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# AMU - Mathematics - 1997

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• ### AMU - Mathematics - 1997

1. 1. AMU –PAST PAPERSMATHEMATICS - SOLVED PAPER - 1997
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 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. 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. 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. 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. 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. 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. 9. 07 Problem If r is the correlation coefficient, then a. r 1 b. r 1 c. |r| 1 d. |r| 1
10. 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. 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. 12. 10 Problem find the median of 18, 35, 10, 42, 21 : a. 20 b. 19 c. 21 d. 22
13. 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. 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. 15. 13 Problem n The value of n pr r 1 r! a. 2n b. 2n – 1 c. 2n –1 d. 2n + 1
16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 30. 28 Problem The modulus of 2i 2i is : a. 2 b. 2 c. 0 d. 2 2
31. 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. 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. 33. 31 Problem 1 is equal to : sin cos 6 a. 2 b. 6 c. 3 3 d. 2
34. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 44. 42 Problem x 1 equals to : lim 1 x x a. e b. e-2 c. e-1 d. e2
45. 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. 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. 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. 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. 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. 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. 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. 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. 53. 51 Problem n The sum of series S (n n)! is : n 0 a. - e2 1 b. e c. e2 d. e
54. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 89. 87 Problem If then x is equal to : a. 6 4 b. 3 5 c. 6 2 d. 3
90. 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. 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. 92. 90 Problem The measure of dispersion is : a. Mean deviation b. Standard deviation c. Quartile deviation d. All a, b and c
93. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET