1. UPSEEPAST PAPERSMATHEMATICS- UNSOLVED PAPER - 2002

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 The value of 2 2 2 .... is equal to : a. 5 b. 3 c. 2 d. none of these 4. 02 Problem If 3x 3x-1 = 6, then xx is equal to : a. 2 b. 4 c. 9 d. none of these 5. 03 Problem a b a b The determinant is equal to zero, if a, b, c are in : b c b c 2 10 a. GP b. AP c. HP d. None of these 6. 04 Problem3n 3n22 The value of 1 21 2is : a. Zero b. 1 c. d. 2 7. 05 Problem The value of i1/3 is :3i a.23i b. 2 c. 1 i321 i3 d.2 8. 06 Problemn Least value of n for which 1 i 3 is an integer, is :1 i 3 a. 1 b. 2 c. 3 d. 4 9. 07 Problem2 6 If A = 5 7, then adj (A) is equal to :7 6 a. 5 22 6 b. 5 77 5 c.6 2 d. none of these 10. 08 Problem1 x2 x3 1 2 x4 x6 If sin x ... cos x ... for 1 | x | 2 , then24242 the value of x is : a. 1/2 b. 1 c. -1/2 d. none of these 11. 09 Problem If I /2 cos x dx , then the value of I is equal to :0 sin x cos x a.2 b. 4 c.6 d. none of these 12. 10 Problem If sin cosec =2, then sinn cosecn is equal to : a. 2 b. 2n c. 2n-1 d. none of these 13. 11 Problem2 48 If the sum of the series 1 ... is a finite number, then :x x2 x3 a. x < 2 b. x > 2 c. x > d. none of these 14. 12 Problem1 1 1 The sum of n terms of the series 1 3 3 5 5 7.. is equal to : a. 2n 12n 1 b.2 c. 2n 1 12n 1 1 d.2 15. 13 Problem A person traveling with a velocity v1 for some time and with uniform velocity v2 for the next equal time. The average velocity v is given by : a. v = v1 v2 2 b. vv1v2211 c. v v1 v211 1 d. v v1v2 16. 14 Problem In the binomial expansion (a + bx)-3 = x + .., then the value of a and b are : a. a = 2, b = 3 b. a = 2, b = -6 c. a = 3, b = 2 d. a = -3, b = 2 17. 15 Problem 22ax a x ax a x1 22 The value of determinant bx b x bx b x1 , is : 22 xcc cx c x1 a. 0 b. 2 abc c. a2b2c2 d. none of these 18. 16 Problem If one root of the equation x2 + px + q = 0 is 2 + , then values of p and q are : a. - 4, 1 b. 4, -1 c. 2, 3 d. - 2, - 3 19. 17 Problem x2 14x 9 If x is real, then value of the expression 2 lies between :x2x 3 a. 5 and 4 b. 5 and 4 c. -5 and 4 d. none of these 20. 18 Problem10 x 3 The coefficient of x4 in the expansion of is equal to : 2 x2405 a.256504 b. 263450 c. 263 d. none of these 21. 19 Problem 2 tan-1 (cos x) = tan-1 (cosec2 x), then x is equal to : a.2 b. c. 6 d. 3 22. 20 Problem Period of sin2 x is equal to : a. b. 2 c. /2 d. none of these 23. 21 Problem If 2 1 sin xdx 4cos ax b c , then the value of (a, b) is equal to :1 a. ,2 4 b. 12 c. 1,1 d. none of these 24. 22 Problem Among 15 players, 8 are batsmen and 7 are bowlers. The probability that a team is chosen of 6 batsmen and 6 bowlers, is :8C6 x 7C5 a.15C1128 b.1515 c.28 d. none of these 25. 23 Problem In the expansion of (1 + x)n, coefficients of 2nd, 3rd and 4th terms are in AP. Then, n is equal to : a. 7 b. 9 c. 11 d. none of these 26. 24 Problem 3 /2 2 The degree and order of the equation dyd2yis :1 kdxdx 2 a. (2, 2) b. (3, 2) c. (2, 3) d. none of these 27. 25 Problem If a and b are two unit vectors inclined at as angle , then sin /2 is equal to : a. 1 b. 1/2 c. - 1/2 d. none of these 28. 26 Problema b c If, ,are in HP, then :b c a a. a2b, c2a, b2c are in AP b. a2b, b2c, c2a are in HP c. a2b, b2c, c2a are in GP d. none of the above 29. 27 Problem ab axb /x Ifbcbxc = 0, then a, b, c are in :ax b bx c0 a. AP b. GP c. HP d. None of these 30. 28 Problem If x2 5x + 6 > 0, then x : a.,23, b. [2, 3] c. (2, 3] d. none of these 31. 29 Problem If x > 0 and x 1, y > 0 and y 1, z > 0 and z 1, then the value of1logx y logx z is equal to :logy x 1logy zlogz x logz y 1 a. 1 b. -1 c. zero d. none of these 32. 30 Problem 3 2 4 If matrix1 1 adj (A), then k is equal to : A 1 21 and Ak 0 1 1 a. 7 b. -7 c. 15 d. 11 33. 31 Problem The projection of the vector ij 2 k on the vectorij 4 4 7k is equal to : a. 5 6 1019 b. 9 9 c.19 6 d. 19 34. 32 Problem If a2ij 4 5k and b ij 2 3k, then| a x b | is equal to : a. 11 5 b. 11 3 c. 11 7 d. 11 2 35. 33 Problem the area of the parallelogram whose diagonals are is equal to : a. 53 b. 52 c. 25 3 d. 25 2 36. 34 Problem1 The most general solution of tan 1 and cos is :2 7 a. n4 7 b. n( 1)n47 c. 2n 4 d. none of these 37. 35 Problem x tan x dx The value of dx is equal to :0 sec x cos x2 a. 42 b.23 2 c. 22 d.3 38. 36 Problemx ex ex ......... dy If y e ,is equal to :dxx a. 1 xy b. 1 yy c. y 11 y d. y 39. 37 Problem cos2x3 1is equal to : lim x 0 sin6 2x 1 a.16 1 b. - 161 c. 321 d. - 32 40. 38 Problemd2 y If x a sin and y b cos , then 2 is equal to :dxa a.2 sec2bb b. sec2ab c.2 sec3ab sec3 d. a2 41. 39 Problem Differential equation of y = sec (tan-1 x) is : a. (1 + x2) dy =y+x dxdy b. (1 + x2)dx=yxdy c. (1 + x2) dx = xydyx d. (1 + x 2) = ydx 42. 40 Problem If H is the harmonic mean between P and Q, then H H is : P Q a. 2 PQ b.P Q c. P Q PQ d. none of these 43. 41 Problem If tan3x tan2x1, then x is equal to :1tan3x tan2x a. b. 4 c. n,n 1, 2, 3,... 4 d. 2n ,n1, 2, 3,... 4 44. 42 Problem The value of 2(sin6 + cos6 ) 3(sin4 + cos4 ) + 1 is equal to : a. 2 b. zero c. 4 d. 6 45. 43 Problem/4 If un = tann d , then un un 2 is equal to :01 a. n 1 b. 1n 1 c. 12n 11 d. 2n1 46. 44 Problem The equation of the normal to the hyperbola x2 y2 1 at (-4, 0) is : 16 9 a. y = 0 b. y = x c. x = 0 d. x = -y 47. 45 Problemx dxis equal to : 1 x4 a. log (1 + x2) + c b. tan-1 x2 + c1 c. 2 tan-1 x2 + c d. none of these 48. 46 Problem2 3 The value of a b 1 a b 1 a b....a 2a3ab a. log a b. log a log b c. log a + log b d. none of these 49. 47 Problem If y = a log x + bx2 + x has its extremum value at x = 1 and x = 2, then (a, b) is : 1 a. 1, 21 b. ,22 c.12, 2 2 1 d., 3 6 50. 48 Problem y2 + xy + Px2 x 2y + P = 0 represent two straight lines, if P is equal to : a. 2 b. 2/3 c. 1/4 d. 1/2 51. 49 Problemaf (x) xf (a) If f(x) is a differentiable function, then lim is equal to :x ax a a. af(a) f(a) b. af(a) f(a) c. af(a) + f(a) d. af(a) f(a) 52. 50 Problem If z = x + iy, then the area of a triangle with vertices z, iz and z + iz is equal to :3 a. 2|z|2 b. |z|21 c. 2|z|21 d. 4 |z|2 53. 51 Problem If sets A and B are defined as A {(x, y); y ex , x R}B {(x, y); y x, x R} , then : a. B A b. A B c. A B d. A B A 54. 52 Problem Number of way in which 7 mean and 7 women can sit on a round table such that no two women sit together are : a. (7!)2 b. 7! X 6! c. (6!)2 d. 7! 55. 53 Problem Let a and b be two equal vectors inclined at an angle , then sin /1 is equal to : 1 |a b| a.2 | a|1 |a b| b. 2 | a| c. |a b| d. | ab| 56. 54 Problem The angle of intersection between the curves x2 = 4(y +1) and x2 = - 4(y +1) is : a. /6 b. /4 c. Zero d. /2 57. 55 Problem If a, b, c are three vectors such that | a | 3,| b | 4,| c | 5,| a b c | 0, then the value of a b b c c a is equal to : a. -20 b. -25 c. 25 d. 50 58. 56 Problem9 The value of [ x 2]dx where [.] is the greatest integer function :0 a. 31 b. 22 c. 23 d. none of these 59. 57 Problem1 is : limcos x 0x a. continuous at x = 0 b. differentiable at x = 0 c. does not exist d. none of the above 60. 58 Problem The work done by the force Fij 2 4k in displacing an object from 5 3k to r2 r1 ij3 8 5k ijis equal to : a. 0 unit b. 20 unit c. 15 unit d. none of these 61. 59 Problem Equation 3x2 + 7xy + 2y2 + 5x + 5y + 2 = 0 represents : a. a pair of straight lines b. an ellipse c. a hyperbola d. none of the above 62. 60 Problem If an angle divided into two parts A and B such that A B = K and tan A : tan B = k: 1, then the value of sin k is :k 1 a. sink 1k b. sink 1k 1 c. sink 1 d. none of these 63. 61 Problem Which is true about matrix multiplication : a. It is commutative b. It is associative c. Both a and b d. None of these 64. 62 Problem Inverse of the function, y = 2x 3 is equal to : a. x 32 b. x 321 c.2x 3 d.none of these 65. 63 Problem In a right angled triangle, the hypotenuse is four times as long as the perpendicular to it from the opposite vertex, one of the acute angles is : a. 150 b. 300 c. 450 d. none of these 66. 64 Problem Angle between two curves x2 = 4(y + 1) and x2 = - 4(y +1) is : a. 0 b. 900 c. 600 d. 300 67. 65 Problem1 If the binomial expansion of (a + bx)-2 is - 3x + , then (a, b) is equal to :4 (2, 12) (2, 8) (-2, -12) none of these 68. 66 Problem The range of a projectile fixed at an angle of 150 is 50 m. If it is fixed with the same speed at an angle of 450, then the range will be : a. 50 m b. 100 m c. 150 m d. none of these 69. 67 Problem1 0 k Matrix A = 2 1 3is not invertible for :k 0 1 a. k = 2 b. k = 2 c. k = 0 d. all real values of k 70. 68 Problem1 1 If A , then A100 is equal to :1 1 a. 2100 A b. 299 A c. 2101 A d. none of these 71. 69 Problem The probability that the two digit number formed by digits 1, 2, 3, 4, 5 is divisible by 4, is a. 1/30 b. 1/20 c. 1/5 d. none of these 72. 70 Problem Let f (x y) f (x)f (y ) x, y R, f(5) = 2, f(0) = 3, then f(5) equals : a. 4 b. 1 c. 1/2 d. 6 73. 71 Problem A particle is projected down from the top of tower 5 m high and at the same moment another particle is projected upward from the bottom of the tower with a speed of 10 m/s, meet at a distance h from the top of tower, then h is equal to : a. 1.25 m b. 2.5 m c. 3 m d. none of these 74. 72 Problem If a couple is acting on two particles of mass 1 kg attached with a rigid rod of length 4 m, fixed at centre acting at the end and the angular acceleration of system about centre is 1 rad/s2 , then magnitude of force is equal to : a. 2N b. 4 N c. 1 N d. none of these 75. 73 Problem The integral factor of expansion (x2 + 1) + 2xy = x2 1, is : a. x2 +1 2x b. 2 x1x2 1 c.x2 1 d. none of these 76. 74 Problem From a pack of cards, 2 cards are drawn at random one by one with replacement. The probability that the first is heart and second is king, is equal to : a. 1/26 b. 1/52 c. 1/13 d. 1/10 77. 75 Problem Eccentricity of the curve x2 y2 = a2 is equal to : a. 2 b. 2 c. 4 d. none of these 78. 76 Problem A weight of 10 N is hanged by two ropes as shown in figure, tensions T1 and T2 are : image* a. 5N, 5 3 N b. 53 N, 5N c. 5N, 5N d. 53 N, 5 3 N 79. 77 Problem Let A and B be two events such that P (A) = 0.3, P( A B ) = 0.8. If A and B are independent events, then P(B) is equal to :5 a.7 5 b. 131 c.31 d.2 80. 78 Problem Two masses are attached to the pulley as shown in figure. Acceleration of centre of mass is : a. g/4 b. - g/4 c. g/2 d. - g/2 81. 79 Problem A particle is thrown with velocity u at an angle of 300 from horizontal line. It bcomes perpendicular to its original position at time : u a. 2g b. 2ug c. u 3 g d. none of these 82. 80 Problem In the following, one-one function is : a. ex2 b. e x c. sin x d. none of these 83. 81 Problem The value of (ex - 1) is always : a. Greater than 1 for all real values b. Less than 1 for all real values c. Greater than 1 for some real values d. None of the above 84. 82 Problem If the force represented by 3j2k is acting through the point 5ij4 3k , then its moment about the point (1, 3, 1) is : a. ij 14 8 12k b. ij 14 8 12k6 9k i j c. d. 6 i j 9k 85. 83 Problem8 Coefficient of x2 in the expansion of X1 is equal to : 2x a. 1/7 b. -1/7 c. -7 d. 7 86. 84 Problem0 3 If A and A 1A , then is equal to :2 01 a. - 61 b. 31 c. - 31 d. 6 87. 85 Problem1 x x2 Minimum value offor al real x is equal to :1 x x2 a. Zero b. 1/3 c. 1 d. 3 88. 86 Problem [ ] [kji ] [ ] ikj jkiis equal to : a. -1 b. 3 c. -3 d. -2 89. 87 Problem If x, 2x + 2, 3x + 3 are in GP, then the fourth term is : a. 27 b. -27 c. 13.5 d. -13.5 90. 88 Problem If nC12 = nCB, then n is equal to : a. 20 b. 12 c. 6 d. 30 91. 89 Problem 1 2 1 4 0 1 If A,Band C , then 5A 3B + 2C is equal to : 3 02 31 08 20 a. 7 9820 b. 798 20 c. 7 987 d. 20 9 92. 90 Problem 11 12 13 is equal to : 12 13 14 13 14 15 a. 1 b. zero c. -1 d. 67 93. 91 Problemx 1 x 1 dy If y sec 1x 1sin 1x,1 dx is equal to : a. 1x 1 b. x 1 c. zerox 1 d. x 1 94. 92 Problem If sin y = x sin (a + y), then dy is equal to : dx sin a a.sin a sin2 aysin2 ay b.sin a c. sin a sin2 (a + y)sin2 ay d. sin a 95. 93 Problem The two curves x3 3xy2 +2 = 0 and 3x2 y3 = 2 : a. Cut at right angle b. Touch each other c. Cut at an angle /3 d. Cut at an angle /4 96. 94 Problem dxis equal to :3/4 2 4 x x1 1/4 a.1 1 c x4 b .(x4 + 1)1/4 + c1/4 1 c. 1c x41/4 d. -1 1 c x4 97. 95 Problem The value of the integral is : a. Zero b. 1/2 c. -1/2 d. none of these 98. 96 Problem A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals : a. 1/2 b. 1/32 c. 31/32 d. 1/5 99. 97 Problem There are four machines and it is known that exactly two of them are faulty. They are tested one in a random order till both the faulty machines are identified. Then, the probability that only two tests are needed, is ; a. 1/3 b. 1/6 c. 1/2 d. 1/4 100. 98 Problem If fifth term of an GP is 2, then the product of its 9 terms is : a. 256 b. 512 c. 1024 d. none of these 101. 99 Problem If p and q are the roots of the equation x2 + px + q = 0, then : a. p = 1 b. p = 0 or 1 c. p = - 2 d. p= - 2 or zero 102. 100 ProblemLet A and B be two finite sets having m and n elements respectively. Then, thetotal number of mappings from A to B is :a. mnb. 2mnc. mnd. nm 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET