = - 16 2003 2a 2b 2c -p -q -r (A) det(B) = - 2 Q. 19 (D) det(B) = 8 (D)9 The digit at the unit place ofthe number (2003) is (A) 1 (B) 3 (C) 7 Q.20 AB AF Let ABCDEFGHIJKL be a regular dodecagon, then the value of Ð + Ð is Ar AB (A) 4 (B)2-s/3 (C) 2V2 (D)2 &Q.21 Urn A contains 9 red balls and 11 white balls. Urn B contains 12 red balls and 3 white balls. One is to roll a single fair die. If the result is a one or a two, then one is to randomly select a ball from urn A. Otherwise one is to ran domly select a ball form urn B. The probability of obtaining a red bail, is (A) 41/60 (B) 19/60 (C) 21/35 (D)35/60 Q.22 L e t / be a real valued function of rea l and positive argument such that / ( x ) + 3x / (l/x) = 2(x + 1) for all real x > 0. The value of /(10099) is (A) 550 Q.23 (B) 505 2 (C)5050 (D) 10010 a \2 / If a and P be the roots of the equation x + 3x + 1 = 0 then the value of 1+P + P a +1 ¹ is equal to Q.24 (A) 15 (B) 18 (C) 21 (D) none The equation (x - l)(x - 2)(x - 3) = 24 has the re al root equal to 'a' and the complex roots b and c. Then the value of b c / a , is (A) 1/5 (B) - 1/5 (C) 6/5 (D) - 6/5 cos m0 · sin n0 Ð then m + n is equal to sin0 (D) 12 If m and n are positive integers satisfying 1 + cos 20 + cos 40 + cos 60 + cOs 80 + cos 100 = (A) 9 (B) 10 (C) 11 Q.25 Q.26 A circle of radius 320 units is tangent to the inside ofa circle ofradius 1000. The smaller circle is tangent to a diameter of the larger circle at the point P. Least distance of the point P from the circumference of the laiger circle is (A )300 (B)360 (C)400 (D) 420 Select the correct alternative. (More than one are co rrect) [8x4 = 32] x e 1/x Q.27 In which of the following cases limit exists at the indicated points. at x = 0 x l + e 1/x where [x] denotes the greatest integer functions. tan-11 x | (C)/(x) = (x - 3)1/5 Sgn(x - 3) at x = 3, (D)/(x) = at x = 0. x where Sgn stands for Sig num function. (A) /(x) [x+|x|] at x = 0 (B)/(x) = &Q.28 Let A and B are two independent events. If P(A) = 0.3 and P(B) = 0.6, then (A) P(A and B) = 0.18 (B) P(A) is equal to P(A/B) (C) P(A or B) = 0 (D) P(A or B) = 0.72 Q.29 Let T be the triangle with vertices (0, 0), (0, c2) and (c, c2) a nd let R be the region between y = cx and y = x2 where c > 0 then (A) Area (R)=c 3 (B) Area of R=Ð 3 In Area (T) Area(T) _ 3 (C) Lim Ð =3 (D) Lim + + c-»o Area (R) c-»o Area(R) 2 ( x+3 Q.30 Q.31 Consider the graph of the function f (x) = e U+i . Then which of the following i s correct. (B) / (x) has no zeroes. (A) range of the function is (1, oo) (D) dom ain of f is ( - oo, - 3) u (-1, oo) (C) graph lies completely above the x-axis. 1 1 x x-1 Let /,(x) = x, / 2 (x) = 1 - x; / 3 (x) = - ,/ 4 (x) = ; / 5 (x) = ; / 6(x) = X I X x-1 Suppose that (A) m = 5 / 6 ( / m ( x ) ) =/ 4 (x) and / n ( / 4 ( x ) ) =/ 3 (x) then (B) n = 5 (C) m = 6 (D) n = 6 Q.32 The graph of the parabolas y = - (x - 2)2 - 1 and y = (x - 2)2 - 1 are shown. Us e these graphs to decide which of the statements below are true. (A) Both functi on have the same domain. (B) Both functions have the same range. (C) Both graphs have the same vertex. (D) Both graphs have the same y-intercepts. Consider the function / ( x ) = f a x + l"\ vbx + 2y where a2 + b2 * 0 then Lim / ( x ) X-»CO Q.33 (A) exists for all values of a and b (B) is zero for a < b (D) is e~ (5/a) or e~ (l/b) if a = b (C) is non existent for a > b Q.34 Which of the following fiinction(s) would represent a non singular mapping. (A) / : R -» R f (x) = | x | Sgn x (B) g : R -> R g(x) = v 3/5 where Sgn denotes Signum functi on 3x 2 - 7 x + 6 (C) h : R R h (x) = x4 + 3x2 + 1 (D) k : R R k (x) : x -x 2 - 2 MATCH THE COLUMN ^^^E^TT-S [4x4 = 16] INSTR UCTIONS: Column-I and column-II co ntains four entries each. Entries of column-I are to be matched with some entrie s of column-II. One or more than one entries of column-I may have the matching w ith the same entries of colurnn-II and one entry of column-I may have one or mor e than one matching with entries of column-II. Q.l (A) (B) Column I Constant fun ction/(x) = c, c e R The function g (x) = P Ð (x > 0), is Column II (P) Bound (Q) periodic Monotonic neither odd nor even Ji t (C) (D) The function h (x) = arc tan x is The function k (x) = arc cot x is (S) (R) Q.2 (A) (B) (C) (D) Q.3 Column I cor (tan(-37 )) cos" (cos(-233°)) A 1 -cos T sin v9, cos - arc cos Column I (A) Number of integral values of x satisfying the inequality x-1 1 1 0 Column II (P) (Q) (R) 143° 127° 3 4 2 3 Column II 1 - 2 - 1 0 (S) 2 (P) 4 x-3 (B) The quadratic equations 2006 x2 + 2007 x + 1 = 0 and x2 + 2007x + 2006 = 0 have a root in common. Then the product of the uncommon roots is (Q) 3 3 (C) Suppose sin 9 - cos 9 = 1 then the value of sin 9 - cos 9 is (9 e R) (R) sin2x-2tanx (D) The value ofthe limit, L l ™ Ð ~ ; 3 ; Ð i s (S) /n(i + x ) Q.4 A qua dratic polynomial / ( x ) = x2 + ax + b is formed with one of its zeros being 4 + 3^3 where a and b 2 + V3 are integers. Also g (x) = x 4 + 2x 3 - 10x2 + 4x - 10 is a biquadratic polynomial such that 8 (A) (B) (C) (D) 4 + 3y3 2 + V3 = + d where c and d are also integers. Column II (P) 4 (Q) 2 (R) -1 (S) -11 13 x 8 = 24] _1 _i Column I a is equal to b is equal to c is equal to d is equal to SUBJECTIVE: Q.l Q.2 Let y = sin"'(sin 8) - tan (tan 10) + cos (cos 12) - sec"'(sec 9) + cor '(cot 6) - cosec "'(cosec 7). If y simplifies to an + b then find (a - b). Suppose a cub ic polynomial / (x) = x3 + px2 + qx + 72 is divisible by both x2 + ax + b and x2 + bx + a (where a, b, p, q are constants and a ^ b). Find the sum of the square s of the roots ofthe cubic polynomial. ~3~ Ð r44Ð Q.3 The set of real values of'x' satisfying the equality V X = 5 (where [ ] denotes the greatest integer ( b function) belongs to the interval a , - where a, b, c e N and ~ is in its lo west form. Find the value of c I c. a + b + c + abc. 4 Q. 1 Q.2 | BANSAL CLASSES | Target IIT JEE 2007 DATE: 26-27//07/2006 MATHEMATICS Daily Practice Problems TIME: 45 Min. DPP. NO.-32 CLASS: XII (ABCD) This is the test paper of Class-XI (J-Batch) held on 23-07-2007. Take exactly 45 minutes. If (sin x + cos x) 2 + k sin x cos x = 1 holds V x e R then find the v alue of k. If the expression r cos X [3] 371 v2 , V 2y is expressed in the form of a sin x + b cos x find the value of a + b. Q.3 r>. 371 + x + sin (327t + x) - 18 cos(19rt - x) + + sin cos(56tc + x) - 9 sin(x + 17tc) [3] 3 statements are given below each of which is either True or False. State whethe r True or False with appropriate reasoning. Marks will be allotted only if appro priate reasoning is given. I (log 3 169)(log 13 243) = 10 II cos(cos 7t) = cos ( cos 0°) III cos x + 3 =T cosx 2 1 S3] Q.4 Q. 5 Prove the identity cos 4 t = ~ + - cos 2t + r cos 4t. o 2 o Suppose that for som e angles x and y the equations · i 3a 0 sin^x + cos^y = Ð and a2 cos x + sin y = Ð J 2 2 2 ¹ 3 1 1 [3] hold simultaneously. Determine the possible values of a. Q. 6 Find the sum of al l the solutions of the equation (log 27 x 3 ) 2 = log 27 x 6 . [3] [3] 7i % 10y-10~y If - Ð < x < Ð and y = log 10 (tan x + sec x). Then the expression E = Ð simplifies to one £ ** JL the six trigonometric functions,findthe trigonometric f unction. 13] Q.8 If log 2 (log 2 (log 2 x))= 2 then find the number of digits in x. You may use log ?0 2 = 0,3010. [3] Q. 9 Assuming that x and y are both + ve satisfying the equation log ( x + y ) = l o g x + l o g y find y in terms of x. Base of the logarithm is 10 everywhere. [3] If x = 7.5° then find the value of cos x ~ cos 3x ~ : . sin 3x - sin x [3] Q.10 Q. 11 Find the solutions of the equation, log ^ sm x (1 + cos x) = 2 in the interval x e [0,2n]. [4] Q. 12 Q, 13 Given that log a2 (a 2 +1) = 16 find the value of log a32 (a + - ) a [4] If cos e = - find the values of (i) cos 36 (ii)tam [4] [5] Q. 14 If log 12 27 = a find the value of log 6 16 in term of a. Q . 15 sin x - c o s x + 1 1 + sinx Prove the identity, Ð r = =tan Ð+Ð , wherever it is defin ed. Starting with left 4 2 sin x + c o s x - 1 cosx hand side only. [5] [5] [6] Q. 16 Q. 17 Find the exact value of cos 24° - cos 12° + cos 48° - cos 84°. S olve the system of equa tions 5 (logxy + log y x) =26 and xy = 64. r=4 Q.18 Prove that £ sin r=l V (2r-l)7c' 8 -2 r=l r=4 cos (2r-l)7t -\ 4 8 [6] Also find their exact numerical value. r i 1a Solve for x: log 2 (4 - x ) + log (4 ·-x). log f x + - 1 - 2 log 2 x + Ð 2, V 2J 0,19 = 0. [6] 4 | BANSAL CLASSES STarget iiT JEE 2007 DATE: 05-06/06/2006 The value of Lim / n x - / n Vx +1 + x X-»00 MATHEMATICS Daily Practice Problems TIME: 50 Min. DPR NO.-28 CLASS : XII (ABCD) (A) 1 (B)/n 7t/4 vw (C) does not exist (D) 0 Evaluate J(tanx-sec 4 x ) d x . (A) 1/4 (B) 1/2 (C) 3/4 (D)l The product of two positive numbers is 12. The smallest possible value of the su m of their squares is (A) 25 Q4 (B) 24 (C) 18 V2 (D) 18 Given that log (2) = 0.3010 number of digits in the number 2000 2000 is (A) 6601 (B) 6602 (C) 6603 (D)6604 , , 1 1 1 Given that a, b and c are the roots of the equation x" - 2x 2 - 1 1 x + 12 = 0, then the value of Ð + Ð + ~ (A) (B) n 12 (C) 13 12 (D) 7 If Jtan x dx = 2, then b is equal to (A) arc cos(2e) t. Q/7 (B) arc sec(2) (C) a rc sec 2 (e) = 8 ( x 2 + 3 x + 2 ) , is (C)-3 (D)-5 (D)none The sum of all values of x so that 16 ( " 2+3x (B)3 (A) 0 A certain function/(x) satisfies f (x) + 2 / ( 6 - x) = x for all real numbers x . The value o f / ( l ) , is (A) 3 (B)2 (C)l (D) not possible to determine Q.9 N umber of ways in which the letters A, B, C and D be arranged in a sequence so th at A is not in position 3, B is not in position 1, C is not in position 2 and D is not in position 4, is (A) 8 (B) 15 (C) 9 (D) 6 Using only the letter from the word WILDCATS with no repetitions allowed in a codeword, number of 4 letter cod ewords are possible that both start and end with a consonant, are (A)360 (B)900 (C) 1680 (D)2204 Q.10 { Q:ll Find j(x/nx)dx (A)- (B)- (C)-l (D)l Q.12 IfP(x) is a polynomial with rational coefficients and roots at 0,1, -Jl P(x) is at least (A) 4 (B) 5 V + (C) 49 (C)6 + 00 anci 1 - \/3 , then the degree of (D)8 > cc ua t0 7 Sum of the infinite series, 4 - ^ + Ð (A) ^ Q.14 (B) 24 l l (D) A florist has in stock several dozens of each of the following: roses, carnation s, and lilies. How many different bouquets of half dozen flowers can be made? 8! ( ) 2!-6! A 9! (B) 12! 3!-6! (D) 56 ^ 1 5 Let/(x) = (A)-9 e3x - 1 . if x * 0 x then/'(0), is 3 if x = 0 (B)9 (C) 9/2 (D) nonexistent n6 I f f "(x) = 10 and f ' (1) = 6 and f ( l ) = 4 then f (-1) is equals (A)-4 (B) 2 ' (C)8 The coefficient of x in the expansion of (A) 97 (B)98 3 (D)12 Q.17 x" v 2 + \12 4 xy ,is (D)100 (C) 99 Q.18 In how many ways can six boys and five girls stand in a row if all the girls are to stand together but the boys cannot all stand together? (A) 172,800 (B) 432,0 00 (C) 86,400 (D)none The composite of two functions f and g is denoted by fog a nd defined by (fog)(x) = f (g(x)). When f(x) 5x and g (x) = Ð which one of the fol lowing is equal to (fog)(x)? x Ð1 x-2 4-x x-2 The equation In k iA (k + 1)i/(k+D 6x 3 Ox > (C) In 1 - x-2 4x + 2 1 +ÐInk k (D) 15x 2x + l = F(k) k+1 is true for all k wherever defined. F(100) has the value equal to (A) 100 (B) 1 101 (C)5050 (D) 1 100 Q.21 Compute f ,_ ^Vx+K/x ill BANSAL CLASSES H Target I I I JEE 2007 CLASS: XII (ABCD) DATE: 28-29/06/2006 Q.l tan 9 = 2Ð+ - MATHEMATICS Daily Practice Problems TIME: 50 Min each DPR DPR NO.-25 TIME: 50 Min DATE: 28-29/06/2006 I > P P - 2 5 1 ~ where 9 e (0,2n), find the possible value of 6. {2] 2 + '--co + Q. 2 Q.3 Find the sum of the solutions of the equation 2e2x _ 5e x 4 = o_ [2] Suppose that x and y are positive numbers for which log 9 x = log 12 y = log 16 (x + y). If the value of --2 cos 9, where 9 e (o, rc/2) find 9. [3] Q. 4 Using L Hospitals rule or otherwise, evaluate the following limit: Limit X->0+ Limit n->eo [l2 (sinx)" j + 22 (sinx)x + n3 + n2 (sinx)x " where [. ] denotes the [4] greatest integer function. Q.5 1 Consider f ( x ) = - ^ = , ~ sin2 x I Ð . V a + htan'x , f o r b > a > 9 & the functions g(x)&h(x) |1 + I 5 ·(V sinx are defined, such that g(x) = [f(x>] - j - ^ J & h(x) = sgn (f(x)) for x e domai n of »f, otherwise g(x) = 9 = h(x) for x £ domain o f ' f , where [x] is the greates t integer function of x & {x} is the fractional 7t part of x. Then discuss the c ontinuity of'g' & *h' at x =Ð and x = 9 respectively. ~ ^ f x 2 tan _1 [5] x , Q. 7 Using substitution only, evaluate: jcosec 3 x dx. [5j DATE: 30-01/06-07/2006 Q.l 12 A If sin A = Ð . Find the value of tan Ð , JIME: 50 Min. [2] Q.2 x v The straight line - ·+ ^ = 1 cuts the x-axis & the y-axis in A& B respectively & a straight line perpendicular to AB cuts them in P & Q respectively. Find the locus of the point of intersection ofAQ & BP. [2] Q.J tan 9 1 cot 9 If - - Ð - Ð Ð = Ð, find the value of - Ð . tan 9 - tan 39 3 cot9-cot39 HI Q.4 If a A ABC is formed by the lines 2x + y - 3 = 0; x - y + 5 = 0 and 3x - y + 1 = 0, then obtain a cubic equation whose roots are the tangent of the interior ang les of the triangle. [4] Integrate Q.5 f Q.6 J a 2 - tan2 x)Vb2 - tan2 x dx (a>b) 15] xsmxcosx I ((a 2 cos 2 x +, b sin x) dx ¹ ¹ T,2 ¹;¹2 \2 [5] Q.7 d dy Let ~Ð (x 2 y) = x - 1 where x ^ 0 and y = 0 when x = 1. Find the set of valu es of x for which Ð dx [5] is positive. DATE: Q. 1 03-04/07/2006 TIME; 50Min. Let x = (0.15) 20 . Find the characteristic and mantissa in the logarithm of x, to the base 10. Assume log 10 2 = 0.301 and log 10 3 = 0.477. [2] Two circles of radii R & r are externally tangent. Find the radius ofthe third circle which is between them and touches those circles and their external common tangent in ter ms of R & r. [2] Q. 2 Q. 3 Let a matrix A be denoted as A = diag. 5 x , 5 5 \ 5 5 S then compute the value ofthe integral j( det A)dx. Q. 4 P] Using algebraic geometry prove that in an isosceles triangle the sum ofthe di stances from any point of the base to the lateral sides is constant. (You may as sume origin to be the middle point of the base of the isosceles triangle) [4] Ev aluate: f1 +-x + x Q.5 J dx Vx + X2 +x 3 fa v 3 [5] a2-3] ; ? Q.6 If the three distinct points, fb3 [b-r b 2 -3^1 b-ij ? r c3 [c-l ' c 2 -3^1 c-lj a-l ' a-1 are collinear then [5] [5] show that abc + 3 (a + b + c) = ab + be + ca. Q.7 Integrate: j^/tanx dx ill BANSAL CLASSES I g l T a r g e t HT JEE 2007 CLASS: XII (ALL) DATE: 23-24/06/2006 Select the correct alternative: (Only one i s correct) Q. 1 MATHEMATICS Daily Practice Problems TIME: 50 Min. DPP. NO.-24 [16 x 3 = 48] A circle of radius 2 has center at (2,0). Acircle of radius 1 has center at (5,0 ). Aline is tangent to the two circles at points in the first quadrant. Which of the following is the y-intercept ofthe line? (A) 3 (B) V2 (Q3 8 (D) 2a/2 Q.2 In a triangle ABC, the length ofAB is 6, the length of BC is 5, and the length o f CAis 4. If K lies on BC BK 3 such that the ratio of length r Ð is Ð, then the leng th ofAK is KC 2 (A) 2V3 Q. 3 (B)4 (C) 3V2 (D) 2, Which one of the following quadrants has the most solutions to the inequality, x - y < 2? (A) I quadrant (B) II quadrant (C) ID quadrant (D) I and III quadrant have same The range of the function / ( x ) = sin _1 x + tan~'x + cos _1 x, is ( A) (0,71) (B) 7t 371 Q. 4 4'T (C) [0,71] (D)R Q.5 The area of the region of the plane consisting of all points whose coordinates ( x, y) satisfy the conditions 4 < x 2 + y 2 < 36 and y < | x is (A) 24n (B) 27TI (C) 20TT (D) 32tc A straight wire 60 cm long is bent into the shape of an L. The shortest possible distance between the two ends of the bent wire, is (A) 30 cm (B) 3 0 V 2 c m (C)10V26 N Q. 6 (D) 20^5 71 Q.7 '7t ·X holds, is Sum of values of x, in (0, n/2) for which tan Ð + X = 9 tan 4' 4 v (A) 0 (B) 71 - tan _1 (2) (C) cor'(O) (D) tan -1 (2) Given/"(x) = cos x, / ' ^ y J = e a n d / ( 0 ) = 1, then/(x) equals. (A)sinx-(e +l)x (B) sinx + (e+ l)x ( C ) ( e + l)x + c o s x (D) ( e + l ) x - c o s x + 2 Q. 8 Q.9 Evaluate the integral: j x e c o s x 2 sin x 2 dx (A) | e c o s x 2 + C (B)- -es mx +C 1 (C) 1 _sin x 2 + C (D)- iecosx2 + C Q.10 The value of Lim x->n (A)0 e~n -e" x is sin x (B)-e- (D)e- (D)nonexitent Q.LL Let/(x) = x e R, is (A) ( - « , , - 2 ] ^" k x + x2-k 1 . The interval(s) of all possible values of k for which/is continuous for every (B)[-2,0) (C) R - ( - 2,2) (D)(-2,2) Q.12 Q, 13 Suppose F (x) = / (g(x)) and g(3) = 5, g'(3) = 3,/'(3) - 1 , / ' ( 5 ) = 4. Then the value of F'(3), is (A) 15 (B) 12 (C) 9 (D)7 From a point P outside of a cir cle with centre at O, tangent segments PA and PB arc drawn. If 1 ( A O ) 2 1 " + ~ Ye ' t 1 b e n l e n t b chord AB is (C) 8 (D) 9 (A) 6 a (B)4 n a Let l2 al3 a23 a33 a21 a31 a22 a32 , Aj * 0 b i, b 12 b22 b32 b13 b23 b33 b21 where b- is cofactor of a^ V i, j = 1,2, 3 b3] c and n C12 C22 C32 C13 C 23 C 33 C 21 C31 where c^ is cofactor of V i, j = 1,2, 3. then which one of the following is always correct. (A) Aj, A2, A3 are in A.P. (B ) Aj, A2, A3 are in G.P. (C) A2 Q. 15 A 3 (D) A, A0 The first three terms of an arithmetic sequence, in order, are 2x + 4,5x - 4 and 3x + 4. The sum of the first 10 terms of this sequence, is (A) 176 (B) 202.4 (C ) 352 (D) 396 The value of V3 i (A) Ð + w 2 2 / n Q. 16 r 4 71 \ wI 71 . . 7n \ 5 T . 7t / 7i cosÐ+ zsin Ð cos Ð +1 sin Ð is equal to 15 15, 8 8 7 L 7t . . K J 2 2 (C) S i (D)-^--i w 2 2 Subjective: Evaluate: Q.l J" dx xV a x - : x 5 q.2 ff s5* s* j 5 x5 rsin j , sm 1 dx Q.3 V x - cos 1 yfx r , r* r dx Ð + COS BANSAL CLASSES 8 T a r g e * I I T JEE 2 0 0 7 CLASS: XII(ALL) DATE: 16-22/06/2006 2 DATE: 16-17/06/2006 Q. 1 For x > 0 and ^ 1 and n e N, evaluate, n-»co MATHEMATICS Daily 1 TIME: 45 Min. 1 logx 2 ~o n_1 Practice Problems DPR NO.-21, 22, 23 Lim 1 1 + + logX 2 . log 4' logx 4 . log 8 V ° =>x · ° · · + .Iog_ -2 n ·~ o x y Q. 2 Show that (a + b + c), (a 2 + b 2 + c 2 ) are the factors of the determinant a2 b2 c2 (b + c) 2 (c + a) 2 (a + b) 2 be ca . Also find the remaining factors. ab Q. 3 Q. 4 Q.5 Prove that a non singular idempotent matrix is always an involutaiy matrix. Find an upper triangular matrix A such that A 3 = 8 0 -57 27 d2 ^ y ¹ dy I f ' y' is a twice differentiable function of x, transform the equation, (1 - x 2 ) -Ð7 - x Ð- + y = 0 by dx dx means of the transformation, x = sin t, in t erms of the independent variable' t'. Atangent line is drawn to a circle of radius unity at the point A and a segment AB is laid offwhose length is equal to that of the arc AC. A straight line BC is drawn to intersect the extension of the diameter AO at the point P. Prove that: 9 (1 - cos 0) (ii)L^tpA=3. (i) PA = e - sin e Use of series expansion or L1 Hos pital's rule prohibited. Q. 6 DATE: Q. 1 19-20/06/2006 TIME: \ l-x\ 45Min. Without using any series expansion or L' Hospital's rule, Evaluate: Lim x la e| 1 + x/ VT3+V3 Find the value of the determinant V15+V26 3 + V65 2V5 5 VPS V5 V10 5 / Q. 2 Q. 3 / ( x ) is a diffrentiable function satisfy the relationship f2 (x) + f 2(y) + 2 (xy - 1 ) = f 2 (x + y) V x, y e R. Also f (x) > 0 V x e R , and f (V2 )= 2. De termine f (x). Q.4 Let,y = t a n - | j Ð 5 + tan" x 2.3 + x j z + tan - 1 j 3.4 + x^ + upto n terms. dy Find Ð- expressing your answer in two terms, dx Q. 5 0 x+a Without expanding the determinant show that the equation x+b root. x-a 0 x+c x-b x-c 0 : 0 has zero as a Q.6 Let a j , a 2 & p j, (3, be the roots of ax 2 +bx + c = 0 & px 2 + qx + r = 0, r espectively. If the system b ac of equations a , y + a 2 z = 0 & p t y + p 2 z = 0 has a non-trivial solution, then prove that Ð = Ð . D r » P - 2 3 X + ^X 0 ~1 TIME: 45 Min. Asi ~ Zs>-J j-2.. DATE: Q. 1 21-22/06/2006 Compute x in terms x 0 , x,, and n. Also evaluate Lim x n = Q.2 AÐ 2 vb a 5 c 8 2 d is Symmetric and B = b - a -2 3 e 6 a - 2b - c is Skew Symmetric, then find AB. -f Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer. x +1 Q.3 Let f ( x ) = e x , x<0 = 0,7 x= 0 2 =x , i x> 0 Discuss continuity and differen tiability of f (x) at x = 0 . 1 0 Show that the matrix A = 2 1 can be decomposed as a sum of a unit and a nilpotent marix. Hence evaluate the matrix 1 0 2 1 2007 Q. 4 Q. 5 dv Find Ð , if (tan"1 xV + y cotx = 1. dx ·f w -)_ bY^) f If / is differentiate and Lim ^ h L'Hospital's rule. 1 + e" x <0 ^n = 'thenfmd the value Q.6 Without using Q.7 Consider the function / ( x ) = x + 2 , 0 < x <3 (a) (b) (c) x >3 x Find all points where f (x) is discontinuous. Find all points when f (x) is not differentiable. Draw the graph, showing clearly the points of discontinui ty or non derivability. 6Ð 3 | h B A N S A L C L A S S IES V S T a r g e t I I T JEE 2 0 0 7 CLASS: XII (EXCEPT A-2) Q. 1 TIME: 60 Min. MATHEMATICS Daily Practice Problems DPR NQ.-20 3 10 The set of all x for which Ð > Ð2Ð7 consists of the union of a finite and an infi nite interval. The length x x +1 ofthe finite interval is (A) 3 (B)2 1 C I O (D)2 T Q.2 Five persons put their hats in a pile. When they pick up hats later, each one ge ts some one else's hat. Number of ways this can happen, is (A) 40 (B)44 (C) 96 ( D) 120 Suppose the origin and the point (0,5) are on a circle whose diameter is along the y-axis and (a, b) lies on the circle. Let L be the line that passes th rough the origin and (a, b). If a 2 + b 2 = 16 and a > 0 then the equation of th e line L is (A) 3 x - 4 y = 0 (B) 2 0 x - 3y = 0 (C)2x-y = 0 (D)4x-3y = 0 If 1 l ies between the roots of the equation y 2 - my + 1 = 0 then the value of has the value equal to (Here [x] denotes gratest integer function) (A) 0 (B) 1 (C) 2 4[ x] VxeR Q.3 Q.4 IxI +16 (D) none 3 2 Q.5 Q.6 The sum of the squares of the three solutions to the equation x + x + x + 1 = 0, is (A)~ 1 (B)0 (C)l (D)2 Let / ( x ) = 1 + x 3 . If g (x) = / _ 1 ( x ) , i.e. if g is the inverse / , then g'(9) equal to (A) 1/12 (B) 1/243 (C) 1/8 (D) 1/24 .Lim j V x - V x - Vx + Vx x-»oo v (A) equal to 0 is (C) equal t o - 1 (D) equal t o - 1/2 Q.7 (B) equal to 1 Q. 8 Suppose f is a differentiable function such that / ( x + y ) = / ( x ) + / ( y ) + 5xy for all x, y and f'(0) = 3. The minimum value of f (x) is (A) - 1 (B) Ð9/10 (C) - 9/25 (D)none x-1 Q-9¹ f i i n Jfg . x + l = 3x then the value of g (3), is v y (A)Q. 10 Q. 11 15 V2 (B)- (C)9 (D) V3 9 sin( A + B) For acute angles A and B if (tan A)(cot B) = - then the value of Ð Ð e qual to 5 sin(A Ð d) (A) 7/4 (B) 2/7 (C) 4/7 (D) 7/2 The value of this product of 98 numbers ! (A) 3y 1 1 - - 1-2 5y 1 - - 98 1- 99 1 - - 100 ,is Q. 12 10 5050 2 If T = 3 /n(x + £x) with £ > 0 and x > 0, then 2x + £ is equal to 98 (B) 100 (C) (D) 1 4950 (A) V-f'2 + 4eT/3 (B) V^2 + 4e-T/3 (C) (D) V^2-4eT/3 Q.13 Evaluate: -2 - , ^ U -12X + 35 (A)-1.25 (B)-1.5 (C)-1.75 (D)-2 Q.14 Q.15 Q.16 Let/be a polynomial function such that for all real x f(x 2 + 1) = x 4 + 5x 2 + 3 then the premitive o f / ( x ) w.r.t. x, is 3 2 x 3 3x 2 x 3 3x 2 x 3 3x 2 x + C vD)Ð+ ^ + x + C ( ( AJ ) Ð + Ð Ð x + C w (B)з- Ð + x + C (C) ¹ K 3 2 3 2 3 2 3 Number of r ular polygons that have integral interior angle measure, is (A) 20 (B)21 (C) 22 (D)23 Suppose/ is a differentiable function such that for every real number x, / ( x ) + 2 / ( - x ) = sin x, then f'(n/4) has the value equal to (A)l/V2 Q.17 (B)-l/V2 (B) -1/2V2 (D)V2 The number of permutation of the letters A A A A B B B C i n which the A's appea r together in a block of four letters or the B's appear in a block of 3 letters, is (A) 44 (B) 50 (C) 60 (D)none If {x} denotes the fractional part function the n the number x = TT^a (A) 1/2 (B)0 (C) - 1 / 2 Q.18 {sf-iyif f/Ð)2 simplifies to (D)none Q.19 Which one of the following is wrong? 2 (A) JtanOsec2 0dO = tankx x +C (B) JtanOsec2OdO = (D)none +C (C) Jxsinxdx = s i n x - x c o s x + C for x < 0 Q.20 Let/(x) = (A) none Q.21 Find L™ y->2 (A)0 Q.22 1 . I f / ( x ) is continuous at x = 0 then the number of values of k is (B) 1 1 ( B)/nx 3 2 3x + 2k for x > 0 (C) 2 1 x/ (C)-l/x2 (D) does not exist (D) more than 2 y-2^x +y-2 Let p(x) be the cubic polynomial 7x - 4x + K. Suppose the three roots of p(x) fo rm an arithmetic progression. Then the value of K, is (A) 21 (B) 16 147 (C) 16 441 (D) 128 1323 Q.23 The sum (in radians) of all values of x with 0<x<2n which satisfy V2 (cos 2x - s in x - 1 ) = 1 + 2 sin x, is (A) 2 T T (B) 3n (C) 4ti The value of Lim n-»°o (D) 671 Q.24 (A) 1 Q.25 Tn is n=0 v 2 y (B) 2 (C) 4 (D) none If sin(x + 2y) = 2x cos y, the the value of dy/dx at the point (0,71) must be (A )-l (B) - 3/2 (C)0 (D)2 | | | BANSAL CLASSIES Target I I T JEE 2 0 0 7 CLASS: XII (ALL) Take approx. 40 to 45 min. for each Dpp. DATE: 02-03/06/2006 MATHEMATICS Daily Practice Problems DPP. NO.-l 7, 18,19 -X Q.l Ify = , dy n . then Ð at x = Ð is 1 + cotx 1 + tanx dx 4 (A)0 (C)l (B)-l W f \f 3k-tan Mequals The value of cot v 3J) + sin x cos x ~ 7 (D)2 Q. 2 ( A ) ( i o ( B ) ( Q.3 Lim 3 + v i o y equals (C) (3 + VI0) (D)(l0 + V3) sm ^ yfH-ylcos l x x->-l+ x (B)l f (A)0 (C) ijn Q.4 1_x If/(x)=3 + l + 7 then (B) v (A) v Lim f (x) = 4 ' x->r Lim f (x) = 3 (C) L ^ f ( x ) = 5 Q.5 (D)/has irremovable discontinuity at x = 1 f(1 h) I f / ( x ) = 3 x 1 0 - 7 x 8 + 5 x 6 - 2 1 x 3 + 3 x 2 - 7 t h e n t h e v a l u e o f Lim x^r (A)53 (B)22 (C) 53 h 3 +3h 22 (D) Ð 'is Q.6 If the triangle formed by the lines x 2 - y2 = 0 and the line Ix + 2y = 1 is iso sceles then /= (A) 1 (B)2 (C)3 (D)0 . (e 2x - l - 2 x z ) ( c o s x - l ) Using L'Hospital's rule or otherwise evaluate the limit, Lim x->o (sin 3x - /n(l + 3x) )x 4 Q. 7 Q.8 Q.9 Q. 10 e x - / n ( x + e) Evaluate the limit Lim . Use of L'Hospital's rule or surd exp ansion not allowed. x x-»o e - l Find all real numbers t satisfying the equation ( 3 t - 9 ) 3 + ( 9 t - 3 ) 3 = (9t + 3 t - 12)3. Find g'(3) if g (x) = x · 2h<x> w here h (3) = ·- 2 and h'(3) = 5. P Q.l Q.2 F P - 1 8 3 \6 ' 2 Find the value of the expression log 4 (2000) log 5 (2000) Let f (x) = a cos(x + 1) + b cos(x + 2) + c eos(x + 3), where a, b, c are real. Given that f (x) has at least two zeros in the interval (0, n), find all its rea l zeroes. 1 . V63 Calculate, sin Ð arc sin V In an infinite pattern, a square is p laced, inside a square, as shown, such that each square is at a constant angle 0 to its predecessor. The largest, outermost square is of side unity. Find the su m of the areas of all the square in the infinte pattern as a function of 0. If 0 is eliminated from the equations, a cos 0 + b sin 0 = c & a cos 2 0 + b sin 2 0 = c, show that the eliminant is, (a - b) 2 (a - c) (b - c) + 4 a 2 b 2 = 0. A t riangle has side lengths 18,24 and 30. Find the area of the triangle whose verti ces are the incentre, circumcentre and centroid of the triangle. Q.3 Q. 4 Q.5 Q.6 - 1 Q. 1 Find the real solutions to the system of equations log 10 (2000xy) - lo g 10 x · log10y = 4 log10(2yz) - log10y · log 10 z = 1 and log 10 (zx) - log1Qz · log 10 x = 0 1-cosx 12cosx + 13 1 9 Q.2 Q.3 Q.4 Q.5 Prove that, cos 1 = 71 - 2 cot 1 - t a n Ð where x e (0, n). 5 2 X 1 _i 24 1 Compute the value of cos - t a n Ð 4 7 If g (x) = x 3 + px 2 + qx + r wh ere p, q and r are integers. If g (0) and g (-1) are both odd, then prove that t he equation g (x) = 0 cannot have three integral roots. Sum the series, c o r ' ( 2 a + a) + cot" 1 (2a"1 + 3a) + c o r ! ( 2 a - 1 + 6a) + cor 1 (2a _ 1 + 10a) + Also find the sum of infinite terms, (a>0). 44 + to ' n ' terms. ^Tcosn Q.6 Let x = Ð 44 ^sinn n=l find the greatest integer that does not exceed 1 OOx. c l | | BANSAL CLASSES v B Target ilT JEE 2007 CLASS: XII (ALL) 2 cos x - sin 2x DATE: 12-13/05/2006 This DPP will be discussed on Friday & Saturday. Q.l MATHEMATICS Daily Practice Problems TIME: 60 Min. DPP. NO.-14 -1 e g(*) = 8x - 47t (7t-2x)z ' ; f (x) for x < 7t/2 h(x) = g (x) for x > 7t/2 t hen which ofthe following holds? (A) h is continuous at x = n!2 (B) h has an irr emovable discontinuity at x = 7t/2 f(x) = (C) h has a removable discontinuity at x = tc/2 ( D ) / (% \ Q. 2 V2 J Two balls are drawn from a b ag containing 3 white, 4 black and 5 red balls then the number of ways in which the two balls of different colours are drawn i s / 2 =g ( _" N 71 (A) 94 Q.3 A B C (B) 47 (C) 38 A B (D) 19 C If A ABC if cosA, cosB, cosC areinA.P. then which ofthe following is also an A. P.? (A) tan Y , tanÐ, tanÐ (C)(s-a)(s-b),(s-c) Q. 4 Q.5 (B) c o t y , c o t y , cotÐ (D) none (D)xG(j) is (D) x = nrc - ( - l ) n 71 12 1 4 If tan"1 ± + (A) x =(x) 1 tan" (2x) + t a n = 0 = n, then (B) x (3x) (C) x = 1 3 sin2x+ 2 The most general solutions of the equation x = ^ (A)x = n7t + ( - l ) " - | ( B ) x = y - C - l ) " ^ where n e I Q. 6 (C)x = 0 Q. 7 The sum of the square of the length of the chords intercepted by the line x + y = n, n e N on the circle x 2 + y 2 = 4 is (A) 11 (B) 22 (C) 33 (D) none y y=f(x) Which one of the following statements about the function I y = f (x), graphed h ave is true? , . o\ y 1 1. . (A) L i m f ( x ) = 0 (B) Lim f (x) - 1 i / x-»0 x-»l +Y \ I " (C) Lim f (x) exists at eveiy point x 0 is ( - 1 , 1 ) (D) Lim f (x) = 1 " X->1 Q.8 a s i n b x - b sinax , Lim Ð Ð (a ^ b) is x-+o tan bx - tan ax (A) 1 Q.9 2x 2 L im is x->0 3 - 3 c o s x (A) 2 (B) a-b a+b (C) a+b a~-b (D)nonexistant (a). 4 3 < >4 C CD) Q. 10 Let / = LimX x2 - 9 ^3VX2+7-4 " and m = Lim x2 - 9 then (D) / = m x->-3yjx2 + 7 - 4 (A)Um Q. 11 (B) I-2m x-»0 (C)/ = - m If (2 - x 2 ) < g (x) < 2 cosx for all x, then Lim g (x) is equal to (A)l (B)2 ( C) 1/2 + (D)0 Q 1 2 Le, (A)-5 x-1 (B)2 + 5 then Lim f (x) is x-»l (C)-6 (D) non existent i Q. 13 xлe+ Lim (/nx) x_e is (B)e e (C) e(D)e £ (A) e e Q.14 Lim (e x + x ) x x->0 Q. 15 (A) e1 /n(x + l) Lim - 7 Ð is X> O log 2 x -Q ( B ) e 1/2 (C) e 3 / 2 (D) e 2 (A) log2e Q. 16 Let / = Lim O)0 3 0 + 71 71 (C)/n2 (D) non existent Ðj ^ then [ I ] is, where [ ] indicates greatest integer function A e->-- sin 0 + I 3y (D) none existent (C) equal to 1 (B) equal to 2 (A) equal to 3 Lim x->0 Q.17 e - sin x - e x (A) e + 1 Q.l 8 (B)e- 3 + l (C) e 3 + 1 (D) e 3 - 1 Lim cos(tan _1 (sin (tan -1 x)]) is equal to (A) - 1 (B)V2 aX + (C) x-»oo 1 S (D) 4 Q. 19 Let f (x) = (A) 1 , if Lim f (x) = 1 and Lim f (x) = 1 then f (2) + f (-2) is equal to (B) 2 x-»0 X +1 (C)0 Q.20 If Lim xÐ>3 x n Ð3n X - 3 108 (n e N) then the value of n is (B)4 (C)5 (D)6 (A) 3 SUBJECTIVE _ , tan2x-2sinx Q. 21 Find the limiting value of ~ Ð Ð as x tends to zero . Q. 22 Without using series expansion or L' Hospital's rule evaluate, ^ ™ Q. 23 S how that the sum of infinite series, 4 4 4 4 tan - 1 Ð + tan" 1 Ð + tan -1 Ð + tan - 1 Ð + Lim l n Q + * x 2 PI + x 4 ) (e - l ) > [3] 71 0 = Ð +cot _ 1 3. 0 [3] Jig BANSAL CLASSES v S T a r g e t IIT JEE 2007 CLASS: XII (ALL) Q. 1 DATE: 22-23/05/2006 3X 4-x , -1<X<1 , l<x<4 MATHEMATICS Daily Practice Problems TIME: 55 to 60 Min. DPP. NO.-15 Draw the graph of the function/ (x) = at x = 1. & discuss its continuity & defferentiability Q.2 Given f : [0, a] Ð S, such that f (x) = 3 cosÐ. Find the largest value of'a', for wh ich fhas an inverse » function f _ 1 . Find f _ 1 . State the domain and the range of f & f - 1 . Find the gradient of the curve y = f _ 1 (x) at the point where the curve crosses the y - axis. Q. 3 Given/(x) = [x] tan (71 x) where [x] denotes greatest integer function, find the LHD and RHD at x = k , where k e I. Examine the continuity at x= 0 ofthe sum fu nction of the infinite series: x + 1 (x +1) (2x +1) (2x+l)(3x + l) l + /nt If x = Ð 2 Ð t 3 + 2/nt t .00 Q. 4 Q.5 and y = dv fdvA . Showthat y Ð = 2x Ð +1. dx Vdx j x Q.6 Let f (x) = tan Ð secx + tan X v/ 2 TT 2 sec Ð + tan 2 sec t2t + 2 where x e ^ 71 7CN X X X +tan ~ t 2 sec 2 t z t X X and g (x) = f (x) + tan ~ 2 2 and n e N . Evaluate the following limits. (a) Limit x-»0 'g(x)^ V X y (b) Limit x->0 v A y (c) Limit x-^0 V X y x + / n | V x 2 +1 - x Q. 7 Without using L' Hospitals rule or series expansion evaluate: Lim x-»0 ill BANSAL CLASSES v S T a r g e t BIT JEE 2007 CLASS: XII (ALL) Q.l (a) (b) (c) DATE: 24-25/05/2006 Suppose f(x) = tan (sin -1 (2x)) Find the domain and range of / Express f (x) as an algebaric function ofx. Find f' I v4y MATHEMATICS Daily Practice Problems TIME: 55 to 60 Min. DPP. NO.-16 X (3e 1/x +4) Q. 2 Discuss the limit, continuity & differentiability of the function f (x) = 2-e1/x ,XTK) ,x=0 atx=0. Q.3 Evaluate: Limit x->0 r* a 7t In tan Ðhax 4 sin bx (b*0). Use of series expansion and L'Hospital's rule is not allowed. Q.4 The function/is defined by y=f(x). W h e r e x = 2 t - | t|, y = t 2 + t 11|, t g R. Draw the graph of / for the interval - 1 < x < 1. Also discuss its continui ty & differentiability at x = 0. 1 /nx 1 A Use of series expansion or L' Hospita l's rule is not allowed. x-1 Q.5 Evaluate Un } ' Q. 6 Q. 7 If g is an inverse function of / & /'(x) = Ð 7 , prove that g'(x) = 1 + [g(x)]n. 1 ~ X h Given a real valued functions f(x) as follows x2 + 2 c o s x - 2 f(x) x" f or x < 0 p forx=0 x sinx - ^n(e cosx) for x > 0 6x2 Determine the value of p if possible, so that the function is continuous at x = 0 . Use of power series or L'Hospital's rule is not allowed. ill BANSAL CLASSES V g Target IIT JEE 2007 CLASS: XII (ALL) DATE: 03-04/05/2006 ax-sinax x x-»0 MATHEMATICS Daily Practice Problems TIME: 60 Min. DPP. NO. -11 is equal to a ( O _2 2 If 'a' is a fixed constant then Lim a2 t^ 2 a3 <B)£ a3 (D)j I f / (x) = x 2 + bx + c and/ ( 2 +1) = / ( 2 - 1 ) for all real numbers t, then which of the following is true? (A)/(l)</(2)</(4) (B)/(2)</(l)</(4) (C)/(2)</(4 )</(l) ( D ) / ( 4 ) </(2) < / ( l ) Let P > 0 and suppose AABC is an isosceles right triangle with area P sq. units. The radius of the circle that passes throu gh the points A, B and C, is (A) VP (B) V2P (B) 1 2 (C) 2-s/P (C) 2 (D) (D) infinite Y \O B\ 4 /A " V Q4 Number of real solutions of the equation cos x + cos {^[2 x) = 2, is (A) 0 A qua dratic polynomial y = ax +bx+c has its vertex at (4, -5) and has two x-intercept , one positive and one negative as shown. Which one of the following must be neg ative? (A) only a (B)onlyb (C) only c (D) only b and c X5 -5' Q.6 If in a AABC, the altitudes from the vertices A, B, C on opposite sides are in H .P, then sinA, sinB, sinC are in (A) GP. (B)A.P. (C) A.G.P. (D) H.P. Suppose tha t two circles C[ and C2 in a plane have no points in common. Then (A) there is n o line tangent to both Cj and C2. (B) there are exactly four lines tangent to bo th CL and C2. (C) there are no lines tangent to both Ct and C2 or there are exac tly two lines tangent to both C{ and C2. (D) there are no lines tangent to both C1 and C2 or there are exactly four lines tangent to both Cj and C2. Q8 Number of three term arithmetic progressions which exist in the set {1,2, 3, dif ference d ^ 0, is (A)190 (B)200 (C)380 (D)400 _i x _t 1 The smallest positive in teger x so that tan tan Ð + tan 10 x +1 (A) 8 (B)9 (C)7 constant which is equal to (A) 7 (B)-7 : ,40} and common tanÐ is 4 (D)0 % Q. 10 When (XV4 _ x 2 / 3 ) 7 is multiplied out and simplified, one of its terms has the form kx3 where 'k' is a (C) 35 (D)-35 Q.ll ¹f = - 1 + /V3 and, y = - I - / V 3 , I x where i2 - -1, then which of the followi ng is not correct? (A) x 5 + y5 = - 1 (B) x 7 + y 7 Ð - 1 (C)x 9 + y 9 = l (D) x 1 1 + y n = - 1 J& rX 2 Number of solutions ofthe trigonometric equation cos 3 x - 3 cos x sin^ = cos 3x where x e (0,1), is (A) 0 (B) 1 (C) 2 In which one of the following cases, limi t does not tend to e? i / 1 / x + 4 NX+3 r i x (A) Lim x ~' (B) Lim 1 + (C) Lim X Ð X Ð V x+2 X> Ð1 Xy (D) Lim ( l + / ( x ) ) ? o o when Lim / ( x ) X->CC (D)infhinte 1 3 0 Q. 14 *y The lines L and K are symmetric to each other with respect to the line y = x. If the equation of the line L is y = a x + b where a and b are non zero, then the equation of K is x b x ¹ x b x b (A)y=--(B)y=---b (C)y= - - + (D)y=- + a a a a a a a Domain of definition of the function f (x) = (A) ( - o o , 0 ] (B) [0, oo) Q. 15 Q. 16 3X-4X Vx -3x-4 is (D)(-oo,l)u(l,4) (C) ( - oo, - l ) u [ 0 , 4) The roots of x 2 + bx + c = 0 are both real and greater than 1. If s = b + c + 1 , then's' (A) may be less than zero (B) may be equal to zero (C) must be greater than zero (D) must be less than zero Which one of the following does not reduce to sin x for every x where the expressions are defined? -2 s· x m ^s ^ sin x s e c x ¹ . . (A) Ð 9 5Ð w csc x - cot x cos x (C) (B) (D) all reduce A sin x to v ' sec x - tan x tanx Let/(x) be a fiinction with two properties (a) for any two real nu mber x and y, f ( x + y) = x + / ( y ) and (b) f ( 0 ) = 2. The value o f / ( 1 0 0 ) , is (A) 2 (B) 98 (C) 102 (D) 100 Read the following statements carefully: I If a, b and c are positive numbers not equal to 1 and a < b, then log a c < l og b c. II The equation x 2 - b = 0 has a real solution for x for any real numbe r b. HI The sequence a n defined by a n = 3 (0.2)"n is a geometric sequence. IV cos(cos(x)) < 1/2, V x e R ' Now indicate the correct alternative. (A) exactly o ne is always true (B) exactly two are always true (C) exactly three are always t rue (D) exactly four are always true. If x = a + b/ is a complex number such tha t x 2 = 3 + 4i and x 3 = 2 + 11/ where i = J I \ , then (a + b) equal to (A) 2 ( B)3 (C)4 (D)5 Q. 17 Q' 1 8 Q.19 Q.20 MORE THAN ONE ARE CORRECT Q 21 If x satisfies log 2 x + log x 2 = 4, then log 2 x can be equal to (A) t a n ~ Q.22 (B)coty (C)tan| (D)cot^ In a triangle ABC, altitude from its vertex meet the opposite sides in D , E and F. Thenthe perimeter of the triangle DEF, is abc (A)-F 2A (B)T R(a + b + c) ( C i - ^ Ð l _ 2rs ( D ) T where A is the area of the triangle ABC and all other symbols have their usual m eaning. , Q 23 In a triangle ABC if Z B = 3 0°, b = 3 V2 - a/6 and c = 6 then (A) the triangle ABC is an obtuse triangle (B) angle Z A can be 15° (C) there can be o nly one value for the side BC (D) the value of tanA tanC will be unique. n Let z =(0, l)eC. Where C is the set ofcomplex numbers, then the sum ^ z for n e N can be equal to k=0 Q.24 Q.25 (A) 1 + i (B)i (C)0 (D)-l Value of the expression log 1/2 (sin6° · sin42° · sin45° · sin 66° · sin 78°) (A) lies between nd 5 (B) is rational which is not integral (C) is irrational which is a simple s urd (D) is irrational which is a mixed surd. i l l BANSAL CLASSES \ 8 T a r g e t IIT JEE 2007 CLASS:XII(ALL) , Ifa>0and Lim x-»0 + MATHEMATICS Daily Practice Problems TIME: 50Min. DPP. NO.-12 DATE: 05-06/05/2006 f v ' 1 + ax \l/x l 1 l J- 2x + ^ (B)2 has the value equal to unity then'a'is equal to (C)3 (D)4 (A)l Q2 Thefirstthree terms of a geometric sequence are x, y, z and these have the sum e qual to 42. If the middle 5y term y is multiplied by 5/4, the number x, Ð , z now form an arithmetic sequence. The largest possible value ofx, is (A) 6 (B) 12 (C) 24 (D) 30 , y^f.3 The value of the expression sin2 1° + sin2 2° + sin2 3° + (A) 0 (B)45 + sin2 90°, is (C) 45.5 (D)90 ^JQA In a triangleABC with altitude AD, ZBAC = 45°, DB = 3 and CD = 2. The area of the triangle ABC is (A) 6 (B) 15 (C) 15/4 (D) 12 When the polynomial 5x3 + Mx + N is divided by x 2 + x + 1 the remainder is 0. The value of (M + N) is equal to (A) -3 (B) 5 (C) - 5 (D) 15 Number of real values of x for which the area ofthe tria ngle formed by 3 points A(-2,1) ;B(1,3) and C(3x, 2 x - 3) is 8 sq. units is (A) 0 (B) 1 (C) 2 (D) infinitely many p. 7 Assume that p is a real number. In order for ^/x + 3p + l - ^/x = 1 to have real solutions, it is necessary that (A) p > 1/4 (B) p > Ð 1/4 (C) p > 1/3 (D) p > Ð 1/3 SUBJECTIVE Find the equation of the circle which has its diameter the chord cut off on the line px+qy - 1 = 0 by the circle x 2 + y 2 = a2. [4] Obtain a relatio n in a and b, if possible, so that the function / ( X ) = , Q9 j . x n (a + sin(x n ))+ (b - sin(x n )) n n n ^ n-»oo (l + x )sec(tan (x +x" )j - ^ i n u o u s a t x = 1. [6] QT10 The interior angle bisector of angle A for the triangle ABC whose coordinates of the vertices are A (-8, 5); B(-l5, -19) and C(1, - 7) has the equation ax + 2y + c = 0. Find 'a' and V. [6] i v/ Q.l v/42 v ^ 4 Si BANSAL CLASSES Target I I T JEE 2 0 0 7 DATE: 08-09/05/2006 In AABC (a + b)(a - b) = c(b + c), the measure of angle A, i s (A) 30° (B) 60° (C) 90° TIME: MATHEMATICS Daily Practice Problems 50Min. (D)120° DPP. NO.-13 CLASS: XII (ALL) The point A (sin 9, cos 9) is 3 units away from the point B (2 cos 75°, 2 sin 75°). If 0° < 9 < 369°. Then 9 is (A) 15° (B) 165° (C) 195° (D)255° The radius of the circle inscr ibed in a triangle with sides 12,3 5 and 37, is (A) 4 (B)5 (C)6 (D)7 Consider th e equation 19z2 - 3/z - k = 9, where z is a complex variable and i2 - - 1. Which ofthe following statements is Tme? (A) For all real positive numbers k, both ro ots are pure imaginary. (B) For real negative real numbers k, both roots are pur e imaginary. (C) For all pure imaginary numbers k, both roots are real and irrat ional. (D) For all complex numbers k, neither root is real. The set ofvalues of x for which the function defined as 1-x /(x)= x<l 1<x <2 x>2 (C){1,2) (D)(1) J*' .5 (1 - x)(2 - x) 3-x fails to be continuous or differentiable, is (A)(1) (B) {2} Q.6 The digram shows several numbers in the complex plane. The circle is the unit ci rcle centered at the origin. One ofthese numbers is the reciprocal of F, which i s (A) A (B)B (C)C (D)D A triangle has side a - , the opposite angle a = 69°, and the sum of the two other sides is 3 (B)V2 K)-, (b + c) = 5. The ratio of the longest to the shortest side of the triangle, is ( A) 1 SUBJECTIVE Q.8 Evaluate : Lim -n+Va) (a>o, n e N ) n »o Ðo Use of series expans ion and L'Hospital's rule is not allowed. ¹ # [4] Q.9 Show that the centroid of the triangle of which the three altitudes to its sides lie on the line y = nijx; y = m 2 x & y = m 3 xlieontheline,y(m 1 m 2 + m 2 m3 + m 3 m 1 + 3 ) = (mj +m2 + m3 + 3 m 1 m 2 m 3 )x. [6] Find the equations of the circles which touch the co-ordinate axes and the line, 3 x + 4 y = 12. [6] ^QTlO J s BANSAL CLASSES V g Target ilT JEE 2007 CLASS: XII (ALL) Q. 1 DATE: 26-27/04/2006 -8 MATH EMATICS Daily Practice Problems TIME: 40 to 45 Min each Dpp. DPR NO,-8,9,10 A variable straight line whose length is C moves in such a way that one of its e nd lies on the x-axis and the other on the y-axis. Show that the locus of the fe et of the perpendicular from origin on the variable line has the equation, (x 2 + y 2 ) 3 = C2x2y2. |5] Evaluate: cosx / n ( x - a ) x->a ln(e" - e ) [51 Q.2 Q.3 Let t,, t 2 and t3 be the lengths of the tangents drawn from a point (x,, yj) to the circles x 2 + y 2 = a2, x 2 + y 2 = 2ax and x 2 + y 2 = 2ay respectively. T he lengths satisfy the equation ty = t 2 t 2 + a 4 . Show that locus of (xj, y ^ consists of x + y = 0 and x 2 + y 2 = a(x + y) [5] Q.4 9 . T; 2cos0 + l Let an = 2 c o s9 " - l then show that L i m (a,a,a,....a nii-l n )7 = Ð ,a n,' 0 e R . n~>oov 1 i i l [8] Q. 5 Consider a function f : x Ð > function,findthe following (i) the range of / x +a ; x e R - {1} where a is a real constant. If / is not a constant x Ð1 1 W JJ [8] (ii) f~x, is it exist (iii)/ - / V / V / -9 Q. 1 Given Lim x-»0 ffx) x - 2 then evaluate the following limits, giving explicit reasoning. f(x) X (i) Lim [f(x)l ;(ii)Lim x->0 x-»0 where [x] denotes greatest integer function. [5] Q. 2 Find the sum to n terms of the series Sn = c o t - ^ 2 2 + £ ) + c o r if2 3 + ^ + cor 1 2*+' 23y +, upto n terms [5] Also deduce that Limit S n = c o t _ 1 2 . nÐ Q. 3 The vertices of a triangle are A(x t , x}tan 6j), B(x2, x^an 0 2 ) & CCxg, x3tan 03). Ifthe circumcentre O of the tnangle ABC is at the ongin & H (x,y) beits ort hocentre, then show that Ð= Ð=-. y sinOj+sinOj+sinOj f5] ' . . ^ X COS0, +COS0~ +COS0o Q.4 Q.5 If (1 + sin t)(l + cos t) = - . Find the value of (1 - sin t)(l - cos t). 10 ide ntical balls are distributed in 5 different boxes kept in a row and labled A, B, C, D and E. Find the number ofways in which the balls can be distributed in the boxes ifno two adjacent boxes remain empty. [8] JF»3F»-;l_ Q. 1 Q Tangents are drawn from any point on the circle x 2 + y2 = R 2 to the circle x 2 + y 2 = r 2 Show that if the line joining the points of intersection ofthese ta ngents with the first circle also touches the second, then R = 2r. [5] - /n(2-co s2x) /n (l + sin3x) f o r x < Q Q.2 Let a functionf (x) be defined as f(x) = sin 2 x _ i /n(l + tan9x) for X > 0 [5] Find whether it is possible to define f (0) so that ' f ' may be continuous at x = 0. Q. 3 Find all possible values of a and b so that f (x) is continuous for all x e R if | ax+ 3 | , 13x + a ] , *sia2*-2b, cos2x-3 , if x < - 1 if - 1 < x < 0 if 0 < x <7t if x > 7 t /(x) [5] Q.4 Prove that in a AABC, the median through A divides the angle Ainto two parts who se cotangents are, 2 cot A + cot C and 2 cot A + cot B and it makes an angle wit h the side BC whose cotangent is | (cot B - cot C). [8] / Q.5 Find the value of y = sin cot cos tan x where x - cosec cos V -1 -1 -i J v3 2 cos V6+1 2A/3 > y [8] 1 Q.l Q.2 a BANSAL CLASSES Target 11T JEE 2007 DATE: 24-25/04/2006 MATHEMATICS Daily Practice Problems TIME: 55 to 60Min. DPP. NO.-7 CLASS: XII (ALL) Let T = {1, 2 , 3 , 4 , 5 }. A f u n c t i o n / : T - » T i s said to be one-to-o ne if tj * t 2 implies that/(tj) ^/(t 2 ). Obtain a one-to-one function such tha t t + / ( t ) is a perfect square for every t in T. [4] If a > b > c > 0 then find the value of : cot - 1 C ab+l^i /bc+n / c a + l"] r + cor 1 t Ð + cot""1 Ð ~ . \a-by vb-cy vcÐay [4] Q.3 Find the equation to the locus of the centre of all circles which touch the line x = 2a and cut the circle x 2 + y 2 = a 2 orthogonally. [4] Q.4 Let/(x) = ( * - 4 ) ( X 2 ~ 4 X - 5 ) , F i n d (x - 2 x - 3 ) ( 4 - x ) (a) the domain o f f (x) (c) all x such t h a t / ( x ) > 0 (b) the roots o f f (x) (d) all x such t h a t / ( x ) < 0 [6] Q.5 The points ( - 6 , 1 ) , (6,10), (9,6) and (-3, - 3 ) are the vertices of a rect angle. What is the area of the portion of this rectangle that lies above the x a xis? [6] Q.6 Let/(x)= V ax 2 + bx · Find the set of real values of'a' for which there is at lea st one positive real value of 'b' for which the domain of / a n d the range of / are the same set. [6] Q. 7 Two circles of different radii R and r touch each other externally. The three co mmon tangents form a 2(RrW2 triangle. Show that the area of the triangle is Ð Ð - Ð . R-r f8| 3BANSAL c l a s s e s Target I I T JEE 2 0 0 7 CLASS: XII (ALL) The value of 2-V3 8 ZX4r£: 21-22/04/2006 TIME: is MATHEMATICS Daily Practice Problems 60Min. DPP NO.-6 sin 120° 16 cos 15° · cos3 0° · cos 120° · cos 240° (B) V3-1 (C) (A) 2-V3 (D)2-V3 Let S denote the set of all numbers m such that the line y = mx does not interse ct the parabola y = x 2 +1. S is a bounded interval. The length of S is (A) 3 (B ) 3.5 (C) 4 (D) 4,5 . xQ 3 A line lx has a slope of (-2) and passes through the point (r, - 3). A second line l2, is perpendicular to lx, intersects at (a, b), and passes through the point (6, r). The value of'a' is equal to (A)r VQ.4 (B) 2r (C) 2 r - 3 (D) 5r 7 11 + 1511 when divided by 22 leaves the remainder (A) 0 (B) 1 (C) 7 The coeffi cient of x in the expansion of (1 + x)(l + 2x)(l + 3x) (A) 4950 (B) 5000 (C) 505 0 (D)10 (1 + lOOx), is (D)5100 Suppose AABC is an equilateral triangle and P is a point interior to AABC. If th e distance from P to sides AB, BC and AC is 6, 7 and 8 units respectively, then the area of the AABC, is (A) 147V3 ^M 7 (B) 147V3 , N 21V3 (C) (D) 441 The tune Twinkle Twinkle Little Star' has 7 notes in its first line, CCGGAAG All notes are held for the same length of time. If the notes are rearranged at rand om, number of different melodies that can be composed, is (A) 72 m (B) 105 (B) 3S ( B ) t (C) 210 (D)5040 /Q.8 If the graphs of y = cos x and y=tan x intersect at some value say 9 in the firs t quadrant. Then the value of sin 9 is (A) -1 + V 2 -l + fi (C) - 1 + V5 (D) -1±V5 V Q.9 If S = 1 + - + - + Ð + 4 9 16 S (A)- , then 1 + - + Ð H Ð + Ð 9 25 49 equals (C) > 4 CD)S-i Q 10 How many solutions are there for the equation cos 2 x - sin22x = 0 on [0, 2n]7 ( A) 6 (B)4 (C)2 (D)l Number of ways in which 7 people can be divided into two tea ms, each team having at least one member, is (A) 72 (B) 32 (C) 144 (D)63 Let P b e a point on the complex plane denoting the complex number z. If (z - 2) (z + /' ) is a real number then the locus of P is (A) y = 2x + 1 (B) 2y = 2 - x (C)y = x - 2 (D)y = 2 x - 1 Q. 11 Q. 12 13 The positive value o f x that satisfies VlO = e x + e~x, is (A)|/n(4-Vl5) ( B ) ^ / n ( 4 + Vl5) ( Q ~/n(4 + V l ? ) f n (D) ^ - J r i ) ^/Q.14 L e t / : R - {0} - > R be any function such that/(x) + 2 / /(x)=l,is (A)l (B)2 (C)-l Ð vx; - 3x. The sum of the values of x for which (D)-2 ^ QTl 5 Let r j, r 2 , r 3 , r 4 be the roots of the equation, x 4 - 9x 3 + ax2 + bx + 1 6 = 0 r, + r, + r3 + r. where a, b are constants. Then the difference between th e arithmetic mean - Ð and the 4 geometric mean i ] ^ 2 h r 4 (A) 5/2 the roots, is equal to (C) 17/4 (D) 1/4 (B) 1/2 16 Sum ofthe x and^ intercepts ofthe circle described on the line segment joining ( -2, 1) and (1, 2) as diameter, is (A)l (B)2 (C)3 (D)4 1 If sin lf, a = Ð, then the value of 5 Q. 17 1 cos a H (A) 4 . Q. i 8 (B)6 2 Ð h ÐI 1 + sin a 1 + sin a (C) 8 1 , is 1 + sin a (D) 10 4 A variable circle touches the x-axis and also touches the circle with centre at (0,3) and radius 2. The locus of the centre of the vaiable circle is (A) an elli pse (B) a circle (C) a hyperbola (D) a parabola 1 > -3ÐÐ is X 1 X 2* (B)x<- 1 (C)x<2 andx<- 1 1 19 The set of real numbers x satisfying (A)x>2 (D)-l<x<2 \J$.20 A sequence of three real numbers forms an arithmetic progression with a first te rm of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting number form a geometric progression. The smallest possible valu e for the third term of the geometric progression, is (A) 1 (B)3 (C)4 (D)6 ·.^M.l 1 The curve such that each point P on the curve has equal distances from the point (2, - 2) and from the line y = x has the equation (A)(x + y)2 = 4 (B) (x - y) 2 = 4 (C) (x + y) 2 + 8(y - x) = - 16 (D) (x + y) 2 + 2(x + y) = 16 For any strai ght line, let m and b represent its slope and y-intercept, respectively. Conside r all lines having the property that 2m + b = 3. These lines all have the specif ic point (x l3 y } ) in common. The ordered pair, (Xj, y,) is equal to (A) (2, 3 ) (B)(3,2) (C) (1, 2) (D)(2,l) V.Q'.'23 If the sum of the solutions of the equation sin2x - sin x = cos 2 x on the inter val-[0,2ri] is expressed as a7i/b, where a and b are positive integers, a/b in l owest terms, then (a + b) is (A) 8 (B) 9 (C)10 (D) 11 Q. 24 Quadratic equation with real coefficients whose one root is (2 + /')(3 - z) wher e /' = ^ T J , is (A) x 2 - 14x + 48 = 0 (B) x 2 - 14x + 50 = 0 (C) x 2 + 1 4 x - 4 8 = 0 (D) x 2 + 14x + 49 = 0 Which of the following is equal to sec(t) + tan (t) ? n (A) cot H + 4 [ t (B) cot -2 + Ð 4 r ( C ) - c o t -t+Ð v 2 (D)-cot 71 Q2 5 \^26 If / ( x ) = x + 4 and g(f (x>) = 2x + 1, then the function g (x) is (A) 2x - 7 (B) 8x + 3 (C)2x + 9 (D)2x 2 + 5x + 4 Q.27 In a triangle ABC, Z A = 7 2 ° , b = 2 a n d c = ^ 5 +1 thenthe triangle ABC is (A ) obtuse isosceles (C) right isosceles (B) acute isosceles (D) not isosceles Js BANSAL CLASSES V 8 T a r g e t I1T JEE 2007 CLASS: XII (ALL) DATE: 19-20/04/2006 MATH EMATICS Daily Practice Problems TIME: 50 to 60 Min. DPR NO.-5 [ 6 X 3 = 18] Only one is correct. There is NEGATIVE marking for each wrong answer 1 mark will be deducted. 1 Given an isosceles triangle, whose one angle is 120° and radius of its incircle is ^[j,. Then the area of triangle in sq. units is (A)7+I2V3 (B) 1 2 - 7 ^ 3 (C) 1 2 + 7 ^ 3 (E>) 4n J%2 If 0 < 9 < 2n, then the intervals of values of 9 for which 2sin 2 9 - 5sin9 + 2 > 0, is f i5n ¹ ^ n 5tx u Ð , 2n 0, (A) (B) 8'T J v 6 (C) u 71 571 6'T (D) 41TT 48 -,7t Q.3 If w = a + ip where p values o f z i s (A) {z: |z| = l } w Ð wz and z ^ 1, safisfies the condition that Ð Ð Ð is purely real, then the set of 1Ðz ( B) {z: z = z ) (C) { z : z * l } (D) {z : | z | = 1, z ^ 1} x_ Q.4 Let a, b, c be the sides of triangle. No two of them are equal and X, e R. If th e roots of the equation x 2 + 2(a + b + c)x + 3X(ab + be + ca) = 0 are real, the n (A) (B)^>3 (CH '1 v3'3y (D) X € v3'3y ^ QC5 If r, s, t are prime numbers and p, q are positive integers such that the LCM of p, q is r 2 t 4 s 2 , then the numbers of ordered pair of (p, q) is (A) 252 (B) 254 (C)225 (D)224 A$.6 and t, = (tan0) tan0 , t 2 = (tan9) cote , ^ = (cot9) t a n e , t 4 = (cot0) cot e , then V 4y (A)tj<t2<t3<t4 (B) t 4 > t 3 > t j > t 2 (C)t3<tj <t2<t4 (D)t2<t3< t1<t4 LetOe [ 1 X 5 = 5] One or more than one is/are correct. There is NEGATIVE marking for each wrong an swer 1 mark will be deducted. \JQ.1 Internal bisector of Z A of a triangle ABC meets side BC at D. A line drawn thro ugh D perpendicular to AD intersects the side AC at E and the side AB at F. If a , b, c represent sides of AABC then (A) AE is harmonic mean of b and c 4bc . A ( C)EF=Ðsinb+c 2 (B) A D = v 7 A cosÐ b+c 2 2bc (D) the triangle AEF is isosceles Only one is correct. [ 3 x 5 = 15] There is NEGATIVE marking for each wrong answ er 2 marks will be deducted. Comprehension Let ABCD be a square of side length 2 units. C 2 is the circle through vertices A, B, C, D and C, is the circle touch ing all the sides of the square ABCD. L is a line through A PA2+PB2+PC2+PD2 I f P is a point on C, and Q in another point on C 7 , then 7 - 7 2 , , 1 t^A + v^D + ' V*-' (A) 3/4 - Q. 9 (B) 3/2 (C) 1 (D)9 , JQ. 8 isequalto A circle touches the line L and the circle C, externally such that both the circ les are on the same side of the line, then the locus of centre of the circle is (A) ellipse (B) hyperbola (C) parabola (D) parts of straight line Aline M throug h A is drawn parallel to BD. Point S moves such that its distances from the line BD and the vertex A are equal. If locus of S cuts M at T 2 and T 3 and AC at Tj , then area of ATjT 2 T 3 is (A) 1 /2 sq. units (B) 2/3 sq. units (C) 1 sq. unit (D) 2 sq. units Q. 10 SUBJECTIVE: [ 2 x 6 = 12] There is NO NEGATIVE marking. "4 11 If roots of the eq uation x 2 - 1 Ocx - 1 Id = 0 are a, b and those of x 2 - 1 Oax Ð 1 l b Ð 0, then fi nd the value of a + b + c + d. (a, b, c and d are distinct numbers) 3 3 v3 Q.12 If A = Ð - Ð + Ð +· + V - /l ) n ' Ð and Bn = 1 - A , then find the minimum natural ( n n A AI A 1 4 J f" f' ^4; number n 0 such that B n > A n . V n > nQ. CLASS : XI (P, Q, R, S) Q. 1 Which I II III IV (A) I 4 3BANSAL CLASSES 8Target IIT JE1 2007 DATE: 26/01/2006 of the following sets does NOT represent a function? {(x, y) | y = 2x + 1} {(x, y) j x 2 + y2 = 10, y > 0} {(3, 1) (4, 1),(5, 2), (6, 2), (7, 3 )} {(x, y) | y = 2X + 1} (B) II (C) III MATHEMATICS DaiSy Practice P r o b l e m s TIME: 45 Min. DPP. NO.-53 (D) IV (E) none Q.2 I f f (x) is a function from R R, we say that f (x) has property I if f (f (x) ) = x for all real number x, and we say that f (x) has property II if f (-f(x)) = - x for all real number x. How many linear functions, have both property I and II? (A) exactly one (B) exactly two (C) exactly three (D) infinite The function f (x) is defined by f (x) = cos 4 x + K cos 2 2x + sin 4 x, where K is a constan t. If the function f (x) is a constant function, the value of k is (A) - 1 (B) - 1/2 n-l Q.3 (C) 0 (D) 1/2 (E) 1 Q.4 Define the function f (n) where n is a non negative integer satisfying f (0) = 1 and f (n) is defined for n > 0 as f (n) = n · i=0 . Let 2 m be the highest power of 2 that divides f (20). The value of m is (A) 18 (B) 19 (C) 20 (D) 21 (E)22 Direction for Q.5 and Q.6 The graph of a relation is (i) Symmetric with respect to the x-axis provided that whenever (a, b) is a point on the graph, so is (a, - b) (ii) Symmetric with respect to the y-axis provided that whenever (a, b) is a point on the graph, so is ( - a, b) (iii) Symmetric with respect to the origin provided that whenever (a, b) is a point on the graph, so is ( - a, - b) (iv) Sy mmetric with respect to the line y = x, provided that whenever (a, b) is a point on the graph, so is (b, a) Q.5 The graph of the relation x 4 + y 3 = 1 is symme tric with respect to (A) the x-axis (B) the y-axis (C) the origin (E) both the x -axis and y-axis (D) the line y = x Q.6 Suppose R is a relation whose graph is symmetric to both the x-axis and y-axis, and that the point (1,2) is on the graph of R. Which one of the following points is NOT necessarily on the graph of R? (A) ( - 1 , 2 ) " (B) ( 1 , - 2 ) (C)(-l, -2) (D) (2, 1) (E) all of these points are on the graph of R. Q.7 Suppose that f (n) is a real valued function whose domain is the set of positive integers and that f (n) satisfies the following two properties f (1) = 23 and f (n + 1) = 8 + 3 · f (n), for n > 1 It follows that there are constants p, q and r such that f (n) = p · q n - r, for n = 1, 2, then the value of p + q + r is (A) 1 6 (B) 17 (C) 20 (D) 26 (E)31 x rx Let f (x) = ÐÐ and let g (x) = . Let S be the set of all real numbers r such that 1+x 1-x f (g(x)) = g (f (x)) for infinitely many real number x. The number of elements in set S is (A) 1 (B) 2 (C) 3 (D)5 Let f be a linear function with the properties that f (1) < f (2), f (3) > f (4), and f (5) = 5. Which of the following statements is true? (A) f (0) < 0 (B) f (0) = 0 (C)f(l)<f(0)<f«-1) (D) f (0) = 5 f 1-x If g (x) = 1 - x 2 and f (g(x) ) = Ð w h e n x * 0. then f (1/2) equals x (A) 3/4 (B) 1 (C)3 (D)V2 2 Q.8 Q.9 Q. 10 * Q. 11 A function f from integers to integers is defined as follows - n+3 f(n) = L if n is odd if n is even n/2 Suppose k is odd and f ( f ( f (k))) = 27. The sum of the digit of k is (A) 3 Q. l2 (B) 6 (C) 9 for some (D) 12 positive a. If f | f ( V 2 ) )= - ^ 2 then a Let f be the function defined by f (x) = ax 2 - ^ 2 equals (A) V2 (B) | (C) ^ (D) ^ ill BAN SAL CLASSES I S T a r g e t IIT J E E 2 0 0 7 CLASS: XII (ALL) DATE: 10-11/04/2006 MATHEMATICS Daily Practice Problems TIME: 35 to 45 Min. DPP. NO.-l DPP 1 to 4 complete revision of class XI. This DPP will be discussed on Monday ( 10-04-2006). Q. V ABC is triangle. Circles C p C 2 and C 3 are drawn with sides AB, BC and CA as their diameters. The radical axis between any two circles w.r.t the AABC is o ne of its (A) angle bisector (B) altitude (C) median (D) perpendicular bisector of the sides. The function/ ( x ) defined on the real numbers has the property t hat / ( / (x)) · (l + / (x)) = - / ( x ) for all x in the domain off. If the numbe r 3 is in the domain and range of f, then the value of / ( 3 ) equals (A)-3/2 (B )-3/4 (C) 1/4 (D) 1/2 If m and b are real numbers and mb> 0, then the line whose equation is y=mx + b cannot contain the point (A) (0,2006) (B) (2006,0) (C) ( 0 , - 2 0 0 6 ) (D)(19,-97) Q. 2" Q .3 Q.4 Which is the inverse of the function/(x) = - / n f x + V x 2 + l j ? (A) 3(e 3x + e"3x) sin(a + P) p q ' c o t P has the value equal to p-q p+q (B) ^ (C) v ' p+ q q tan a (B) | (e 3x + e- 3x ) (C) ^ (e~3x - e 3x ) (D) - (e 3x - e~3x) Q-? sin(a - p) p+q (A) v ' p-q if p-q (D) 4 C6/ Number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number, is (A) 26 (B) 33 (C) 32 (D)53 I f / ( x ) = x 4 + ax3 +bx 2 + cx + d be polynomial with real coefficient and real r oots. If | f ( i ) | = 1, where i = ^/CT > then a + b + c + d i s equal to (A) - 1 (B) 1 (C)0 (D) can not be determined t Q-8- Let ABCDE is a regular pentagon with all sides equal to 4. Which one of the foll owing is a correct solution for the length AC? (i) 2 csc(18°) (ii) 2 sec (72°) (iii) V32-32cos(108°) (B) only (ii) and (iii) are correct (D) all are correct (A) only (i) and (ii) are correct (C) only (iii) and (i) are correct 9 A line x = k intersects the graph of y=log 5 x and the graph of y=log 5 (x+4). T he distance between the points of intersection is 0.5. Given k = a + Vb , where a and b are integers, the value of (a + b) is (A) 5 (B)6 (C) 7 (D) 10 , Q.10 Which ofthe foliowing sets ofrestrictions is true for the function / ( x ) = ax2 + bx + c represented by the graph as shown (A) a > 0 , b < 0 , c > 0 (B) a > 0 , b < 0 , c < 0 (C) a > 0, b > 0, c < 0 (D) none of these ^yQ. 11 The radius of the circle passing through the vertices of the triangle ABC, is (A) 8y/l5 5 (B) 3y/l5 / 0 (C) 3^5 Q. 12 CD) 3V2 The area of the region consisting of all points (x, y) so that x 2 + y 2 < 1 < | x | +1 y |, is (A) n (B)n-l (C)tc-2 (D)TC-3 1 3 5 9 17 A^.iJ j Q 13 13 7 15 11 19 21 23 25 27 29 Consecutive odd numbers are arranged in rows as shown. If the row s are continued in the same pattern, then the middle number of row 51, is (A) 26 01 (B) 2500 (C)2704 (D)2401 Q.14 The expression [x + (x 3 -l) 1 / 2 ] 5 + [ x - ( x 3 - l ) 1 / 2 ] 5 is a polynomial of degree (A) 5 (B)6 (C) 7 (D) 8 Let a, b, c, d, e, f , g, h be distinct elements in the set {-7, - 5 , - 3 , - 2 , 2 , 4 , 6 , 1 3 }. The minimum possible value of (a + b + c + d) 2 + (e + f + g + h) 2 is (A) 30 (B)32 (C) 34 (D)40 Q. 15 J | BANSAL CLASSES ^ B T a r g e t IIT JEE 2007 CLASS: XII (ALL) _ Q.l DATE: 12-13/04/2006 If the circles x 2 + y 2 + 2ax + 2by + c = 0 and x 2 + y 2 + 2bx + 2ay + c = 0 MATHEMATICS Daily Practice Problems TIME: 35 to 45 Min. DPR NO.-2 where c > 0, have exactly one point in common then the value of ^ (A)l (/Q.2 2c is (D) 1/2 (B)V2 (C) 2 Suppose/ is a real function satisfying/(x +/(x)) - 4 / ( x ) a n d / ( I ) = 4. Then the value of /(21) is (A) 16 (B)21 (C) 64 (D) 105 100 Q.3 The value of n=0 equals (where i= (B)i ) (C) 96 + i (D) 9 7 + i (A) - 1 >jQ. 4 Given AABC is inscribed in the semicircle with diameter AB. The area of AABC equ als 2/9 of the area of the semicircle. If the measure of the smallest angle in A ABC is x then sin 2x is equal to (A) n (B) 2n (C) 71 18 (D) 71 8 , The value of b > 0 for which the region bounded by both the x-axis and y = - 1 2 x | + b has an area of 72, is (A) 12 (B) 36 (C)6V2 (D) 144 , 0(6 y If 500! = 2 m · N, where N is an odd positive integer, then m is equal to (A) 452 (B) 494 (C)498 (D) none of these L e t / be a linear function for which/(6) - / ( 2 ) = 12. The value o f / ( 1 2 ) - / ( 2 ) is equal to (A) 12 (B) 18 (C) 24 ( D)30 1rv2] U s ) tpB-) Чtn=i , then cot(a - p) ^ - Ð v^r.8 a = tan -1 A/2+1 V2-1 -tan I2 J and P = tan~'(3) - sin -1 I5 J W (A) is equal to 1 v/Q.9 (B) is equal to 0 (C) is equal to J 2 - 1 (D) is non existent There are three teachers and six students. Number of ways in which they can be s eated in a line so that between any two teachers there are exactly 2 students, i s (A) 3 - 3 ! - 6 ! (B) 2 · 6! (C) 2 · 3! · 6! (D)3-6! The average of the numbers n si n n° for n = 2 , 4 , 6 , (A) 1 (B)cot 1c ^QCIO 180 (C)tan 1° (D) A circle with center O is tangent to the coordinate axes and to the hypotenuse o fthe 30°-60°-90° triangle ABC as shown, where AB = 1. To the nearest hundredth, the ra dius ofthe circle, is (A) 2.37 (B) 2.24 C (C) 2.18 (D)2.41 1 B ^ 1 2 If s = l + + + ,then 1 - 1 - + I - - 1 - + (C)"-2 equals (B)^-s (D) sÐ1 ^ j Q . 13 Find the value of x that satisfies the equation log (A)l (B)10 ' X x1/x ^ l/(x+l) 5050 (D)1000 (C) 100 Q. 14 The set of points (x, y) whose distance from the line y = 2x + 2 is the same as the distance from (2,0) is a parabola. This parabola is congruent to the parabol a in standard form y = Kx 2 for some K which is equal to (A) V 12 (B) V5 T 4 12 . 15 The number 2006 is made up of exactly two zeros and two other digits whose sum i s 8. The number of 4 digit numbers with these properties (including 2006) is (A) 7 (B) 18 (C) 21 (D) 24 i l l BANSAL CLASSES Target IIT JEE 2007 CLASS: XII (ALL) Q.l DATE; 14-15/04/2006 MATHEMATICS Daily Practice Problems TIME: 35 to 45 Min. DPP. NO.-3 (x + y + z)(xy + xz + yz) xyz (C)6 Let / be the function defined by/(x, y, z) : y and z. The smallest possible valu e of / , is (A) 9 (B)8 for all positive real numbers x, (D)3 (J3-2 The right-angled triangle has two circles touching its sides as shown. If the an gle at R is 60° and the radius of the smaller circle is 1, then the radius of the larger circle is (A)2V3 (C) 2V2 (B)2 (D)3 R + . If [/(xy)] 2 =x ( / ( y )) 2 for all positive numbers x and y and (C) 40 (D)50 ^3.3 Let 7 be a function defined from R + / ( 2 ) = 6 then/(50) is equal to (A) 10 (B )30 Q.4 An equilateral triangle, with sides of 10 inches, is inscribed in a square ABCD in such a way that one vertex is at A, another vertex on BC and one on CD. The a rea of the square is (A) 2 5 ( 2 - V 3 ) (B) 25(2 + V3) (C) 25 100 Q.5 The coefficient of x 3 in the expansion of (1 + x + x 2 ) 12 , is (A)352 (B)350 (C)342 (D) 332 The radius ofthe inscribed circle and the radii of the three escribed circles of a triangle are consecutive terms of a geometric progression then triangle (A) i s acute angled (B) is obtuse angled (C) is right angled (D) is not possible ^A-l A function/is defined for all positive integers and satisfies / ( I ) = 2005 an d / l ) + / 2 ) + ...+y(n) = n2y(n) for all n > 1. The value of/(2004) is (A) ^ . 8 1 1002 (B) 1 2004 (C) 2004 2005 (D)2004 The line (k + 1 ) 2 x + ky - 2k 2 - 2 = 0 passes through a point regardless of t he value k. Which of the following is the line with slope 2 passing through the point? (A) y = 2x - 8 (B) y = 2x - 5 (C)y = 2 x - 4 (D)y = 2x + 8 If the solutio n set for/(x) < 3 is (0,00) and the solution set for/(x) > - 2 is ( - 00,5), the n the true solution set for ( / ( x ) ) 2 > f ( x ) + 6,is (A) (-oo,+ 00) (B)(-0 0,0] (C) [0,5] (D)(-oo,0]u[5,oo) Q. 10 ABCD is a quadrilateral with an area of 1 and ZBCD - 100°, ZADB = 20°, AD = BD and B C = DC shown in figure. The product (AC) x (BD) is equal to D^ V3 CA)V (C) V3 (B ) (D) 2V3 Ð 4V3 3 , Q. 11 Locus of the feet of the perpendicular from the origin on a variable line passin g through a fixed point (a, b) (where a * 0, b ^ 0) is a circle with x-intercept p and y-intercept q, then (A) p = 0 and q = 0 (B) p = 0 and q * 0 (C) p * 0 and q = 0 (D) p * 0 and q * 0 Two rods AB and CD of length 2a and 2b respectively (a > b) slides on the x and y axes respectively such that the points A, B, C and D are concyclic. The locus of the centre of the circle through A, B, C and D is a conic whose length of the latus rectum is .2 (A) Ð (B) 2aVia 2 - b 2 a (C) 2 a b V a 2 - b 2 1 if x is rational (D) 2 V a 2 - b 2 Q. 12 v_X).13 Let / ( x ) = 0 if x is irrational (D)g(x) = |x A function g (x) which satisfies x f (x) < g (x) for all x is (A) g(x) = sin x ( B)g(x) = x (C)g(x) = x 2 ^ / Q . 14 How many of the 900 three digit number have at least one even digit? (A) 775 (B) 875 (C)450 cot 10° + tan 5 0 equal to (A) sec 10° (B) sec 5° (D)750 15 (C)cosec5° (D)cosecl0° ill BAN SAL CLASSES Target IIT JEE 2007 CLASS: XII (ALL) DATE: 17-18/04/2006 MATHEMATICS Daily Practice Problems TIME: 35 to 45 Min. DPP. NO.-4 (^A A sequence ofequilateral triangles is drawn. The altitude of each is J 3 tim es the altitude ofthe preceding triangle, the difference between the area of the first triangle and the sixth triangle is 968 The perimeter ofthe first triangle is (A) 10 (B) 12 (C) 16 (D) 18 square unit. Two circles with centres at A and B, touch at T. BD is the tangent at D and TC i s a common tangent. AT has length 3 and BT has length 2. The length CD is (A) 4/ 3 (B) 3/2 (C) 5/3 (D)7/4 Q.3 The value of cos 5° + cos 77° + cos 149° + cos 221° + cos 2 93° is equal to (A)0 (B) 1 (C)-l (D) 1/2 Let C be the circle of radius unity centr ed at the origin. If two positive numbers x, and x 2 are such that the line pass ing through (x,, - 1 ) and (x 2 ,1) is tangent to C then (A) x,x 2 = 1 (B) X j X 2 = Ð 1 (C)xj+x2=L (D)4x,x2= 1 Q.5 Suppose that (o and z are complex numbers such that both (1 + 20® and (1 + 2/)z are different real numbers. The slope of the lin e connecting © and z in the complex plane is (A)-2 (B)-1/2 (C) 2 (D) can not be de termined xsec0 + ytan0 = 2cos0 J * 6 If x t a n 0 + y s e c 0 = cot0 cos 20 sin0 then y equals (A) (B)sin0 (C) cos 0 (D) sin 20 The locus of the point of intersection ofthe tangent to the circle x 2 + y 2 = a 2 , which include an angle of 45° is the curve (x 2 + y 2 ) 2 = la2 (x 2 + y 2 - a 2 ). The value of X is (A) 2 (B)4 (C) 8 (D) 16 Consider the circle x 2 + y 2 - 14x - 4y + 49 = 0. Let 1, and 12 be lines through the origin 'O' that are tange nt to the circle at points 'A' and 'B'. If the measure of angle AOB is tan - 1 ( X) then X equal to 2 (A)~ 21 <B>« 28 (D)none The value of the expression, 14 tan tan ' - + tan ' - + tan which is equal to (A ) 2 13 + tan 21 + tan 1 31 is an integer (B)5 (C)7 (D)10 10 If a, b are positive real numbers such that a - b = 2, then the smallest value o f the constant L for which V x 2 + a x - V x 2 + b x < L for all x > 0, is (A) 1 /2 (B) 1/V2 (C)l (D)2 ^Q.ll If every solution of the equation 3 cos 2 x - cos x - 1 = 0 is a solution of the equation a cos 2 2x + bcos2x - 1 = 0 . Then the value of (a + b) is equal to (A ) 5 (B) 9 (C) 13 (D) 14 What is the y-intercept of the line that is parallel to y=3x, and which bisects the area of a rectangle with corners at (0,0), (4,0), (4 ,2) and (0,2)? (A) ( 0 , - 7 ) (B)(0,-6) (C) (0, - 5) (D)(0,-4) Let / ( x ) = x 2 + kx ; k is a real number. The set of values of k for which the equation f (x) = 0 and / ( / ( x ) ) = 0 have exactly the same real solution set is (A) (0,4) (B) [0,4) (C)(0,4] (D)[0,4] 12 13 ^ 1 4 If X'° 83(4) = 27, then the value of x 0 o g 3 4)2 (A) 4 (B) 16 (C) 64 (D) 81 Q^ l 5 If Q is the point on the circle x 2 + y 2 - 1 Ox+6y+29 = 0 which is farthest fro m the point P(-l, -6), then the distance from P to Q is (A)2V5 (B) 2V7 (C)4V5 (P )4y/7 ill BAN SAL CLASSES V 8 T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 14/11/2005 MATHEMATICS Daily Practice Problems TIME: 120 Min. DPP. NO.- 50 DPP OF THE WEEK This is the test paper of Class-XI (J-Batch) held on 13-11-2005. Take exactly 120 minutes. To be discuss on Friday (18-11-2005) Only one alternative is correct. PART-A [20 x 1.5 = 30] There is NEGATIVE marking. For each wrong answer 0.5 mark will be deducted. ZERO for not attempted. Q. 1 If the solutions of the equation sin20 = k ( 0 < k < l ) i n ( 0 , 2 n ) are in A. P. then the value of k is (A)| Q. 2 (B)^ (C)^ (D)i Number of real x satisfying the equation | x - l | = | x - 2 | + | x - 3 | i s ( A) 1 (B)2 (C) 3 (D) more than 3 Q.3 A rectangle has its sides of length sin x and cos x for some x. Largest possible area which it can have, is (A) --1 T (B) 1 (C) ~ Z (D) can not be determined Q.4 Consider an A . P . t j j ^ t g , If 5th, 9th and 16th terms of this A.P. form t hree consecutive terms of a GP. with non zero common ratio q, then the value of q is (A) 4/7 (B) 2/7 (C) 7/4 (D)none The new coordinates of a point (4,5) when t he origin is shifted to the point (1, - 2 ) are (A) (5,3) (B)(3,7) (C)(3,5) (D)n one A particle is moving along a straight line so that its velocity at time t > 0 is v (t) = 3t2. At what time t during the interval from t = 0 to t = 9 is its velocity the same as the average velocity over the entire interval? (A) 3 (B)4.5 (C) 3(3) 1/2 (D)9 Acute angle made by a line of slope - 3/4 with a vertical lin e is (A) cot_1[ (B) tan"1 - I ^ Q.5 Q.6 Q.7 A) (C) tan -1 ! 2 _ -if 3^ .-l S3\ (D) cot v2y Q. 8 If logAB + log B A 2 = 4 and B < A then the value of iog A 8 equals (A)V2-1 (B) 2 V 2 - 2 (C) 2 - V3 (D) 2 - V 2 Q. 9 The sum of 3 real numbers is zero. If the sum of their cubes is 7ccthen their pr oduct is (A) a rational greater than 1 (B) a rational less than 1 (C) an irratio nal greater than 1 (D) an irrational less than 1 Three circle each of area 4 n, are all externally tangent (i.e. externally touch each other). Their centres for m a triangle. The area of the triangle is (A) 8V3 (B) 6v'3 (C) 3^3 (D) 4^3 Q. 10 Q.ll IfthelinesL, : 2 x + y - 3 = 0 , L 2 : 5 x + k y - 3 = OandL 3 : 3 x - y - 2 = 0 , are concurrent, then the value o f k is (A)-2 (B)5 (C)-3 (D)3 Suppose x, y, z is a geometric series with a common ratio of'r' such that x ^ y . Ifx, 3y, 5z is an arithmetic sequence then the value of'r 1 equals (A) 1/3 (B) 1/5 (C) 3/5 (D) 2/3 The radius of the incircle of a right triangle with legs of length 7 and 24 , is (A) 3 (B) 6 (C) 8.5 (D) 12.5 Number ofintegers which simultaneously satisfi es the inequalities | x | + 5 < 7 and | x - 3 | > 2, is (A) exactly 1 (B) exactl y 2 (C) more than 2 but finite (D) infinitely many The value of (VTj) 2959 is (A )l (B)-l (C)V^T (D)-V=l Q. 12 Q. 13 Q. 14 Q. 15 Q. 16 The points Q = (9,14) and R = (a, b) are symmetric w.r.t. the point (5,3). The c oordinates of the point R are (A) v Q. 17 ' 17^ 2, (B) (13,25) (C)(l,-8) (D)none If F (x) = 3x 3 - 2x 2 + x - 3, then F(1 + /') has the value equal to (A) 8 + 3/ (B) 8 - 3 / (C) Ð 8 Ð 3/ ( D ) - 8 + 3/ Q. 18 If x 2 + (A) 18 x = 7 then the value of (B)21 X equals ( x > 0 ) (C) 24 (D) 27 Q. 19 S et of all real x satisfying the inequality ! 4iÐ1 - log 2 x | > 5 is, where i = ^ p l . (A) [4, oo) (B) r i rA i i (C) I 0, Ð 16. (D) ( I ll 16. u [4, °o) Q.20 Let Xj and X2 are two realnumbers such that x 2 + x 2 = 7 and xj* + x 2 = 10. Fi nd the largest possible value of Xj + X2 is (A) 8 (B) 6 (C) 4 (D)2 PART-B Q.l For what values of m will the expression y2 + 2xy + 2x + my - 3 be capable o f resolution into two rational factors? [3] Q.2 If one root of the quadratic equation x 2 + mx - 24 = 0 is twice a root of the e quation x 2 - (m + 1 )x + m = 0 then find the value of m. [3] I f x is eliminate d from the equation, sin(a+x) = 2b and sin(a-x) = 2c, then find the eliminant. [ 3] r Q.3 Q.4 Solve the logarithmic inequality, logj 2(x - 2) N ,(x + l ) ( x - 5 ) , P] Q.5 Find allx such that ^Tk-x k=l =20. [3] Q.6 Find the area of the convex quadrilateral whose vertices are (0,0); (4, 5); (9,2 1) and (-3,7). P] Q.7 Find the direction in which a straight line must be drawn through the point (1,2) so that its point of intersection with the line x + y = 4 may be at a dist ance ^ y[6 from this point. Q. 8 [4] We inscribe a square in a circle of unit radius and shade the region between the m. Then we inscribe another circle in the square and another square in the new c ircle and shade the region between the new circle and the square. If the process is repeated infinitely many times, find the area of the shaded region. [4] In a AABC, if a, b, c are in A.P, then prove that cos(A - C) + 4cosB = 3 Q.9 [4] i l l BAN SAL CLASSES v B T a r g e t IIT JEE 2007 CLASS: XI(P, Q, R, S) Q. 1 DATE: 10/10/2005 Daily Practice Problems TIME: 50Min. DPP. NO.- 49 MATHEMATICS Identify whether the statement is True or False. There can exist two triangles s uch that the sides of one triangle are all less than 1 cm while the sides of the other triangle are all bigger than 10 metres, but the area of the first triangl e is larger than the area of second triangle. Number of positive integers x for which/ ( x ) = x 3 - 8x2 + 20x - 1 3 , is a prime number, is (A) 1 (B)2 (C)3 (D) 4 The value of m for the zeros of the polynomial P(x) = 2x 2 - mx - 8 differ by (m - 1 ) is 10 (A)4,-y 10 (B)-6,Ð 10 (C)6, Ð 10 (D)6,-y A Q.2 Q. 3 Q. 4 Each side of triangle ABC is divided into 3 equal parts. The ratio of the area o f hexagon UVWXYZ to the area of triangle ABC i s u, 5 (A) 1 (C) 2 2 (B) j 3 (D) 4 Q.5 If cos A, cos B and cos C are the roots of the cubic x 3 + ax 2 + bx + c = 0 whe re A, B, C are the angles of a triangle then (A) a 2 - 2 b - 2c = 1 (B) a 2 - 2b + 2c - 1 (C) a 2 + 2 b - 2c = 1 (D) a 2 + 2b + 2c = 1 What quadrilateral has th e points (-3,6), (-1, -2), ( 7 , - 4 ) and (5,4) taken in order in the xy-plane as its vertices? (A) Square (B) Rhombus (C) Parallelogram but not a rhombus (D) Rectangle but not a square Which of these statements is false? (A) A rectangle i s sometimes a rhombus. (B) A rhombus is always a parallelogram. (C) The digonals of a parallelogram always bisect the angles at the vertices. (D) The diagonals of a rectangle are always congurent. Points P and Q are 3 units apart. A circle centre at P with a radius of 3 units intersects a circle centred at Q with a rad ius of ^ 3 units at point A and B. The area of the quadrilateral APBQ is (A)V99 a/99 (B) - f [99 (C) ^ f (D) 199 ^ Q.6 Q. 7 Q. 8 Directions for Q.9 to Q . l l : A straight line 4x + 3y = 72 intersect the x and y axes at A and B respectively. Then Q.9 Distance between the incentre and the orthocentre of the triangle AOB is (A)2V6 Q. 10 (B)3V6 (C)6V6 (D) 6V2 The area of the triangle whose vertices are the incentre, circumcentre and centr oid of the triangle AOB in sq. units is (A) 2 (B) 3 (C) 4 (D) none The radii of the excircles of the triangle AOB (in any order) fonn (A)anA.P. (B)aG.P. (C)anH. P. Q.ll (D) none Directions for Q.12 to Q.15: Consider two different infinite geometric progressions with their sums S j and S 7 as S ] = a + ar + ar 2 + ar3 + 00 S 2 = b + bR + bR 2 + bR 3 + 00 If Sj = S 2 = 1. ar = bR and ar2 = Ð then answer the following: Q.12 The sum of their common ratios is (A) Q. 13 1 (B) 4 (C)l (D>2 The sum of their first terms is (A)l (B) 2 Common ratio ofthe first G.P. is (A) 1 (B) l-x/5 (C)3 (D)none Q. 14 (C) V5-1 4 (D) V5+1 Q.15 Common ratio of the second G.P. is (A) 3 + V5 (B) 3-V5 (C) (D)none ill BAN SAL CLASSES V 8 T a r g e t I1T JEE 2007 CLASS: XI (P, Q, R, S) DATE: 03/09/2005 MATHEMATICS Daily Practice Problems TIME: 120 Min. DPP NO.-48 This is the test paper of Class-XI (J-Batch) held on 02-10-2005. Take exactly 2 hours. PART-A Only one alternative is correct. There is NEGATIVE marking. For each wrong answer 0.5 mark will be deducted. Q. 1 If n arithmetic means are between two quantities 'a' and 'b' then the /7th arit hmetic mean is b + na (A) v ' n+1 Q. 2 (B) v 7 a + nb n (C) w n ( b - a )2 p n+1 v(D) [20 x 1 = 20] a + nb ' n+1 If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (log b c) 3 + (logca)3 is (A) an odd pri me (B) an even prime (C) an odd compo site (D) an irrational number Sum to n ter ms of the sequence + ^21 + T>4l+ . Q. 3 77(3" - 1 1 ) (C) Ð ^ Q.4 (D) none of these Ifthearcsofthe same length in two circles S t and S 2 subtend angles 75° and 120° re spectively at the S, centre. The ratio Ð is equal to S 2 , 1 (A) J Q. 5 81 CB)- 64 ( O - 25 (D)- Ifthe roots of the cubic, x 3 + ax2 + bx + c = 0 are three consecutive positive integers. Then the value of a2 b+1 (A) 3 is equal to (B) 2 (C)l (D) none of thes e Q.6 2 3 2 3 2 3 Ð + Ð + Ð + Ðr + Ðr + + s 5 5 5 5 5 5 15 (A)^ 13 (B)^ 00 isequalto 3 (C)? 4 (D)? Q. 7 Number ofprincip al solution of the equation tan 3x - tan 2x - tan x = 0, is (A) 3 (B)5 (C)7 (D) more than 7 Q. 8 If the mth, nth and pth terms of G P. form three consecutive terms of another G. P. then m, n and p are in (A)A.P. (B)GP. (C)H.P. (D)A.GP. Each of the four state ments given below are either True or False. I. m. 1 sin765° = - ^ 1371 1 tanÐ = ^ II . IV. cosec(-1410°) = 2 cot 1571 Q. 9 4 . = -1 Indicate the correct order of sequence, where'T' stands for true and 'F' stands for false. (A) F T F T (B)FFTT (C)TFFF (D)FTFF Q. 10 o 2k+2 o The sum ^T ÐÐ equal to k=i 3 (A) 12 (B) 8 (C) 6 (D)4 Q.ll The value of p which satisfies the equation 122p_1 = 5(3p -7p) is /n5-/nl2 /n21- /nl2 ^ / n l 2 + /n5 /nl2-/n21 /n5 + /nl2 /nl44-/n21 ^ /nl2 /nl2-5/n21 Q. 12 If 0 is eliminated from the equations x = a cos(0 - a ) and y = b cos (0 - P) th en 2xy JL cos(a - P) is equal to a bl ab (A) sec2 ( a - P) (B) cosec2 ( a - P) + x2 y2 (C)cos2(a-p) (D)sin2(a-p) Q. 13 Which of the following is the largest? (A)2 1 o 8 s 6 (B) 3 log 6 5 (C) 3log56 ( D)3 Q. 14 The quadratic equation X 2 - 9X + 3 = 0 has roots r and s. If X 2 + bX + c = 0 h as roots r 2 and s2, then (b, c) is (A) (75,9) (B) (-75,9) (C)(-87,4) (D)(-87,9) ^ . tan220°-sin220° . The expression T ; simplifies to tan 2 20°-sin 2 20° (A) a ration al which is not integral (C) a natural which is prime (B) a surd (D) a natural w hich is not composite n i r Q.15 Q.16 2024 571 971 If sin2x= r r r r , where Ð < x < Ð , the value of the sin x - cos x is equal to H 2025 ' 4 4 Q. 17 If a, b, c are real numbers such that a 2 + 2b = 7, b2 + 4c = - 7 and c 2 + 6a = - 14 then the value of a 2 + b2 + c 2 is (A) 14 (B)21 (C) 28 (D) 35 Q. 18 Q.19 The value of x that satisfies the relation x = l - x + x2-x3 + x4-x5 + 0 0 (A) 2 cos36° (B) 2 cos 144° (C)2sinl8° (D)none 2 If sin 0 and cos 9 are the roots of the eq uation ax - bx + c = 0, then (A) a 2 - b2 ^ 2 a c (B)a 2 + b 2 = 2ac (C) a2 + b 2 + 2ac = 0 ( D ) b 2 - a 2 = 2ac The equation, | sin x | = sin x + 3 in [0, 2tc ] has (A) no root (B) only one root (C) two roots Q.20 (D) more than two roots More than one alternative are correct. There is NO negative marking. Q.21 [5x2 = 10] Thevalue(s) of 'p' for which the equation a x 2 - p x + ab = 0 and x 2 - a x - b x + ab = 0 may have a common root, given a, b are non zero real numbers, is (A) a + b 2 (B) a 2 + b (C)a(l+b) (D)b(l+a) If ax 2 + b x + c = 0 , b * l be an equ ation with integral co-efficients and A > 0 be its discriminant, then the equati on b 2 x 2 - Ax - 4 a c = 0 has : (A) two integral roots (B) two rational roots (C) two irrational roots (D) one integral root independent of a, b, c. FortheAP. given by a t , a^, (A) aj+ 2a2 + % = 0 (C) a, + 3a2 - 3a3 - a 4 = 0 , an, , the equations satisfied are (B) ^ +%=0 (D) aj + - 4a 4 + a 5 = 0 Q.22 Q.23 Q. 24 It is known that sin P = Ð and 0 < P < % then the value of 5 5 (A) independent of a for all p in (0,7t/2) (7 + 24cota) (C) Ð - Ð Ð for tan P < 0 (B) (D) none for tan p > 0 · V3sin(a + P) cosItc 6) v sin a ' ' r-rTcos(a + P) is: Q. 25 The sum of the first three terms of the G.P. in which the difference between the second and the first term is 6 and the difference between the fourth and the th ird term is 54, is (A) 39 (B) - 1 0 . 5 (C) 27 (D)-27 PART-B Q. 1 Q.2 Q. 3 If cos(a + p) + s i n ( a - p ) = 0 a n d t a n p = ^ ^ . F i n d t a n a . If a , p are the roots of ax2 + bx + c = 0, find the value of (aa + b) ~3 + (ap + b)~3. Find the largest integral value ofx satisfying the inequality l og 2 ( 3 - 2 x ) > l . [3] [3] [3] Q.4 If between any two positive quantities there be inserted two arithmetic means A p A^; two geometric means G t , G 2 and two harmonic means Hj, F^, then show tha t GjG 2 : H , H 2 = A 1 +A2 : Hj + U 2 . P] Q. 5 Find all the values of the parameter'm' for which every solution of the ine quality 1 < x < 2 is a solution of the inequality x 2 - mx + 1 < 0. [3] Find the general solution of the equation, sin 4 2x+cos 4 2x = sin 2x cos 2x. Find the s um of the series, ^ 1.2.3 H ^ I 2.3.4 + I +ÐÐÐ Ð . n(n + l)(n + 2) [3] [41 Q, 6 Q. 7 Q. 8 Show that the triangle ABC is right angles if and only if sinA+ sinB + sinC = co sA+ cosB + cosC +1. [4] Q. 9 Find the real solutions to the system of equations log 10 (2000xy) - log 10 x · log 10 y = 4 log10(2yz)-log10ylog10z=l and log 1 0 (zx)-log 1 0 z-log 1 0 x = 0. [4] i l l BANSAL CLASSES Target NT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 26-27/09/2005 MATHEMATICS Daily Practice Problems TIME: 60 Min. DPR NO.-47 OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1 TEST [3 x 20 = 60] In a triangle ABC, R(b + c) = a Vbc where R is the circumradius of the triangle. Then the triangle is (A) Isosceles but not right (B) right but not isosceles (C ) right isosceles (D) equilateral Starting with a unit square, a sequence of squ are is generated. Each square in the sequence has half the side length of its pr edecessor and two of its sides bisected by its predecessor's sides as shown. Thi s process is repeated indefinitely. The total area enclosed by all the squares i n limiting situation, is 5 (A) - sq. units 75 (C) Ð sq. units 79 (B) Ð sq. units 1 ( D) Ð sq. units Q.2 Q.3 1 1 1 + Thesum Ð Ð Ð Ð Ð Ð H Ð : Ð Ð Ð Ð Ð . s m 4 5 sin46° sin47°sin48° sin49°sin50° (A) sec ot(l)0 1 + . . is equal to M sin 133°sin 134° (D)none Q.4 8 _ Number of real values of x e (0, n) for which Ð Ð Ð ^ 3 sin 2 x < 5, is d sin x. s in J X (A) 0 (B) 1 (C) 2 (D) infinite If f (x) = x 2 + 6x + c, where 'c' is an i nteger, then f (0) + f (-1) is (A) an even integer (B) an odd integer always div isible by 3 (C) an odd integer not divisible by 3 (D) an odd integer may or not be divisible by 3 If abed = 1 where a, b, c, d are positive reals then the minim um value of a 2 + b 2 + c 2 + d 2 + ab + ac + ad + be + bd + cd is (A) 6 (B) 10 (C) 12 (D) 20 Minimum vertical distance between the graphs o f y = 2 + s i n x a n d y = cosx is (A) 2 (B)l (C)V2 (D)2-V2 Q.5 Q.6 Q.7 Q. 8 A square and an equilateral triangle have the same perimeter. Let Abe the area o f the circle circumscribed A about the square and B be the area of the circle ci rcumscribed about the triangle then the ratio ~ is B 9 (A).jg 3 (B) 27 (C) (log] 0n)-l (D) 3V6 - f Q.9 Iflog 10 sinx + l o g 1 0 c o s x = - 1 and log 10 (sinx + c o s x ) = (A) 24 (B ) 36 (C) 20 then the value of 'n'is (D)12 Q. 10 Let f (x) = x 2 +x 4 + x 6 + x 8 + oo for all real x such that the sum converges . Number of real x for which the equation f (x) - x = 0 holds, is (A) 0 (B) 1 (C )2 (D)3 Find the smallest natural 'n' such that tan( 107n)° = (A) n = 2 (B) n = 3 cos 96° +sin 96° Ð . Ð. cos96 - s i n 9 6 (C)n = 4 Q.ll ^ Q. 12 (D)n = 5 ABC is an acute angled triangle with circumcentre 'O' orthocentre H. If AO - A H then the measure of the angle A is (A)71 <B)7 7t (C)j 71 (D)~ 571 Q.13 Let a, b, c be the three roots of the equation x 3 + x 2 - 333x - 1002 = 0 then the value of a3 + b3 + c3. (A)2006 (B)2005 (C)2003 (D)2002 44 cosn Q. 14 Let x = j 4 4 y n=l Zsin n° then the greatest integer that does not exceed 1 OOx is equal to (A) 240 Q. 15 (B) 241 (C) 242 (D)243 The numbers b, c, d are all integers. The parabola y = x 2 + bx + c and the line y = dx have exactly one point in common. With these assumption, which one of th e following statement is necessarily True? (A) b = 0 (B) d - b is even (C) | a | 2 = | b | 2 (D)c = 0 The number of solutions to the system of equations y 2 - x y - | x | y + x | x | = 0 and x 2 + y 2 = 1 is (A)l (B)2 (C) 3 Q. 16 (D)4 Answer the following questions on the basis of the information given below: (Q.1 8 to Q.21) Triangle ABC has vertices A (0,0), B (9,0) and C (0,6). The points P and Q lie o n the side AB such that AP = PQ = QB. Similarly the points R and S lie on the si de AC so that AR = RS = SC. The vertex C is joined to each of the points P and Q in the same way, B is joined to R and S. Also the line segment PC and RB inters ect at X and the line segments QC and SB intersect at Y. Q. 17 Equation of the l ine AX is (A)y=|x Q.18 Q. 19 (B)y = x (C)y=|x (D)y=|x Equation ofthe line XY is (A) 3 x - 4 y = 0 (B)y = x + 1 (C)4x-4y+3=0 (D)none Radius of the circle inscribed in the triangle APS is (A) 4 (B) 1 (C)j (D) 2 Q. 20 Distance between centroid and circumcentre of the triangle ABC is Jl3 2J13 Ju Ju J s B A N S A L CLASSES y j T a r g e t IIT JEE 2007 CLASS: XI (PQRS) Take approx. 50 min. for each Dpp. DATE: 12-15/09/2005 MATHEMATICS Daily Practice Problems DPR NO.- 43, 44 DPP Q.l - 43 A B C In a triangle ABC, prove that, t a n y + t a n Ð + t a n Ð > ^ 3 Q. 2 Q.3 Find the general solution of the equation, cos ( 1 0 x + 1 2 ) + 4V2 sin(5x + 6) = 4. If p, q, r be the roots of x 3 - ax 2 + bx - c = 0, show that the area of the triangle whose sides are p, q & 1 r is Ð [a(4ab - a 3 - 8 c ) ] m . Q- 4 Q.5 If t a n ( a + f3-y) _ tany s h o w t h a t e i t h e r s i n tan ( a - P + y) ~ ta np ' ^ ~ ^ = 0 ' o r ' s i n 2 a + sin2^ + sin2y = 0 In the triangle A' B' C, ha ving sides B' C = a ' , A' C = b' and A' B' = c', a circle is drawn touching two of its sides a' & b' and having its diameter on the side c'. If A' is the area of the triangle A' B' C , find the radius of the circle. Further, a line segment parallel to A' B' is drawn to meet the sides C A ' , CB' (produced) in points A & B respectively and to touch the given circle forming a triangle ABC with side s BC = a, AC = b and AB = c. If A denotes the area of the triangle ABC, show tha t; C O a b c 77 ~ 17 = 77 a' b' c' = a' + b' + c' 7T~u~r a' + b' md , ¹ w (a) A A' ( a ' + b' + c ' ^ V a' + b PPP Q. 1 Q. 2 - 44 If log 10 (l 5) = a and log 20 (50) = b then find the value of log 9 (40) Find t he general and principal solution ofthe trigonometric equation sec x - 1 = ( ^ 2 _ i ) t a n x Q.3 The ratios of the lengths of the sides BC & AC of a triangle ABC to the radius o f a circumscribed circle are equal to 2 & 3/2 respectively. Show that the ratio of the lengths of the bisectors of the interior angles B & C is, 7(V7-l) 9V2 Q.4 Q.5 If two vertices of a triangle are (7,2) and (1,6) and its centroid is (4,6) find the third vertex. If A , B , C are the angles of a triangle & sin 3 6 = sin (A - 0 ) . sin (B - 9). sin (C - 0), prove that cot 0 = cot A + cot B + cot C. J a BANSAL CLASSES Target IIT JEE 2007 CLASS: XI(PQRS) Take approx. 50 min. for each Dpp. DATE: 16-17/09/2005 Daily Practice Problems DPR NO.-45~46 MATHEMATICS DPP Q. 1 - 45 If the sum of the pairs of radii of the escribed circle of a triangle taken in o rder round the triangle be denoted by, sl, s 2 , s 3 and the corresponding diffe rences by d j , d 2 , d 3 , prove that, dj d 2 d 3 + d, s 2 s 3 + d ? s 3 s, + d 3 s, s 2 = 0; Find the general solution of the trigonometric equation cosec x - cosec 2x = cosec 4x Let the incircle ofthe A ABC touches its sides BC, C A & A B at A j , Bj & Cj respectively. If p j , p 2 & p 3 are the circum radii of the triangles, Bj I C j , Cj I A, and A, IB, respectively, then prove that, 2 p, p 7 p 3 = Rr 2 where R is the circumradius and r is the inradius ofthe A ABC. If th e area ofthe triangle formed by the points (1,2); (2,3) and (x, 4) is 40 square units, find x. If a , p, y are angles, unequal and less than 2n, which satisfy t he equation b Ð+ + c = 0, then prove that sin(a + P) + sin(P + y) + sin (y + a ) = 0 cosB sinQ a Q. 2 Q.3 Q.4 Q. 5 DPP Q. 1 - 46 If dp d 2 , d 3 are diameters of the excircles of AABC, touching the sides a, b, c respectively then prove Q.2 Q.3 Q. 4 ^ Show that for any triangle 2r < R (where R is the inradius and R is the circumra dius) Find the least positive angle satisfying the equation cos 5a = cos 5 a. Fi nd the equation of the straight line which passes through the point (1,2) and is such that the given point bisects the part intercepted between the axes. Q.5 In a A ABC, if cosA+ cosB = 4sin 2 -y, prove that tan y . t a n ^ = ^ . Hence de duce that the sides of the triangle are in A.R J j B A N S A L CLASSES v S T a r g e t ||T JEE 2007 CLASS: XI (PQRS) Q. 1 log a b-c then prove that ab Daily Practice Problems TIME: 60Min. DPP. NO.- 42 MATHEMATICS DATE: 09/09/2005 logb c-a logc a-b c If a, b, c are positive real number such that +c + b c +a + a + b > 3 Q.2 Find all values of k for which the inequality, 2x2 - 4k2x Ð k 2 + 1 > 0 is valid f or all real x which do not exceed unity in the absolute value. Find the values o f'p' for which the inequality, ( 2 Q.3 - ( p £ r ) ) x2 + 2 x (1 + 1o & ph) -2( ! + pfr) >0 is valid for all real x. 1 Ð Ð Ð 8 If positive square root of, a* . ( 2 a ) 2 a . ( 4 a ) 4 a . (8a) 8 a .... .... °° is Ð , find the value of 'a1. Q.4 Q.5 Provethat 2 4 - + - = Ð + ^ 4Ð + x + 1 x + 1 x +l 1 2" 2 1 + Ð2 = ÐÐr^T"-; 2 (x +1) 1-x Q.6 Q.7 x Find the general solution of the equation (1 + c o s x ) i j t a n Ð - 2 + sinx = 2 cos x Ifp,q,r be the lengths of the bisectors of the angles of a triangle AB C from the angular pointsA,Band C respectively, prove that w A 1 B 1 C 1 1 1 c o s Ð i Ð cos Ð HÐ cos Ð = Ðl bÐ and p 2 q 2 r 2 a b c 1 K pqr abc(a + b + c) nil ^ ^ = > 4A (a + b)(b + c)(c + a) Q.8 If x, y, z are perpendicular distances of the vertices of a A ABC from the oppos ite sides and A is the area of the triangle, then prove that Ð7 + Ðr- + -r- = Ð (cotA + cotB + cotC) x2 y2 z2 Av ill BANSAL CLASSES V S Targe* IIT JEE 2007 CLASS : XI (PQRS) Q. 1 Q. 2 DATE: 05/09/2005 Daily Practice Problems TIME: 60 Min. DPP. NO.- 41 MATHEMATICS Solve the inequality, ^j\ogy 2 x + 41og 2 Vx < V2 (4 - log^x 4 ). Find the set o f real values of 'a' for which there are distinct reals x, y satisfying x=a-y2 a nd y = a-x2. A polynomial in x of degree greater than 3 leaves the remainder 2, 1 and - 1 when divided by (x - 1) ; (x + 2) & (x + 1) respectively. Find the rem ainder, if the polynomial is divided by, (x2 - 1) (x + 2). Find the general solu tion of the equation sin6x + cos 6 x = Ð . 4 Q.3 Q.4 Q.5 If pj, p 2 are the roots of the quadratic equation, ax2 + bx + c = 0 and q ]5 q 2 are the roots of the quadratic equation cx2 + bx + a = 0 such that Pj, qj, p 2 , q 2 is an A.P. of distinct terms, then prove that a + c = 0 where a, b, c e R . 88 1 cosk Let k = 1 t h e n prove that T \ ÐÐ: 7 Ð = ÐxÐ ^ cosnk · cos(n + l)k sin^k Q.6 r Q. 7 S olve the equation, 1 x 2 2 2 J + V2 4 cos J 1 UJ cosx 2 y Q. 8 Let al, a 2 , a 3 , a 4 and b be real numbers such that 4 b + XaK =8 K=1 4 ; b + Z4=16 2 K=1 Find the maximum value of b. ? J j S A N S A L CLASSES ^ B T a r g e t l i t JEE 2006 CLASS: XI (PQRS) DATE: 29/08/2005 Daily Practice Problems Max. Marks: 60 DPR NO.-40 MATHEMATICS PPP OF THE W E E K This is the test paper of Class-XI (J-Batch) held on 28-08-2005. Take exactly 12 0 minutes. 22 x If sec x + tan x = Ð , find the value of tanÐ. Use it to deduce the value of cosec x + cot x. [3] Q. 1 Q.2 Simplify the expression -r + -r. log 4 (2000) 6 log 5 (2000) [3] Q.3 1 1 1 1 Prove that . · + Ð Ð r ~ + ~ Ð T ~ + + . = cot x - cot 2 n x for any natural num ber sin2x sin2 x sm2 x sin2 x n and for all real x with sin 2 r x ^ 0 where r = 1,2, n. [3] Let X = sin 2 72° - sin 2 60° and Y = cos 2 48°-sin 2 12° Find the value of XY. If A + B + C = ^ then prove that £ s i n 2 A + 2 ] ~ [ s i n A = 1PI Q.4 [3] Q.5 Q. 6 (a) (b) Q.7 The position vector of a point P in space is given by r = 3 cos t i + 5 sin t j + 4 cos t k Show that its speed is constant. Show that its velocity vector v , i s perpendicular to r . Find the value o f k for which the graph of the quadratic polynomial P (x) = x 2 + (2x + 3)k + 4(x + 2) + 3k - 5 intersects the axis of x at two distinct points. Let u = 1 0 x 3 - 13x2 + 7x and v = l l x 3 - 1 5 x 2 - 3 . du Find the integral values of x satisfying the inequality, ™ > [3] [3] Q.8 dv ^ · V6 [3] 42. Q. 9 Let a and b are two real numbers such that, sin a+sin b = - y and cos a+cos b = - - - . Find the value of (i)cos(a-b) and (ii) sin(a + b). [3] Q.10 Let a and b be real numbers greater than 1 for which there exists a positive rea l number c, different from 1, such that 2(logac + logbc) = 91ogabc Find the larg est possible value of logab. [5] Q, 11 Find the product ofthe real roots of the equation x2 + 18x + 30 = 2a/x2 +18x + 45 ix [5] Q.12 If a . p be two angles satisfying 0 < a, P < Ð and whose sum is a constant k¹ find t he maximum value of (i) cos a · cos p and (ii) cos a + cos p. [5] Q. 13 Find a quadratic equation whose sum and product of the roots are the values o f t h e expressions (cosec 10° - 7 3 sec 10°) and (0.5 cosec 10° - 2 sin70°) respectively. Also express the roots of this quadratic in terms of tangent of an angle lying in (n A ~ . [6] Q. 14 Q.15 x +2x-3 If y = Ð5 then find the interval in which y can lie for every x e R wherev er defined. x + 2x Ð 8 Prove the inequality, 1 1 sinx+ - sin2x+ - sin3x>0 for0<x<1 80° [6] [6] CLASS: XI(P, Q,R,S) RAKSHA.BANDHAN Take approx. 50 to 55 min. for each Dpp. HOLI DAY 4 gBANSAL CLASSES B Target NT JEE 2007 MATHEMATICS Holiday Assignment DPP. NO.-37, 38, 39 ASSIGNMENT These DPP will be discussed on the very first day after vacation. DPP For 9 = 1°, prove that 2 sin20 + 4 sin40 + 6 sin60 + Q. 2 Find all the solutions o f the equation which satisfy the condition x € [0, 2n] Q.3 Solve the equation, l + l o g x =[log !0 (log, 0 p ) - l ] log x 10- - 37 + 180 sinl 800 = 90 cot0 - x) = Vcosx V 10 y How many roots does the equation have for a given value of p? Q. 4 Find the set of values of'a' for which the equation, ( 2 \2 2 X x 3 a Ði H 4 a = 0 have real ro ots. (1+a) 2 x2 + 1 VX + 1 , Find four numbers, such that the first three form a G.P and the last three an A.P., while the sum ofthe first and last terms is 14 and the sum of the inner terms is 12. Q. 5 DPP Q. 1 - 38 59 Each angular of a regular r-gon is Ð times larger than each angle of a regular s-gon. Find the largest 58 possible value of s. i Q.2 Q. 3 Q. 4 Solve for '0' satisfying cos(0) · cos (7t0) = 1. Find the solution set ofthe inequ ality 31x1-2 |x|-l > 2 The sum of an infinite GP is 2 & the sum of the GP made from the cubes of the te rms ofthis infinite series is 24. Find the series. A circle is inscribed in an e quilateral triangle ABC ; an equilateral triangle in the circle, a circle again in the latter triangleand so on; in this way (n + 1) circles are described; if r , Xj, x 2 , , x n be the radii of the circles, show that, r = x t + x2 + x3 + + x n _ } + 2 xn. Q. 5 DPP Q. 1 - 39 If the equation sin4x + cos 4 x = a has real solutions then find the range of va lues of'a'. Find the general solution of the equation when a : 1 2' Q.2 Find the complete set of real values of 'a' for which both roots ofthe quadratic equation ( a2 - 6a + 5) x 2 - y a 2 + 2a x + (6a - a 2 - 8) = 0 lie on either s ide of the origin. n(n - 1 ) , _ up to n terms) = In In 2 + Ð - Ð 3 Q.3 Show that In (4 x 12 x 36 * 108 Q. 4 Find the value of x satisfying the equation x -x +4 - 2 <2 + x - 12. Q.5 Show that tan f 7 r s·i n x \ v 4 sin y y + tan ^TtCOSX ^ v4cosyj 7Z TC 7t > 1 for 0 < x < - and - < y < - . 2 6 i ill BANSAL CLASSES Target I8T JEE 2007 CLASS:XI(P, Q, R, S) DATE: 10-13/08/2005 Take approx. 50 min. for each Dpp. Daily Practice Problems DPP. NO.-35 MATHEMATICS DPP Q. 1 Q. 2 - 35 If sec(a - 2P), seca and sec ( a + 2p) are in arithmetical progression, show tha t cos 2 a = 2 cos2|3 (P & nrc, n e l ) Show that the sum to n terms of the serie s : sma cos 3 a + sin 3 a cos 5 a + . . sin2(n + l)a.sin2na n . + sm(2n~ l)a.cos (2n+ l ) a = 2sin^a ~ ~2 s a Q. 3 Find the set of real values of p for which the equation, VP c o s x - 2 sinx = V 2 + V 2 - P possess solutions. Q.4 Q. 5 Solve the equation cos0 1 + sinB : Ð+ Ð Ð l + sm0 cos0 2 Ð. cos0 Prove that if (ac)' 08a b = c 2 , then the numbers log a N, log b N and log c N are three successive terms of an arithmetic progression for any positive value o f N ^ 1. x2 - a x - 2 Find all values o f ' a ' for which Ð5 lies between -3 and 2 for all real values of x. x + x +1 Solve the inequality for every a e R x 2(a - 1 ) a 2 < Ð (x + 1). 3a Q.6 Q.7 DPP Q. 1 - 36 Let A j , ^ , A 3 A n are the vertices of a regular n sided polygon inscribed in a circle of radius R. If (Aj A 2 ) 2 + (Aj A3)2 + + (Aj A n ) 2 = 14 R 2 , find the number of sides in the polygon. 3 + cosx Showthat Ð Ð - Ð V x e R can not have an y value between - 2 V 2 and 2V2 . What inference can you draw about the values o f sinx 3 + cosx ? Q.2 Q.3 Q.4 Q.5 Q.6 Find the solution set of the equation, log _ x 2_ 6 x (sin 3 x + sinx) = log x2_ 6x (sin 2x). 10 10 Find the set of values o f x satisfying the equation sin |x| tan5x = cosx Find t he general solution of the equation, tan 2 (x + y) + cot 2 (x + y) - 1 - 2x - x2 . If p, q are the roots of the quadratic equation x 2 + 2bx + c = 0, prove that 2 log (i/y-~i) + \/y~~q) = log 2 + log f y + b + ^jy 2 + 2by + c Q. 7 Find all real values of x for which the expression Jlogl/2 [ ^ | is a real number. J a B A N S A L CLASSES JEE V S Target IIT JE 2007 CLASS:XI(P, Q. 1 Q,R,S) DATE: 03-04/08/2005 TIME: 70Min. DPP. NO.-3 4 If (xj, yj ) is the solution of the equation, log 225 (x) + log 64 (y) - 4 and (x2, y 2 ) a s the solution of logx(225) - log y (64) = 1 then show that the value of log 30 (x 1 y 1 x 2 y 2 ) =12. Q.2 Let P (x) = x 2 + bx + c, where b and c are integer. If P (x) is a factor of bot h x 4 + 6x2 + 25 and 3x4 + 4x 2 + 28x + 5, find the value o f P ( l ) . Q.3 Given a, b, c are +ve integer forming an increasing geometric sequence, b - a is a perfect square, and log 6 a + log 6 b + log 6 c = 6. Find the value of a + b + c. Q. 4 S olve the inequality, 2 log,/2 (x - 1 ) < ^ - Q. 5 Let there be a quotient of two natural numbers in which the denominator is one l ess than the square of the numerator. If we add 2 to both numerator & denomenato r, the quotient will exceed 1/3 & if we subtract 3 from numerator & denomenator, the quotient will lie between 0 & 1/10. Determine the quotient. Q. 6 The number of terms of an A. P. is even; the sum of the odd terms is 310; the su m of the even terms is 340; the last term exceeds the first by 57. Find the numb er of terms and the series. Q. 7 Find the two smallest po sitive values of x for which sin x° = sin (xc) J | BANSAL CLASSES y S T a r g e t NT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 01-02/08/2005 Daily Practice Problems TIME: 60 Min. DPP. NO.-33 MATHEMATICS TEST [3 x 25 = 75] OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1 Solution set of the inequali ty, 2 - log, (x2 + 3x) > 0 is: (A) [ - 4 , - 3 ) u (0,1] ' " (B) [ - 4 , 1 ] (C) ( - 0 0 , - . 3 ) U (1,00) ( D ) ( - 0 0 , - 4 ) U [1,00) Q.2 If A + B + C = 7r & cosA = cosB . cosC then tanC . tanB has the value equal to : (A) 1 (B) 1/2 (C) 2 (D) 3 If a, b, c be in A.P., b, c, d in G.P. & c, d, e in H .P., then a, c, e will be in: (A)A.P. (B)G.P. (C)H.P, (D) none of these If the r oots of the equation x 3 - px 2 - r = 0 are tan a , tan (3 and tan y then the va lue of sec 2 a · sec2|3 · sec2y is (A) p 2 + r2 - 2rp + 1 (B) p2 + r 2 + 2rp + 1 (C) p 2 - r 2 - 2rp + 1 (D)None 1 3 7 15 1S Q.3 Q.4 ^ ¹ Q. 5 The sum to n terms of the series, (A) 2 n - n - 1 equal to: (D)2n-1 (B) 1 - 2 " n (C) 2~n + n - 1 Q. 6 sinx - cos2x - 1 assumes the least value for the set of values of x given by: (A ) x - tm + (~l) n+1 (n/6) (B) x = nn + ( - l ) n (n/6) (C) x = n% + (-l) n (ti/3) where n e l Q.7 (D) x = nw - (-l) n (tt/3) If the equation a (x - l) 2 + b(x 2 - 3x + 2) + x - a 2 = 0 is satisfied for all x e R then the number of ordered pairs of (a, b) can be (A) 0 (B) 1 (C) 2 (D) i nfinite * Q. 8 The base angles of a triangle are 22.5° and 112.5°. The ratio of the base to the hei ght of the triangle is: (A)V2 (B)2V2-1 2x+i) ^ ' 1 (C)2V2 (D)2 Q.9 If 1) ^ 1 are in Harmonical Progression then (B) x is an integer (D) x is a negative rea l (A) x is a positive real (C) x is rational which is not integral Q. 10 The absol ute term in tile quadratic expression ' t l (A) zero X 3k+ 1 A x- 1 ^ 3k-2y as n л oo is 2 (C) (D) 1 (B) 1 Q. 11 Given four positive number in A.P. If 5 , 6 , 9 and 15 are added respectively to these numbers, we get a G.P., then which of the following holds? (A) the common ratio of G.P. is 3/2 (B) common ratio of G.P. is 2/3 (C) common difference of t he A.P. is 3/2 (D) common difference of the A.P. is 2/3 Q.12 x The equation, sin 2 9 - Ð 3 r Ð ' = 1 r H . sin 0 - 1 (A) no root (B) one root sin3 0 - 1 has : (D) infinite roots (C) two roots Q.13 The equation (x e R) (A) has no root + 1Ð, x? =x : (C) two roots (D) four roots 4 V '- i (B) exactly one root Q.14 " Q. 15 If x s i n 9 = y c o s 9 then Ð L Ð 4 Ð is equal to sec29 cosec29 (A) x (B)y (C)x2 (D) y2 An H.M. is inserted between the number 1 /3 and an unknown number. If we diminis h the reciprocal of the inserted number by 6, it is the G.M. of the reciprocal o f 1/3 and that of the unknown number. If all the terms of the respective H.P. ar e distinct then (A) the unknown number is 27 (B) the unknown number is 1/27 (C) the H.M. is 15 (D) the G.M. is 21 The number of integers 'ri such that the equat ion nx 2 + (n + l)x + (n + 2) = 0 has rational roots only, is (A)l (B)2 (C) 3 (D )4 The roots of the equation, cot x - cos x = 1 - cot x . cos x are : (A)mi+j (C ) mt + where n e I or 2 nn±n (B) (D) ( 4 n + l ) ^ or (2n+l)n Q. 16 Q. 17 Q. 18 If x 2 + Px + 1 is a factor of the expression ax 3 + bx + c then (A) a 2 + c 2 = - ab (B) a 2 - c 2 = - ab (C) a 2 - c 2 = ab The expression (tan49 + tan29) ( 1 - tan 2 39 tan 2 9 ) is identical to (A) 2 cot 39 . sec 2 9 (B) 2 sec 39. tan 2 9 (C)2tan39. sin29 x2~3x + c X "i" i X H C ~ (D) none of these Q. 19 (D) 2 tan39. sec 2 9 Q.20 If the maximum and minimum values of y = c is equal to (A) 3 i are 7 and Ð respectively then the value of / (B)4 (C)5 (D)6 Q. 21 The general value ofx satisfying the equation 2cot 2 x + 2 V3 cotx + 4 cosecx + 8 = 0 is (A) nn n (B) nn + 71 (C) 2nTX - n (D) 2mc + 7t 6 Q. 22 If the sum of n terms of a G.P. (with common ratio r) beginning with the p* term is k times the sum of an equal number of terms of the same series beginning wit h the q111 term, then the value of k is: (A) rp/q (B) r^P (C)rP^ (D)rP + i The s um ofthe roots ofthe equation (x + 1) = 2 log 2 (2 x + 3) - 2 log 4 (l 980 - 2"x ) is (A) 3954 (B)log 2 ll (C)log 2 3954 (D) indeterminate If the expression, 2 ( ^ 2 _ i ) sin x - 2 cos 2x + 2 (0,2n) is: (A) is negative then the set of value s of x lying in f 5n llTl") U ' 6 J u V 6/ Q.23 Q.24 W 6, 'it 57r" / u r57i l b O U (5n Ð ' 6 J Ð (B) (C) 71 7T 1 v6 ' 2 J 3T^ 1 4 2J (D) 571 5tc T ' T u In ,2n Q.25 Solution set of the inequality log 3 x - log? x < : log, .4 is %/2j2) ( C ) f - o o , i u [ 9 , oo) (D) u (1,9] (A) [3,9] (B) (H u[9,oo) J j B A N S A L CLASSES v B Target I IT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 28/07/2005 Daily Practice Problems TIME: 60 Min. DPP. NO.-32 MATHEMATICS TEST [3 x 25 = 75] OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1 A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is (A) 1 - i j , then the side of the hexagon is: (B) 2 ( C) V2 (D) 2V2 Q.2 Q.3 If 3 + ^ (3 + d) + ^ y (3 + 2d) + (A) 9 (B) 5 + upto 0 = 8, then the value of d is : 0 (C)l (D) none of these If in a A ABC, sin 3 A + sin 3 B + sin 3 C = 3 sinA · sinB · sinC then (A) A ABC may be a scalene triangle (B) A ABC is a right triangle (C) A ABC is an obtuse angl ed triangle (D) A ABC is an equilateral triangle The value of (0.2) loSvI ^ + » + ^ (A) 4 (B) 6 + Q.4 Q.5 ^ is equal to (C)8 (D)2 The set of angles btween 0 & 2n satisfying the equation 4 cos 2 0 - 2 - ^ 2 cos 0 Ð 1 Ð 0 is (A) 23n J j L ^L ] I12 ' 12 ' ~12 ' T T j (B) ! JL l2L ]2JL 2 3 n 1 12 ' 12 ' 12 ' 12 f 7 1% 1771 2371 C (C) f ^ L l^ZL 1 9 7 t ] I 1 2 ' 12 ' 12 J 1 1 1 + + + 1-4 4.7 7.10 (B)y = 2x 0 and 0 Q.6 Let x= 1 1 1 y = ~ + 7TT + T T 4 " ' 1-2 2-3 3-4 ( C ) x + >>=1 °° then (D)x+y=^ (A)y = 3x Q.7 If cos a = "" c o s PÐ- then tan ^1 cot^- has the value equal to, where(0 < a < n and 0 < B < 71) 2 - cosp 2 2 (A) 2 (B)V2 0 is equal to 0 (C) 1/2 then: (C)xyz + x + y + z = 1 (D) none (D) 1/4 (C) 3 (D) Vs Q.8 Q.9 2m. 4 1/8 . 8 1/16 .16 1/32 . 32 1/64 (A) 2 (B) 1 If xsin0 = y s i n | e + y j = z sin^e + (A) x + y + z = 0 (B)xy + yz + zx = 0 Q. 10 Q. 11 If x AM's are inserted between xr and 1 then the value of the xth arithmetic mea n is (A) I- x (B)l+x (C)x 2 Ð x + 1 (D)x If a cos 3 a + 3a cos a sin 2 a = m and a sin 3 a + 3a cos 2 a sin a = n . Then (m + n) 2/3 + (m - n) 2/3 is equal to : ( A) 2 a 2 (B) 2 a 1/3 (C) 2a 2 / 3 (D) 2 a 3 ,a n ; the G.P. b>, b 2 , 9 such tha t a 1 = bj = 1 ; a 9 = b 9 and ]!Ta r = 369 then r=l (A) b 6 = 27 (B) b ? = 27 ( C)b g = 81 Consider the A.P. a t , s^ , , bn Q. 12 (D)b9=18 Q. 13 If tan A & tan B are the roots of the quadratic equation x 2 - ax + b = 0, then the value of sin2 (A + B) is: (A) a2 « a + (1-b) 2 , 2 (B V a2 a 2 + b2 _ a2 ( Q - ^ - T2 (b + a) (D) a2 b 2 (1 - a) 2 Q. 14 Q.15 v If a, b, c are distinct positive reals in G. P., then; log a n , log b n , log c n (n > 0, n * 1) are in: (A) A. P. (B) G. P. (C)H.P. (D) none A If A = 3 4 0 ° th en 2 sin Ð is identical to 2 (A) Vl + sinA + Vl - sin A (C) Vl + sinA - -\/l - sin A (B) - V 1 + sinA - V l - s i n A ^ D ) - Vl + sinA + V l - s i n A gn such th at g 1 + g 2 + g 3 = 13 and gj + g 2 + g 3 = 9 1 (B) 3g4 = g 3 (D)g 2 = 3 Q.16 Consider a decreasing G.P.: g 1 ,g 2 ,g 3 , then which of the following does not hold? (A) The greatest term ofthe G.P. is 9. (C)g, = l Q. 17 *\/3 + 1 "J3 Number of roots of the equation cos 2 x + Ð - Ð sinx - Ð - 1 = 0 which li e in the interval [-71, tt] is (A) 2 (B)4 (C) 6 (D) 8 Q.18 The sum ofthe first three terms of an increasing G.P. is 21 and the sum oftheir squares is 189.Then the sum of its first n terms is (A) 3 ( 2 - 1) n 1 \ 1 (B)12 1 - ^ r V 2 / r (C)6 \ 1 1-^r) (D)6(2«-l) Q.19 Ifsin(6 + a ) = a & sin(G + p) = b (0 < a , p9 0 < tc/2) then cos2 ( a - (3) - 4 ab cos(a - P) = (A) 1 - a 2 - b 2 (B) 1 - 2a 2 - 2b 2 (C.) 2 + a 2 + b 2 J.fS=4 - + - r Ð r + n (D)2-a2-b2 Then S¹ is not greater than n S Q.20 V + (B) 1 2 *r \ ' 1 +2 +3 + 2 - , n = 1,2, 3, +n (C) 2 l3 1 +2 (A) 1/2 Q.21 Q.22 (D)4 (D) 1 The exact 73° + cos (A) 1/4 value of cos (B) 1/2 47° + (cos73°. cos47°) is (C)3/4 Let Sj , S 2 , S 3 be the sums of the first n , 2n and 3n terms of an A.P. respe ctively. If S 3 = C (S 2 - S,) then, 'C' is equal to (A) 4 " (B)3 (C)2 (D)l \ Q.23 Maximum value of the expression cos6 · sin ® v 1 (A) j V3 (B)^ 6y 1 4 V 9 e R, is (C) (D)l Q.24 The value of the expression (sinx + cosecx) 2 + (cosx + secx) 2 - (tanx + cotx) 2 wherever defined is equal to (A) 0 (B)5 (C)7 (D) 9 The roots of the equation 2 + cotx = cosec x always lie in the quadrant number (A) I only (B) I and II (C) II and IV (D) II only Q.25 i k BANSAL CLASSES ^ S T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 24/07/2005 Dally Practice Problems TIME: 60 Min. DPP. NO.-31 [3 x 25 = 75] J ( x - 8 ) (2-x) tÐ -y > 0 and iogo.3 ("T (log2 5 - 1)) MATHEMATICS TEST OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For each wrong answer 1 mar k will be deducted. Q.l The set of values of x satisfying simultaneously the ine qualities 2 X - 3 - 31 > 0 is : (A) a unit set (C) an infinite set Q.2 (B) an empty set (D) a set consisting of exactly two elements. The roots of the equation (xÐl) 2 Ð 4 | x Ð 1 | + 3 = 0, (A) form an A.P. (B) form a G P. (C) form an H. P (D) do not form any progression. The perimeter of a certain sector of a circle is equal to the length of the arc of a semicircle having the same radius. The angle of the sector in radians is: (A) 2 (B) 7i - 1 (C) 7i - 2 (D) none If the roots of the equation, x3 + Px2 + Qx - 19 = 0 are each one more than the roots of the equaton, x3 - Ax2 + Bx - C = 0 where A, B, C, P & Q are co nstants then the value of A+B+C = (A) 18 / (B) 19 (C) 20 (D) none Number of orde red pair(s) of (x, v) satisfying the system of simultaneous equations I x2 - 2x j + y = 1 and x2 + | y f = 1 is (x, y e R) : (A) 1 (B) 2 (C) 3 (D) infinitely ma ny Given log2x · log,xyz = 1 0 log 2 y-log 2 xyz = 40 log2z · log2xyz = 50 where x > 0 ; y > 0 ; z > 0 then which of the following inequalities may be true? (A)x<y< z&z<x<y (B)x>y>z & z<x<y (C)x>y>z&x<y<z (D) x > z > y & z < x < y The quadratic equation whose roots are the A.M. and H.M. between the roots of the equation, 2x 2 - 3x + 5 = 0 is : (A) 4x2 - 25x + 10 = 0 (B) 12x2 - 49x + 30 = 0 (C) 14x2 - 12 x + 35 = 0 (D) 2x2 + 3 x + 5 = 0 If the sum of the first n natural numbers is 1/ 5 times the sum oftheir squares, then the value of n is : (A) 5 (B) 6 (C) 7 (D) 8 A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, l/2unitup, l/4unittotheright, 1/8 unit down, 1/1 6 unit to the right etc. The length of each move is half the length of the previ ous move and movement continues in the 'zigzag'manner indefinitely. The co-ordin ates of the point to which the 'zigzag' converges is: (A) (4/3, 2/3) (B) (4/3, 2 /5) (C) (3/2, 2/3) (D) (2, 2/5) Q. 3 Q.4 Q.5 Q. 6 Q 7 Q.8 Q. 9 1/4 CJ 1 0 " s U 1/16 v X Q.10 A quadratic equation defined over rational coefficient whose one root is sin237i /10 is: (A) 16x2 + 12x - 1 = 0 (B) 4x2 + 2x - 1 = 0 2 (C) x - 3x + 1 = 0 (D) 16x 2 - 12x+ 1 = 0 Let an be the nth term of a G.P. of positive numbers . Let X a * p. Then the common ratio of the G.P. is : W fp C B )a £ ( C ) \j f jp ( D )Vj ai 1 00 n 1 00 Q. 11 a2n = a & X n = 1 = P such that n =1 Q. 12 Given a sequence a p a2, a3, an, in which the sum of the first m terms is 2 Sm = m - 5m then which of the following is not true? (A) a5 = 0 (B) a5 = 4 (C)a 6 = 6 (D)itisanAP. l°g 2 x + l°S 4 y + l ° g 4 z = 2 log3y + log9z + log9x = 2 log4z + log 16x + log16y = 2 then which ofthe following is true? (A) y > z (B) x > y Given Q.13 (C)x>y>z (D)x<y<z Q. 14 The number of integral values of m, for which the roots ofx2 - 2mx+m 2 - 1 = o w ill lie between - 2 and 4 is (A) 2 (B) 0 (C)3 (D)l Given a regular triangle with side 'a', a new regular triangle is formed by the length ofits altitudes. This process is repeated. This procedure being repeated n times. The limit of the sum of areas of all the triangle as n Ð oo is > (A) 3a2 r (B)V3a2 · · 2 s i n x + sin2x 1 -cosx \2/3 The interval in which y can lie for V x e R . (C)[0,1] Q.15 (C) 2a2 (D)V^a2 Q.16 Let y = v2cosx+sin2x 1-sinx^ (A) [l,oo) Q. 17 (B)(-a,, co) (D)[0,oo) A horse is teethered to a stake by a rope 9 m long. Ifthe horse moves along the circumference of a circle always keeping the rope tight then the distance traver sed by the horse when the rope has raced an angle of 70°, is (Assume n = 22/7) f ( A) 7 m (B)9m (C)llm (D)22m If x g R, the numbers ( 51+x + 51"*), a/2, (25x + 25' x ) form an A.P. then 'a' must lie in the interval (A) [6,o)) (B)[12,oo) (C) [2 4, oo) (D)[24,oo) If 7 times the 7th term of an A.P. is equal to 11 times its el eventh term then the 18th term of the A.P. is (A) 0 (B) 7 (C) 11 (D) 18 ABC is a triangle such that, sin ( 2 A + B ) = sin (C - A) = - sin (B + 2C) = ^ · If A, B, C are in A.P, A B & C are respectively. (A) 30°, 60°, 90° (B) 45°, 60°, 75° (C) 15°, 60°, 105 ) none of these Q.18 Q. 19 Q.20 3 2 Q.21 Set ofintegral solution of the equation x (A) 1 (B) 2 +iog2X 5 4 _ ^ js (D) 0 (C) 3 y Q. 22 If sin (x - y), sin x and sin (x+y) are in H. P., then sin x . sec acute angles) (A) 2 . 10 has the value equal to (x, y, z are +ve (B) V? 1025 imi - (C)l (D) none Q.23 If P anc * ^ = Q then th e value of log 10 4100 in terms of p and q is equal to Q.24 Q.25 (A)p + 9q (B) p + lOq (C)12p + q 2 (D)p + 12q cos2x C0S x Number ofvalues of x s atisfying the equality 2 = 3.2 - 4 and the inequality x2 < 30 is (A) 0 (B) 1 (C) 2 (D) 3 Let a, P, y be the roots of the equation x3 + 3 ax2 + 3 bx + c = 0 . If a , p, y are in H.P. then P is equal to: (A) a (B) c/b (C) - a (D) - c / b J j B A N S A L CLASSES v B Target IIT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 18/07/2005 MATHEMATICS Daily Practice Problems TIME: 60Min. DPP. NQ.-30 OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1 If a > b > 0 are two real nu mbers, the value of, ib + (a - b) -,/ab + (a - b) ^/ab + (a - b) -v/ab + (A) ind ependent of b (C) independent of both a & b Q. 2 TEST [3 x 25 = 75] is : (B) independent of a (D) dependent on both a & b . If a , P are the roots of the equation ax2 + bx + c = 0 , then the roots of the equation ax2 + bx(x+l) + c (x +1) 2 = 0 are (A) a - 1, 0 - 1 (B) a + 1, P + 1 (C ) ~ , ^ (D) ^ , ^ VQ 3 ^ Q.4 The number of the integral solutions for the equation x + 2 y ~2xy is (A) 2 (B) 1 (C) 4 ' (D) infinitely many Given a, b, c are non negative real numbers and if a 2 + b 2 + c 2 = 1, then the value of a + b + c is (A) > 3 (B) > 2 + (C)<V3 " + (D)< V2 Q. 5 Q. 6 Q. 7 Q. 8 The value of (0.2) ! ° S ^ & + * (A) 2 (B) 4 i s equal to (C) 5 is (D)652 (D) none The maximum value of the sum of the A.P. 50, 48, 4 6 , 4 4 , (A)325 (B)648 (C)65 0 Solution set of the inequation, glo§2x - 2x 2 > x - 2 is (A) (0, 1) (B) (2, oo) (C) (0, 2) (D) (0, 1) u (2, oo) The quadratic equation (3 + sin9)x2 + (2 cos9)x + 2 - sinQ = 0 has (A) equal roo ts for all 0 (B) real and distinct roots for all 8 (C) complex roots for all 0 ( D) real or complex roots depending upon 0 If the roots of the quadratic equation ax2 + bx + c = 0 are imaginary then for all values of a, b, c and x e R , the e xpression a 2 x 2 + abx + ac is (A) positive (B) non - negative (C) negative (D) may be positive, zero or negative If \ 2 + -V + 4 " + l 22 32 (A)~ u Q. 9 Q. 10 Pt0 00 = then 6 TT + \ I2 3 (C)~ + ~7 + 5 equal to (B) (D) none Q.ll In a potato race, 8 potato es are placed 6 metres apart on a straight line, the first being 6 metres from the basket which is also placed in the same line. A co ntestant starts from the basket and puts one potato at a time into the basket. F ind the total distance he must run in order to finish the race. (A) 420 (B) 384 (C)432 (D)none Q.12 Q.13 Q. 14 The sum ofthe first 100 terms common to the series 17,21,25, (A) 101100 (B) 1110 00 (C) 110010 and 16,21,26, (D) 100101 is If x,y,z e N then the number of ordered triplets of (x,y, z) satisfying the equa tion x + y + z = 1 0 2 i s (A) 4950 (B) 5050 (C) 5150 (D)None Consider an A.P. w ith first term 'a' and the common difference d. Let S k denote the sum of the fi rst K terms. Let "77" is independent of x, then (A) a = d/2 (B) a = d (C) a = 2d (D)none SjQJ Q. 15 If p, q, r in harmonic progression and p & r be different having same sign then the roots ofthe equation, px 2 + q x + r = 0 are : (A) real and equal (B) real a nd distinct (C) irrational (D) imaginary. The numbers (A) an A. P. 1 Q. 16 log32 , Ð-Ð & Ð - Ð consitute log 6 2 j log122 (B)aG.P. (C)aH.P. (D)None Q. 17 1 If log2, l o g ( 2 x - 1) and log(2 X + 3 ) are in A.P,, then xis equal to : (A) 5/2 (B) log3 2 (C) log 2 5 (D)none Complete set of the values of x satisfying t he inequality, log7 x - log x (l/7) < - 2 is (A) (0,1) (B) (1, 2) (C) (1, co) (D )(2,co) If the equation sm4 x - (k + 2) sin 2 x - (k + 3) = 0 has a solution the n k must lie in the interval: (A) ( - 4 , - 2 ) (B)[-3,2) ' (C)(-4,-3) (D) [ - 3 , - 2 ] The first term of an infinitely decreasing GP. is unity and its sum is S . The sum ofthe squares ofthe terms of the progression is: V ; Q. 18 Q.19 Q. 20 (A) Ð ( 2S-1 B )V S2 ; 2S-1 (C) 2-S K (D) S 2 ' Q.21 The expression sin(a + 0) - sin(a - 0) c o s ( p _ e ) _ c o s ( p + e ) is (B) independent of |3 (D) independent of a and p (A) independent of a (C) independent of 0 Q. 22 Q. 23 Q.24 Number of values of 0 e [ 0,2 n ] satisfying the equation cotx - c o s x = 1 - c otx. cosx (A) 1 (B)2 (C)3 (D)4 The solution set ofthe inequality log}/3 x + 2Iog I 9 (x - 1) < log1/3 6 i s (A) ( - « , - 2 ] U [3, *>) (B) [-2, 3] ' ( C ) R - [- 2, 3] (D) [3, oo) 4 sin5° sin55° sin65° has the values equal to V3 + 1 V3-1 · S - l _ 3(V3 - l) Q.25 The values ofx smaller than 3 in absolutevalue which satisfy the inequality log (2a _ x 2 ( x - 2 a x ) > 1 for a l l a > 5 is (A) - 2 < x < 3 (B) - 3 < x < 3 ( C)-3<x<0 (D)-3<x<-l fi BAN SAL CLASSES Target IIT JEE 2 0 0 7 CLASS : XI (P, Q, R, S) DATE: 16/07/2005 TIME: MATHEMATICS Daily Practice Problems 60Min. DPR NO.-29 [3 x 25 = 75] OBJECTIVE PRACTICE Select the correct alternative. Only one is correct. For eacii wrong answer 1 mark will be deducted. Q. 1 TEST a2 p2 If a, p are the roots o f t h e equation ax + 3x + 2 = 0 (a < 0) then Ð + Ð is 2 (A) > 0 Q.2 (B) > 1 (C) < 1 (D)<0 The value of f (x) = x 2 + (p - q)x + p 2 + pq + q 2 for real values of p, q and x (A) is always negative (B) is always positive (C) is some times zero for non zero value of x (D) none of these For an increasing A.P. a,, a 2 , a 3 ,an,.... if a! + a 3 + a 5 = - 12 ; a,a 3 a 5 = 80 then which o f t h e following does no t hold? (D) a, = 2 1 (A) a,= - 10 (B)a 2 (C) a, = - 4 The solution set of the in equality log< o 2 - x 2x 1 1 3 > 1 is Q.3 Q. 4 (A) u (B) \ (C) Q.5 2' 4 u U ) (D) -co. J u f 3 ^ T'00 J If a , p are the roots of the quadratic equation (p2 + p + l)x 2 + (p - 1 )x + p 2 = 0 such that unity lies between the roots then the set of values of p is (A)* (B)peO,-l)U(0,oo) (C)p e ( - 1 , 0 ) (D) ( - 1 , 1 ) If cos9 + cos(|) = a and s inB + sincj) = b, then the value of cos9-cos(|) has the value equal to I2-4a2 (A ) 41 hb 2 ) |( a 2 + b 2 ) ! 2 - 4 b 2 2(a2+b2)I ^ 2 ( 2 2 1 a +b )1 - Q.6 4a 2 |( a 2 + b 2 ) |2 - 4 b 2 (U) " ( 41 a 2 + b 2 ) 1 2(a2+b2) Q.7 If both roots of the equation (3X + l)x 2 - (21 + 3 p)x + 3 = 0 are inifinte the n (A) A, = p 1 3 1 2 (C)X = - - ; p = 1 (D)?i = --;n = -- SPACE FOR ROUGH WORK Q.8 If p & q are distinct reals, then 2 { ( x - p ) ( x - q ) + ( p - x ) ( p - q ) + ( q - x ) ( q - p ) } = (p - q)2 + (x - p)2 + (x - q)2 is satisfied by: (A) no value of x (B) exactly one value of x (C) exactly two values of x (D) infinite values of x. The expression cot 9° + cot 27° + cot 63° + cot 81° is equal to (A)Vl6 (B)V 64 (C)V80 (D) none of these If the quadratic equation ax2 + bx + 6 = 0 does not have two distinct real roots, then the least value of 2a + b is (A) 2 (B)-3 (C)- 6 (D)l The set of values of'p' for which the expression x2 - 2 px + 3 p + 4 is n egative for atleast one real x is: (A)<1) (B) ( - 1 , 4 ) (C) ( - 0 0 , - 1 ) 1 ^ ( 4 , 0 0 ) (D) { - 1 , 4 } The equation 5log" X+ 1 Q.9 <x Q. 10 Q.ll Q.12 + 5l o g o 25 X " 1 = y has (A) no integral solution (C) one irrational solution Q. 13 (B) only one rational solution (D) two real solutions If a, b, c are positive reals and b2 < 4ac, then the difference between the maxi mum and minimum values of the function, f (0) = a sin20 + b sin0-cos0 + c cos20 V 9 e R, is (A) 0 (B) a + c (C)V^V (D)ac Q. 14 Let a > 0, b > 0 & c > 0. Then both the roots of the equation ax2 + bx + c = 0. (A) are real & negative (B) have negative real parts (C) are rational numbers (D ) none Greatest integer less than or equal to the number log 2 15 . log |/f) 2. log 3 1/6 is: (A) 4 (B)3 (C) 2 " ' (D) 1 Integral value of x satisfying the equa tion (x2 + x + 1) + (x2 + 2x + 3) + (x2 + 3x + 5) +....+ (x2 + 20 x + 39) = 4500 is (A) .10 (B) - 1 0 (C) 20.5 (D) None Given a2 + 2a + cosec2 ~ ( a + x) = 0 th en, which of the following holds good? V.2 y (A)a = l ; | e l (C) a e R ; x e ^ (B)a = - l ; | el Q.15 Q. 16 Q.17 (D) a, x are finite but not possible to find SPACE FOR ROUGH WORK Q.18 If a, P are the roots of the equation, x2 + (sin<j)- l)xÐ Ð cos2{j) = 0 then the max imum value of the sum ofthe squares of the roots is : (A) 4 (B) 3 (C) 9/4 (D) 2 Q. 19 If In2 x + 3 In x - 4 is non negative then x must lie in the interval: (A) [e, c o) (B) (-oo, e~4)u[e, °o) ( C ) ( l / e , e ) (D) none Q.20 If the quadratic polynomial, y = (cot a)x 2 + 2 (V sin a ) x + ^ tan a, a e [0, 2 TT| can take negative values for all x e R , then the value of a must in the i nterval: (A) a e , it) (B) a Q. 21 If a, p are roots of the equation x2 - 2mx + m2 - 1 = 0 then the number of integ ral values of m for which a, p € (-2, 4) is (A) 0 (B)l (C)2 (D)3 P.T.O. X X Q.22 The value of the expression, log4 v (A) - 6 (B) - 5 4 y - 2 log4 (4 x4) when x = - 2 is : (C) - 4 1 (D) meaningless Q.23 If x, and x 2 are the roots of the equation x2 + px xj1 + x 2 (A)V2 is equal to (B)V2(2-V2) = 0 , (p e R ) then the minimum value of (C)2+V^ { ^ \6x+10-x2 (D) 2 + 2^/2 Q. 24 Number of integral values of x satisfying the inequality (A) 6 (B) 7 (C) 8 v4y 27 . < Ð is 64 (D) infinite a a +1 a Q.25 If the roots of the equation ax2 + bx + c = 0 are real and of the form (a + b + c) 2 is (A) b 2 - 4ac (B) b 2 - 2ac (C) b 2 + 4ac r and a-1 then the value of (D) b 2 + 2ac X Class - XI Roll No. ANSWER KEY Date: 16-07-2005 Max. Time: 1 Hr. - X Max. Marks: 75 Only alternative is correct. There is NEGATIVE Marking. For each wrong answer 1 mark will be deducted. [25x3 = 75] D 3 o 4 o 5 o 2 1 A O O o o B o o o 6o 7 o 8o 9 o o o o o o o o o 1o 0 1o 1 12 13 c o o o o o o o o o o o o o o o 1 o 4 17 o A o o o o o o o 1 o o 5 1 o o 6 1 o 8 19 O O O o B c o o o o o o o o o 2 0 2 1 o 2O o 2 23 O o 24 o o 25 o o o o o o o o o o o o o D o o o o · o o o o o o o o o o · X J j B A N S A L CLASSES ySTarget MATHEMATICS Daily Practice Problems TIME: 40 Min. DPP. NO.-27 IIT JEE 2007 DATE: 08-09/07/2005 CLASS: XI (P, Q, R, S) Q. 1 Select the correct alternative : (Only one is correct) Which of the statement is false. (A) If 0 < p < n then the quadratic equation, (cos p - 1) x 2 + cos px + sinp = 0 has real roots. (B) If 2a + b + c = 0 (c * 0) then thequadratic equati on, ax 2 + bx + c = 0 has no root in (0,2). (C) The necessary & sufficient condi tion for the quadratiic function f(x) = ax 2 + bx + c to take both positive & ne gative values is, b 2 > 4ac. (D) The sum of the roots of the equation cos2x = 1 which liei in the interval [0,314] is 49 5071. For every x e R, the polynomial x 8 - x 5 + x 2 - x + 1 is : (A) positive (B) never positive (C) positve as well as negative (D) negative If x ] 5 x 2 & x 3 are the three real solutions of the equation; 2 3 xlog 0x+log10x +3 = Q.2 Q.3 2 w h g r e X ] > X 2 > X 3 ; t h £ n ^x+T-i ,/x+i+i (A) Xj + x 3 = 2 x 2 Q.4 (B) x , . x 3 = x 2 2 (C) x 2 = 2Xl X, + x 2 *2 (D) x f 1 + x f 1 = x f 1 If exactly one root of the quadratic equation x 2 - (a +1 )x + 2a = 0 lies in th e interval (0,3) then the set of values 'a' is given by (A) (-co , 0) u (6,oo) ( C) (-00 , 0] vj [6,oo) (B)(-oo,0]u(6,oo) CD)(0,6) Q.5 Three roots of the equation, x 4 - px 3 + qx 2 - rx + s = 0 are tan A, tanB & ta n C where A, B, C are the angles of a triangle. The fourth root of the biquadrat ic is : (A)-P^_ 1-q+s ( B ) - ^ 1+q-s ( C ) - ^ ' 1-q+s (D) ÐPJlIÐ y 1+q-s ¹ , Q.6 . . The value of the expression sin 8x cos x - sin 6x cos 3x n . when x = Ð is sin 3x sin 4x - cos x cos 2x 24 (C) V2-1 (D)V2+1 (A)V3-2 Q. 7 (B)^ If the roots ofthe quadratic equation ( 4 p - p 2 - 5 ) x 2 - (2p - 1 )x + 3p = 0 lie on either side of unity then the number of integral values of p is (A)0 ' (B) 1 (C)2 " (D) infinite Q.8 « The inequalities y ( - 1) > - 4, y( 1) < 0 & y(3) > 5 are known to hold for y = ax 2 +bx + c then the least value of 'a' is : (A) - 1 / 4 Q.9 (B) - 1 / 3 (C ) 1/4 (D) 1/8 Number of ordered pair(s) satisfying simultaneously, the system of equations, 2 r x + f y - 2 5 6 & log 10A /xy - l o g 1 0 1 . 5 = 1, i s : (A) z ero (B) exactly one (C) exactly two Q. 10 (D) more than two Find the values of 'a' for which one of the roots of the quadratic equation,x2 + (2 a + 1) x + (a2 + 2)=0 is twice the other root. Find also the roots of this e quation for these values of 'a'. i l l BANSAL CLASSES v 8 T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 11-12/07/2005 (Only one is correct) Select the corr ect alternative: Q.l Daily Practice Problems TIME: 40 Min. DPP. NO.-28 MATHEMATICS If a, b, p, q are non-zero real numbers, the two equations, 2 a 2 x 2 - 2 ab x + b 2 = 0 and p 2 x 2 + 2 pq x + q 2 = 0 have : (A) no common root (B) one common root if 2 a 2 + b 2 = p 2 + q 2 (C) two common roots if 3 pq = 2 ab (D) two com mon roots if 3 qb = 2 ap The equations x 3 + 5 x 2 + p x + q = 0 and x 3 + 7 x 2 + p x + r = 0 have two roots in common. If the third root of each equation is r epresented by Xjand x 2 respectively, then the ordered pair (x 15 x 2 ) is: (A) ( - 5 , - 7 ) (B)(1,-1) (C)(-1,1) (D)(5,7) If the roots of the quadratic equatio n x 2 + 6x + b = 0 are real and distinct and they differ by atmost 4 then the ra nge of values of b is : (A) [ - 3 , 5 ] (B) [5,9) (C) [6,10] (D) none The expres sion sec 4 x - 4 tan 3 x + 4 tanx is always : (A) positive (B) negative (C) non- positive Q. 2 Q.3 Q.4 (D) non-negative Q.5 The value of the biquadratic expression, x 4 - 8 x 3 + 1 8 x 2 - 8 x + 2 when x = 2 + V3 is (A) 1 (B) 2 (C) 0 (D) none If one root o f t h e quadratic equation px 2 + qx + r = 0 (p ^ 0) is a surd where p, q, r ; a, b are all rationals then the other root is Va Ja(a-b) a + ,/a(a-b) Va - V a - b Ð7 Va + Y a Ð b Q.6 Q. 7 The minimum value of cos (cos x) for every x e R i s : (A) 0 (B) - cos 1 (C) cos 1 (D) - 1 Q.8 The area of the circle in which a chord of length 2a makes an angle 9 at its cen tre is (A) 7i a 2 c o t 2 | (B) 2 7i a2 (l + cot 2 1) ( Q % a2 (l + cot2 | J (D) 4 71 a 2 (l + cot 2 1) Q.9 The equation a sinx + cos2x = 2a - 7 has a solution, if (A) a > 2 Subjective: (B ) a < 2 (C)2<a<6 (D)a<2ora<6 Q. 10 Solve the inequality, log2x (x2 - 5x + 6) < 1. JjBANSAL CLASSES V B Target IIT JEE 2007 CLASS: XI (P, Q, R, S) DATE: 06-07/07/2005 Select the correct alternative : (Onl y one is correct) Q. 1 x 2 + 2x + c If x is real, then Ðcan take all real values i f: x + 4x + 3c (A) 0 < c < 2 (B) 0 < c < 1 27c 3T T 28 671 28 (C) - 1 < c < 1 9n Daily Practice Problems TIME: 40 Min. DPP. NO.-26 MATHEMATICS (D) none 18C T 28 27 n 28 Q.2 T h e e x a c t v a l u e o f c o s Ð c o s e c Ð + c o s Ð c o s e c Ð + c o s Ð cosecÐÐ iseq ualto (A)-1/2 Q.3 28 (B) 1/2 28 (C)l (D) 0 If a,b,c are real numbers satisfying the condition a + b + c = 0thenthe roots of the quadratic equation 3ax 2 + 5bx + 7c = 0 are : (A) positive (B) negative (C) real & distinct (D) imaginary 3 Q.4 Let, N = (A) 0 8l'° g59 +3 ^ Ð 409 log _3 f l r-\ 2 , . (V7 H 7 -125 log25 6 (B) 1 then log 2 N has the value = y (C) - 1 (D) none ^ , Q.5 ¹ It . 271 . 47t . 8rc A A = sin Ð + s i n Ð + s m Ð ec ua 2n 4TT 8C T and B = cos Ð + c o s Ð + c o s Ð then 7 a 2 + B 2 (A)l Q.6 ZX ~ I X 4" 4 I l to (B) V2 (C)2 (D) V3 x- 3 x- 2 If Ð Ð - < then the most general values are : (A) ( x < - 4 ) (B>(x>{l ( C ) (" 4 < X < 1) (D) ( x < - 4 ) u ( x > | Q.7 Q. 8 The equation | s i n x | = s i n x + 3 has in [0,2 71] : (A) no root (B) only on e root (C) two roots The number of solution of the equation, log(- 2x) = 2 log ( x + 1) is : (A) zero (B) 1 (C) 2 (D) more than two roots. (D) none Q. 9 IfA and B are complimentary angles, then : (A) [l + t a n | - ] [ l + t a n | j = 2 (C) + secyj (l + cosec|j = 2 (B) [l + c o t ^ j (l + cot|j = 2 (D) f l - tan~j f l - t a n | j = 2 Subjective: Q.10 I f un = sin"0 + cos n 0, prove that ^ u3-u5 u, U5-U7 =~ u3 i l l BANSAL CLASSES ™ T a r g e t l i t JEE 2007 CLASS: XI (P, Q, R, S) Q. 1 Q.2 DATE: 04-05/07/2005 Daily Practice Problems TIME: 40 Min. DPR NO.-25 . MATHEMATICS Fill in the blank : If (x + 1 ) 2 is greater then 5x - 1 and less than 7x - 3 th en the integral value of x is equal to If x 2 - 4x + 5 - sin y = 0, y e (0, 2n) then x = & y= . Q.3 If the vectors, p =(log 2 x) i Ð 6 j Ð k and q =(log 2 x) i + 2 j +(log 2 x) k are p erpendicular to each other, then the value of x is . Select the correct alternative : (Only one is correct) Q.4 The equation, 7tx = - 2 x 2 + 6 x - 9 has : (A) no solution (B) one solution (C) two solutions (D) infinite solutions Q.5 Q. 6 cos a is a root of the equation 25x 2 + 5x - 12 = 0, - 1 < x < 0, then the value of sin 2 a is : (A) 12/25 (B) - 1 2 / 2 5 (C) - 2 4 / 2 5 (D) 20/25 Number of o rdered pair(s) (a, b) for each of which the equality, a (cos x - 1) + b 2 = cos (ax + b 2 ) - 1 holds true for all x e R are : (A) 1 (B) 2 (C) 3 Let y = cos x ( cos x - cos 3 x) . Then y is : (A) > 0 only when x > 0 (C) > 0 for all real x (D) 4 Q.7 (B) < 0 for all real x (D) < 0 only when x < 0 x 2 Q. 8 For V x e R , the difference between the greatest and the least value of y = (A) l (B)2 (C)3 (D)| ^ is Q.9 In a triangle ABC, angle A = 36°, AB = AC = 1 & BC = x. If x = (p, q) is : (A)(l,- 5) Subjective: (B) ( 1 , 5 ) (C)(-l,5) t h e n t h e ordered pair (D) (-1 , - 5 ) n Q.10 Find the value(s) ofthe positive integer n for which the quadratic equation, ^ ( x + k - l ) ( x + k) = 10n k=l has solutions a and a + 1 for some a . J j BANSAL CLASSES V S Target I IT JEE 2007 CLASS:XI(P, Q,R, S) DATE:20-21/06/2005 Take approx. 40 min. for each Dpp. Q.l MATHEMATICS Daily Practice Problems TIME:40Min. DPR NO.-22 PPP - 22 Q.2 Q.3 Q.4 Q.5 If 0 is eliminatedfromthe equations asec0-xtan0=y and bsec0+ytan9=x thenfindthe relation between x and y, where a, b are constants. 2T T 4TI .6% 7r.37i.57t Prov ethat: s i n Ð + s i n Ð - s i n - y = 4 s i n y s i n Ð s i n Ð IfA, B, C denote the an gles of a triangle ABC then prove that the triangle is right angled if and only if sin4A + sin4B + sin4C = 0. 1 1 1 1 Solve the inequality: - Ð ^ Ð XTI X X I Z Q.6 Let p & q be the two roots ofthe equation, mx + x (2 - m) + 3 = 0. Let m t , m 2 be the two values of m p q 2 nil m2 satisfyingÐ + Ð =-.Determine the numerical valu e of m + m j . Ð Ð q p 3 2 i 17 Jgj Find the value ofthe continued product ] ~ [ s i n Ð 18 k=i $ $ $$ * # * * * * * * * * * * * # # * * * * * * * * 2 PPP Q.l Q.2 Q. 3 4 4 6 - 23 If 15 sin a + 10cos a = 6, evaluate 8cosec a + 27sec 6 a Prove that the function y = (x 2 + x + l)/(x 2 + 1 ) cannot have values greater than 3/2 and values sma ller than 1/2 for V x eR. If a, |3 are the roots of the equation (tan2135°)x2 - (c osecl0° - V3 secl0°)x + tan2240° = 0 then prove that the quadratic equation whose root s are (2a + (3) and (a + 2P) is x 2 - 12x + 35=0. Q.4 John has 'x' children by his first wife. Mary has x + 1 children by herfirsthusb and. They many and have children oftheir own. The whole family has 24 children. Assuming that the children ofthe same parents do notfight,find the maximum possi ble number offightsthat can take place. Q.5V Solve the following equation for x, 3x 3 = [x2 + Vl8 x + a/32] [x2 - Vl8 x - V32] - 4x 2 , where x e R. Q.6** If cosA = tanB, cosB = tanC and eosC=tanA, th en prove that sinA = sinB = sinC=2 sin 18°. ** * * * t *** *** **** * **** PPP Q. 1 Q.2 Q.3 Q.4 Q.5 Q.6 - 2JU Find the minimum value of the expression 2 log 10 x - logx0.01 ; where x > 1. If x,y,z be all positive acute angle thenfindthe least value of tanx (cot y + cot z)+tany (cot z + cot x) + tanz (cot x + cot y) r, ,, . ... sinx - 1 , 1 . Prove the mequality + - > smx ~ 2 2 2 - sinx w ¹ V x e R. 3 - sinx Prove that: 5 sin x = sin(x + 2y) =>2 tan(x + y) = 3 tan y. If cos 0 + cos < = a and sin 0 + sin < = b, prove that: tanÐ+tanÐ = ¹ ^v b b . 2 2 a + b + 2a Find the sum of (n - 1 ) terms ofthe series: . it . 2% 371 sinÐ + s i n Ð + s i n Ð + n n n _ Dedu ce the value of n if this sum is equal to 2 + J 3 . n y CLASS: XI (P, Q, R, S) Q.l « abansal classes Target IIT JEE 2007 DATE: 17-18/06/2005 Identify whether the statement is True or False. tan 2 x sin 2x Ð tan 2 x - sin 2 x o o MATHEMATICS Daily Practice Problems TIME: 40Min. DPP. NO.-21 o a sin 8 2 - .cos 3 7 - and sin 1 2 7 - .sin 9 7 - have the same value. 2 2 2 2 (ii i) Ov) (v) (vi) If tan A VI VI then tan (A - B) must be irrational. & tanB = 4-^3 4+ S 1 c sB ° sinB ,' then t a n 2 A = t a n B . If tanA = 1, tanB = 2 and tanC = 3 then A, B, C can not be the angles of a trian gle. If t a n A = There exists a value of 0 between 0 & 2n which satisfies the equation, sin 4 0 - sin2 0 - 1 = 0 . Select the correct alternative : (More than one are correct) Q.2 If x = sec (|) - tan <|) & y = cosec <j) + cot < then : > j (A) x = Z j t i ( (C ) x = - Ð w v B ) y - Ð N y-l ' " l1-- xx ' y +' l1 (D) xy + x - y + 1 = 0 Q.3 If the sides of a right angled triangle are {cos2a + cos2p + 2cos(a + P)} and {s in2a+ sin2p + 2sin(a + p)}. then the length of the hypotenuse is: (A)2[l+cos(a-P )] ( B ) 2 [ l - c o s ( a + P)] ( C ) 4 c o s 2 - ^ ^ (D)4sin 2 - a + f3 Q.4 Which of the following functions have the maximum value unity ? sin2x - cos2x (A ) s i n 2 x - c o s 2 x (B) V2 ( Q _ s i n 2 x - cos2x (D) l M5 . I sinx + Ðp^cosx ' V3 (2 cos 2n~10 - 1). Then: (D) f 5 (ti/1 28) = V2 Q.5 For a positive integer n , let f n ( 0 ) = (2 cos0 + l) (2 cos0 - l) (2 cos 2 0 - I) (2 cos 2 2 0 - l) (A) f 2 (ti/6) = 0 (B) f 3 (?t/8) = - 1 (C) f 4 (tt/32) = 1 Q.6 Two parallel chords are drawn on the same side of the centre of a circle of radi us R . It is found that they subtend an angle of 0 and 2 0 at the centre ofthe c ircle. The peipendicular distance between the chords is Q . . . _ _ , 30 . 0 (A) 2 R sm Ð sin Ð (B) 1 - cos 1 + 2 cos-1 R 2 2 2j 30 0 (C) (l + cos^J fl - 2 c o s ^ j R (D) 2 R sin -Ð sin Ð 4 4 Subjective Determine the smallest positive value o f x (in degrees) for which tan(x + 100°) = tan(x + 50°) tan x tan (x - 50°). Let sin Q.7 Q.8 Q.9 ( 9 - a ) = -, & c o s ( 9 - ° 0 Ð sin (0 - P) b cos (0 - P) 0- 4 then rprove that cos ( a - p) = a ° ' b d d ad + be 3tt 12 , Y=cos 0 + 7rc 12 + cos 0 71 If X = sinf© + Y ^ j + s i n then prove that X Y Y X 71 12 + sin 0 + 12 + COS 0 + 3tt 12 -2 tan20. t Q.2 fit BANSAL CLASSES Target I1T JEE 2007 MATHEMATICS Daily Practice Problems DPP. NQ.-20 CLASS: XI (P, Q, R, S) DATE: 15-16/06/2005 TIME: 40Min. Select the correct alter native : (Only one is correct) c 71X \ : 2 Q. 1 The number of solutions ofthe eq uation cos x + 2A/3X + 4 is 2V3 (A) more than 2 (C)l (D)0 (B)2 r r il r The valu e of cot 7 Ð + tan 67 Ð - cot 67Ð -tan7Ð is: 2 2 2 2 (A) a rational number (B) irrationa l number (C) 2(3 + 2 v 3 ) If t a n a x2 - x X - x + 1 (D)2(3-V3) 71 Q.3 and tan p : (B)-l 1 2 ZX -j ZX. 7 ( X5fc 0, l ) , w h e r e 0 < a , P < Ð, then tan (a + P) has z (C)2 (D) 3/4 the value equal to : (A)l Q.4 The value of (A) 1 Q.5 10 4 cos Ð - 3 secÐ - 2 tan Ð 10 10 is equal to (C) V5 + 1 (D) zero (B) V5 - 1 The maximum value of (A) 25 ( 7 cosO + 24 sinO ) x ( 7 sinG - 24 cos9 ) for every 0 e R . 625 625 (B) 625 (C ) Q.6 As shown in the figure AD is the altitude on BC and AD produced meets the circum circle of AABC at P where DP = x. Similarly EQ = y and FR - z. If a, b, c respec tively a b c denotes the sides BC, CA and AB then Ð + -Ð + Ð 2x 2y 2z has the value eq ual to (A) tanA + tanB + tanC (B) cotA + cotB + cotC (C) cosA + cosB + cosC (D) cosecA + cosecB + cosecC The graphs of y = sin x, y = cos x, y = tan x & y = cos ec x are drawn on the same axes from 0 to u/2. A vertical line is drawn through the point where the graphs of y = cos x & y = tan x cross, intersecting the othe r two graphs at points A & B. The length ofthe line segment AB is: (A) 1 (B) nsi nAcosA 1-ncos2 A sin A V5-1 Q.7 (C) V2 (D) V5 + 1 Q.8 If tanB then tan(A + B) equals (B) K±JJ (n - 1 ) cos A sinA (C) v w sin A (n-l)cosA v(D) sin A (A) ( l - n ) c o s A Q.9 ' (n + l)eosA In a triangle ABC, angle A is greater than angle B. If the measures of angles A & B satisfy the equation, 3 sinx - 4 sin 3 x - K = 0, 0 < K < 1 , then the measu re of angle C is (A) n/3 (B) TI/2 (C) 2tc/3 (D) 5TC/6 The value of sin 9 +cos 9 , for all permissible vlaues of 9 sin9-cos9 tan 9 - 1 (B) is greater than 1 (A) is less than - 1 (D) lies between- J 2 a n d (C) lies between - 1 and 1 includin g both 2 Q.10 sin 2 9 including both Q. 11 The number of solution ofthe equation log 3x (3/ x) + log x = l is (B)2 (C)l (A) 3 (D)0 ft BAN SAL CLASSES 4 MATHEMATICS Paiiy Practice Problertis TIME: 40Min. DPP. NO.-19 Target IIT JEE 2007 DATE: 13-14/06/2005 CLASS: XI (P, Q, R, S) Select the correct alternative : (Only one is correct) Q.l 5rc If Ð < x < 371, then the value of the expression x (A) -cot-; Q.2 The exac t value of (A) 12 Q.3 (B)cot| 96 sin80 V l - s i n x + v l + sinx V l - s i n x - VI + sinx is (Qtan| (D)-tan| ° s i n 6 5 ° s i n 3 5 ° 0 is equal to sin20° + sin50° + sinllO (B) 24 (C)-12 (D) 48 3 - 9tan x 3tanx - tan 3 x The value of cot x + cot (60° + x) + cot (120° + x) is equal to (A) cot3x (B) tan3x 3 + cot 76° cot 16° (C) 3tan3x (D) Q.4 Q.5 is: cot 76 + cot 16 (D)cot46° (A) cot 44° (B) tan 44° (C) tan 2° a , p, y & 5 are the sm allest positive angles in ascending order of magnitude which have their sines eq ual to the positive quantity k . The value of a p Y 8 4 sin Ð + 3 sinÐ + 2 sinÐ + sinÐ i s equal to : 2 2 2 2 (A) 2 v n (B) 2VTTk (D) 2 k (C) 2Vk In A ABC, the minimum v alue of 2 > t 2A +2B Ð,cot Ð 2 2 is 2A 2 (C)3 3tt 9 , l + cos 571 9 l+cos 7tc is (D) non existent The value of Q. 6 n«* (A) 1 Q.7 (B)2 71 \ i The value of l+cosÐ 9y r V l + cos Q. 8 (B)l? (C)12 16 16 For each natural number k , let C k denotes the circle with ra dius k centimeters and centre at the origin. On the circle C k , a particle move s k centimeters in the counter- clockwise direction. After completing its motion on C k , the particle moves to C k+1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1,0).lf the particle crosses the positive direction of the x- axis for the first time on the circle C n then n equal to (A) 6 (B) 7 (C) 8 (D) 9 ( n) ^(x-f) The set of values of x s atisfying the equation, 2taa\x'V _ 2 (0.25)" cos2x + 1 = 0 , is : (A) an empty s et (B) a singleton (C) a set containing two values (D) an infinite set If 0 = 3 a and sin 0 = (A) 1 Va + b 2 2 Q-9 Q.10 Va2 + b . The value of the expression, a cosec a - b sec a is (C) a + b (D) none (B) 2i/a 2 + b2 ,|i BANSAL CLASSES gTarget i l T JEE 2007 CLASS: XI (P, Q, R, S) Fill in the blanks : Q.l Q.2 If log 2 14 = athen log49 32 in terms of'a'is equal to The value of 1 10 § cosiht cos 6 97 Daily Practice Problems TIME: 50 Min. . . 27t 3 MATHEMATICS DPP. NO.-17 DATE: 08-09/06/2005 ^ is equal to Q.3 The solution set of the system of equations, x + y = Ð , cos x + cos y = Ð , where x & y are real, is . + xj < b then the ordered pair (a, b) is . If a < sinx. s i n - xj . sin Q.4 Select the correct alternative : (Only one is correct) Q. 5 Which of the followi ng conditions imply that the real number x is rational? I x 1/2 is rational II x 2 and x 5 are rational III x 2 and x 4 are rational (A) I and II only Q.6 (B) I and III only (C) II and III only (D) I, II and III If a 3 + b3 and a + b * 0 th en for all permissible values of a, b ; log (a + b) equals 1 (A) - (log a + log b + log 3) 1 (B) - (loga + logb + log2) f (C) log(a2 - ab + b 2 ) 3 Q.7 (D)log ' a + b 3ab i he number of all possible triplets ( a p a^ a 3 ) such that a, + a 2 cos2x + a 3 sin 2 x = 0 for all x is : (A) 0 (B) 1 ~ (C) 3 (D) infinit e (E)none Q-8 T - L ^ r + V3sin250° cos 290° (A) ~ (B) ^ (C) V3 (D) none Q.9 The product cot 123°. cot 133°. cot 137°. cot 147°, when simplified is equal to : (A)-l (B) tan 37° (C) cot 33° (D) 1 V x e R the greatest and the least values of y = j cos 2x + sin x are respectively 3 1 (A)-,~ (B) 3 3 1 3 (C)-,-(D) 3 1,-- Q. 10 Q.ll Given sinB= ~ sin (2A+B) then, tan(A+B) = ktanA, where k has the value equal to (A) 1 (B) 2 (C) 2/3 (D) 3/2 Q.12 ( c\ C A B If A + B + C = 7i & sin A + Ð = k s i n 2 , then tanÐ t a n - = V 2J 2 2 (A) k+1 (B) i l l k- 1 (C) - A k+1 (D) k ^ , | i BANSAL CLASSES Target NT JEE 2007 CLASS: XI (P, Q, R, S) Q.l DATE: 10-11/06/2005 MATHEMATICS Daily Practice Problems TIME: 50 Min. DPR NO.-18 If tan A & tan B are the roots of the quadratic equation, ax 2 + b x + c = 0 the n evaluate a sin 2 (A + B) + b sin (A + B). cos (A + B) + c cos 2 (A + B). 7i 3n 571 7T T Find the exact value of tan 2 Ð + t a n 2 Ð + t a n 2 Ð +tan 2 Ð 16 16 16 16 Q.2 Q.3 If A + B + C = 7i:,provethat r tanA ^ v tanB.tanC y X (tan A ) - 2 X (cot A). Q.4 If a cos (x + y) = sin (y - x) then prove that, 1 l + asin2x 1 l-asin2y 2 1-a2 Q. 5 In any triangle, if (sin A + sin B + sin C) (sin A + sin B - sin C) = 3 sin A si n B, find the angle C. Q. 6 If cos9 + coscj) = a and sinG + sintj) = b then prove that, (a2-b2)(a2+b2-2) cos 29 + cos2(j) Q.7 2 7t If a = Ð , prove that, sec a + sec2a + sec4a = - 4. ,|i BANSAL CLASSES 9Target 1IT JEE 2007 CLASS: XI (P, Q, R, S) Fill in the blanks : Q.l Q.2 DATE: 06-07/06/2005 MATHEMATICS Daily Practice Problems TIME: SO Min. DPP. NO.-16 A rail road curve is to be laid out on a circle. If the track is to change direc tion by 28° in a distance of 44 meters then the radius of the curve is . (use n = 22/ 7) If'm' is the number of integers whose logarithms to the base 10 have the characteristic 5, and 'n' is the number of integers the logarithms of whose reci procals to the base 10 have the i H characteristic (-3) then Iogio ~~ has the va lue equal to Vn ) . Q.3 Q.4 Q. 5 ln(ab)-ln|b| simplifies to The least value of the expression ^x 1 + sin cot tan2x f o r Q < x < _ j s - 8xj ^ The greatest value of the expression sin 2 f · 4xj - sin 2 · - 4xj for 0 < x < is . Select the correct alternative : (Only one is correct) 1 0 9 Q.6 If £ ka k = 22 and j ] pa p = 32 then a ] 0 = k=l (A)-10 Q.7 Q. 8 (A) 9 p=l (B)-l (B) 19 (C)10 (C) 39 (D) 1 (D) none Let x + y = 1 and x 3 + y 3 = 19 then the value of x 2 + y 2 is equal to The sid e of a regular dodecagon is 2 cm. The radius of the circumscribed circle in cms. is: (A)4(V6-V2) ( B ) V 6 + V2 (C) V37T 7 Fl (D) V6-V2 Q.9 In a triangle ABC, angle A is greater than angle B. Ifthe measures of angles A a nd B satisfy the equation 2 tanx - k(l+tan 2 x) = 0,where k e (0,1), then the me asure of the angle C is (A) 7 1 6 (B)f = p where 0 e (C)12 51 7 7 1 (D)~2 Q.10 Let sin 30 cos20 Us 2371^1 0 sin 3(3 & cos2p " ' 48 J where p e (A) p > 0 and q > 0 Q.ll (B)p>0andq<0 (C)p<0andq<0 Then t 48 ' 48 (D)p<0andq>0 (I3n 1471 Select the correct alternative : (More than one are correct) Which of the follow ing statement(s) does/do not hold good ? (A) log 10 ((1.4)2 - 1 ) is positive ^ (B) log 1 + log 2 + log 3 = log (1 (C) log0.1 Q.12 T_ cot 3) then n = m 4 . 3% Ð is negative 8 (D) If m = 4>°S4 and n = 7 f i V2'0637 sin39 11 . 0 , , . ^ = Ð then tan Ð can have the value equal to : smO 25 2 (A) 2 (B) 1/2 (C) - 2 If '(D)-1/2 BANSAL CLASSES 8 Target I IT JEE 2007 CLASS: XI (P, Q, R, S) Q.l DATE: 03-04/06/2005 If secA - tanA = p, p # 9, find t he value of sinA. 2n-l MATHEMATICS Daily Practice Problems TIME: 45 Min. DPP. NO.-15 Q.2 Evaluate the product | " | t a n ( r a ) where 4 n a = 7t. r=l 3 Q.3 If cos ( y - z ) + cos ( z - x ) + cos ( x - y ) = - ~ , prove that cosx + cosy + cosz = 0 = sinx + siny + sinz 0 V '4 If sinx = sin3x sin5x aj-2a,+a5 a3-3aj = a thenshowthat Ð a Ð - Ða a3 5 3 } Q.5 If 9 = At i 1 , prove that 2" cos9 cos29 cos229 71 cos2n"19 = 1. What the value of the product whould be if 9 = 7. Q.6 Find the exact value of cosecl9 0 + cosec50°-cosec70 0 . Q.7 Prove that from the equalities, * ^ x + 2 ~*>-y(z + X " log x log y y ) = 2 (* + y~2> log z follows yyX-yZzy=zx xz CLASS: XI (P, Q, R, S) Q.l cos4 a sin4 a i f Ð t w ^ -2d cos p sin P = 4 g BAN SAL CLASSES 8Target i l l JEE 2007 DATE: 01-02/06/2005 cos 4 B sin4 3 1 men rind the value o f Ð · ? · cos a sin a MATHEMATICS Daily Practice TIME: 50 Min. Problems DPR NO. -14 Q.2 ' . (3n "l (3% If [1 - sin (7t + a ) + cos (rc + a)] 2 + 1 - sin| Ð + a j + C 0 S |/7,Ð value of a and b. 2 a = a + b sin 2a then find the Q.3 If cos(A-B) cos(C + D) ^ ~ g7 + _ j)) = 0 then prove that tanA · tanB · tanC · tan D = - 1 Q.4 Q.5 Prove that tan 80 = ( 1 + sec20) (1 + sec40) (1 + sec80) Express sin 2 a + sin 2 p - sin2 y + 2 sina sinP cosy as a product of two sines and two cosines. Q.6 Find the solution set ofthe equation 5. Yln x 25 + 4 5 COS2X = (25) sin2x 2 where x e [ 0,2n] Q.7 Show that x = ^J 2 cos 36° is the only solution of the equation log x (x 2 + l) = A /log^(x 2 (l + x 2 ) ) + 4 ,|i BANSAL CLASSES pTerget SIT i i i 2007 Fill in the blanks : Q. 1 Q.2 CLASS: XI (P, Q, R, S) DATE: 30-31/05/2005 Dally Practice Problems. TIME: SO Min. . DPP. NO.-13 MATHEMATICS The exact value of cos 4 9 + cos 4 29 + cos 4 30 + cos 4 40 if 0 = 7t/3 is The e xpression sin 4 1 + cos 4 1 - 1 Ð when simplified reduces to sin 6 1 + cos 6 1 - 1 Q.3 sin24 0 cos6 0 -sin6°sin66 Q The exact value of s i n 2 i° C os39°-cos51 0 sin69° is. co s Ð f . 3 Ð + logjL 3j when simplified reduces to j^log 1 Exact value of tan200° (cot 10° - tan 10°) is ^ . T Ð C 96 V3 sm Ð If cosa = tan a + p") 1 COS Q.4 Q.5 Q.6 Q.7 . va 71 48 COS 7t 24 C0S 7t 7t , L2 C0S "6 n a S *ue = · and sin(3 = 1 where a e 4 th quadrant and p e 2 n d quadrant then cot f V a-p 2 Select the correct alternative: Q. 8 (More than one are correct) Identify the statements) which is/are incorrect (A) Vl + s i n a - V l - s i n a =2sinÐ 2 (% \ (n ) ifaefet > (B) sin 2 a + c o s ! y ~ a j · cos [Ð+ a J is independent of a . u/(C) log, (cos2 (8 + <j>) + cos2 (0 - (j>) - cos 20-cos2<))) is equal to 1. Q. 9 3 (D) If tana = Ð where a e 3rdQ then cos3a is positive. Which ofthe following whe n simplified reduces to unity? , . (A) l-2sin2a T T v 7 2cot Ð+ a cos Ð a U J U ) ¹ (B ) sin(Ti-a) * J Ð +cos(7i-a) s m a - c o s a tan Ð 2 4 sm a c o s ' a Q. 10 4 tan a (sma + c o s a ) The equation logx+1 (x - 0.5) = log x _ 0s (x +1) has (A) no real solution (C) a n irrational solution (B) no prime solution (D) no composite solution BANSAL CLASSES P Target SIT JEE 2 0 0 7 DATE: 27-28/05/2005 CLASS: XI (P, Qj R, S) Fill in the blanks : Q.l Exact value of tan 12° tan 24° tan 3 6° tan 12°+ tan 24° - tan 36° Q.2 iS £ q U a l t 0 MATHEMATICS Dally TIME: 40 Min. DPR NO.-12 Practice Problems. ' The logarithm of 32.5 to the base 10 is 1.5118834. The number whose logarithms t o the same base is 4.5118834 is . . Q.3 Q.4 If cos0 = log9log log log3273 then the set of values of 6 lying in [0, 2TC] is L et a and (3 be the solution ofthe equation logx2 · lo gx/16 2 = logx/642 where ( a > (3) then a = & P= Select the correct alternative ; (Only one is correct) Q.5 If 7 < 29 < Ð-, then v ; 2 + V2 + 2 cos 40 equals : T (A) - 2 cos 0 (B) - 2 sin 0 (C) 2 cos 0 (D) 2 sin 0 Q.6 In a right angled triangle the hypotenuse is 2 times the perpendicular drawn fro m the opposite vertex. Then the other acute angles ofthe triangle are 7T ( A 7T & 71 ( B ) 3ir & 7T ( c ) 71 & 71 <D) 371 > I i i i i I i » Q.7 If sin 0 + cosec 0 = 2 , then the value of sin8 0 + cosec8 0 is equal to : (A) 2 (B) 2 8 (C) 24 (D) none of these If the expression 4sin5acos3acos2a is expresse d as the sum of three sines then two of them are sin4a and sinl 0a.The third one is {A) sin 8a (B) sin 6 a (C) sin 5 a (D) sin 12a Given a system of simultaneou s equations 2x.5y = 1 and 5x+1.2y = 2. Then (A) x = log 1 0 5andy = log102 (C) x - log1Q and y = log102 Q. 8 Q. 9 (B)x = log 1 0 2andy = log105 (D) x - log i0 5 and y = log10 ( j Q.10 Ifae J then the expression ( . ·\Asin 4 a + s i n 2 2 a + 4cos2( ^ - ^ J equals (A) 2 (B) 2 - 4cosa (C)2-4sina (D )none CLASS: XI (P, Q, R, S) 4!Target I1T JEE 2007 If cosQ = ^ ( a + fit BANSAL CLASSES DATE: 25-26/05/2005 Select the correct alternative : (Only one is correct) MATHEMATICS Daily Practice Problems TIME: 45 Min. DPP. NO.-ll Q. 1 t^11 cos terms °f ' a ' = (C)4(a3 + - i (D)none (A) ( B ) j ( ^ Q.2 The expression reduces to: (A) 1 1 + sin 2a cos (2a - 2%) . tan (a - 4r) (B) 0 4 a '3* sin 2 a cotÐ + cot 2 2 2 )_ when simplified (C) sin2 (a/2) (D) sin 2 a Q.3 sin3 e - c o s 3 9 sinG - cosG (A)9e|0,|) COS0 - 2 tan 0 cot 0 ~ Ð 1 if ^ i + cot 2 9 ( B ) e e ( ^ ] (0)0 6 ^ (D) 9 e Q.4 Exact value of cos 20° + 2 sin 2 55° - V2 sin 65° is : (A) 1 (B) -jL . ( C ) V2 (D) ze ro Q.5 If cos (0 + <> = m cos (0 - (j>), then tan 0 is equal to : j) (A) f \ + m^l , , tan<)> 1 - my f l - m) tanij) (B) U + my (C) 1- m cot<j) 1 + my (D) fi + m N 1, " 1 - my COt<j) Subjective: Q.6 Q.7 Solve the equation, 3 . 2 ,og * (3x ~ 2) + 2 . 3 log * (3x " 2> = 5 . 6 '°8*2 ( 3 x " 2) Prove the identity, c o s ^ y - + 4 a ] + sin (3tt - 8a) - sin (47t - 12a) = 4 cos 2 a cos 4 a sin 6a. Q.8 Solve the following equation for x : -i . a A - 3 B = 9 C where A = l o g a x . l o g 1 0 a . l o g a 5 , B = log 10 (x/10) & C = log 1 0 0 x + log 4 2. cos5x + cos4x Prove that: 2cos3x- 1 ~°°S Q.9 X+ °° S 2x ' Q. 10 ^ Prove the identity, sin 2 a (1 + tan 2 a . tan a ) + * + s m a - = tan 2a + tan 2 ^ + ^ 1 - sma V4 2 J g B A N S A L CLASSES Target l!T JEE 2007 CLASS: XI (P, Q, R, S) Q. 1 DATE: 23-24/05/2005 Daily Practice Problems TIME: 45 Min. DPP. NO.-IO MATHEMATICS Select the correct alternative : (Only one is correct) Exact value of cos 2 73° + cos 2 47° - sin 2 43° + sin 2 107° is equal to : (A) 1/2 (B) 3/4 (C) 1 (D) none Theexp ression sin22° cos8° + cosl58°cos98° sin23 cos7 + cos 157 cos97 (A) 1 (B) - 1 whensimplified Q.2 reduces to : (D) none (C) 2 Q. 3 The tangents of two acute angles are 3 and 2. The sine of twice their difference is : (A) 7/24 (B) 7/48 (C)7/5Q (D)7/25 If ~ s i n 3 a + s i n 4 a = tan k a is an identity then the value k is equal to: cos 2a - cos 3a + cos 4a (B)3 (C)4 + l og_5 1 + 2~42~ (C)4 w sin2a Q.4 (A) 2 Q.5 (D)6 ^ e n simplified has the value log 3 fl + ^ ) + l o g 3 f 1 + ~ ] + l o g 3 f l + ^ j + equal to (A) 1 (B)3 (D)5 Select the correct alternative : (More than one are correct) 5 Q.6 99 . The cosi ne of the third angle is: The sines of two angles of a triangle are equal to Ð & Ð 245 <A> T3T3 Q.7 255 735 765 l <B> a n <c> T3T3 ^ oi 17 5 If secA = Ð and cosecB = Ð then sec(A + B) can have the value equal to (B)-f (C)-£ (D)| (A) | Subjective: Q. 8 Q.9 Q.10 If log 6 (l 5) = a and log 12 (l 8) = p, then compute the value of log 25 (24) i n terms of a & p. Solve the equation : 2>x + 1 1 - 2X = 12X - 11 + I. If sinx + cosx + tanx + cotx + secx + cosecx = 7 then sin 2x = a - W 7 where a, b e N. Fin d the ordered pair (a, b). CLASS : XI (P, Q, R, S) Fill in the blanks. 0.1 Q.2 ( Iftana = 2 a n d a e ! v # fit BANSAL CLASSES Target I1T JEE 2007 DATE: 20-21/05/2005 MATHEMATICS Daily Practice Problems TIME: 40 Min. DPP. NO.-9 3« cosa I then the value ofthe expression Ð s 3Ð is equal to r 2J sin a + cos a If log,47 = a and log 14 = b then the value of log17556 is equal to 3% 4 Q.3 logo.75 logj cos Ð - l o g j cosec i ) 75 The value of (cos 15° - cos75 0 ) 8 is. has the value equal to Q.4 Select the correct alternative ; (Only one is correct) Q. 5 Number of roots of t he equation tan2x + cot2x = 2 which lie in the interval (0,4n) is (A) 4 (B) 6 (C ) 8 (D) 10 \ - e and cosO - cos| Ð - 0 . The length of its 2 / V2 /-5¹ 371 hypotenuse is (A)l Q.7 · 4 3n If f (x) = 3 sm I 2 J (B)2 (C) V2 (D) some function of 0 ^ f 6 X + sin (3TC + X) - 2 sin Ð + X + sin (57t-x) then, for all permissible J ) f 7~ C U J values of x, f (x) is (A)-l Q. 8 (B) 0 (C)l (D) not a constant function Select the correct alternative; (More than one are correct) Which ofthe followin g numbers are positive? (A) log log32l 2 (B) log. (C) log (log 9) (D) log (sinl25°) Q. 9 Identify the correct statement (A) If f(x) - sinx - cosx then f(l) < 0 .371 sin . 5n smÐ sec 7 T 5n tanÐ have the same value wУÐt 71 ^7 C COS H COS cosecÐ+ cotÐ 4 3 4 4 ( q 5 iog5 2+iog5 3 is equal to 6 (D) log:j5 + logs3 is greater than 2 Subjective: Q.10 Solve the equation, | x - 1 | + | x + 2 | - | x - 3 | = 4. CLASS : XI (P, Q, R, S) Q.l 4 | BANSAL CLASSES pTorget SST JEE 2007 DATE: 18-19/05/2005 MATHEMATICS Daily Practice Problems TIME: 45 Min. DPR NO.-8 Select the correct alternative : (Only one is correct) Number of solutions of th e equation. log C0 , J£ sinx = 1 when x e [ - 2%,2 %] is : (A) 4 (B) 3 * (C) 2 (D) 1 If tanG = JÐ where a, b are positive reals and 0 e 1st quadrant then the value of sinG sec70 + cos0 cosec79 is (a (A) + Q.2 b)3(a4+b4) (ab)7'2 (a + b) 3 (a 4 - b 4 ) (ab)7'2 (a + b) 3 (a 4 + b 4 ) (D) (a + b) 3 (b 4 - a 4 ) (C) (ab)7'2 (ab)7'2 = 2 is : (D) {10, 102, 104} Q.3 The solution set of the equation, 3 ^/log10 x + 2 log10 (A) {10, 102} (B) {10,10 3 } (C) {10, 104} Select the correct alternative : (More than one are correct) Q.4 If J -ÐÐÐ + - Ð-Ð, for al l permissible values ofA, then A belongs to Vl + sinA cos A cos A (A) First Quad rant (B) Second Quadrant (C) Third Quadrant (D) Fourth Quadrant Q.5 The solution set of the system of equations, log12 x log 2 x . (log 3 (x + y)) = 3 log 3 x is: (A) x = 6 ; y = 2 (B)x = 4 ; y = 3 ( i logx2 > + log2 y = log 2 x and / (C)x = 2 ; y = 6 (D)x = 3 ; y = 4 Q.6 The equation ^l + logx V27 log 3 x + 1 = 0 has : (A) no integral solution (C) tw o real solutions (B) one irrational solution (D) no prime solution Q. 7 Which of the following are correct ? (A) log 3 19. log 1/7 3 . log 4 HI >2 (C) l og10 cosec (160°) is positive Subjective: (B) log5 (1/23) lies between - 2 & - 1 (D) log^ sinj Ð . log V5 - 5 simplifies to an irrational number Q. 8 If an equilateral triangl e and a regular hexagon have the same perimeter then f ind the ratio oftheir areas. tan 3 0 cot 3 0 _ 1 - 2 s i n 2 Qcos 2 0 = sin 0 co s 0 ' Q.9 Q.10 Prove the identity, Solve the equation, + | x - 1 | - 2 j x - 2 | + 3 | x - 3 | = 4. CLASS · XI (P, Q, R, S) Fill in the blanks. Q.l Q.2 4 ;BANSAL CLASSES B Target i l T JEE 2007 DATE: 16-17/05/2005 MATHEMATICS Dally Practice Problems TIME: 45 Min. DPR NO.-7 If 3 logl0 x = 5 4 - x l08l ° 3 , then x has the value equal to . . If log7 2 = m, then log 49 28 in terms of m has the value equal to Q.3 1 3 5108,5 + · simplifies to y V - l o g " (0.1) . Q.4 If x 2 - 5x + 6 = 0 and log 2 (x + y) = log 4 25, then the set of ordered pair(s ) of (x, y) is Select the correct alternative : (Only one is correct) . Q.5 Let m = tan 3 and n = sec 6, then which one of the following statement holds goo d? (A) m & n both are positive (B) m & n both are negative (C) m is positive and n is negative (D) m is negative and n is positive. x +1 x Ð1 Q.6 Solution set of the equation = 1 is (B)<j) (C) a set consisting of more than 2 elements x i (A) a singleton \ (C) a set consisting of two elements Q.7 Number of values o f ' x ' in (-2%,2%) satisfying the equation 2 s i n " x +4.2 C0S (A) 8 (B)6 2 2 =6 is (C) 4 +x+6 (D)2 Q.8 Solution set ofthe equation 3 2 x -2.3 X (A) { - 3 , 2 } (B) {6,-1} Subjective: + 3 2 ( x + 6 ) = 0 is J C ) {Ð2, 3} (D){l,-6} Q.9 Q. 10 Find the set of values o f ' x ' satisfying the equation ^ 6 4 - ^ 2 3 x + 3 + 1 2 = 0. Find 'x' satisfying the equation 4 log io X+1 - 6 log io x - 2.3logio x 2 + 2 - 0. CLASS: XI (P, Q, R, S) Time: Take approx. 40 min. Fill in the blanks. 0.1 Q.2 Q. 3 « and &BANSAL CLASSES Target I IT JEE 2007 DATE: 13-14/05/2005 MATHEMATICS Daily Practice Problems DPR NO.-6 ^ . tan(l 80° - a)cos(l 80° - a)tan(90° - a ) , .. , . , . l4 The expression Ð ^ { } r-Ð-h ^ wherever it is defined, is equal to sin(90° + a)cot(90° - a)tan(90° + a ) If 2 cos2 (7i + x) + 3 sin (tt+x) vanishes then the values of x lying in the interval fro m 0 to 2tt are If tan 25° = a then the value of tan2Q5 tan115 tan245° + tan335° in terms of 'a' is Q.4 The product, (log 2 17) x (log1/5 2) * (log3 - ) lies between two successive int egers which are _ Q.5 The value of the sum, Ð - Ð + Ð - Ð + -Ð-Ð + log2 N log3 N log4 N continued product of first 2000 natural numbers. + - Ð i s l°g2oooN , where N denotes the Q.6 Select the correct alternative : (Only one is correct) If Xj and x 2 are two sol utions of the equation log3 j 2x Ð 7 j = 1 where Xj < x 2 , then the number of int eger(s) between Xj and x 2 is/are: (A) 2 (B) 3 (C) 4 (D) 5 The value of the expr ession, log4 (A)-6 (B)-5 fX V . 4 Q.7 - 2 l o g 4 ( 4 x 4 ) w h e n x = - 2 is: (C) - 4 x (D) meaningless is: (D) none of these Q.8 The number of real solution(s) of the equation, sin (2X) = nx + it (A) 0 (B) 1 ( C) 2 f TT^ (3n ^ . 3 (lit 1Z tan X Ð .COS + x -sin I 2 12 ; I 2) The expression fH ) n) COS f x Ð .tan Ð + X I 2 J I 2j (A) ( 1 + cos 2 x) (B) sinhi 1 2 + - X 1 J simplifies to Q.9 (C) - (1 + cos 2 x) (D) cos2x Q.10 Let y = 3+- 1 1 1 3 + .. , then the value of y is 2+- (A) Vl3+3 (B) Vl3-3 .(C) V15+3 (D) Vl5~3 d (b) jBANSAL CLASSES 8 T a r g e t SIT JEE 2007 DATE: 11-12/05/2005 MATHEMATICS Daily Practice Problems DPR NO.-5 CLASS: XI (P, Q, R, S) Time: Take approx. 40 min. True and False : Q. 1 State whether the following sta tements are True or False. (a) sec2 8 . cosec 2 9 = sec 2 0 + cosec 2 9. There e xist natural numbers, m & n such that m 2 = n2 + 2002. Fill in the Q.2 Q.3 Q.4 b lanks. If the eighteen digit number A 3 6 4 0 5 4 8 9 8 1 2 7 0 6 4 4 B i s divisible b y 99 then the ordered pair of digits (A, B) is . . The positiveintegersp,q&r are all primes. If p 2 - q 2 = r then the set ofall possible values of ris The solu tion set ofthe equation x loga x = ( a K )log*x is , (where a > 0 & a * 1) . Q.5 Select the correct alternative : (Only one is correct) The number of real soluti on ofthe equation log10 (7x - 9) 2 + log ]0 (3x - 4) 2 = 2 is (A) 1 (B) 2 (C) 3 (D) 4 2 3 2 log 2 ,/4a_ 3 i og27 ( a + i) _ 2a Q.6 The ratio (A) a 2 - a - 1 4fo^a _ 1 7 4iog4 9 a_ a _| Ð " simplifies to: (C)a2-a+l (D)a2 + a + l (B) a 2 + a - 1 Q. 7 Which one of the following denotes the greatest positive proper fraction? /|\l°S26 (A) 7 /lY°g3 5 (B) V-V i (C) 3 3 (D)8 -log, 2 Select the correct alternative : (More than one are correct) Q. 8 Which of the f ollowing when simplified, vanishes ? 1 2 (A) t Ð r + log3 2 log9 4 3 log27 8 (B) l o g ^ f j + l o g 4 ( j (C) - log8 log 4 log 2 16 (D) logjQ cot 1° + logjq c ot 2° + logjQ cot 3° + Q. 9 Which ofthe following numbers are positive (A) log 9 (2. 7) -0 - 3 Subjective: Q. 10 Compare the numbers log 3 4 and log56. (B) log 1/2 ( l/3) (C) logvT5 VlT (D) log1/2 ~ - 2 + log 10 cot 89° c K BANSAL CLASSES Target SIT JEE 2007 DATE : 09-10/05/2005 M A I HfctVlAS DPP. NO.-4 Daily Practice Problems CLASS: XI (P, Q, R, S) Time : Take approx. 40 min. Fill in the blanks. Q. 1 The expression -Jlog0 5 8 has the value equal to + 2 - Q.2(a) Solution set of the equation 1 - !ogi x 3 - log. x is a b (b) If (a2 + b2)3 = (a3 + b 3 ) 2 and ab * 0 then the numerical value of Ð + Ð i s equal to b a Q.3 A mixture of wine and water is made in the ratio of wine : to tal = k ; m. Adding x units of water or removing x units of wine (x * 0), each p roduces the same new ratio of wine : total. The numerical value ofthe new ratio is A polynomial in x of degree three which vanishes when x = 1 & x = - 2 , and h as the values 4&28when x " Ð 1 and x = 2 respectively is . The solution set ofthe equation 4 !og 9 x - 6.x log 9 2 + 2 log 3 27 = 0 is . Q.4 Q.5 Q.6 The smallest natural number of the form 1 2 3 X 4 3 Y, which is exactly divisibl e by 6 where 0 < X, Y< 9, is . Select the correct alternative : (Only one is cor rect) Q.7 x+1 1 The equation, log2 (2x 2 ) + log 2 x . x. logx 0°S2 ) + - log 4 2 (x 4 )+ 2" 3 1 o g , ' 2 ( l o g 2 x ) ^ has (A) exactly one real solution (C) 3 real solutions (B) two real solutions (D) no solution. Select the correct alternative : (More than one are correct) Q.8 The equation = 3 has : (B) one natural solution (D) one irrational solution (log 8 x)- (A) no integral solution (C) two real solutions Subjective Q.9 Which is smaller ? log,-or log, u s + v: Q. 10 8 ax It is known that x = 9 is a root ofthe equation log. (x2 + 15 a 2 ) - log^ ( a - 2 ) = log,, -Ð ; -. Find the other root(s) of this equation. K BANSAL CLASSES Target SIT JEE 2007 CLASS: XI (P, Q, R, S) Time: Take approx. 30 min. Fill in the blanks : Q.l Q.2 T he value of b satisfying log ^- b = 3 - is DATE: 02-03/05/2005 MATHEMATICS Daily Practice Problems DPR iva-i The number of integral pair(s) (x ,y ) whose sum is equal to their product is Se lect the correct alternative : (only one is correct) The number of values of k f or which the system of equations ( k + l ) x + 8y = 4k ; kx + (k + 3)y = 3k - 1 has infinitely many solutions is (A) 0 (B) 1 (C) 2 (D) infinite In a triangle AB C, 3 coins of radii 1 cm each are kept so that they touch each other and also th e sides ofthe triangle as shown. The side of the triangle is (A) 3 + V3 (C) 2 ( 1 + 7 3 ) (B) 3V3 (D)3(3-V3) = x Q.3 Q-4 Q.5 The equation (A) two natural solution (C) no composite solution _ 2 has (B) one prime solution (D) one integral solution Q.6 116 people participated in a knockout tennis tournament. The players are paired up in the first round, the winners of the first round are paired up in the secon d round, and so on till the final is played between two players. If after any ro und, there is odd number of players, one player is given a bye, i.e. he skips th at round and plays the next round with the winners. The total number ofmatches p layed in the tournament is (A) 115 (B) 53 (C) 232 (D)116 Subjective: Q.7 Q.8 (i) (Hi) Q.9 Q.l 0 Prove that x 4 + 4 is prime only for one value of x e N. Establish tricotomy in each of this following pairs of numbers (ii) log4 5 and logI/16 (1 / 25) 3 log27 3 and2 log 4 2 4 and log 3 10 + log Compute the value of 10 81 ^ ^log53 (iv) log 1/5 (l / 7) and !og 1/7 (l / 5) 4 log 9 36 + 27 3 79 The length of a common internal tangent to two circles is 7 and a common externa l tangent is 11, Compute the product of the radii of the two circles. 1st bpp OM rue mn-i of succee& J^f CLASS: XI (P, Q, R, S) Time: Take approx. 40 min. Fill in the blanks : Q.l t fit BANSAL CLASSES T a r g e t I T JEE 2 0 0 7 1 DATE: 04-05/05/2005 MATHEMATICS Daily Practice Problems DPR NO.-2 1 1 1 Ð Ð + Ð + Ð has the value equal to log _ abc log ^ abc log ab abc ca J be v' V (As sume all logarithms to be defined) ^Q-2 Q.3 Solution set of the equation, log,20 x + log.0x2 = logj20 2 - 1 is log (o T) The expression (0.05) " v 2 0 ' ' ' is a perfect square of the natural number (wher e o.T denotes 0.111111 oo) Q.4 OL The line AB is 6 meters in length and is tangent to the inner one of the two con centric circles at point C. It is known that the radii of the two circles are in tegers. The radius of the outer circle is _ _ _ _ _ _ Q.5 The expression, xlo»" ~ loez · y 1 ^ - log* · zlo°x - w h e n simplified reduces to Sele ct the correct alternatives : (More than one are correct) Q.6 If p, q G N satisfy the equation (A) relatively prime (C) coprimc = jVxj then p & q are : (B) twin prime (D) if logqp is defined then logpi is not & vice versa where p > 2, p e N, when simplified is : (B) independent of n, but dependent on p j(D) negative. 21og2 + log3 log48 - log4 Q.7 The expression, log log n radical sign v(A) independent of p, but dependent on n _(€) dependent on both p & n Q. 8 ^ Which ofthe following when simplified, reduces to unity ? (A) log ]0 5 . log1Q20 + log20 2 (C)-log5log3^ Q.9 The number N : -1- logs 2 when simplified reduces to · (B) an irrational number (D) a real which is greater than log76 (i + iog 3 2) (A) a prime number (C) a real which is less than log,7t Subjective : Q.10 Given, log712 = a & log1224 = b . Show that, log54168 = 1 + ab a (8 - 5 b) J|BANSAL CLASSES H g T a r g e t 1ST JEE 2007 CLASS: XI (P, Q,R,S) Time: Take approx. 40 min. Fill in the blanks Q. 1 Q.2 The solution set of the equation 4/|x - 3jx+1 = If x = ?/? + 5V2 1 - 3|x~2 is DATE: 06-07/05/2005 MATHEMATICS Pally Practice Problems DPR NO.-3 · , then the value of x3 + 3x - 14 is equal to Q.3 Iflog x Iog1B(V2 + Vs) = - . Then the value of 1000 x is equal to Select the cor rect alternative : (Only one is correct) . Q. 4 Which one of the following when simplified does not reduce to an integer? 2log6 *°g2 3 2 log, 16-log, 4 ( 1N"2 Q. 5* Let m denotes the number of digits in 264 and n denotes the number of zeroes bet ween decimal point and the first significant digit in 2 _ 6 4 5 then the ordered pair (m, n) is (you may use log i0 2 = 0.3) (A) (20.21) (B) (20.20) (C) £19. 19) CD) (20.19) PQRS is a square. SR is a tangent (at point S) to the circle with ce ntre O and TR = OS. Then, the ratio of area of the circle to the area ofthe squa re is 7i 11 3 7 (A) j (B) (C) (D) - Q. 6 Q.7 V Let u = (logjx) 2 - 6 log2x + 1 2 where x is a real number. Then the equat ion xli - 256 has (A) no solution for x (B) exactly one solution for x (C) exact ly two distinct solutions for x (D) exactly three distinct solutions for x Subje ctive: Q. 8 If x, y, z are all different real numbers, then prove that 1 (x-y) 2 Q.9 Q.10 Solve 1 1 ( 1 + 1 + r = 2 2 (y-z) (z-x) Vx-y y - z 3 X + 1 - 13* - 11 = 2 log5 j 6 - x j. 1 z-x) If log )g 36 = a & log2472 = b, then find the value of 4 ( a + b ) - 5 a b . IIQBANSAL CLASSES 1 8 T a r g e t SIT JEE 2 0 0 7 CLASS : XII (ABCD) DATE: 28-29/06/2006 Q.l tan 9 = 2+ Ð YJI K k..!^ . 7 MATHEMATICS Daily Practice Problems TIME: 50 Min each DPR DPR NO.-25 TIME: 50 Min. DATE : 28-29/06/2006 Ðj where 0 e (0,2%), find the possible value of 0. [2] 2 + '--oo Q. 2 Q.3 Find the sum of the solutions of the equation 2 e 2 x ~ 5e x + 4 = 0. [2] Suppose that x and y are positive numbers for which log 9 x = log 12 y = log 15 (x + y). If the value of - =2 cos 0, where 0 e (0,n/2) find 0. [3] Q. 4 Using L Hospitals rule or otherwise, evaluate the following limit: Limit + x->0 Limit n->«> l 2 (sinx)* ] + [22 (sinx)x ] + I n3 + [n2 (sinx)x ] where [ . ] denotes the [4] greatest integer function. Q.5 1 Consider f ( x ) = - j = Vb , VT~ /b-a . S1 »2x . \2 . I Va + b t a n x , , f o r b > a > 0 & t h e functions g(x)&h(x) M v -jÐ sinx I b-a 1 are defined, such that g(x) = [f(x)] - j-^y-j & h(x) = sgn (f(x» for x e domain o f f , otherwise g(x)=0=h(x) for x < domain of''f, where [x] is the greatest inte ger function of x & {x} is the fractional £ 7 C part of x. Then discuss the contin uity of 'g' & 'h' at x=Ð and x = 0 respectively. Q.6 Q.7 J f ^ d x Using substitut ion only, evaluate: jcosec 3 xdx. [5] [5] [5] TIME : 50 Min. [2] DATE: 30-01/06-07/2006 Q.l Q.2 12 A If sin A = Ð . Find the value of tan ~ . x y The straight line - + - = 1 cuts the x-axis & the y-axisinA&B respectively & a s traight line perpendicular to AB cuts them in P & Q respectively. Find the locus of the point of intersection ofAQ & BP. [2] tanO 1 cot0 If -ÐÐÐ-ÐÐ = Ð, find the value of . tan 0 - t a n 30 3 cot0~cot30 Q.3 [3] 1 J Q.4 If a A ABC is formed by the lines 2x + y - 3 = 0 ; x - y + 5 = 0 a n d 3 x - y + l = 0 , then obtain a cubic equation whose roots are the tangent of the interio r angles of the triangle. |4] r dx Integrate: J / 2 2 x f~r~ 2 (a - t a n xK/b - tan x (a>b) [5] Q.5 Q.6 J xsmxcosx z (a cos^ x + b z sin 2 x) 2 dx [5] Q.7 d dy Let Ð (x 2 y) = x - 1 where x ? 0 and y = 0 when x = 1. Find the set of value s of x for which Ð dx [5] is positive. S 3S 55 sj? ^ jjc ^ $ J C S j c ^ DATE : 03-04/07/2006 Q. 1 TIME; 50Min. Let x = (0.15) 20 . Find the characteristic and mantissa in the logarithm of x, to the base 10. Assume log in 2 = 0.301 and Iog 10 3 = 0.477. K O l o c / 3 f Z P I ^ o - M ^ Z PI Two circles of radii R & r are externally tangent. Find the r adius ofthe third circle which is between them and touches those circles and the ir external common tangent in terms of R & r. [2] p Q.2 Q.3 Let a matrix A be denoted as A=diag. 5 X ,5 ,5 then compute the value ofthe integral J( det A)dx. Q. 4 [3] Using algebraic geometry prove that in an isosceles triangle the sum ofthe d istances from any point of the base to the lateral sides is constant. (You may a ssume origin to be the middle point of the base of the isosceles triangle) [4] E valuate: Q.5 J' - x dx Vx + X 2 + X3 + x [5] a2-3 a-1 / Q.6 If the three distinct points, 'a \ 3 f U3 b-1 ' b a-1 -3 b-1 ( ¹3 cÐ1 c-1 are collinear then [5] [5j 2 vAV show that abc + 3 (a + b + c) = ab + be + ca. Q.7 Integrate: j\/tanx dx 3 jVv^ot d 2{.l & AX* 1 4 -t G
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