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AMU - Mathematics - 1998

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  • 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 b x 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 3 1 d. 10 2 metres
  • 8. 06 Problem 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 r r , then the triangle is : 1 1 1 1 2 r2 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 z arg 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 x 51 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 Problem 1 The probability of safe arrival of one ship out of five is 5 . The probability of safe arrival of atleast 3 ship is : a. 3 52 1 b. 31 184 c. 3125 181 d. 3125
  • 20. 18 Problem 10 The coefficient of x4 in the expansion of x 3 is : 2 x2 405 a. 256 504 b. 259 450 c. 263 540 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 x2 x b. 1 x2 2x c. 1 x2 x d. 1 x2
  • 25. 23 Problem tan 2x x is equal to : lim x 0 3x sin x a. 1 2 1 b. - 2 3 c. 2 3 d. - 2
  • 26. 24 Problem x If a function f(x) is defined as , x 0 f (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 exist x 0 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. <x<0 d. 1 < x < 3
  • 31. 29 Problem The value of function for which the function f(x) = 1 + 2 sin 2x + 3 cos 2x has maximum value : a. 3 13 b. 3 c. 13 d. 0
  • 32. 30 Problem If the line ax + by + c = 0 is a normal to the curve xy = 1, then : a. a < 0, b > 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 Problem dx equal to : x x e e 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 4 5 2 log tan b. 3 12 4 5 c. log tan 3 12 d. none of these
  • 36. 34 Problem for any integer n the integral 1 2 ecos 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 = 0 d2y b. - 4y = 0 dx 2 c. d2y = y2 dx 2 d. d2y -y=0 dx 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 5 125 b. 4 52 c. 4 15 d. 4
  • 41. 39 Problem m 3 2n m 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, d be the position vectors of four points A, B, C, D such that :         (a d ).(d c) (b d ).(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 A 3ˆ j ˆ k, B ˆ i 2 ˆ are adjacent sides of a parallelogram j then its area is : a. 17 b. 41 c. 14 d. 7
  • 44. 42 Problem  A force F 2ˆ i ˆ j ˆ 5k is applied at the point A (1, 2, 5). If moment about the point ˆ 6ˆ ˆ B (-1, - 2, 3) is (16i j 2 k ) , then is equal to : a. 2 b. - 1 c. 0 d. - 2
  • 45. 43 Problem The value of loga 1 y2 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. 5 5 d. 6
  • 47. 45 Problem 1 A and B are two independent events the probability that both A and B occurs is 6 1 and the probability that neither of them occur 3 is , then probability of the occurane of A is : 1 a. 5 1 b. 3 1 c. 4 1 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. 7 3 b. 7 2 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. 3 2 b. 3 1 c. 4 2 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 n 2n 1 1 b. n 1 2n 1 c. n 2n 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 1 n2 n c. n 2n 1 d. n 1
  • 53. 51 Problem The observation which occur most frequently is known as : a. Mode b. Median c. Weighted mean d. Mean
  • 54. 52 Problem i 0 If A = 0 i then 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) < r 1 b. 2 (bxy + byx) = r 1 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 by y y yyx (x x) andx x bxy (y y), then bxy x byx is : a. x * y x b. y c. x y x 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) 3 4 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 y 2 3 b. 3x y 2 3 c. x 3y 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 z 1 a. 1 2 3 x y z b. 1 3 6 9 x 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. a b2 c2 2 a1 b1 c1 b. a2 b2 c2 a1 b1 c1 c. a2 b2 c2 a1 b1 c1 d. a2 b2 c2
  • 65. 63 Problem The value of loga 1 y3 is : 1 y y2 a. loga (1 - y) b. loga (1 + y) c. loga (1 + y2) d. loga (1 - y2)
  • 66. 64 Problem 2 If the circles x y2 2ax 2b y 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. 1 4 1 b. 2 3 c. 4 5 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 ] ,1 defined 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 5 3 1 b. , 5 5 3 1 , c. 5 5 2 1 , d. 5 5
  • 71. 69 Problem The eccentricity of the conic x2 – 4x + 4y2 = 12 is : 3 a. 2 2 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 function x2 bc has no real values between : 2x b 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. 8a 3 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 Let b the range being all real numbers except a, and b = (x) ax x 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 Problem 1 tan x is equal to ; lim x 4 1 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 Problem d {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 Problem 1 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 India’s second win occurs at the third test is : a. 1 2 b. 1 3 c. 1 4 1 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 xdy 3 3x 2 y 2e x dx 0 is : x 3 a. ex c y b. x ex 3 0 y x 3 c. - ex c y 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 of a 4ˆ i 6 ˆ along j b 3ˆ j ˆ 4k is : 1 ˆ ˆ a. (3 j 4k ) 5 18 ˆ ˆ (3 j 4k ) b. 25 18 ˆ (3ˆ j 4k ) c. 13 18 ˆ (3ˆ j 4k ) 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. 8P 6 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 Problem dy If x sin cos , y cos cos 2 then the value of at is : dx 4 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/ 3 2 /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 to is : 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 0 n 0 a. xyz = xz + y b. xyz = xy – z c. x + y + z = xyz d. xyz = yz + x
  • 102. 100 Problem In the group G = {0, 1, 2, 3, 4, 5} under addition modulo 6, (2 + 3-1 + 4)-1 is equal to : a. 0 b. 2 c. 3 d. 5
  • 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET