1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER - 1998

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 If the probability that A and B will die within a year are p and q respectively, then the probability that only one of them will be alive at the end of they year, is : a. p+q b. p + q 2pq c. p + q pq d. p + q + pq 4. 02 Problem Ten different letters of alphabet are given. Words with five letters are formed from these given letters. Then, the number of words which have atleast one letter repeated, is : a. 69760 b. 30240 c. 99748 d. none of these 5. 03 Problem If (1 + x + x2)n = C0 + C1x + C2x2 + then the value of C0C1- C1C2 + C2C3 a. 0 b. 3n c. (-1)n d. 2n 6. 04 Problem Given positive integers r > 1, n > 2 and that the coefficient of (3r)th and (r + 2)th term in the binomial expansion of (1 + x)2n are equal, then : a. n = 2r b. n = 3r c. n = 2r + 1 d. none of these 7. 05 Problem A ball of mass 1 kg moving with velocity 7 m/s, overtakes and collides with a ball of mass 2 kg moving with velocity 1 m/s in the same direction. If 2 = , the velocity of lighter ball after impact is : a. 6 m/s3 b. 2m/s c. 1 m/s d. 0 m/s 8. 06 Problem A bullet of 0.05 kg moving with a speed of 120 m/s penetrates deeply into a fixed target and is brought to rest in 0.01 s. The distance through which it penetrates is : a. 3 cm b. 6 cm c. 30 cm d. 60 cm 9. 07 Problem a, b, c are real a b, the roots of the equation (a - b)x2 5 (a + b) x 2 (a - b) = 0 are : a. real and equal b. complex c. real and unequal d. none of these 10. 08 Problem /2 The value of 0 | sin x cos x | dx is equal to : a. 0 b. 2( 2 -1) c. 22 d. 2( 2 + 1) 11. 09 Problem1 The value of 1x | x | dx is equal to a. 2 b. 1 c. 0 d. none of these 12. 10 Problemx sin1 x The value of 1 x2dx is equal to : a. (1 x 2 )sin-1 x + c b. x sin1 x + c c. x - (1 x 2 )sin-1 x + c d. (sin-1 x)2 + c 13. 11 Problem cos3 2n 1 xdx has the value2 For any integer n, the integral esin x 0 a. -1 b. 0 c. 1 d. 14. 12 Problem The cube roots of unity when represented argand diagram from the vertices of : a. an equilateral triangle b. an isosceles triangle c. a right angled triangle d. none of the above 15. xa13 Problem (ax The value of / x )dx is equal to :x a. a loge a + c b. 2 a x log10 a + cx c. 2 a loga e + cx d. 2 aloge a + c 16. 14 Problem If +b then1 a. a = 32 b. a =31 c. a = - 32 d. a = - 3 17. 15 Problema If f(a - x) = f(x), then x f(x)d0x is equal to :a a2 0 a.f(x) dx a b. a 0 f(x)dxa2 a c.2 0 f(x)dx d. none of these 18. 16 Problem1 If z 5 i Then z lies , where | | 1. z a. a circle b. a parabola c. an ellipse d. none of these 19. 17 Problem If log 2, log (2x - 1), log (2x + 3) are in AP Then x is equal to : a. 5/2 b. log2 5 c. log3 2 d. log5 3 20. 18 Problem Points D, E are taken on the side BC of a triangle such that BD = DE = EC. If BAD , DAE , EAC , then the value of sin sin is sin sin equal to : a. 1 b. 2 c. 4 d. none of these 21. 19 Problem The value of cos (2 cos-1 x + sin-1 x) at x = 1/5 is equal to :2 6 a. 52 5 b. 62 6 c. - 52 5 d. -6 22. 20 Problem /2 If f(x) is an odd function of x, then / 2f (cos x )dx is equal to : a. 0 /2 b. 0 f (cos x )dx /2 c. 0 f (cos x )dx d. f cos x dx 23. 21 Problem y = cos-1 [sin (1 x) / 2] + xx, then dy/dx at x = 1 is equal to : a. 3/4 b. 0 c. 1/2 d. -1/2 24. 22 Problem f(x) = (sin x + cos 2x), (x > 0) has minimum value for x is equal to : n a. 23 b. n 1 2 1 c.2n 1 2 d. none of these 25. 23 Problem The point P on curve y2 = 2x3 such that the tangent at P is perpendicular to the line 4x 3y + 2 = 0 is given by : a. (2, 4) b. (1, 2)11 c. 2 , 21 1 d., 8 16 26. 24 Problem Let I1 = 2dx 2 dx then :11 x2 and I2 1x , a. I1 > I2 b. I2 > I1 c. I1 = I2 d. I1 > 2I2 27. 25 Problem The value of the integral is equal to : a. 01 b. 2 c. 12 d. none of these 28. 26 Problem In ABC, the value of cosec A (sin B cos C + cos B sin C) is equal to :c a. a b. ac c. 1 d. none of these 29. 27 Problem The value of tan 90 tan 270 cot 270 + cot 90 is : a. 2 b. 3 c. 4 d. none of these 30. 28 Problem In ABC, 3 sin A = 6 sin B = 2 3 sin C, then the value of A is equal to : a. 00 b. 450 c. 600 d. 900 31. 29 Problem If the side of the triangle are 5k, 6k, 5k and radius of incircle is 6, then the value of k is a. 4 b. 5 c. 6 d. 7 32. 30 Problem The angle of depression of the top and the foot of the chimney as seen from the top of second chimney which is 150 m high and standing on the same level as the first are and respectively. The distance between their tops when tan4 5 is equal to : and tan 3 2 50 m 100 m 150 m none of these 33. 31 Problem The medians AD and BE of the triangle with vertices A (0, b), B (0, 0) and C (a, 0) are mutually perpendicular, if : a. b = 2 a b. a = b 2 c. b = - 2 a d. a = - 2b 34. 32 Problem The distance between the chords of contact of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 from the origin and the point (g, f) is : a. g2 + f2 b. (g2 + f2 + c)g2 f 2 c c.2 g2 f 2 d.g2 f 2 c 2 g2 f 2 35. 33 Problem A line is drawn through a fixed point p (h, k) to cut the circle x2 + y2 = a2 at Q and R. Then PQ . PR is equal to : a. (h + k)2 a2 b. h2 + k2 a2 c. (h k)2 + a2 d. h2 + k2 + a2 36. 34 Problem The locus of the mid point of a focal chord of a parabola is : a. Circle b. Parabola c. Ellipse d. Hyperbola 37. 35 Problem 2 2 The straight line x + y = c will be tangent to the ellipse x y 1 then c is equal9 16 to : a. 8 b. 5 c. 10 d. 6 38. 36 Problem The length of the subnormal of the curve y2 = 2ax is equal to a. a b. 2a c. a/2 d. -a 39. 37 Problem2 2 If is the angle between the asymptotes of the hyperbola x y 1 witha2b2 eccentricity e, then sec 2 is equal to : a. 0 b. e c. e2e d. 2 40. 38 Problem The latus rectum of the hyperbola 9x2 16y2 + 72x 32y 16 = 0 is :9 a. 2 b. - 92 c. 32 3 d. - 32 3 41. 39 Problem x2 1 The value of p and q from lim px q 0 are : x 1 x a. p = 0, q = 0 b. p = 1, q = -1 c. p = -1, q = 1 d. p = 2, q = - 1 42. 40 Problem Which of the following functions is an even function ? ax 1 a. f x x x a 1 b. f(x) = tan x ax a x c. f x x xa 1ax 1 d.f x xa 1 43. 41 Problem x 2 sin1 / x, x 0 If f(x) = , then : 0,x 0 a. f and f are continuous at x = 0 b. f is derivable at x = 0 c. f is derivable at x = 0 and f is not continuous at x = 0 d. f is derivable at x = 0 44. 42 Problemp q r Let p, q, r be positive and not all equal, then the value of the determinant q r pr p q is equal to : a. Positive b. Negative c. 0 d. none of these 45. 43 Problem Suppose n 3 person are sitting in a row. Two them are selected at random. Then probability that they are not together, is L: a. 1- 2/n b. 2/(n -1) c. 1 1/n d. none of these 46. 44 Problem The minimum value of x2 3x + 3 in the interval (-3, 3/2) is equal to : a. 3/4 b. 5 c. - 15 d. - 20 47. 45 Problem 100 100 Let tn be the nth term of the GP of positive numbers. Let t n1 2n x and t2n1 yn1 such that x y , then the common ratio isx a. yx b.y c. yx d. none of these 48. 46 Problem If every element of third order determinant value of is multiplied by five, then the value of the new determinant is : a. b. 5 c. 25 d. 125 49. 47 Problem In the expansion of (1 + x)50, the sum of the coefficients of odd powers of x is : a. 0 b. 249 c. 250 d. 251 50. 48 Problem The equation sin6 cos6 a has a real solution, if : a. 1/2 a 1 b. 1/4 a 1 c. -1 a 1 d. 0 a 1/2 51. 49 Problem (1 x2) sin x cos2 x dx is equal to : a. 0 b. - 3/3 c. 2 - 37 d. 2 32 52. 50 Problem The derivative of f x x3dt (x 0) is equal to : x2 log t a. 113log x 2log x 1 b. 3log x3x 2 c. 3 log x x 1 x d. log x 53. 51 Problem4 41 If f x dx 4 and [3 f (x)]dx 7 , then the value of f x dx is equal122 to : a. 2 b. - 3 c. - 5 d. none of these 54. 52 Problem e [ (x) (x)]dx is equal to : x a. ex (x)dx b. e x ( x ) c c. ex (x) c d. none of these 55. 53 Problem 1 The value of | sin2 x |0dx is equal to : a. 01 b. - 1 c. 2 d. 56. 54 Problem A house has multi-storeys. The lowest storey is 20 ft high. A stone which is dropped from the top of the house passes the lowest storey in 1/4 s, then the height of the house is a. 110.00 ft b. 110.2 ft c. 110.25 ft d. none of these 57. 55 Problem A particle was dropped from the top of the tower and at the same time another body is thrown vertically upwards from the bottom of the tower with such a velocity that it can just reach the top of the tower, then they will meet at the height of : a. h/4 b. 3h/4 c. h d. h/2 58. 56 Problem A particle starts with a velocity of 200 cm/s and moves in a straight line with a retardation of 10 cm/s2. Then the time it takes to describe 1500 cm is : a. 10 s, 30 s b. 5 s, 15 s c. 10 s d. 30 s 59. 57 Problem The area bounded by the curve y = x3, the x axis and the ordinates x = -2 and x = 1 is : a. -9 sq unit b. -15/4 sq unit c. 15/4 sq unit d. 17/4 sq unit 60. 58 Problem Two balls are projected respectively from the same point in direction inclined at 300 and 600 to the horizontal. If they attain the same height, then the ratio of their velocities of projection is a. 1 : 3 b.3 :1 c. 1 : 1 d. 1 : 2 61. 59 Problem A particle is projected under gravity (g = 9.81 m/s2) with a velocity of 29.43 m/s at an angle of 300. The time of flight in seconds to a height of 9.81 m are : a. 5, 1.5 b. 1, 2 c. 1.5, 2 d. 2, 3 62. 60 Problem The path of a projectile in vacuum is a : a. A straight line b. Circle c. Ellipse d. Parabola 63. 61 Problem A particle is projected with initial velocity u making an angle with the horizontal, its time of flight will be given by : a. 2u sin g b. 2u 2 sin g u sin c.gu2 sin d.g 64. 62 Problem If x2 + px + 1 is a factor of x2 + bx + c, then : a. a2 + c2 = - ab b. a2 c2 = - ab c. a2 c2 = ab d. none of the above 65. 63 Problem The probabilities of occurrence of two events E and F are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither E occurs nor F occurs : a. 0.39 b. 0.25 c. 0.11 d. none of these 66. 64 Problem A sphere impinges directly one an equal sphere which is at rest. Then the original kinetic energy lost is equal to : 1 e2 a.2times the initial KE1 e2 b.21 e2 c.times the initial KE2 d. none of these 67. 65 Problem A hockey stick ball is at rest for 0.01s with an average force of 5 N. If the ball weight 0.2 kg, then the velocity of the ball after being pushed is equal to : a. 2.5 m/s b. 2 m/s c. 3.0 m/s d. 5 m/s 68. 66 Problem A given force is resovled into components P and Q is equally inclined to it, then : a. P = 2Q b. P = Q c. 2P = Q d. none of these 69. 67 Problem If the forces of 12,5 and 13 unit weight balance at a point, two of them are inclined at : a. 300 b. 450 c. 900 d. 600 70. 68 Problem ABC is a triangle. Forces P, Q, R act along the lines OA, OB and OC and are in equilibrium, if O is incentre of ABC, then : P Q R A B C a. cos cos cos2 2 2 P QR b.OA OBOC P QR c. A BC sin sinsin 2 22 d. none of the above 71. 69 Problem If two equal perfectly elastic balls impinges directly, after impact : a. Their velocities are not effected b. They interchange their velocities c. Their velocities changes their direction d. Their velocities get doubled 72. 70 Problem In triangle ABC (sin A + sin B + sin C) (sin A + sin B sin C) = 3 sin A sin B then : a. A = 600 b. B = 600 c. C = 600 d. A = 900 73. 71 Problem If sin A p, cos A q then :sin B cos Bp q2 1tan A a. q 1 p2 2 b. tan A p q 12q 1 pq2 1 c.tan B 1 p2 d. all are correct 74. 72 Problem In a triangle ABC, A and AD is median then :3 a. 4AD2 = b2 + bc + c2 b. AD2 = b2 + bc + c2 c. 2AD2 = b2 + bc + c2 d. 4AD2 = b2 bc + c2 75. 73 Problem If the radius of the circumcircle of isoseceles triangle ABC is equal to AB = AC, then angle A is equal to : a. 300 b. 600 c. 900 d. 1200 76. 74 Problem The focus of the parabola y2 x 2y + 2 = 0 is : a. (1/4, 0) b. (1/2) c. (3/4, 1) d. (5/4, 1) 77. 75 Problem If A (3, 1), B (6, 5) and C (x , y) are three points such that the angel CAB is a right angle and the area of CAB = 7, then the number of the point C is ; a. 0 b. 1 c. 2 d. 4 78. 76 Problemdy Solve, dx = (4x + 3y - 1)2. 79. 77 Problemlim Evaluate x 0 [sin (x + a) + sin (a - x) 2sin a]/x sin x 80. 78 Problem A rod is moveable in a vertical plane about a hinge at one end, another end is fastened to a weight equal to half the weight of the rod, this end is fastened by a string of length l to a pint at a height c vertically over the hinge find the tension in of the string. 81. 79 Problem Sun of infinity of the series 12 22 12 22 321 ........2!3! 82. 80 Problem Evaluate,cos x dx 1 sin x 2 sin x 2 83. 81 Problem Find the equation to the chord of the hyperbola 25x2 16y2 = 400 having mid point at (6, 2). 84. 82 Problemij i j i j Find the vector moment f the three vectors 2 3k,2 3 4k, k acting on a particle at point P (0, 1, 2) about the point A (1, -2, 0). 85. 83 Problem If the equation k(6x2+3) + rx+2x2 -1 =0 and 6k (2x2 +1 ) + px +4x 2 -2 =0, have both the common roots find the value of (2r p) 86. 84 Problem Find the equation to the common tangnt to the parabola y2=2x and x2 = 16y 87. 85 Problem x 5 Determine the value of x in the expansion of x xlog 10 if the third term in the expansion is 10,00,000. 88. 86 Problem In parallelogram ABCD the interior bisectors of the consecutive angles B and C intersect at P, then find . BPC 89. 87 Problem Find the area bounded by the curve y = 2x x2 and the straight line y = - x. 90. 88 Problem b2 c 2 a2 a2 Find the value of determinant b2 c 2 a2 b2 c2 c2 a2 b2 91. 89 Problem The probability of getting sum more than 15 in three dice will be 5/108. Prove it. 92. 90 Problem If tan 2 tan =1, then find the value of . 93. 91 Problem If A = [1 2 3] and B = 5 4 0 , then find AB. 0 2 1 1 3 2 94. 92 Problem Find the length of tangent of circle x2 + y2 + 6x 4y 3 = 0 from point (5, 1). 95. 93 Problem Find the value of cos 200 cos 400 cos 600 cos 800. 96. 94 Problem x 3 y 4 z 5 Find the distance from point (3, 4, 5) of that point where line 1 2 2 cut the plane x + y + z 17 = 0. 97. 95 Problem Find the point on the curve 9y2 = x3, where normal to the curve makes equal intercepts with the axes. 98. 96 Problem A ladder 15 m long leans against a wall 7 m high and a portion of the ladder protrudes over the wall such that its projection along the vertical is 3 m. How fast does the bottom start to slip away from the wall, if the ladder slides down along the top edge of the wall at 2 m/s ? 99. 97 Problem Solve the equation sin [2 cos-1 {cot (2 tan-1 x) }]= 0 100. 98 Problem The side AB, BC, CD and DA of a quadrilateral have the equations x + 2y = 3, x = 1, x 3y = 4, 5x + y + 12 = 0 respectively. Find the angle between the diagonals AC and BD. 101. 99 Problem The angles of top of the tower from the foot and top of a building are . Find the height of tower. 102. 100 ProblemIf lines px2 pxy y2 = 0, make the angle from x-axis, then find the value of tan . 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET