1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER 2005

2. SECTION -I Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them iscorrect. Indicate you choice of the correct answer for each part in your answer-book bywriting the letter (a), (b), (c) or (d) whichever is appropriate 3. 01 Problem Of a total of 600 bolts, 20% are too large and 10% are too small. The remainder are considered to be suitable. If a bolt is selected at random, the probability that it will be suitable is :1 a.5 7 b. 10 1 c. 10 3 d. 10 4. 02 Problem The area enclosed within the curve |x| + |y| = 1 is : a. 1 sq unit b. 2 2 sq unit c. 2 sq unit d. 2 sq unit 5. 03 Problem 3 11 If P(B) , P( A B C) and P(A B C) , then P(B c) is : 4 33 a. 1/12 b. 1/6 c. 1/15 d. 1/9 6. 04 Problem Two masses are projected with equal velocity u at angle 300 and 600 respectively. If the ranges covered by the masses be R1 and R2, then : a. R1 > R2 b. R1 = R2 c. R1 = 4R2 d. R2 > R1 7. 05 Problem1 1 The value of sin sin 13sec 13 cos tan 12tan 12 is : a. 1 b. 2 c. 3 d. 4 8. 06 Problem It is given that f(a) exists, then x f (a) af (x) is equal to : lim x a (xa) a. f(a) af(a) b. f(a) c. -fa d. f(a) + af(a) 9. 07 Problem/2 cot xis equal to : dx0 cot x tan x a. 1 b. -1 c. 2 d.4 10. 08 Problem Area bounded by the curve y = log2 x, x = 0, y 0 and x axis is : a. 1 sq unit b. sq unit c. 2 sq unit d. none of these 11. 09 Problem If | a x b |2 | a b |2 144 and | a| 4, then| b | is equal to : a. 12 b. 3 c. 8 d. 4 12. 10 Problem Given that | a| 3, | b | 4,| a x b | 10, then| a b |2 equals : a. 88 b. 44 c. 22 d. none of these 13. 11 Problemlim x log sin x is equal to :x 0 a. zero b. c. 1 d. cannot be determined 14. 12 Problem If x = 1 + a + a2 + to infinity and y = 1 + b + b2 + .. to infinity, where a, b are proper fractions, then 1 + ab + a2b2 + to infinity is equal to : xy a. x y 1xy b.x y 1xy c.x y 1xy d. x y 1 15. 13 Problemcos4 sin4 is equal to :1 2 sin2 a.2 b. 2 cos -1 c. 1 2 sin2 2 d. 1 + 2 cos2 16. 14 Problem x 2 If y f (x ) , then : x 1 a. x = f(y) b. f(1) = 3 c. y increases with x for x < 1 d. f is a rational function of x 17. 15 Problem If two like parallel forces of PQhave a resultant 2N, then :N and NQP a. P = Q b. 2P = Q c. P2 = Q d. P = 2Q 18. 16 Problem A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 600. When he retreats 20 ft from the bank, he finds the angle to be 300. The breadth of the river in feet is : a. 15 b. 153 c. 10 3 d. 10 19. 17 Problem If are the cube roots of a positive number p, then for any real x, y, z the expression x y z equals :x y z a. 13i213i b.21 3i c. 21 3i d. 2 20. 18 Problemm1 If tanand tan, then is equal to : m 1 2m 1 a.3 b. 4 c. zero d. 2 21. 19 Problem If f(x) = x , [ x x 1] then a. f(x) is continuous but not differentiable at x = 0 b. f(x) is not differentiable at x = 0 c. f(x) is differentiable at x = 0 d. none of the above 22. 20 Problem Tan+ 2 tan 2 + 4 tan 4 + 8 cot 8 is equal to : a. tan 16 b. 0 c. cot d. none of these 23. 21 Problem A book contains 1000 pages numbered consecutively. The probability that the sum of the digits of the number of a page is 9, is : a. Zero 55 b. 1000 33 c.1000 44 d.1000 24. 22 Problem The value of [a bbcca] is : a. 2 [a b c] b. [a b c] c. 1 d. none of these 25. 23 Problem A number is chosen at random among the first 120 natural numbers. The probability of the number chosen being a multiple of 5 or 15 is :1 a.81 b.51 c. 24 1 d.6 26. 24 Problem Let A, B and C be n x n matrices. Which one of the following is a correct statement ? a. If AB = AC, then B = C b. If A3 + 2A2 + 3A + 5I = 0, then A is invertible c. If A2 = 0, then A = 0 d. None of the above 27. 25 Problem a2i j k, b i 2 j k, c i j k , then a x (b x c ) If equals : a. 5 i 7j3k b. 5 i 7j3k c. 5 i 7j3k d. zero 28. 26 Problem If AB x AC2i 4 j 4k , then the area of ABC is : a. 3 sq unit b. 4 sq unit c. 16 sq unit d. 9 sq unit 29. 27 Problem10 The coefficient of x4 in the expansion of x 3 is : 2 x2504 a. 259450 b.263 c.405 256 d. none of these 30. 28 Problem Equation of the ellipse whose foci are (2, 2) and (4, 2) and the major axis is of length 10, is :2 2x3y 2 a. 124252 2x3y 2 b. 124252 2x3y 2 c. 1 25 242 2 x 3y2 d. 1 25 24 31. 29 Problem The volume of the solid generated by the revolution of the curve ya3a2x2 about x-axis is :1 3a2 a. 2 b. 3 a21 c. 2a32 d. 2 a3 32. 30 Problem The radius of the circle z i =5 is given by :z i13 a. 12 5 b. 12 c. 5 d. 625 33. 31 Problem If a (1, p,1), b (q,2,2), a b r and a x b = (0, -3, -3), then p, q, r are in that order : a. 1, 5, 9 b. 9, 5, 1 c. 5, 1, 9 d. none of these 34. 32 Problem The circle passes through the point (a, b) and cuts the circle x2 + y2 = k2 orthogonally, then the locus of its centre s given by : a. 2ax + 2by (a2 + b2 + k2) = 0 b. 2ax + 2by + (a2 + b2 - k2) = 0 c. 2ax + 2by + (a2 + b2 + k2) = 0 d. none of the above 35. 33 Problem The foci of an ellipse are (0 4) and the equations for the directrices are y = 9. the equation for the ellipse is : a. 5x2 + 9y2 = 4 b. 2x2 - 6y2 = 28 c. 6x2 + 3y2 = 45 d. 9x2 + 5y2 = 180 36. 34 Problem The straight lines x + y = 0, 3x + y 4 = 0 and x +3y 4 = 0 from a triangle which is : a. Right angled b. Equilateral c. Isosceles d. None of these 37. 35 Problem The eccentricity of the hyperbola 9x2 16y2 18x 64y 199 = 0 is :16 a.95 b.4 c. 2516 d. zero 38. 36 Problem A four-digit number is formed by the digits 1, 2, 3, 4 with no repetition. The probability that the number is odd, is ; a. Zero1 b. 31 c.4 d. none of these 39. 37 Problem The coefficient of xn in the expansion of (a bx) is :ex( 1)n a. (a bn)n! b. ( 1)n(b an)n! c. ( 1)n 1(a bn)n! d. none of these 40. 38 Problem A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is ;5 a.6 b. 16 c. 12 d. none of these 41. 39 Problem1 41 The value of cot 9 cosec-1is given by :4 a. 0 b. 4 c. tan-1 2 d. 2 42. 40 Problem Let a, b, c be distinct non-negative numbers. If the vectors ai ajck , i k and cicj bk lie in a plane, then : a. c2 = ab b. a2 = bc c. b2 = bc d. none of these 43. 41 Problem The greatest coefficient in the expansion of (1 + x)2n is : a. 2nCn b. 2nCn+1 c. 2nCn-1 d. 2nC2n-1 44. 42 Problemex log(1x) (1 x) 2 The value of limis equal to :x 0x2 a. 0 b. -3 c. -1 d. infinity 45. 43 Problem The values of k for which the equations x2 k x- 21 = 0 and x2 3k x + 35 = 0 will have a common roots are : a. k = 4 b. k = 1 c. k = 3 d. k = 0 46. 44 Problem are two non-zero vectors, then (a b) (a b) is equal to : a. a + b b. (a - b)2 c. (a + b)2 d. (a2- b2) 47. 45 Problem If sin x + sin2 x = 1, then cos6 x + cos12x + 3 cos10 x + 3 cos8 x is equal to : a. 1 b. cos3 x sin3 x c. 0 d. 48. 46 Problem The integrating factor of the differential equation is : a. x b. ln x c. 0 d. 49. 47 Problem/2 x sin2 x cos2 x dx is equal to :02 a. 322 b.16 c.32 d. none of these 50. 48 ProblemH H If H is harmonic mean between P and Q, then the value of P Qis ; a. 2PQ b. (P Q)(P Q) c.PQ d. none of these 51. 49 Problem The value of p for which the equation x2 + pxy + y2 5x 7y + 6 = 0 represents a pair of straight lines is :5 a.2 b. 5 c. 22 d. 5 52. 50 Problem Angle between the vectors is : 3(a x b) and b (a b)a a. 2 b. 0 c. 4 d.3 53. 51 Problem The equation of the circle passing through (4,5) having the centre (2, 2) is : a. x2 + y2 + 4x + 4y 5 = 0 b. x2 + y2 - 4x - 4y 5 = 0 c. x2 + y2 - 4x = 13 d. x2 + y2 - 4x - 4y + 5 = 0 54. 52 Problem n the smallest positive integer n for which 1 i is : 1 1 i a. n = 8 b. n = 12 c. n = 16 d. none of these 55. 53 Problem The equation of tangents drawn from the origin to the circle x2 + y2 2rx 2hy + h2 = 0 are : a. x = 0, y = 0 b. x = 1, y = 0 c. (h2 - r2) x 2 rhy = 0, y = 0 d. (h2 r2) x 2 rhy = 0, x = 0 56. 54 Problem The value of 91/3 x 91/9 x 91/27 x .. is : a. 9 b. 1 c. 3 d. none of these 57. 55 Problem If a, b, c are any three coplanar unit vectors then : a. a (b x c )=1 b. a (b x c )=3 c. (a x b) c0 d.c x a) b 1 58. 56 Problem Let 0 < P(A) 0, b > 0 b. a > 0, b < 0 c. a < 0, b < 0 d. data is unsufficient 63. 61 Problem If f(x) = cos(log x), then f (x )f (y ) 1 x has the value : f f (xy ) 2 y a. -11 b. 2 c. -2 d. zero 64. 62 Problem If y = 3x-1 + 3- x 1 (x real), then the least value of y is : a. 2 b. 6 c. 2/3 d. none of these 65. 63 Problem the value of lying between 0 and and satisfying the equation21sin2 cos24 sin 4sin21cos2 4 sin 4 0are :sin2cos214 sin 4 a.7 24 b.5 24 11 c.2 d. 24 66. 64 Problem100 1001 3 1 3 is equal to :2 2 a. 2 b. zero c. - 1 d. 1 67. 65 Problem If ,be the two roots of the equation x2 + x + 1 = 0, then the equation whose roots areand is : a. x2 + x + 1 = 0 b. x2 - x + 1 = 0 c. x2 - x - 1 = 0 d. x2 + x - 1 = 0 68. 66 Problem In a binomial distribution, the mean is 4 and variance is 3. Then, its mode is : a. 5 b. 6 c. 4 d. none of these 69. 67 Problem If a force F 3 i2 j 4k is acting at the point P(1, -1,2), then the moment ofF about the point Q(2, -1, 3) is : a.57 b.39 c. 12 d. 17 70. 68 Problem the equation of a line passing through (-2, -4) and perpendicular to the line 3x y + 5 = 0 is ; a. 3y + x 8 = 0 b. 3x + y + 6 = 0 c. x + 3y + 14 = 0 d. none of these 71. 69 Problem (cosec x)1/logx is equal to : a. 0 b. 1 c. 1/e d. none of these 72. 70 Problem the minimum value of f(x) = sin4 x + cos4 x, 0 x is :2 1 a.2 21 b. 4 1 c. 21 d.2 73. 71 Problem Which of the following is a true statement ? a. {a}{a, b, c} b. {a}{a, b, c} c. {a, b, c} d. none of these 74. 72 Problem A vector of magnitude of 5 and perpendicular to (i 2 j k ) and (2 i j 3k )5 3 a.(i j k) 3 b. 5 3 (i jk) 35 3 c.(i jk) 35 3 d. ( i jk) 3 75. 73 Problem /3x sin x is : dx /3cos2 x a. 1(4 1)3 b. 4 5 2log tan3 124 5 c.log tan3 12 d. none of these 76. 74 Problemm n r Cn is equal to :r0 a. n + m + 1C n+1 b. n + m + 2C n c. n + m + 3C n-1 d. none of these 77. 75 Problem The angle between the lines 2x = 3y = - z and 6x = -y = -4z is : a. 900 b. 00 c. 300 d. 450 78. 76 Problem A particle is projected vertically upwards at a height h after t1 seconds and again after t2 seconds from the start. Then h is equal to : a. 1 g(t t2) 12 1 b.g(t1 + t2) 2 c. 1 Gt1t22 d. None of these 79. 77 Problem If sin + cosec =2, then sin2 + cosec2 is equal to : a. 1 b. 4 c. 2 d. none of these 80. 78 Problem/2 sin x The value ofdx , is :0 sin x cos x a.2 b.4 c.8 d. 6 81. 79 Problemsin2 y 1 cos ysin y The value of expression 1 is equal to : 1 cos y sin y 1 cos y a. 0 b. 1 c. - sin y d. cos y 82. 80 Problema1 0 If f(x) = ax a1 , then f(2x) f(x) equal to : ax 2 ax a a. a (2a + 3x) b. ax (2x + 3a) c. ax (2a + 3x) d. x (2a + 3x) 83. 81 Problem 2 2 1 2 3 2 3 3 If is a non-real cube root of unity, then 2 2 is equal 2 3 3 3 2 to : a. -2 b. 2 c. - d. 0 84. 82 Problem ab If in a ABC , cos Acos B then : a. sin2 A + sin2 B = sin2 C b. 2 sin A cos B = sin C c. 2 sin A sin B sin C = 1 d. none of the above 85. 83 Problem The graph of the function y = f(x) has a unique tangent at the point (a, 0)loge {1 6f (x)} through which the graph passes, Then lim is :x a 3f (x) a. 0 b. 1 c. 2 d. none of these 86. 84 Problemna is equal to : lim 1sinn n a. ea b. e c. e2a d. 0 87. 85 Problem3c If the equation ax2 + 2bx 3c = 0 has no real roots and4 < a + b, then : a. c < 0 b. c > 0 c. c0 d. c = 0 88. 86 Problem The line 3x 4y = touches the circle x2 + y2 4x 8y 5 = 0 if the value of is : a. - 35 b. 5 c. 20 d. 31 89. 87 Problem If OA i 2j 3k, OB 3i j2k, OC 2i 3 j k. Then AB AC is equal to : a. 0 b. 17 c. 15 d. none of these 90. 88 Problem The value of tan2 (sec-1 2) + cot2 (cosec-1 3) is a. 15 b. 13 c. 11 d. 10 91. 89 Problem The sum of all proper divisor of 9900 is : a. 29351 b. 23951 c. 33851 d. none of these 92. 90 Problem The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented 2x2 xy y2 = 0 is : a. 2x2 xy y2 4x y = 0 b. 2x2 xy y2 4x + y + 2 = 0 c. 2x2 + xy + y2 2x + y = 0 d. none of the above 93. 91 Problem a 1 2 If a, b, c are in AP, then , ,are in :bc c b a. AP b. GP c. HP d. None of these 94. 92 Problem A particle is in equilibrium when the forces , u u F1 10k, F2 (4 i 12 j 3k), F2(4i 12j3k)13 13v F3( 4i j 12 3k) and F4 (cos sin ) act on it, then : i j 1365 v 65 cot a. 3 b. u = 65 (1 3 cot ) c. w = 65 cosec d. none of the above 95. 93 Problem There are 10 points in a plane out of these 6 are collinear. The number of triangles formed by joining these point is : a. 100 b. 120 c. 150 d. none of these 96. 94 Problem Ifx and yare two unit vectors and is the angle between them, then1 |x y| is equal to :2 a. 0 b.2 sin c.2 cos d.2 97. 95 Problem a b c a b x a c If a, b and c are three non-coplanar vectors, thenis equal to : a. 0 b. [a b c ] c. 2 [a b c ] d. - [a b c ] 98. 96 Problem The coefficient of x5 in the expansion of (1 + x2)5 (1+ x)4 is : a. 30 b. 60 c. 40 d. none of these 99. 97 Problem The function f(x) = x3 3x is : a. Increasing on (- , -1)(1, ) and decreasing on (-1, 1) b. Decreasing on (- , -1)(1, ) and increasing on (-1, 1) c. Increasing on (0, ) and decreasing on (- , 0) d. decreasing on (0, ) and increasing on (- , 0) 100. 98 Problem A man in a balloon rising vertically with an acceleration of 4.9 m/s2, releases a ball 2 s after the balloon is let go from the ground. The greatest height above the ground reached by the ball, is : a. 19.6 m b. 14.7 m c. 9.8 m d. 24.5 m 101. 99 Problem A bag contain n + 1 coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and 7 tossed. If the probability that toss results in heads is, then the value of n is :12 a. 3 b. 4 c. 5 d. none of these 102. 100 ProblemxIf (x)sin t 2dt , then(1) is equal to : 1/ xa. sin 1b. 2 sin 1 3c. 2 sin 1d. none of these 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET