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AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
AMU - Mathematics  - 1997
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AMU - Mathematics - 1997

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    • 1. AMU –PAST PAPERSMATHEMATICS - SOLVED PAPER - 1997
    • 2. SECTION – I CRITICAL REASONING SKILLS
    • 3. 01 Problem The probability that a card drawn from a pack of 52 cards will be a diamond or king is : a. 1 52 2 b. 13 c. 4 13 d. 1 13
    • 4. 02 Problem N cadets have to stand in a row if all possible permutations are equally likely, the probability of two particular cadets standing side by side is : a. 4 N b. 3 N2 c. 1 2N d. 2 N
    • 5. 03 Problem In a simultaneous throw of 2 coins, the probability of having 2 heads is : a. 1 4 1 b. 2 1 c. 8 1 d. 6
    • 6. 04 Problem The probability of getting more than 7 when a pair of dice are thrown is : 7 a. 36 5 b. 12 c. 7 12 d. none of these
    • 7. 05 Problem If sets A and B are defined as A = {(x, y) | y = ex , x R} B = {B = {(x, y)| y = x, x R}, then : a. B A b. A B c. A B d. A B A
    • 8. 06 Problem The co-efficient of variation is computed by : mean a. standard deviation standard deviation b. mean mean c. standard deviation x 100 standard deviation d. mean x 100
    • 9. 07 Problem If r is the correlation coefficient, then a. r 1 b. r 1 c. |r| 1 d. |r| 1
    • 10. 08 Problem The reciprocal of the mean of the reciprocals of n observation is the : a. Geometric mean b. Median c. Harmonic mean d. Average
    • 11. 09 Problem Find the mode from the data given below : Marks obtained 0-5 5-10 10-15 15-20 20-25 20-30 No. of students 18 20 25 30 16 14 a. 16.3 b. 15.3 c. 16.5 d. none of these
    • 12. 10 Problem find the median of 18, 35, 10, 42, 21 : a. 20 b. 19 c. 21 d. 22
    • 13. 11 Problem The quartile deviation from the following data x 2 3 4 5 6 f 3 4 8 4 1 is 1 a. 2 1 b. 4 3 c. 4 d. 1
    • 14. 12 Problem If z k k , then z1z2z3z4 is equal to : k cos i sin 10 10 a. 1 b. -1 c. 2 d. -2
    • 15. 13 Problem n The value of n pr r 1 r! a. 2n b. 2n – 1 c. 2n –1 d. 2n + 1
    • 16. 14 Problem the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines : a. 6 b. 9 c. 18 d. 12
    • 17. 15 Problem The probability that in a random arrangement of the letter of the word ‘UNIVERSITY’, the two I’s do not come together is : a. 4 5 1 b. 10 9 c. 10 1 d. 5
    • 18. 16 Problem The coefficient of x4 in expansion of (a + x + x2 + x3)n is ; a. nC n b. nC + n C2 n c. nC + n C1 + n Cn + n C2 n d. none of these
    • 19. 17 Problem n 2n If 1, , 2 are the cube roots of unity, the 1 has the 2n n 1 value : n 2n 1 a. 1 b. c. 2 d. 0
    • 20. 18 Problem If x2 x2 y2 z2 z2 for all positive value of x, y and z then : y x y z x a. x < y < z b. x < y > z c. x < y > z d. x > y < z
    • 21. 19 Problem If A and B are independent events such that P (A) > 0, P (B) > 0, then : a. A and B re mutually exclusive b. A and B are independent c. A and B are dependent d. P(A/B) + P ( A /B) = 1
    • 22. 20 Problem The least value of the expression 2 log10 0.001- logx0.01 for x > 1 : a. 2 b. 1 c. 4 d. 3
    • 23. 21 Problem If p and q are respectively the sum and the sum of the square of n successive integers beginning with a, nq – p2 is : a. Independent of a b. Independent of n c. Dependent of a d. None of these
    • 24. 22 Problem If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c has : a. atleast on root in (0, 1) b. one root is (1, 2) other in (-1, 0) c. both imaginary d. none of these
    • 25. 23 Problem If the root of the equations (x - c) (x - b) – k = 0 are c and b, then roots of the equation (x - a) (x - d) + k are : a. a and c b. b and c c. a and d d. a and b
    • 26. 24 Problem If f :R R be a mapping defined by f(x) = x3 + 5, then f-1 (x) is equal to : a. (x + 3)1/3 b. (x - 5)1/3 c. (5 – x)1/3 d. (5 – x)
    • 27. 25 Problem If ax = b, by = c and cz = a, then xyz is equal to : a. 1 b. 2 c. -3 d. -1
    • 28. 26 Problem If the natural numbers are divided into groups as (1, 2, 3), (4, 5, 6).., then 1st terms of the 10th group will be : a. 40 b. 45 c. 46 d. 48
    • 29. 27 Problem The value of 1 3 is : cos sin 5 3 a. 5 4 b. 5 4 c. 6 5 d. 3
    • 30. 28 Problem The modulus of 2i 2i is : a. 2 b. 2 c. 0 d. 2 2
    • 31. 29 Problem The value of [ 2 {cos (560 15’) + i sin (560 15’)}]8 is : a. 4i b. 8i c. 16i d. -16i
    • 32. 30 Problem The sum of two irrational number is always : a. An irrational number b. A rational number c. Both rational number and irrational d. None of these
    • 33. 31 Problem 1 is equal to : sin cos 6 a. 2 b. 6 c. 3 3 d. 2
    • 34. 32 Problem The family of curves represented by dy x2 x 1 and the family dx y2 y 1 represented by dy y 2 y 1 2 0 : dx x x 1 a. Touch each other b. Are orthogonally c. Are one and the same d. None of these
    • 35. 33 Problem dy The family of curves represented by the differential equation x dx = cot y is : a. x cos y = log x b. x cos y = constant c. log (x cos y) = x d. cos y = log x
    • 36. 34 Problem The differential equation of all parabolas having their axis of symmetry coinciding with the axis of x is : 2 d2y dy y 0 a. dx 2 dx 2 d2 x dx x 0 b. dy 2 dy d2y dy c. y 0 dx 2 dx d. none of these
    • 37. 35 Problem d3y d2y For which of the following functions does the property holds : dx 3 dx 2 a. y = ex b. y = e-x c. y = cos x d. y = sin x
    • 38. 36 Problem The domain of definition of the function 1 is : f (x) |x| x a. R b. (0, ) c. (- , 0) d. none of these
    • 39. 37 Problem 1 The range of the function for real x of y is : 2 sin3 x 1 y 1 a. 3 1 y 1 b. - 3 1 c. - y 1 3 1 d. - y 1 3
    • 40. 38 Problem The period of the function f(x) = sin 2x 3 is : 6 a. 2 b. 6 c. 6 2 d. 3
    • 41. 39 Problem A, B, C are three consecutive milestone on a straight road from each of which a distant spine is visible, the spine is observed to bear with north at A, east at B and 600 east of south at C. Then the shortest distance of the spine from the road is : 7 9 3 a. miles 7 5 3 b. 13 miles 7 5 3 c. 15 miles 7 5 3 d. 17 miles
    • 42. 40 Problem The smallest positive value of x in tan (x + 1000) = tan (x + 500) tan x .tan (x - 500) : a. 150 or 300 b. 300 or 800 c. 300 or 450 d. 300 or 550
    • 43. 41 Problem 1 2 n is equal to : lim ... n 1 n2 1 n2 1 n2 a. 2 1 b. - 2 c. e-1 d. e2
    • 44. 42 Problem x 1 equals to : lim 1 x x a. e b. e-2 c. e-1 d. e2
    • 45. 43 Problem If f(x) = x ( x x 1) then : a. f(x) is continuous but not differentiable at x = 0 b. f (x) is differentiable at x = 0 c. f(x) is differentiable but not continuous at x = 0 d. f(x) is not differentiable at x = 0
    • 46. 44 Problem the value of cos2 x dx equals : 1 1 x sin2x c a. 2 2 1 1 b. x sin 2x +c 2 2 1 1 c. x sin2x 2 2 d. (x + sin 2x) + c
    • 47. 45 Problem is equal to : a. x – log | 1 - ex| + c b. x – log |1 - ex| + c c. log |1 - ex| + ex + c d. none of these
    • 48. 46 Problem Two vectors are said to be equal if : a. They originate from the same point b. They meet at the same point c. They have same magnified and direction d. None of these
    • 49. 47 Problem The solution of the differential equation dy 1 ex y is : dx a. (x + c)ex + y = 0 b. (x + c)ex - y = 0 c. (x - c)ex + y = 0 d. (x + c)e- x + y = 0
    • 50. 48 Problem       If axb c and b x c a, then a. a = 1, b = c b. a = 1, c = 1 c. b = 1, c = a d. b = 2, c = 2a
    • 51. 49 Problem If the vectors ˆ (ai ˆ j ˆ i k ),(ˆ ˆ bj ˆ k ) and ˆ i ˆ j ˆ ck(a b, c 1) are coplanar, then the value of 1 1 1 is : 1 a 1 b 1 c a. 1 b. 2 c. 0 d. none of these
    • 52. 50 Problem 1 1 1 The following consecutive terms of a series are in : 1 x1 x1 x a. H.P. b. G.P. c. A.P. d. A.P., G.P.
    • 53. 51 Problem n The sum of series S (n n)! is : n 0 a. - e2 1 b. e c. e2 d. e
    • 54. 52 Problem If 1, a1, a2, …. an-1 are n roots of unity, then the value of (1 – a1) (1 – a2) …..(1 – an - 1) is : a. 0 b. 1 c. n d. n2
    • 55. 53 Problem Let P (x) = a0 + a1x2 + a2x4 + ….. anx2n be a polynomial in a real variable x with 0 < a0 < a1 …. < an. The function P(x) has : a. Neither maximum nor minimum b. Only one maximum c. Only one minimum d. Both maximum and minimum
    • 56. 54 Problem If A = [aij] is a skew-symmetric matrix of order x, then aij equal to : a. 0 for some i b. 0 for all i = 1, 2, ….. c. 1 for some i d. 1 for all i = 1, 2, …, n
    • 57. 55 Problem If x and y are matrices satisfying x +y = I and 2x – 2y = I where I is the unit matrix of order 3, then x equals : 3/4 0 0 0 3/4 0 a. 0 0 0 3 0 4 0 3 0 b. 0 0 0 1 0 1 0 0 0 c. 1 1 1 1 0 0 0 1 0 d. 0 0 1
    • 58. 56 Problem a b If A and A2 , then b a a. a2 b2 , = ab b. a2 b2 , = 2ab c. a2 b2 , = a 2 – b2 d. 2ab, a2 b2
    • 59. 57 Problem If A is an invertible matrix and B is a matrix, then : a. rank (AB) = rank (A) b. rank (AB) = rank (B) c. rank (AB) > rank (B) d. rank (AB) > rank (A)
    • 60. 58 Problem Three lines ax + by + c = 0, cx + ay + b = 0 and bx + cy + a = 0 are concurrent only when a. a + b + c = 1 b. a2 + b2 + c2 = ab + bc + ca c. a3 + b3 + c3 = abc d. a2 + b2 + c2 = abc
    • 61. 59 Problem When number x is rounded to P, decimal digits, then magnitude of the relations error cannot exceed : a. 0.5 x 10-P+1 b. 0.05 x 10P+2 c. 0.5 x 10P+1 d. 0.05 x 10-P+1
    • 62. 60 Problem sin2 x cos2 x 1 equals to : cos2 x sin2 x 1 10 12 2 a. 0 b. 12 cos2 x – 10 sin2 x c. 12 cos2 x – 10 sin2 x -2 d. 10 sin x
    • 63. 61 Problem The equation of the sphere passing through the point (1, 3, - 2) and the circle y2 + x2 = 25 and x = 0 is : a. x2 + y2 + z2 – 11x + 25 = 0 b. x2 + y2 + z2 + 11x - 25 = 0 c. x2 + y2 + z2 + 11x + 25 = 0 d. x2 + y2 + z2 – 11x - 25 = 0
    • 64. 62 Problem For which of the following function does the property hold y d2y : dx2 a. e-3x b. y = ex c. e-2x d. y = e2x
    • 65. 63 Problem the length of common chord of the circle x2 + y2 + 2x + 3y + 1 = 0 and x2 + y2 + 4x + 3y + 2 = 0 is : a. 2 2 b. 4 c. 2 d. 3 2
    • 66. 64 Problem The radical centre of the circles x2 + y2 = 1, x2 + y2 – 2y = 1 and x2 + y2 – 2x = 1 is : a. (1, 1) b. (0, 0) c. (1, 0) d. (0, 1)
    • 67. 65 Problem The natural numbers are grouped as follows 1, (2, 3), (4, 5, 6), (7, 8, 9, 10) ….. the 1st term of the 20th group is : a. 191 b. 302 c. 201 d. 56
    • 68. 66 Problem If the two pairs of lines x2 – 2mxy – y2 = 0 and x2 – 2nxy – y2 = 0 are such that one of them represents the bisector of the angles between the other, then : a. mn + 1 = 0 b. mn – 1 = 0 1 1 0 c. m n 1 1 0 d. m n
    • 69. 67 Problem Solution of the equation tan x + tan 2x + tan x . tan 2x = 1 will be : n a. x 3 12 b. x n 4 c. x n 4 x n d. 4
    • 70. 68 Problem At what point on the parabola y2 = 4x the normal makes equal angles with the axes ? a. (4, 3) b. (9, 6) c. (4, -4) d. (1, -2)
    • 71. 69 Problem The equation x3 + y3 – xy (x + y) + a2 (y - x) represents : a. Three straight lines b. A straight line and a rectangular hyperbola c. A circle and an ellipse d. A straight line and a ellipse
    • 72. 70 Problem 2 The eccentricity of an ellipse x y2 1 whose latusrectum is half of its a2 b2 major axis is : 1 a. 2 2 b. 3 3 c. 2 5 d. 2
    • 73. 71 Problem If cos , cos , cos are direction cosine of a line then value of sin2 sin2 sin2 is : a. 1 b. 2 c. 3 d. 4
    • 74. 72 Problem The curve y – exy + x = 0 has vertical tangent at the point : a. (1, 1) b. at no point c. (0, 1) d. (1, 0)
    • 75. 73 Problem x 2 y 3 z 1 The length of perpendicular from the point (3, 4, 5) on the line 2 5 3 is : a. 17 3 b. 17 17 c. 2 17 d. 5
    • 76. 74 Problem The area bounded by f (x) x2 , 0 x 1, g(x) x 2,1 x 2 and x-axis is : 3 a. 2 4 b. 3 8 c. 3 d. none of these
    • 77. 75 Problem The foot of the perpendicular from P ( , , ) on z-axis is : a. ( , 0, 0) b. (0, , 0) c. (0, 0, ) d. (0, 0, 0)
    • 78. 76 Problem In a parabola semi-latusrectum is the harmonic mean of the : a. Segment of a chord b. Segment of focal chord c. Segment of the directrix d. None of these
    • 79. 77 Problem The plane 2x – 2y + z + 12 = 0 touches the sphere x2 + y2 + z2 – 2x – 4y + 2z – 3 = 0 at the point : a. (1, 4, 2) b. (-1, 4, 2) c. (-1, 4, -2) d. (1, -4, - 2)
    • 80. 78 Problem If sin2 x. sin 3x is an identity in x where C0, C1, C2, …. Cn are constant and then the value of n is : a. 6 b. 17 c. 27 d. 16
    • 81. 79 Problem xf (a) af (x) If f’(a) = 2 and f(a) = 4, then lim equals : x x a a. 2a – 4 b. 4 – 2a c. 2a + 4 d. 4a – 2
    • 82. 80 Problem If y = cex/(x - a), then dy equals : dx a. a (x - a)2 ay b. - (x a)2 c. a2 (x - a)2 d. none of these
    • 83. 81 Problem If cos( ).sin( ) cos( ).cos( ) , then the value of cos .cos .cos is : a. cot b. cot cot( ) c. d. cot
    • 84. 82 Problem If f(x) = loga loga x the f’(x) is : loga e a. x loga x log ea b. x loga x loga a c. x x d. loge a
    • 85. 83 Problem The equation of the tangent to the curve y = 1 – ex/2 at the point of intersection with the y-axis is : a. x + 2y = 0 b. 2x + y = 0 c. x – y = 2 d. none of these
    • 86. 84 Problem The vectors 2ˆ i 3ˆ, 4ˆ j i ˆ and 5ˆ j i ˆ yj have their initial points at the origin. The value of y so that the vectors terminate on one straight line is : a. -1 1 b. 2 c. 0 d. 1
    • 87. 85 Problem Let f(x) = ex in [0, 1]. Then, the value of c of the mean value theorem is : a. 0.5 b. (e- 1) c. log (e - 1) d. none
    • 88. 86 Problem 1 1 1 If r, r1, r2, r3 have their usual meanings, the value of is : r1 r2 r3 a. 1 b. 0 1 c. r d. r
    • 89. 87 Problem If then x is equal to : a. 6 4 b. 3 5 c. 6 2 d. 3
    • 90. 88 Problem The distance between the foci of a hyperbola is 16 and its eccentricity is 2, then equation of hyperbola is : a. x2 + y2 = 32 b. x2 - y2 = 16 c. x2 + y2 = 16 d. x2 - y2 = 32
    • 91. 89 Problem 4R sin A . sin B . sin C is equal to : a. a + b + c b. (a + b + c)r c. (a + b + c)R r d. (a + b+ c) R
    • 92. 90 Problem The measure of dispersion is : a. Mean deviation b. Standard deviation c. Quartile deviation d. All a, b and c
    • 93. 91 Problem The circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 4x + 6y + 4 = 0 : a. Touch externally b. Touch internally c. Intersect at two points d. Do not intersect
    • 94. 92 Problem If x = my + c is a normal to the prabola x2 = 4ay, then value of c is : a. - 2am – am3 b. 2am + am3 2a a c. m m3 d. 2a a m m3
    • 95. 93 Problem A dice is tossed twice. The probability of having a number greater than 3 on each toss is 1 a. 4 1 b. 3 1 c. 2 d. 1
    • 96. 94 Problem If f(x) = 3x -1 + 3 - (x - 1) for real x, then the value of f(x) is : 2 a. 3 b. 2 c. 6 7 d. 9
    • 97. 95 Problem If a function f .[2, ] B defined by f (x) = x2 – 4x + 5 is a bijection, then B is equal to : a. R b. [1, ) c. [2, ) d. [5, )
    • 98. 96 Problem The minimum value of px + qy when xy = r2 is a. 2r pq b. 2pq r pq c. -2r d. none of these
    • 99. 97 Problem The area cut off from parabola y2 = px by the line y = px is : a. p3/3 1 b. 2 P2 1 c. 6p p d. 6
    • 100. 98 Problem The graph of y = loga x is reflection of the graph of y = ax in the line : a. y + x = 0 b. y - x = 0 c. ayx + 1 d. y – ax – 1 = 0
    • 101. 99 Problem Let Q+ be the set of all positive rational numbers. Let* be an operation on Q+ defined by ab a*b= a, b Q . Then, the identity element in Q+ for the operation * 2 is : a. 0 b. 1 c. 2 1 d. 2
    • 102. 100 Problem the complex number 1 2i lies in : 1 i a. I quadrant b. II quadrant c. III quadrant d. IV quadrant
    • 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET

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