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# AMU - Mathematics - 1999

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### AMU - Mathematics - 1999

1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 1999
2. 2. SECTION – I CRITICAL REASONING SKILLS
3. 3. 01 Problem If A and B are non-zero square matrices of the same order such that AB = 0, then : a. Adj A = 0 or adj B = 0 b. | A | = 0 or | B | = 0 c. adj A = 0 and adj B = 0 d. | A | = 0 and | B | = 0
4. 4. 02 Problem 3 3 4 If A 2 3 4 , then A-1 equal to : 0 1 1 a. A b. A2 c. A3 d. A4
5. 5. 03 Problem If A, B, C are square matrices of the same order, then which of the following is true ? a. AB = AC b. (AB)2 = A2B2 c. AB = 0 A = 0 or B = 0 d. AB = I AB = BA
6. 6. 04 Problem The value of is a. 0 b. abc c. 4a2b2c2 d. none of these
7. 7. 05 Problem A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other narrating the same incident ? a. 35% b. 45% c. 15% d. 5%
8. 8. 06 Problem The matrix (a1x1+a2x2+a3x3) is of order : a. 1 x 3 b. 1 x 1 c. 2 x 1 d. 1 x 2
9. 9. 07 Problem Which of the following correct for A – B a. A B b. A’ B c. A B’ d. A’ B’
10. 10. 08 Problem If S denotes the sum to infinity and Sn the sum of n tersm of the series 1 1 1 1 1 ......, such that S Sn , then the least value of n is : 2 4 4 1000 a. 8 b. 9 c. 10 d. 11
11. 11. 09 Problem The series ( 2 + 1), 1, ( 2 -1)…. is in : a. A.P. b. G.P. c. H.P. d. None of these
12. 12. 10 Problem 2 sin2 3x is equal to : lim x 0 x2 a. 12 b. 18 c. 0 d. 6
13. 13. 11 Problem sin m2 is equal to : lim 0 a. 0 b. 1 c. m d. m2
14. 14. 12 Problem Let f(x + y) = f(x) + f(y) and f(x) = x2g(x) for all x, y R, where g(x) is continuous function. Then f’(x) is equal to : a. g(x) b. g(0) c. g(0) + g’(x) d. 0
15. 15. 13 Problem 1 1 1 1 The value of is equal to : r2 r12 r22 r33 a2 b2 c2 a. a2 b2 c2 b. 2 a2 b2 c2 c. 3 a2 b2 c2 d.
16. 16. 14 Problem The function x2 1; x 1 f (x) x 1; x 1 2; x 1 a. Continuous for all x b. Discontinuous at x = -1 c. Discontinuous for all x d. Continuous x = -1
17. 17. 15 Problem In the expansion of (1+ x)n, then binomial coefficients of three consecutive terms are respectively 220, 495 and 792. The value of n is : a. 10 b. 11 c. 12 d. 13
18. 18. 16 Problem The number of roots of the quadratic equation 8 sec - sec + 1 = 0 is : a. Infinite b. 2 c. 1 d. 0
19. 19. 17 Problem If 12Pr = 11P6 + 6. 11P5 then r is equal to : a. 6 b. 5 c. 7 d. none of these
20. 20. 18 Problem 1 The value of the expression ( 3 sin 750 cos 750 ) is : 2 a. 1 b. 2 c. 2 d. 2 2
21. 21. 19 Problem the number of numbers consisting of four different digits that can be formed with the digits 0, 1, 2, 3 is : a. 16 b. 24 c. 30 d. 72
22. 22. 20 Problem For the curve y = xex, the point : a. x = -1 is a point of local minimum b. x = 0 is a point of maximum c. x = -1 is a point of maximum d. x = 0 is a point of maximum
23. 23. 21 Problem the function y = x – cot-1 x – log (x x2 1) is increasing on : a. (- , 0) b. ( , 0) c. (0, ) d. (- , )
24. 24. 22 Problem If x denotes displacement in time t and x = a cos t, then acceleration is given by : a. - a sin t b. a sin t c. a cos t d. - a cos t
25. 25. 23 Problem Let f differentiable for all x. If f (1) = - 2 and f’(x) 2 for all x [1, 6], 2 for all x [1, 6], then : a. f(6) < 8 b. f(6) 8 c. f(6) 5 d. f(6) 5
26. 26. 24 Problem 0 1 The matrix 1 0 is the matrix of reflection in the line : a. x = 1 b. y = 1 c. x = y d. x + y = 1
27. 27. 25 Problem Let A and B be two matrices then (AB)’ equals : a. A’B’ b. A’B c. - AB d. 1
28. 28. 26 Problem If at any point on a curve the subtangent and subnormal are equal, then the tangent is equal to : a. Ordinate b. 2 ordinate c. 2(ordinate) d. none of these
29. 29. 27 Problem If f(x) = (x + 1) tan-1 (e-2x), then f’(0) is : a. 1 2 b. 1 4 c. 5 6 d. none of these
30. 30. 28 Problem dy If y = x log x, then dx is : a. 1 + log x b. log x c. 1 – log x d. 1
31. 31. 29 Problem If xy + yz + zx = 1, then : a. tan-1 x + tan-1 y + tan-1 z = 0 b. tan-1 x + tan-1 y + tan-1 z = c. tan-1 x + tan-1 y + tan-1 z = 4 d. tan-1 x + tan-1 y + tan-1 z = 2
32. 32. 30 Problem The order of the differential equation whose solution is : y = a cos x + b sin x + ce- x is : a. 3 b. 2 c. 1 d. none of these
33. 33. 31 Problem If y = a cos px + b sin px, then : d2y a. dx 2 + p2y = 0 d2y b. dx 2 - p2y = 0 d2y c. dx 2 + py2 = 0 d2y d. dx 2 - py = 0
34. 34. 32 Problem 1/2 1 x cos x log dx is equal to : 1/2 1 x 1 a. 2 1 b. - 2 c. 0 d. none of these
35. 35. 33 Problem dx equals : 3 x 1 x 1 log( 1 x3 ) c a. 3 1 1 x3 1 log c b. 3 1 x 3 1 2 1 log c c. 3 1 x 3 2 1 x3 1 d. log c 3 1 x3 1
36. 36. 34 Problem ex (sin h x + cos h x) dx equal to : a. ex sec h x + c b. ex cos h x + c c. sin h 2x + c d. cos h 2x + c
37. 37. 35 Problem A man can row 4.5 km/hr in still water and he finds that it takes him twice as long to row up as to row down the river. The rate of the stream is : a. 1.5 km/hr b. 2 km/hr c. 2.25 km/hr d. 1.75 km/hr
38. 38. 36 Problem 3 4 10 If m n , then : 4 3 11 a. m = - 2, n = 1 b. m = 22, n = 1 c. m = - 2, n = -23 d. m = 9, n = -10
39. 39. 37 Problem The area between the curve y = 2x4 – x2, the x-axis and the ordinates of two minima of the curve is : a. 7 120 9 b. 120 11 c. 120 15 d. 120
40. 40. 38 Problem If each of the variable in the matrix a b is doubled, then the value of the c d determinant of the matrix is : a. Not changed b. Doubled c. Multiplied by 4 d. Multiplied by 8
41. 41. 39 Problem A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals : a. 1 32 1 b. 2 3 c. 2 1 d. 5
42. 42. 40 Problem The reciprocal of the mean of the reciprocals of n observations is the a. G.M b. H.M c. Median d. Average
43. 43. 41 Problem If the area bounded by the parabola x2 = 4y, the x-axis and the line x = 4 is divided into two equal area by the line x = , then the value of is : a. 21/3 b. 22/3 c. 24/3 d. 25/3
44. 44. 42 Problem         (a 2b c ) {(a b x (a b c )} is equal to : a.   [abc ]   b. 2 [abc ]   c. 3 [abc ] d. 0
45. 45. 43 Problem The unit vector perpendicular to the plane determined by A (1, -1, 2), B (2, 0, -1) and R (0, 2, 1) is : 1 i j ˆ (2ˆ ˆ k ) a. 6 1 ˆ b. (2ˆ i ˆ j k) 3 1 ˆ c. (2ˆ i ˆ j k) 32 d. none of these
46. 46. 44 Problem The probability of occurance of an even A is 0.3 and that of occurance of an event B is 0.4. If A and B are mutually exclusive, then the probability that neither occurs nor B occurs is : a. 0.2 b. 0.35 c. 0.3 d. none of these
47. 47. 45 Problem the probability that a man who is x years old will die in a year in P. Then amongst n persons A1, A2,…., An each x years old now, the probability that A1 will die in one year is 1 a. n2 b. 1 – (1 - P)n 1 c. n2 [1 – (1 - P)n] 1 d. n2 [1 – (1 - P)n]
48. 48. 46 Problem    the vector a x (b x c )is :  a. parallel to a  b. perpendicular to a c. parallel to d. perpendicular to
49. 49. 47 Problem the next term of the series 3 + 7 + 13 + 21 + 31 + …. a. 43 b. 45 c. 51 d. 64
50. 50. 48 Problem If log3 2, log3 (2x - 5) and log3 7 are in A.P., then x is equal to: 2x 2 a. 2 b. 3 c. 4 d. 2, 3
51. 51. 49 Problem If the radius of a spherical balloon increases by 0.2%. Find the percentage increase in its volume : a. 0.8% b. 0.12% c. 0.6% d. 0.3%
52. 52. 50 Problem 3 5 6 x 10 5 If 7 8 9 , then 5 3 6 equal to : 10 x 5 8 7 9 a. b. - c. x d. 0
53. 53. 51 Problem 1 1 5 The positive value of sin sin is : 2 3 a. 5 6 3 b. 5 2 c. 5 2 d. 5
54. 54. 52 Problem three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is : a. 2 - 3 b. 2 + 3 c. 3 -2 d. 3 + 3
55. 55. 53 Problem The nth term of the series 1 (1 2) (1 2 3) ….. is equal to : 2 3 a. n2 (n -1) (n 1)(2n 1) b. 2 n 1 c. 2 n(n 1) d. 2
56. 56. 54 Problem Two finite sets have m and n element. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The value of m and n are : a. m = 7, n = 6 b. m = 6, n = 3 c. m = 5, n = 1 d. m = 8, n = 7
57. 57. 55 Problem 1 The domain of f ( x) 1 x2 is : 2x 1 1 a. ,1 2 b. [- 1, [ c. [1, [ d. none of these
58. 58. 56 Problem sec2 (log x) The value of dx is : x a. tan (log x) + c b. tan x + c c. log (tan x) + c d. none of these
59. 59. 57 Problem The period of f(x) = cos (x2) is : a. 2 b. 4 2 2 c. 4 d. none of these
60. 60. 58 Problem The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is : 1 1 a. , 2 2 1 1 b. , 3 3 c. (0,0) 1 1 , d. 4 4
61. 61. 59 Problem The is acute angle and 4 x 2 sin2 1 = x, then tan is : 2 a. x2 1 b. x2 1 c. x 2 d. none of these
62. 62. 60 Problem The equation of the locus of a point whose abscissa and ordinate are always equal is : a. y + x = 0 b. y – x = 0 c. y + x – 1 = 0 d. y – x + 1 = 0
63. 63. 61 Problem The distance between the parallel lines y = 2x + 4 and 6x = 3y + 5 is : a. 17/ 3 b. 1 c. 3/ 5 d. 17 5 /15
64. 64. 62 Problem The equation y2 – x2 + 2x – 1 = 0, represents : a. A pair of straight lines b. A circle c. A parabola d. An ellipse
65. 65. 63 Problem The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 on x-axs and y- axis are respectively: a. 5,15 b. 6,15 c. 9,15 d. none of thes
66. 66. 64 Problem The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 which are perpendicular to 3x – 4y –1 = 0 are : a. 3x + 4y = 3, 3x + 4y + 25 = 0 b. 4x + 3y = 5, 3x + 4y - 25 = 0 c. 3x - 4y = 5, 3x - 4y + 25 = 0 d. none of these
67. 67. 65 Problem three identical dice are rolled. The probability that the same number will appear on each of them as : a. 1 6 1 b. 18 1 c. 9 1 d. 36
68. 68. 66 Problem 3 The principal value of sin is : 2 a. - 6 b. 6 2 c. - 3 2 d. 3
69. 69. 67 Problem 1 3 cos x 4 sin x dy If y cos then equals : 5 dx 1 a. 1 x3 b. 1 1 c. 1 x3 d. - 1
70. 70. 68 Problem The point on y2 = 4ax nearest to the focus has its abscissa equal to : a. a b. - a a c. 2 d. 0
71. 71. 69 Problem The vertex of the parabola x2 + 8x + 12y + 4 = 0 is : a. (- 4, 1) b. (4, - 1) c. (- 4, -1) d. (4, 1)
72. 72. 70 Problem The standard deviation for the data : 7, 9, 11, 13, 15 is : a. 2.4 b. 2.5 c. 2.7 d. 2.8
73. 73. 71 Problem While dividing each entry in a data by a non-zero number a, the arithmetic mean of the new data : a. Is multiplied by a b. Does not change c. Is divided by a d. Is diminished by a
74. 74. 72 Problem Two circles which passes through the points A (0, a) and B (0, -a) an touch the line y = mx + c will cut orthogonally if : a. c = a 2 m2 b. a = a 2 m2 c. m2 = a2 (1+ c2) d. m = - a 1 c2
75. 75. 73 Problem 2 2 If , are the roots of ax2 + bx + c = 0, then equals : a. c(a b) a2 b. 0 bc c. a2 d. abc
76. 76. 74 Problem The maximum value of 5 sin 3 sin 3 is : 3 a. 11 b. 10 c. 9 d. 12
77. 77. 75 Problem If x= y = 15, x2 = y2 = 49 xy = 44 and x = 5, then byx is equal to: a. 1 3 2 b. 3 1 c. 4 1 d. 2
78. 78. 76 Problem The number of terms which are free from radical sings in the expansion of (x1/5 + y1/10)55 is : a. 5 b. 6 c. 11 d. 9
79. 79. 77 Problem The sum of the co-efficient in the expansion of (x + 2y + x)10 is : a. 10C x+y b. x+yC 10 c. 26.4Cx d. none of these
80. 80. 78 Problem There are 10 points in a plane, out of which 4 points are collinear. The number of triangles formed with vertices as there points is : a. 20 b. 120 c. 40 d. 116
81. 81. 79 Problem If the co-ordinate of the centroid of a triangle are (3, 2) and co-ordinates of two vertices are (4, 1) and (2, 5), then co-ordinates to the third vertex are : a. (6, 8) b. (2, 8/3) c. (0, - 4) d. (6, 0)
82. 82. 80 Problem the argument of 1 i 3 is : 1 i 3 4 a. 3 2 b. 3 7 c. 6 d. 3
83. 83. 81 Problem In how many ways can a constant and a vowel be chosen out of the word COURAGE ? a. 7C 2 b. 7P 2 c. 4P x 3P1 1 d. 4P x 3P1 1
84. 84. 82 Problem The length of the latusrectum of the ellipse 5x2 + 9y2 = 45 is : a. 5 3 b. 10 3 c. 2 5 5 5 d. 3
85. 85. 83 Problem The projections of a line segment on the coordinate axes are 12, 4, 3. The direction cosine of the line are : 12 4 3 a. , , 13 13 13 12 4 3 b. , , 13 13 13 12 4 3 c. , , 13 13 13 d. None of these
86. 86. 84 Problem n The least positive value of n if i(1 3) is positive integer, is : 1 i2 a. 1 b. 2 c. 3 d. 4
87. 87. 85 Problem lim sec loge (2x ) is equal to : x 1 4x 2 a. 0 b. 2 2 c. 4 d. 2
88. 88. 86 Problem The distance between the planes gives by ,  ˆ  ˆ r .(ˆ i 2ˆ j 2k ) 5 0 and r .(ˆ i 2ˆ j 2k ) 8 0 is : a. 1 unit 13 b. 3 units c. 13 units d. none of these
89. 89. 87 Problem If the coefficient of correlation between X and Y is 0.28, covariance between X and Y is 7.6 and the variance X is 9, then the standard deviation of Y series is : a. 9.8 b. 10.1 c. 9.05 d. 10.05
90. 90. 88 Problem 1 1 3 If sin tan , then equals : 4 3 a. 5 b. 1 2 c. 5 3 d. 4
91. 91. 89 Problem (x 1) 2 If 4 , then the value of x 1 is : x x 2 a. 4 b. 10 c. 16 d. 18
92. 92. 90 Problem (x 1) x3 1 If 2 cos , then equals : x x3 1 cos 3 a. 2 b. 2 cos c. cos3 1 cos 3 d. 3
93. 93. 91 Problem The mode of the given distribution is : Weight (in kg) 40 43 46 49 52 55 Number of children 5 8 16 9 7 3 a. 40 b. 55 c. 49 d. 46
94. 94. 92 Problem The factors of x a b are : a x b a b x a. x – a, x – b and x + a + b b. x + a, x + b and x + a + b c. x + a, x + b and x - a - b d. x – a, x – b and x - a - b
95. 95. 93 Problem 7 1 The equation of a curve passing through 2, and having gradient 1 at (x, y) 2 x2 is : a. y = x2 + x + 1 b. xy = x2 + x + 1 c. xy = x + 1 d. none of these
96. 96. 94 Problem The general value of x satisfying is given by cos x = 3 (1 – sin x ) : a. x n 2 b. x n x x m ( 1)n c. 3 6 x n d. 3
97. 97. 95 Problem The angle of elevation of the tops of two vertical tower as seen from the middle point of the line joining the foot of the towers are 600 and 300 respectively. The ratio of the height of the tower is : a. 1 : 2 b. 2 : 4 c. 4 : 2 d. 2 : 1
98. 98. 96 Problem If an angle is divided into two parts A and B such that A – B = x and tan A : tan B = k : 1, then the value of sin x : a. k 1 sin k 1 k sin b. k 1 k 1 sin c. k 1 d. None of this
99. 99. 97 Problem In triangle ABC and DEF, AB = DE, AC = EF and A 2 E . Two triangles will have the same area if angle A is equal to : a. 3 b. 2 2 c. 3 5 d. 6
100. 100. 98 Problem the even function is : a. f(x) = x2 (x2 + 1) b. f(x) = x (x + 1) c. f(x) = tan x + c d. f(x) = sin2 x + 2
101. 101. 99 Problem The middle term in the expansion of (1 + x)2n will be : a. (n + 1)th b. (n - 1)th c. nth d. (n + 2)th
102. 102. 100 Problem For the equation | x |2 | | x | - 6 = 0 a. There is only one root b. There are only two distinct roots c. There are only three distinct roots d. There are four distinct roots
103. 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET