1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER - 1999

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 f(x) = cos (log x), f(x)f(y) -1 fxf ( xy ) is equal to : 2y a. 0 b. 1 c. - 1 d. none of these 4. 02 Problem If [x] stands for the greatest integer functions, then the value of 1 11 2 1999.....is equal to : 2 1000 2 10002 1000 497 498 500 502 5. 03 Problem ex e x The inverse of the function f x ex e x1 is :x a. log10 x 2x b. logex 21 x c. loge22 y d. none of these 6. 04 Problem If f(x) = (x + 1)cot x be continuous at x = 0, then f(x) is equal to : a. 0 b. -e c. e d. none of these 7. 05 Problem 1x If f(x) = f x[4t 2 2ft ]dt, then f(4) is equal to : x2 4 a. 32 9 b. 20 320 c. -332 d. 9 8. 06 Problem If f(x) = kx sin x is monotonically increasing, then : a. k > 1 b. k = 1 c. k < 1 d. k < - 1 9. 07 Problem A stone is dropped into a quiet lake and waves move in a circle at a speed of 3.5 cm/s. At the instant when the radius of the circular wave is 7.5 cm. The rate at which the enclosed area is increasing, is : a. 32.5 cm2/s b. 31.5cm2/s c. 52.5 cm2/s d. none of these 10. 08 Problem If , are the roots of the equation 2x2 + 6x + b = 0, (b < 0), then is less than : a. 1 b. -1 c. 2 d. -2 11. 09 Problem If the roots of the equation x2 px + q = 0 differ by unity, then : a. q2 = 4p b. p2 = 4q + 1 c. q2 = 4p + 1 d. p2 = 4q - 1 12. 10 Problem If ecosx e-cosx = 4, then cos x equal to : a. log (2 + 5 ) b. log5 c. log (2 - 5) d. none of these 13. 11 Problem A horse is tied to a post by a rope. If it moves along a circular path, keeping the rop tight and describes 132 when it has traced out an angle of 1080 at the centre, then the length of the rope is : a. 68 m b. 70 m c. 69 m d. 71 m 14. 12 Problem The value of sin6 cos6 3sin2 .cos2 is : a. 0 b. 1 c. 3 d. 2 15. 13 Problem If 1 sin 1 sin1 sin 1 sin 1 sin 1 sin k, then k is equal to : a. 2 cos cos cos b. - cos cos cos c. cos cos cos d. 2 sinsinsin 16. 14 Problem If tan tan p sec 2 , then the value of p is equal to :4 4 a. 2 b. 3 c. 1 d. 4 17. 15 Problem The maximum value of sin x6cos x6in the interval 0, 2 is attained at : a.6 b.12 c.3 d.4 18. 16 Problem 2 1 x 3 x 3 The determinant 2 vanishes for : 1 x 4 x 4 2 1 x 5 x 5 a. 3 values of x b. 2 values of x c. 1 values of x d. no values of x 19. 17 Problem 1 1 1 If f x1 x x2 , then f(x) vanishes at : 1 x2x3 a. x = -1 b. x = 0 c. x = 1 d. none of these 20. 18 Problem Two persons throw a dice. The probability that outcome of the both throws, will be some is : a. 261461 b.6362 c.146 d. None of these 21. 19 Problem A bag contains n balls. It is given that the probability that among these n balls exactly r balls are white is proportional to . r2 0 r n A ball is drawn at random and is found to be white. Then, the probability that all the balls in the bag are white, will be :2n a. 2n14n b. 2n 12n c. 2n3 d.4n2n 3 22. 20 Problem n letters to each of which corresponds one addressed envelope are placed in the envelope at random. Then, the probability that n letter is placed in the right envelope, will be :11 11n 1 a.....11! 2! 3!4! n!11111 b.....2! 3! 4! 5! n!1111 n 1 c. .... 12! 3! 4! 5!n! d. none of the above 23. 21 Problem n The smallest positive integer n for which 1 i is : 1 1 i a. 3 b. 2 c. 4 d. none of these 24. 22 Problem If z + z-1 = 1, then z100 + z-100 is equal to : a. i b. - i c. 1 d. -1 25. 23 Problem1 1 1 If x > 1, y > 1, z > 1 are in GP, then, ,are in :1 log x 1 log y 1 log z a. AP b. HP c. GP d. None of these 26. 24 Problem The value of a for which 2x2- 2(2a + 1) x + a (a + 1) = 0 may have one root less than a and other root greater than a are given by a. 1 > a > 0 b. - 1 < a < 0 c. 1 a d. a > 0 or a < - 1 27. 25 Problem The values of k for which one of the roots of x2 x + 3k = 0 is double of one, the roots of x2 x + k = 0 is : a. 1 b. - 2 c. 2 d. none of these 28. 26 Problem If A is the AM of the roots of the equation x2 2ax + b = 0 and G is the GM of the roots of the equation x2 2bx + a2 = 0, then : a. AM > GM b. AM GM c. AM = GM d. None of these 29. 27 Problem The straight lines l1, l2, l3 are parallel an lie in the same plane. A total number of m points are taken on l1, n points on l2, k points on l3. The maximum number of triangles formed with vertices at these points are : a. m+n+kC 3 b. m+n+kC mC3 nC3 kC3 3 c. mC + nC3 + kC3 3 d. none of these 30. 28 Problemn If the coefficient of x7 and x8 in x are equal, then n is equal to :23 a. 56 b. 55 c. 45 d. 15 31. 29 Problem In the expansion of (1 + x)50, the sum of the coefficient of odd power of x is : a. 0 b. 249 c. 250 d. 251 32. 30 Problema xcb If a + b + c = 0, then one root ofcb x a= 0 is : b ac x a. x =1 b. x = 2 c. x = a2 + b2 + c2 d. x = 0 33. 31 Problem If a2i2jk and c i 2k , then |c | a is equal to : a. 2 5 i2 5 j 5k b. 2 5 i2 5 j 5k c. 5 i5 j 5k d. 5 i2 5 j 5k 34. 32 Problem If a3i j4k , b 2i 4 j 3k and c i 2 j k, then| 3a 2b 4c | is equal to : a. 298 b. 198 c. 398 d. 498 35. 33 Problem A unit vector parallel to the sum of the vectors is : i j3 6 2k a.5 b. 3 i 6j 2k5 c. 3 i 6 2k j7 d. none of these 36. 34 Problem If a i jk, b 4i 3 j 4k and c i j k are linearly dependent vectors and|e|3, then : a. 1,1 b. 1, 1 c.1, 1 d.1,1 37. 35 Problem If ab c 0,| a | 3,| b | 5,| c | 7, then the angel between a and b is : a.4 b.32 c. 3 d. 6 38. 36 Problem Let A and B be two non-empty subsets of set X such that A is not a subset of B, then : a. A is subset of the complement of B b. B is a subset of A c. A and B are disjoint d. A and the complement of B are non-disjoint 39. 37 Problem The number of real roots of the following equation |x|2 - 7 |x| + 12 = 0 is : a. 1 b. 2 c. 3 d. 4 40. 38 Problem The fourth, seventh and tenth terms of a GP are p, q, r respectively, then : a. p2 = q2 + r2 b. q2 = pr c. p2 = qr d. pqr + pq + 1 = 0 41. 39 Problem 6 The term independent of x in expansion of 1 is : 2x3x160 a.980 b.9160 c.27 80 d. 3 42. 40 Problem A unit vector perpendicular to the vector 4i j 3k and 2i j 2k is :1 i 2 j2k a. 31 b. i 2 j2k31 c. 2i j2k31 d. 2i 2j2k3 43. 41 Problem 11 4 tan-1 5 - tan-1 is equal to : 239 a. b. 2 c. 3 d. 4 44. 42 Problem The two circles x2 + y2 2x 3 = 0 and x2 + y2 4x 6y 8 = 0 are such that : a. They touch each other b. They intersect each other c. One lies inside the other d. Each lies outside the other 45. 43 Problem A point moves in a plane so that its distance PA and PB from two fixed points A and B in the plane satisfy the relation : PA PB = k (k 0), then the locus of P is : a. A parabola b. An ellipse c. A hyperbola d. A branch of a hyperbola 46. 44 Problem The axis of the parabola 9y2 16x 12y 57 = 0 is : a. 3y = 2 b. x + 3y = 3 c. 2x = 3 d. y = 3 47. 45 Problem The straight linex x0 yy0 z z0, is parallel to the plane ax + by +lm n cz = d, then :a b c a.l m n b. al = bm = cna b c c. 0l m n d. al + bm + cn = 0 48. 46 Problem The number of points at which the function f(x) = |x 0.5| + |x -1|+ tan x does not have derivative in the interval (0, 2) is : a. 1 b. 2 c. 3 d. 4 49. 47 Problem The function f(x) = tan x x : a. Always increases b. Always decreases c. Never decreases d. Some times increases and some times decreases 50. 48 Problem The maximum value of the xy subject to x + y = 8 is a. 8 b. 16 c. 20 d. 42 51. 49 Problem 1 The value of the following integral sin11 x dx is : 110 8 6 4 4 2. . . . . a. 11 9 7 7 5 3 b. 10 8 6 4 2. . . . .11 9 7 5 3 2 c. 1 d. 0 52. 50 Problem2 The volume of the solid obtained by rotating the ellipse x y2 1 about thea2b2 axis of x is a. a2 b cu unit b. - b3 cu unit4 c. a2 b cu unit3 d. 4 ab2 cu unit3 53. 51 Problem x2y2 The ellipse1 and the straight line y = mx + c intersect in real points a2b2 only, if : a. a2m2 < c2 b2 b. a2m2 > c2 b2 c. a2m2 c2 - b2 d. 0 cb 54. 52 Problem The resultant of two forces sec B and sec C along sides AB, AC of a triangle ABC is a force acting along AD when D is : a. Middle point of BC b. Foot of perpendicular from A on BC c. D divides BC in the ratio cos B : cos C d. D divides BC in the ratio cos C : cos B 55. 53 Problem There is a system of coplanar forces acting on a rigid body is represented in magnitude direction and sense by the sides of a polygon taken in order, then the system is equivalent to : a. A single non-zero force b. A zero force c. A couple where moment is equal to the area of polygon d. Acouple where moment is twice the area of polygon 56. 54 Problem A boby is in equillbrum on a rough inclined plane of where the coefficient of 1 friction is 3 gradually increased. The body will be on the point of sliding downwards when the inclination of the plane reaches : a. 150 b. 300 c. 450 d. 600 57. 55 Problem A body consists of a solid cylinder with radius a and height h together with a solid hemisphere of radius a placed on the base of a cylinder. The centre of gravity of the complete body is : a. Inside the cylinder b. Inside the hemisphere c. On the interface between the two d. Outside both 58. 56 Problem A body starts from rest and moves in a straight line with uniform acceleration f, the distance covered by it in second, fourth and eighth seconds are : a. In arithmetic progression b. In geometrical progression c. In the ratio 1 : 3 : 7 d. In the ratio 3 : 7 : 15 59. 57 Problem If the roots of the equation x3 12x2 + 39x 28 = 0 are in AP, then their common difference will be a. 1 b. 2 c. 3 d. 4 60. 58 Problem Let R be a reaction defined by a R b, a b where a and b are real numbers, then R is : a. Reflexive, symmetric and transitive b. Reflexive, transitive but not symmetric c. Symmetric, transitive nor reflexive but symmetric d. Neither transitive nor reflexive but symmetric 61. 59 Problem3 The probability that a man hits a target is. If tried 5 times, the probability4 that he will hit the target at least three times, is : a. 471364371 b.464471 c. 582459 d. 512 62. 60 Problem The sum of the series log4 2 log8 2 + log16 2 - . Is : a. e2 b. loge 2 + 1 c. loge 3 2 d. 1 loge 2 63. 61 Problem Let the function f(x) x2 + x + sin x cos x + log (1 + 1 x 1)be defined over the interval (0, 1). The odd extension of f(x) to interval [-1, 1] is : a. x2 + x + sin x + cos x log (1+ 1 x 1) b. - x2 + x + sin x + cos x log (1 + 1 x 1) c. - x2 + x + sin x cos x + log (1 + 1 x 1) d. none of the above 64. 62 Problem If sets A and B are defined as A = {(x, y), y = ex ,x e R}, B = {(x , y), y = x, x }, then : a. B A b. A B c. A B d. A B 65. 63 Problem Let y = x2e-x, then the interval in which y increases with respect to x is ; a. (-1, 1) b. (-2, 0) c. (2, 1) d. (0, 2) 66. 64 Problem1 1 1 If sin sin cos x = 1, then x is equal to :5 a. 1 b. 04 c. 5 d. 15 67. 65 Problem In a right angled triangle the hypotenuse is 2 2 times the length of perpendicular drawn from the opposite vertex on the hypotenuse. Then, the other two angles are : a.,3 6 b. ,4 43 c. ,8 8 5, d. 12 12 68. 66 Problem The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 600. If the third side is 3, the remaining fourth side is : a. 2 b. 3 c. 4 d. 5 69. 67 Problem The locus of the mid point of the chord of the circle x2 + y2 2x 2y 2 = 0 which makes an angle of 1200 at the center is : a. x2 + y2 2x 2y + 1 = 0 b. x2 + y2 x + y = 0 c. x2 + y2 2x 2y 1 = 0 d. none of the above 70. 68 Problem The angle between the tangents drawn from the origin to the parabola y2 = 4a(x - a) is : a. 900 b. 300 1 c. tan-1 2 d. 450 71. 69 Problem The volume of solid cylinder obtained by revolving about y-axis and the area enclosed between the ellipse x2 + 9y2 = 9 and the line x + 3y = 3 in first quadrant is a. 3 b. 4 c. 6 d. 9 72. 70 Problem x 3 x 2 lim is equal to : x x 1 a. 1 b. e c. e2 d. e3 73. 71 Problem The slope of the tangent to the curve x = t2 + 3t 8, y = 2t2 2t 5 at the point (2, -1) is :22 a. 7 6 b.7 c. -6 d. none of these 74. 72 Problem1x2 If ex dx 0, then :0 a. 1 < < 2 b. 0 c. f and f are not continuous at x = 0 d. f is continuous at x = 0 but f is not so 76. 74 Problem A tetrahedron has vertices at O (0, 0, 0), A (1, 2, 1), B (2, 1, 3) and C (-1, 1, 2). Then, angle between the forces OAB and ABC will be :1 19 a. cos3511 b. cos 3 c. 300 d. 900 77. 75 Problem In a triangle ABC three forces of magnitudes 3 AB, 2 AC and 6CB are acting along the sides AB, AC and CB respectively. If the resultant meets AC at D, then the ratio DC : AD will be equal to : a. 1 : 1 b. 1 : 2 c. 1 : 3 d. 1 : 4 78. 76 Problem ABC is a uniform triangular lamina with centre of gravity at G. If the portion GBC is removed, the centre of gravity of the remaining portion is at G. Then, GG is equal to :1 a. AG31 b. AG41 c. AG51 d. AG6 79. 77 Problem A circular cylinder of radius r and height h rests on a horizontal plane with one of its flat ends on the plane. A gradually increasing horizontal forces is applied through the centre of upper end. If the coefficient of friction is , the cylinder will topple before sliding is : a. r < h b. r > h c. r 2 h d. r = 2 h 80. 78 Problem From the top of a tower of height 100 m, a ball projected with a velocity of 16 m/s. It takes 5 s to reach the ground. If g = 10 m/s2, then the angle of projection is ; a. 300 b. 450 c. 600 d. 900 81. 79 Problem A body is moving in a straight line with uniform acceleration. It covers distance of 10 m and 12 m in third and fourth seconds respectively. Then, the initial velocity in m/s is : a. 2 b. 3 c. 4 d. 5 82. 80 Problem To a man running at a speed of 20 km/h the rain drops appear to be falling at an angle of 300 from the vertical. If the rain drops are actually falling vertically downwards, their velocity in km/h is ; a. 10 3 b. 10 c. 20 3 d. 40 83. 81 Problem The relation less than in the set of natural numbers is : a. Only symmetric b. Only transitive c. Only reflexive d. Equivalence relation 84. 82 Problem Let f : R R be define by f(x) = 3x 4, then f-1(x) ;1 a. 3 ( x + 4)1 b. 3(x - 4) c. 3x + 4 d. not defined 85. 83 Problems3n Let Sn denote the sum of first n terms of an AP. If S2n = 3Sn, then the ratio of Sn is equal to : a. 4 b. 6 c. 8 d. 10 86. 84 Problem The coefficient of x4 in the expansion of (1 + x + x2 + x3)n is : a. nC 4 b. nC + n C4 4 c. nC + n C4 + n C2 4 d. nC + n C2 . n C1 + n C2 4 87. 85 Problem Two dice are thrown simultaneously, then the probability of obtaining a total score of 5 is 1 a.18 1 b.121 c.9 d. none of these 88. 86 Problem7 The probability that Krishna will be alive 10 years hence, isand that Hari 15 7 will be alive is. The probability that both Krishna and Hari will be dead 1010 years hence, is : 21 a.150 24 b. 150 49 c. 150 56 d. 150 89. 87 Problem The set of values of x for which is : a. b. {4} n:n1, 2, 3.... c. 4 d. 2n :n 1, 2, 3.... 4 90. 88 Problem The incorrect statement is :1 a. sin5 b. cos = 1 c. 1sec2 d. tan = 1 91. 89 Problem The radius of the incircle of a triangle whose sides are 18, 24 and 30- cm is : a. 2 cm b. 4 cm c. 6 cm d. 9 cm 92. 90 Problem The eccentricity of the ellipse 9x2 + 5y2 30y = 0 is :1 a. 3 b. 23 c. 34 d. none of these 93. 91 Problem Equation of the tangent to the hyperbola 2x2 3y2 = 6 which is parallel to the line y = 3x + 4, is : a. y = 3x + 5 b. y = 3x 5 c. y = 3x + 5 and y = 3x 5 d. none of the above 94. 92 Problem If the angle between the two lines represented by 2x2 + 5xy + 3y2 + 6x + 7y + 4 = 0 is : tan-1 m, then m is equal to :1 a.5 b. 17 c. 5 d. 7 95. 93 Problem The value of the derivative of |x -1| + |x - 3| at x = 2 is : a. -2 b. 0 c. 2 d. not defined 96. 94 Problem The largest value of 2x3 3x2 12x + 5 for occurs at x is equal to : a. -2 b. -1 c. 2 d. 4 97. 95 Problem x A x iix A x jj x A x k is equal to :k a. A b. 2 A c. 3 A d. 0 98. 96 Problem A cart of 100 kg is free to move on smooth rails and a block of 20 kg is resting on it. Surface of contact between the cart and the block is smooth. A force of 60 N is applied to the cart. Acceleration of 20 kg block in m/s2 is : a. 3 b. 0.6 c. 0.5 d. 0 99. 97 Problem A 15 kg block is moving on ice with a speed of 5 m/s when a 10 kg block is dropped onto it vertically. Then both move with a velocity in m/s is : a. 3 b. 15 c. 5 d. indeterminate 100. 98 Problem A particle is projected vertically upwards and it is at a height h after t1 second and after t2 second from the start, h is equal to : a. g(t1 t2) b. g(t1 + t2) c. gt1t2 d. none of these 101. 99 Problem A square hole is punched out of a circular lamina of diameter 4 cms the diagonal of the square being a radius of the circle centroid of the remainder from the centre of the circle, is a distance : a. 12 11 b.11 c.2 11 d.1 102. 100 ProblemA body is pulled up an inclined rough plane. Let be the angle of friction. Therequired force is least when it makes an angle k with the inclined plane where kis equal to : 1a. 3 1b. 2c. 1d. 2 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET