1. AMU PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 1998

2. SECTION I CRITICAL REASONING SKILLS 3. 01 Problem If a, b,c are in A.P., then the value of x 1 x 2 x a is :x 2 x 3 x bx 3 x 4 x c a. 3 b. - 3 c. 0 d. none of these 4. 02 Problem The system of simultaneous equations kx + 2y z = 1, (k - 1) y 2z = 2 and (k + 2) z = 3 have a unique solution if k equals : a. -1 b. -2 c. 0 d. 1 5. 03 Problem If A and B are Hermition matrices of the same order, then (AB - BA) is : a. A null matrix b. A Hermitian matrix c. A Skew-Hermitian matrix d. None of these 6. 04 Problem Let A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5. Then A B is given by : a. {15, 30, 45, .} b. {3, 6, 9, ..} c. {15, 10, 15, 20, } d. {5, 10, 20, ..} 7. 05 Problem A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 450 with the ground. The entire length of the tree is a. 15 metres b. 20 metres c. 10 (1 +2 ) metres 31 d. 102 metres 8. 06Problem The length of the shadow of a pole is times of the length of the pole. The length of elevation of the sun is : a. 450 b. 300 c. 900 d. 600 9. 07 Problem If sin A = sin B, cos A = cos B, then the value of A in terms of B is : a. n+B b. n + (-1)n B c. 2n + B d. 2n-B 10. 08 Problem Cos. cos (90 - ) sin sin (90 - ) equals : a. 1 b. 2 c. - 1 d. 0 11. 09 Problem If in a triangle rr, then the triangle is :11 112r2 r3 a. Right angled b. Isosceles c. Equilateral d. None of these 12. 10 Problem The maximum value of 3 cos x + 4 sin x + 5 is : a. 5 b. 9 c. 7 d. none of these 13. 11 Problem A box contains 10 mangoes out of which 4 are rotten. 2 mangoes are taken out together. If one of them is found to be good, the probability that the other is also good is :2 a.3 5 b. 13 8 c. 13 7 d.13 14. 12 Problem Ten different letters of an alphabet are given words with five letters are formed with three letters. The number of words which atleast one letter repeated is : a. 69760 b. 30240 c. 99748 d. 37120 15. 13 Problem arg zarg z; z 0 is equal to : a. 4 b. c. 0 d. 2 16. 14 Problem If a,b,c,d,e,f are in A.P., then e-c is equal to : a. 2(c - a) b. 2 (d -c) c. 2 (f - d) d. (d - c) 17. 15 Problem a 51 x51 x , , 52 x 5 2x are in A.P, then the value of a is : 2 a. a < 12 b. a 12 c. a 12 d. none of these 18. 16 Problem The harmonic mean and geometric mean of two positive number be in the ratio 4 : 5, then two numbers are in the ratio is : a. 1 : 4 b. 4 : 1 c. 3 : 2 d. 2 ; 3 19. 17 Problem1 The probability of safe arrival of one ship out of five is 5. The probability of safe arrival of atleast 3 ship is : a.352 1 b.31184 c.3125181 d. 3125 20. 18 Problem10 The coefficient of x4 in the expansion of x 3 is : 2 x2405 a.256504 b.259450 c. 263540 d. 276 21. 19 Problem The product of n positive number is unity, then their sum is : a. Divisible by n b. A positive integer 1 c. Equal to n n d. Never less than n 22. 20 Problem A sum of money lent on simple interest becomes double in 8 years the same sum will triple in : a. 24 years b. 16 years c. 32 years d. 12 years 23. 21 Problem The period of the function f (x) = sin4 x + cos4 x is : a. b.2 c. 2 d. none of these 24. 22 Problem 2x Let f ( x) sin 1 , where 0 < x < 1 < f (x) < , then f(x) is equal to :1 x2 2 2 a. 1 x2x b.1 x22x c.1 x2x d.1 x2 25. 23 Problemtan 2x x is equal to : lim x0 3x sin x a. 121 b. - 23 c. 23 d. - 2 26. 24 Problemx If a function f(x) is defined as , x 0f (x) x 2 then :0,x 0 a. f(x) is continuous at x = 0 but not differentiable at x = 0 b. f(x) is continuous as well as differentiable at x = 0 c. f(x) is discontinuous at x = 0 d. none of these 27. 25 Problem Let [x] denotes the greatest inter function and f(x) = [tan2 x,] then : a. lim f(a) does not existx0 b. f(x) is continuous at x = 0 c. f(x) is discontinuous at x = 0 d. f(0) = 1 28. 26 Problem If f(x) = (x + 1) tan-1 (e-2x), then f(0) is : a. 2 +1 b. 4 -1 c. 6 +5 d. none of these 29. 27 Problem The angle of intersection to the curve y = x2 , 6y = 7 x3 at (1, 1) is : a. 2 b. 4 c. 3 d. 30. 28 Problem y = [x (x - 3)]2 increases for all values of x lying in the interval :3 a. 0 < x < 2 b. 0 < x < c. 1 b. a < 0, b < 0 c. a > 0, b > 0 d. a > 0, b < 0 33. 31 Problem The greatest value of f (x) cos(xe( x ) 7x 2 3x), x [ 1, ) is : a. - 1 b. 1 c. 0 d. none of these 34. 32 Problemdx equal to :xxee a. log (ex + 1) + c b. log (ex + e-x) + c c. tan-1 ex + c d. sin-1 ex + c 35. 33 Problem/3 x sin x 2 dx is equal to :/3 cos x 1 a.(4 1) 3 452 log tan b.312 45 c. log tan 312 d. none of these 36. 34 Problem for any integer n the integral 12ecos x [cos3 (2x 1)]x dx has the value :1 a. 0 b. c. 1 d. 2 37. 35 Problem The differential equation of y = Ae2x + Be-2x is :dy a. dx - 4y = 0d2y b.- 4y = 0dx 2 c. d2y= y2dx 2 d. d2y-y=0dx 2 38. 36 Problem The compound interest on Rs. 800 at 8% per annum compounded annually for 2 years is : a. Rs. 133.12 b. Rs. 137.38 c. Rs. 130.15 d. Rs. 125. 25 39. 37 Problem The area of the figure bounded by y = sin x, y = cos x in the first quadrant is : a. 2( 2 - 1) b. 3+ 1 c. 2 (3 + 1) d. none of these 40. 38 Problem The ratio dose the x- axis divide the area of the region bounded by the parabola y = 4x x2 and y=x2-x is a. 12 5125 b. 452 c. 415 d.4 41. 39 Problemm 3 2nm 0 7 If, then the value of m, n, p, q are p 1 4p 6 3 22 a. 3, - 4, 2, - 3 b. 4, 2, 3, - 3 c. - 3, - 2, 4, 5 d. - 4, 2, 3, - 3 42. 40 Problem If a, b, c, dbe the position vectors of four points A, B, C, D such that : (ad ).(d c)(bd ).(c a) 0, then D is the : a. centroid of ABC b. incentre ABC c. circumcentre of ABC d. orthocentre ABC 43. 41 Problem The vectors A3 j k, B i 2 are adjacent sides of a parallelogramj then its area is : a. 17 b.41 c. 14 d.7 44. 42 Problem A force F 2i j5kis applied at the point A (1, 2, 5). If moment about the point 6 B (-1, - 2, 3) is (16ij 2 k ) , then is equal to : a. 2 b. - 1 c. 0 d. - 2 45. 43 Problem The value of loga1y2 is :2 (1 y y ) a. loga (1 - y) b. loga (1 + y) c. loga (1+ y2) d. loga (1 y2) 46. 44 Problem In ABC the angle B is greater than angle A. If the value of the angles A and B satisfy, the equation 3 sin x 4 sin3 x x = 0. Then the value of angle C is :2 a.3 b. 3 c. 55 d.6 47. 45 Problem 1 A and B are two independent events the probability that both A and B occurs is61 and the probability that neither of them occur 3 is , then probability of the occurane of A is :1 a.51 b. 31 c. 41 d. 6 48. 46 Problem Pair of dice is rolled together till a sum of either 5 or 7 is obtained, then the probability that 5 comes before 7 is :4 a.73 b. 72 c. 5 5 7 d. 49. 47 Problem A father has 3 children with atheist one he The probability that he has 2 boys and one girl is :1 a.32 b.31 c. 42 d. 5 50. 48 Problem If a, b, c are any real number, then : a. max (a, b) < max (a, b, c) b. min (a, b) = (a + b + |a - b|) c. max (a, b) < min (a, b) d. max (a, b) < max (a, b, c) 51. 49 Problem The A.M. of the series 1, 2, 4, 8, 16, . , 2n is : a. 2n 1 n2n 1 1 b. n 12n 1 c.n2n 1 d. n 1 52. 50 Problem The variance of first n natural numbers is :n2 1 a. 12 b. (n 1)(2n 1) n 1n2 n c. n2n 1 d. n1 53. 51 Problem The observation which occur most frequently is known as : a. Mode b. Median c. Weighted mean d. Mean 54. 52 Problemi 0 If A = 0 ithen A4n when n is a natural number equals : number equals : a. I b. - A c. - I d. A 55. 53 Problem The standard deviation of 35, 40, 42, 36, 27 : a. 25.8 b. 26.9 c. 26.8 d. 27.8 56. 54 Problem Which one of the following is a true statement :1 a. 2 (bxy + byx) < r1 b. 2 (bxy + byx) = r1 c. 2(bxy + byx) > r d. none of these 57. 55 Problem If A and B are finite sets then (A - B) (B - A) equals : a. (A B) A b. (A - B) B c. (A B) (A B) d. (A - B) A 58. 56 Problem Two lines of regression between x and y are given byy y yyx (x x) andx x bxy (y y), then bxy x byx is : a. x * y x b. y c.x yx y d. 59. 57 Problem The equation of a circle two of whose diameters are 2x 3y + 12 = 0 and x + 4y 5 = 0 and whose are a is 154 sq. units, is : a. x2 + y2 6x + 46 36 = 0 b. x2 + y2 + 6x - 46 36 = 0 c. x2 + y2 6x + 46 + 25 = 0 d. none of these 60. 58 Problem Which of the lines are coplanar ? (x 1) (y 2) (z 3) (i)2 3 4 (x-2) (y 3) (z 4) (ii)34 3(x-3)(y 4) (z 5) (iii)4 5 6 a. (i) only b. (i) only c. (i) only d. all the lines are coplanar 61. 59 Problem If a line joining two points A (2, 0) and B (3, 1) is rotated about A I anti-clockwise direction through an angle 150, then the equation of the line in the new position is : a. 3x y2 3 b. 3x y2 3 c. x3y 2 3 d. None of these 62. 60 Problem Area of the quadrilateral formed by the lines | x | + | y | = 1 is : a. 4 b. 2 c. 8 d. none of these 63. 61 Problem If a plane meets the coordinate axes at A, B and C, in such a way that the centroid of ABC is at the point (1, 2, 3), the equation of the plane is :x y z1 a. 1 2 3x y z b. 13 6 9x y z 1 c. 1 2 3 3 d. none of these 64. 62 Problem The planes a1x + b1y + c1z = 0 and a2x + b2y + c2z + d2 = 0 and parallel of :a1 b1 c1 a. ab2 c2 2a1 b1 c1 b. a2 b2 c2a1 b1c1 c.a2 b2c2a1b1 c1 d. a2b2 c2 65. 63 Problem The value of loga 1y3is : 1 y y2 a. loga (1 - y) b. loga (1 + y) c. loga (1 + y2) d. loga (1 - y2) 66. 64 Problem2 If the circles xy22ax 2by c 0 and 2x2 + 2y2 + 2ax + 2by + c = 0 intersect orthogonally, then : a. aa + bb = c + c c b. aa + bb = c +2 c c. aa + bb = 2+ c d. none of these 67. 65 Problem The lines 3x 4y + 4 = 0 and 6x 8y 7 = 0 are tangents to the same circle. Then its radius is : a. 141 b.23 c. 45 d. 6 68. 66 Problem If tan2 A = 2 tan2 B + 1, then cos 2A + sin 2B equals : a. - 1 b. 1 c. 0 d. 2 69. 67 Problem A function f : R[ 1 ],1defined by f ( x) sin x, R,where R is the subset of real numbers in one-one and onto if R is the interval : a. [0,2 ] 2 b.,2 2 c. [ , ] d. [0, ] 70. 68 Problem On the ellipse 4x2 + 9y2 = 1, the points at which the tangent are parallel to the line 8x = 9y are :2 1 a.,5 53 1 b.,5 531, c. 552 1 , d. 5 5 71. 69 Problem The eccentricity of the conic x2 4x + 4y2 = 12 is :3 a.22 b. 3 c. 3 d. none of these 72. 70 Problem The number of solutions of the equations | x | - 3 | x | + 2 = 0 is : a. 4 b. 1 c. 3 d. 2 73. 71 Problem If x is real the functionx2bc has no real values between : 2xb c a. b and c b b. bc and c c. b2 and c d. b and c2 74. 72 Problem an equilateral triangle is inscribed in the parabola y2 = 4ax whose vertex is at the vertex of the parabola the length of side the triangle is : a. 12a 3 b. 8a3 c. 6a 3 d. 10a 3 75. 73 Problem Focus of the middle points of all chords of the parabola y2 = 4x which are drawn through the vertex is : a. y2 = 8x b. y2 = 2x c. x2 + 4y2 = 16 d. x2 = 2y 76. 74 Problem The equation of the conic with focus at (1, -1), directrix along x y + 1 = 0 and with eccentricity is : a. xy = 1 b. x2 y2 = 1 c. 2xy 4x + 4y + 1 = 0 d. 2xy + 4x - 4y - 1 = 0 77. 75 Problem The slope of the tangent at the point (h, k) of the circle x2 + y2 = a2 is : a. 0 b. 1 c. -1 d. depends on h 78. 76 Problem Letbthe range being all real numbers except a, and b =(x) axx a a2 . Then its inverse is : a. (ax - b)/(x - a) b. (ax - a)/(ax - b) c. (bx - a)/(x - a) d. (a - bx)/(1 - ax) 79. 77 Problem1 tan x is equal to ;limx41 2 sin x a. -1 b. 1 c. 2 d. - 2 80. 78 Problem The value of b for which the function f(x) = sin x bx + c is decreasing in the interval is ( , ) given by : a. b > 1 b. b < 1 c. b 1 d. b 1 81. 79 Problemd {log2(x2 1)} is :dx a. x/(x2 + 1) log2 b. x log 2/x2 + 1 c. log2 e/x2 + 1 d.1(x 2 1) 82. 80 Problem The function f(x) = x4 62x2 + ax + 9 attains its maximum value on the interval [0, 2] at x = 1. Then the value of a is : a. 120 b. - 120 c. 52 d. none of these 83. 81 Problem If a1, a2, a3, . an-1 are positive numbers in A.P. and d is their common difference then an-1 a1 equals : a. nd b. (n - 2)d c. (n + 1)d d. (n - 1)d 84. 82 Problem1 The probability of India winning a test match against Australia is 2 . Assuming independence from match to match to match the probability that in a 5 match series Indias second win occurs at the third test is : a. 12 b. 13 c. 141 d. 5 85. 83 Problem The degree of the differential equation y32/3 + 2 + 3y2 + y1 = 0 is : a. 1 b. 2 c. 3 d. none of these 86. 84 Problem The solution of ydx xdy3 3x 2 y 2e x dx 0 is :x3 a. ex cy b. x ex3 0yx3 c. - ex cy d. none of these 87. 85 Problem For a moderately skewed distribution mean = 34, median = 36, then the mode is : a. 35 b. 45 c. 30 d. 40 88. 86 Problem The component ofa 4i 6 alongj b 3 j 4k is :1 a. (3 j 4k )518 (3 j4k ) b. 25 18 (3 j 4k ) c. 1318 (3 j4k ) d. 10 13 89. 87 Problem The equation of the sphere passing through the origin and the points A (a, 0, 0), B (0, b, 0) and C (0, 0, c) is : a. x2 + y2 + z2 + ax + by + cz = 0 b. x2 + y2 + z2 - ax - by - cz = 0 c. x2 + y2 + z2 - 2ax - 2by - 2cz = 0 d. none of these 90. 88 Problem A quadratic equation with rational coefficient can have : a. Both roots equal and irrational b. One root real and other imaginary c. Both roots real and irrational d. None of these 91. 89 Problem How many words beginning with T and ending with E can be made (with no letter repeated) out of the letters of the word TRIANGLE ? a. 1440 b. 8P6 c. 720 d. 722 92. 90 Problem If f(x) = ex (a cos x + b sin x) where a, b are constant then f(x) + 2f(x) is equal to : a. f(x) b. 2f(x) c. 3f(x) d. 0 93. 91 Problemdy If x sin cos , y cos cos 2 then the value ofat is :dx4 a. 2 b. 1 c. 3 d. 0 94. 92 Problem 1 1 1 If be a complex cube root of unity, then the value of 1 2 1 2 1 2 is : a. 0 b. 1 c. -1 d. 2 95. 93 Problem If A, then Adj. A is equal to : a. b. c. d. 96. 94 Problem The A.M. between two quantities a and b is twice as large as the G.M. then a, b is : a. 3 /2 b. 2 + 3/2 - 2 c. 2 + 3/2 - 3 d. 2/3 97. 95 Problem The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is equal to : a. 2 b. 3 c. 1/ 32 /3 d. 98. 96 Problem The quartile deviation of daily wages of 7 persons which are RS. 12, 7, 15, 10, 17, 17, 26 is : a. 7 b. 14.5 c. 9 d. 3.5 99. 97 Problem x 2y 2, x 2y 8, x, y 0 The maximum value of z = 3x + 2y subjected tois : a. 32 b. 24 c. 40 d. none of these 100. 98 Problem If sin + cosec = 2, sin2 + cosec2 is equal to : a. 1 b. 4 c. 2 d. none of these 101. 99 Problem For , 0 and x cos2n y sin2n .z cos2n .sin2n 2 n 0 n 0n 0 a. xyz = xz + y b. xyz = xy z c. x + y + z = xyz d. xyz = yz + x 102. 100 ProblemIn the group G = {0, 1, 2, 3, 4, 5} under addition modulo 6, (2 + 3-1 + 4)-1 is equalto :a. 0b. 2c. 3d. 5 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET