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AMU - Mathematics - 2000

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    • 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2000
    • 2. SECTION – I CRITICAL REASONING SKILLS
    • 3. 01 Problem Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3,), (4, 4), (1, 2)} be a relation on A. Then R is a. Reflexive b. Symmetric c. Transitive d. None of these
    • 4. 02 Problem The equation of family of curve for which the length of the normal is equal to the radius vector is : a. y2 x2 = k2 b. y x=k c. y2 = kx d. none of these
    • 5. 03 Problem dy y = eax cos bx, dx equals : a. eax (a cos bx + b sin ax) b. eax (a cos bx - b sin ax) c. eax (a sin bx + b sin ax) d. eax (a sin ax - a cos ax)
    • 6. 04 Problem dy ax h The solution of represents a parabola when : dx by k a. a = 1, b = 2 b. a = 0, b = 0 c. a = 0, b 0 d. a = 2, b = 1
    • 7. 05 Problem 2 The equation x 2a a(1 ) where a is constant in the parametric 2 ,y 2 1 1 equation of the curves : a. x2 + y2 = - a2 b. x2 - y2 = a2 c. x2 + y2 = a2 d. x2 + y2 - 2a2 = 0
    • 8. 06 Problem Let z be the set of integers and 0 be binary operation of z defined as a 0 b =a + b - ab for all a, b z. The inverse of an element a( 1) z is : a a. a 1 b. 1 1 a c. a 1 a d. none of these
    • 9. 07 Problem Which term of the G.P. 2,2 2 , 4, ….. is 64 : a. 9th term b. 7th term c. 4th term d. 11th term
    • 10. 08 Problem If two events a and b such that a – b = 6, then the solution of m x a = b for m is : a. Unique b. Does not exist c. Exist when a b d. None of these
    • 11. 09 Problem x3 The value of sin x x is : lim 6 x 0 x5 a. 0 b. 1 60 c. 1 120 d. 1
    • 12. 10 Problem cos x equals : lim x 2 x 2 a. - 6 b. - 1 c. d. -
    • 13. 11 Problem kx 2 , if x 2 If (x) is continuous at x = 2, then the value of k : f (x) 3, if x 2 a. 2 b. 3 2 c. 3 3 d. 4
    • 14. 12 Problem f (m) f (n) For which of the following function m n is constant for all numbers m and n m n: a. f(x) = log x b. f (x) = cos x c. f(x) = 4x + 7 d. f(x) = x2 + 1
    • 15. 13 Problem If x and y are two unit vectors and is the angle between them, then |x y| 2 is equal to a. | sin | sin b. 2 c. | 2 sin | cot 2 d.
    • 16. 14 Problem          a.(b x c ) b.(a x c ) If a, b, c are non-coplanar vector, then       is equal to : (c x a).b c.(a x b) a. 0 b. 1 c. 2 d. 13
    • 17. 15 Problem the general solution of the equation, 3(sin cos ) (sin cos ) 2 is : a. 2n 4 3 b. 2n 4 12 c. 2n 4 3 d. 2n 6 12
    • 18. 16 Problem 1/2 x sin 1 x The value of the integral dx, is 0 2 1 x a. 1 3 2 2 1 b. 2 12 3 c. 1 3 2 12 1 3 d. 2 2
    • 19. 17 Problem Suppose that the velocity of a moving particle is = 30 – 2t m/sec. The total distance in metres it travels between the times t = 0 and t = 20 seconds is : a. 200 b. 225 c. 250 d. 275
    • 20. 18 Problem 9 2 The least value of the function f (x) 4x sin x is : x a. 10 -1 b. 11 -1 c. 12 -1 d. 14 -1
    • 21. 19 Problem The distance between the line 3x + 4y = 9 and 6x + 8y = 15, is : 3 a. 2 3 b. 10 c. 6 d. none of these
    • 22. 20 Problem The angel between the tangent from the point (4, 3) to the circle x2 + y2 –2x – 4y = 0 is : a. 300 b. 450 c. 600 d. 900
    • 23. 21 Problem The value of dx is : 2 4 3/4 x (x 1) 1/4 a. 1 1 c x4 b. (x4 + 1)1/4 + c 1/4 1 c. - 1 c x4 1/4 1 d. 1 c x4
    • 24. 22 Problem If standard deviation of a variate x is , then standard deviation of ax b c where a,b,c are constant is : a. a c c b. a 2 c. c a b d. c
    • 25. 23 Problem The value of the determinant x 1 x 2 x 4 is : x 3 x 5 x 8 x 7 x 10 x 14 a. - 2 b. x2 + 2 c. 2 d. 3
    • 26. 24 Problem Given 12 points in a plane, no three of which are collinear. Then number of line segments can be determined, are : a. 76 b. 66 c. 60 d. 80
    • 27. 25 Problem There are 10 true-false questions in a examination. Then these questions can be answered in : a. 100 ways b. 20 ways c. 512 ways d. 1024 ways
    • 28. 26 Problem 1 2 The value of ex dx lies in the interval : 0 a. [0, 1] b. [1, 2] c. [1, e] d. [1, 3]
    • 29. 27 Problem If 30Cn + 2 = 30Cn - 2, then n equals : a. 8 b. 15 c. 30 d. 32
    • 30. 28 Problem |x a | equals : lim x a x a a. 2 b. - 1 c. 1 d. 0
    • 31. 29 Problem If x, y, z are positive integers then (x + y) (y + z) (z + x) is : a. < 8xyz b. = 8xyz c. > 8xyz d. none of these
    • 32. 30 Problem The nth term of the series, 1 + 3 + 6 + 10 …….. is : n(n 1) a. 2 n 1 b. 2 c. n(n 1) 2 d. n 1 2
    • 33. 31 Problem If cos , cos , cos are direction cosines of line, then value of sin2 sin2 sin2 is : a. 1 b. 2 c. - 1 d. 3
    • 34. 32 Problem The line lx + my + n = 0 touches the circle x2 + y2 = 1 if : 1 a. l2 + m2 = n2 b. l2 + m2 = 2n2 n2 c. l2 + m2 = 2 d. l2 + m2 = n2
    • 35. 33 Problem The value of /2 ( tan x cot x )dx, is 0 a. 2 b. 2 c. 2 d. 2
    • 36. 34 Problem   If a 3i ˆ k, b ˆ i 2ˆ j are and joint sides of a parallelogram, then its area is : a. 1 17 2 1 b. 7 2 c. 41 1 d. 41 2
    • 37. 35 Problem Forces acting on a particle are represented in magnitude and direction by the       sides AB,BC ,CD, and DE , of regular pentagon ABCDE. The resultant of there forces is : a. EA b. AE c. AE 5 d. EA 5
    • 38. 36 Problem The value of a third order determined is 5, then this value of the square of the determinant formed by its co-factors will be : a. 125 b. 250 c. 25 d. 5
    • 39. 37 Problem Out of 40 consecutive integers, two are chosen at random, the probability that their sum is odd is : a. 14 29 21 b. 29 22 c. 39 20 d. 39
    • 40. 38 Problem an anti-aircraft gun takes a maximum of four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits the plane is : a. 0.2412 b. 0.21 c. 0.16 d. 0.6976
    • 41. 39 Problem The area enclosed by the curve y2 = x2 (1 – x2) is : 1 a. 3 sq. units b. 2 sq. units 3 c. 1 sq. units 4 d. 3 sq. units
    • 42. 40 Problem The value of cos 200 - 2 cot 200 is : a. 0 b. -1 c. 2 d. 3
    • 43. 41 Problem The function f(x) = x4 – 62x2 + ax + 9 attains its maximum value in the interval [0, 2] at x = 1. Then the value of a is : a. 120 b. - 120 c. 52 d. 102
    • 44. 42 Problem If , are the roots of the quadratic equation ax2 + bx + c = 0, then 2 2 2 equals : a. 0 bc b. a2 c. Abc c(a b) d. a2
    • 45. 43 Problem Two equals circle of radius r intersect such that each passes through the centre of the other. The length of the common chord is : a. 2 b. 2r c. 3 r d. 3
    • 46. 44 Problem The angle of intersection of the curves y = x2, 6y = 7- x3 at (1, 1) is : a. 4 b. 3 c. 2 d.
    • 47. 45 Problem The maximum value of sin x cos x in the interval 0, is attained 6 6 2 at : a. 12 b. 6 c. 3 d. 2
    • 48. 46 Problem Origin is a limiting point of a coaxial system of which x2 + y2 – 6x – 8y + 1 = 0 is a member. The other limiting point is : a. (- 2, - 4) 3 4 , b. 25 25 3 4 , c. 25 25 4 3 d. , 25 25
    • 49. 47 Problem A vector has constant magnitude but its direction varies with time. The derivative of such a vector is always : a. 0 b. perpendicular to itself c. parallel to itself d. a unit of vector
    • 50. 48 Problem 2 3 a b 1 a b 1 a b Sum of the series : ..... is : a 2 a 3 a a. log a – log b b. log (a - b) c. e(a - b)/a –1 d. e1 – b/a
    • 51. 49 Problem If one vertex of an equilateral triangle is at (2, - 1) and the base is x + y – 2 = 0, then the length of each side is : a. 3 2 b. 2 3 c. 2 3 d. 3 2
    • 52. 50 Problem The eccentricity of an ellipse whose pair of a conjugate diameter are y = x and 3y = -2x is a. 2 3 b. 1 3 1 c. 3 2 d. 3
    • 53. 51 Problem (8C1 + 8C2 + 8C3 - 8C4 + 8C5 - 8C6 + 8C7 ) equals : a. 0 b. 1 c. 70 d. 256
    • 54. 52 Problem The equation of the circle which has its centre at (a, b) and which touches the y- axis is : a. x2 + y2 = b2 b. (x - a)2 + (y - b)2 = b2 c. x2 + y2 = a2 d. (x - a)2 + (y - b)2 = a2
    • 55. 53 Problem The focus of the parabola y2 – 4y – 8x + 4 = 0 is : a. (1, 1) b. (1, 2) c. (2, 1) d. (0, 2)
    • 56. 54 Problem Two dice are tossed 6 times. Then the probability that 7 will show an exactly four of the tosses is : 225 a. 18442 116 b. 20003 125 c. 15552 117 d. 17442
    • 57. 55 Problem The standard deviation of 7, 9, 11, 13, 15 is : a. 2.82 b. 2.4 c. 2.7 d. 2.5
    • 58. 56 Problem x The period of the function f(x) = 2 sin 2 is : a. b. 2 c. 3 d. 6
    • 59. 57 Problem If A is a square matrix of order n x n and is scalar. Then Adj (A ) is equal to : a. (Adj.A) n b. (Adj.A) -n c. (Adj.A) n-1 d. none of these
    • 60. 58 Problem cos 1 x The domain of f (x) is : [x] a. [-1, 1] b. [-1, 0] c. [-1, 0] {1} d. [- 1, - 1]
    • 61. 59 Problem If z is a complex number, then arg z + arg ( z)is equal to : a. 0 b. 2 c. 2 d. 4
    • 62. 60 Problem If p, q are the roots of the equation. x2 + mx + m2 + a = 0, then p2 + pq + p2 + a will be equal to : a. 0 b. 1 c. - m d. m2 + a
    • 63. 61 Problem The co-ordinate of the centre of the sphere, 2x2 + 2y2 + 2z2 – 4x + 6y – 8z – 10 = 0 are : a. 3 ,1, 2 2 3 b. 1, ,2 2 3 c. 1,2, 2 3 d. ,2,1 2
    • 64. 62 Problem A point moves so that its distance from the x-axis half of its distance from the origin. The equation of its locus is ; a. x2 = 2y2 b. x2 = 3y2 c. x = 2y d. 2x = y
    • 65. 63 Problem x x2 x3 If x y z and y y2 y3 0 , then xyz is equal to : z z2 z3 a. 1 b. -1 c. 0 d. x + y + z
    • 66. 64 Problem On the set I, binary operation * is defined as follows : a*b = a + b + 1 Then identity element of the group (I, *) is : a. 1 b. -1 c. 0 d. 2
    • 67. 65 Problem If n is a positive integer, then (n + 1) (n + 2) (n + 3) ………..(2n) is a multiple of : a. 2n b. 2(n + 1) c. 2(n + 1) d. 2n
    • 68. 66 Problem P.I. of the differential equation (D2 – 4D + 3) y = ex, is : a. b. ex 1 c. 2 ex e 4e 3 d. 1 xe x 2
    • 69. 67 Problem x 1 y z 1 x 4 y z z 5 The value of for which the lines and , 2 3 4 3 3 are perpendicular is : a. 6 b. 1 6 c. - 6 1 d. - 6
    • 70. 68 Problem The area bounded by the parabola y = 2 - x2 and the line x + y = 0 is : 9 a. 2 7 b. 2 17 c. 6 34 d. 7
    • 71. 69 Problem If f (x) x (1 t) then f(x) is : log dt, 0 (1 t) a. An odd function b. A period function c. A symmetric function d. None of these
    • 72. 70 Problem The pedal equation of the curve r 2 a2 cos 2 is : a. p = ar3 b. a2p = r3 c. p2 = ar3 d. p = a2r3
    • 73. 71 Problem n px 1 5 If the 4th term in the binomial expansion of is , then : x 2 a. n = 8, p = 6 1 b. n = 8, p = 2 1 c. n = 6, p = 2 d. n = 6, p = 6
    • 74. 72 Problem If is the angle between the plane 4x – y – 12 = 1 and the line whose direction ratio’s are (1, -1, 1) then sin given by : a. 3 6 6 b. 3 3 c. 2 3 d. 6
    • 75. 73 Problem      A straight line r a b meets the plane r n 0 in P. The position vector of P is :    a n a. a  b b n   b.  a n a  b b n   c.  a n a  a b n    a n  d. a  a b n
    • 76. 74 Problem The arithmetic mean of a set of observations is . If each observation is divided by then is increased by 10, then the man of the new series is : a. x b. x 10 c. x 10 d. x 10
    • 77. 75 Problem The maximum area of rectangle inscribed in a circle of diameter R is : a. R2 R2 b. 2 R2 c. 4 R2 d. 8
    • 78. 76 Problem       Let holds a (ˆ i ˆ j ˆ) and b pk (ˆ i j ˆ ˆ k ) then| a b | | a | | b | for : a. p = - 1 b. p = 1 c. all real p d. no real p
    • 79. 77 Problem The equation whose roots are twice the roots of the equation, x2 – 3x + 3 = 0 is : a. 4x2 + 6x + 3 = 0 b. 2x2 - 3x + 3 = 0 c. x2 - 3x + 6 = 0 d. x2 - 6x + 12 = 0
    • 80. 78 Problem x 3 7 If (x + 9) = 0 is a factor of 2 x 2 = 0, then the other factor is : 7 6 x a. (x - 2) (x - 7) b. (x - 2) (x - a) c. (x + 9) (x - a) d. (x + 2) (x + a)
    • 81. 79 Problem If cos sin 2 cos , then cos sin is equal to : a. 2 sin b. 2 cos c. 2 tan d. 2 sec
    • 82. 80 Problem The total number of ways of selecting six coins out of 20 one rupee coins, 10 fifty paise coins an 7 twenty five paise coins is : a. 37C 6 b. 56 c. 28 d. 29
    • 83. 81 Problem The sum of the coefficients of the polynomial (1+x3x2)2143is : a. 1 b. -1 c. 0 d. 2
    • 84. 82 Problem The radius of the incircle triangle whose sides are 18, 24 and 30 cm is: a. 2cms b. 4cms c. 6cms d. 9cms
    • 85. 83 Problem The equations of tangent to the hyperbola 4x2-3y2=24 which make an angle of 600 with x-axis are: a. y 3x 10 b. y 10x 3 c. y 10x 3 y 3x 3 d.
    • 86. 84 Problem Suppose n people enter a chess tournament in which each person is to play one game against each of the others. The total number of games that will be played in the tournament is : n n 1 a. 2 n n 1 b. 2 c. n(n+1) d. n(n-1)
    • 87. 85 Problem If the sides of a triangle are 7cm, 4 3 cm and 13 cm, then the smallest angel of the triangle is : a. 150 b. 450 c. 300 d. 600
    • 88. 86 Problem A curve has the parametric equation x- t2 1 and y= b t 2 1 , then 2t 2t its equation in rectangular Cartesian co-ordinate is : x2 y2 a. a2 14 b2 b. x2+y2=a2b2 c. b2x2-a2y2=a2b2 d. none of these
    • 89. 87 Problem 1 4 20 The solution set of the equation 1 2 5 0 is : 1 2x 5x 2 a. {0,1} b. {1,2} c. {1,5} d. {2,-1}
    • 90. 88 Problem If a square matrix satisfies the relation A2+A-I=0 then A-1 a. Exists and equals I+A b. Exists and equals I-A c. Exists and equals A2 d. None of these
    • 91. 89 Problem   2   2 equals : axb b a a. 0   a.b b. 2 2 c. 2 a .b 2 2 d. a .b
    • 92. 90 Problem If x then the (r+1)th term the expansion of (1-x)2 is : a. (r+1)xr b. rxr-1 c. rx-r+1 d. (r+1)xr-1
    • 93. 91 Problem When m varies, the locus of the point of intersection of the straight lines x y x y 1 is : m and a b a b m a. A parabola b. A hyperbola c. An ellipse d. A circle
    • 94. 92 Problem 1 sin x cos x The differential coefficient of tan w.r.t x is : cos x sin x a. 0 1 b. 2 c. 1 d. 2
    • 95. 93 Problem The coefficient of correlation between x and y ; x : 65 66 67 67 69 70 72 y : 67 68 65 68 72 69 71 is given by : a. 0.5 b. 0.53 c. 0.6 d. 0.7
    • 96. 94 Problem the length of the subnormal at the point (1, 3) of the curve, y = x2 + x + 1 is : a. 1 b. 3 c. 9 d. 3 10
    • 97. 95 Problem The differential equation y dy x a (a is any constant) represents : dx a. A set of circles having centre on the y-axis b. A set of circles centre on the x-axis c. A set of ellipse d. None of these
    • 98. 96 Problem 1 2 3 4 The value of the infinte series ……. Is : 2.3 2.5 2.7 2.9 2 a. 3 b. 2e e c. 2 1 d. 2e
    • 99. 97 Problem Distance between the parallel planes 2x – y + 3z + 4 = 0 and 6x – 3y + 9z –3 = 0 is : 5 a. 3 4 b. 6 5 c. 14 3 d. 2 3
    • 100. 98 Problem Three numbers m + 2, 4m –6, 3m – 2 are in A.P. in m equals to : a. 3 b. 2 c. 1 d. 0
    • 101. 99 Problem The first derivative of the expression (xx + ax) is : a. xx log x + ax log e b. xx log x + ax log a c. xx log x - ax log a d. xx log x - ax log e
    • 102. 100 Problem Counters marked 1, 2, 3 are placed in a bag and one is drawn and replaced. The operation is repeated three times. The chance of obtaining a total of 6 is : 7 a. 27 20 b. 27 13 c. 27 14 d. 27
    • 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET