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# AMU - Mathematics - 2006

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• 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2006
• 2. SECTION – I CRITICAL REASONING SKILLS
• 3. 01 Problem If 4a2 + 9b2 + 16c2 = 2 (3ab + 6bc + 4ca), where a, b, c are non-zero numbers, then a, b, c are in : a. AP b. GP c. HP d. None of these
• 4. 02 Problem 1 1 1 4 1 1 1 It is given that  4  4  ...  to   Then 4  4  4  ...  to  is equal to : 14 2 3 90 1 3 5 4 a. 96 4 b. 45 c. 89  4 90 d. none of these
• 5. 03 Problem The polynomial (ax2 + bx + c) (ax2 – dx - c), ac  0 , has : a. Four real roots b. At least two real roots c. At most two real roots d. No real roots
• 6. 04 Problem If (3 + i)z = (3 - i) z , then the complex number z is : a. a(3  i), a  R a b. (3  i) , a  R c. a(3  i), a  R d. a(3  i), a  R
• 7. 05 Problem If e  cos   i sin  , then for the ∆ ABC eiA eiB eiC is : i a. i b. 1 c. - 1 d. none of these
• 8. 06 Problem Numbers lying between 999 and 10000 that can be formed from the digits 0, 2, 3, 6, 7, 8 (repetition of digits not allowed) are : a. 100 b. 200 c. 300 d. 400
• 9. 07 Problem In a club election the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be 126, then the number of contestants is : a. 4 b. 5 c. 6 d. 7
• 10. 08 Problem 1 1 1 ……. is equal to :    1!(n  1) 3!(n  3)! 5!(n  5)! 2n 4 a. for even value of n only n! 2n4  1 b.  1 for odd values of n only n! c. 2n1 for all values of n n! d. none of these
• 11. 09 Problem + ………+ to  is equal to : 2 3 a  b 1a  b 1a  b     3 a  a 2 a    a. log ab a b. log b b c. log a d. none of these
• 12. 10 Problem 1 1 1   The sum of infinite terms of the series (1  a)(2  a) (2  a)(3  a) (3  a)(4  a)+ …..+ to  , where a is a constant, is : 1 a. 1 a b. 2 1 a c.  d. none of these
• 13. 11 Problem 21 .. 98. The value of log2 log3…log 10099 is equal to : log100 a. 0 b. 1 c. 2 d. 100!
• 14. 12 Problem The value of tan 200 + 2 tan 500 – tan 700 is : a. 1 b. 0 c. tan 50 d. none of these
• 15. 13 Problem A circular ring of radius 3 cm is suspended horizontally from a point 4 cm vertically above the centre by 4 string attached at equal intervals to its circumference. If the angles between two consecutive strings be θ , then cos θ is : 4 a. 5 4 b. 25 c. 16 25 d. none of these
• 16. 14 Problem The number of positive integral solutions of the equation y 3 tan1 x  cos1  sin1 is : 1  y2 10 a. One b. Two c. Zero d. None of these
• 17. 15 Problem sin(ax 2  bx  c) If α is a repeated root of ax2 + bx + c = 0, then lim is : x  (x   )2 a. 0 b. a c. b d. c
• 18. 16 Problem log(1  x3 ) is equal to : lim x 0 sin3 x a. 0 b. 1 c. 3 d. none of these
• 19. 17 Problem the domain of the function f(x) = loge(x – [x]) is : a. R b. R – Z c. (0 +  ) d. Z
• 20. 18 Problem If f(x + y, x - y) = xy, then the arithmetic mean of f(x, y) and f(y, x) is : a. x b. y c. 0 d. none of these
• 21. 19 Problem The equations of the three sides of a triangle are x = 2, y + 1 = 0 and x + 2y = 4. The coordinates of the circumecentre of the triangle are : a. (4, 0) b. (2, -1) c. (0, 4) d. (- 1, 2)
• 22. 20 Problem If the point (a, a) falls between the lines | x + y| = 4, then : a. | a | = 2 b. | a | = 3 c. | a | < 2 d. | a | < 3
• 23. 21 Problem The equation of the image of the pair of rays y = | x | by the line y = 1 is : a. y = | x | + 2 b. y = | x | - 2 c. y = | x | + 1 d. y = | x | - 1
• 24. 22 Problem Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is minimum, is : 5  a.  3 , 0    b.  1 , 0    3  c. (3, 0) d. (5, 0)
• 25. 23 Problem L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through : a. (1, 1) b. (2, 1) c. (1, 2) d. (2, 2)
• 26. 24 Problem C1 is a circle of radius 2 touching the x-axis and the y-axis. C2 is another circle of radius > 2 and touching the axes as well as the circle C1. Then the radius of C2 is : a. 6 – 4 √2 b. 6 + 4 √2 c. 6 – 4 √3 d. 6 + 4 √3
• 27. 25 Problem the locus of a point represented by x  a  t  1  y  a  t  1  is : 2 t    2 t    a. an ellipse b. a circle c. a pair of straight lines d. none of these
• 28. 26 Problem the locus of the centre of the circle for which one end of a diameter is (1, 1) while the other end is on the line x + y = 3, is : a. x + y =1 b. 2 (x - y) = 5 c. 2x + 2y = 5 d. none of these
• 29. 27 Problem The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola, is : a. A circle b. An ellipse c. A parabola d. A hyperbola
• 30. 28 Problem   x 1 y  2 z  3 x  4 y 1 The point of intersection of the lines   and  z is : 2 3 4 5 2 a. (0, 0, 0) b. (1, 1, 1) c. (-1, -1, -1) d. (1, 2, 3)
• 31. 29 Problem The equation of the plane which meets the axes in A, B, C such that the centroid of the triangle ABC is is given by : a. x + y + z = 1 b. x + y + z = 2 c. x + y + z =
• 32. 30 Problem The image of the point (5, 4, 6) in the plane x + y +2z – 15 = 0 is : a. (3, 2, 2) b. (2, 3, 2) c. (2, 2, 3) d. (-5, - 4,- 6 )
• 33. 31 Problem The radius of the circle x + 2y + 2z = 15, x2 + y2 + z2 – 2y – 4z = 11 is : a. 2 b. √7 c. 3 d. √5
• 34. 32 Problem A straight line which makes angle of 600 with each of y and z – axes, is inclined with x – axis at angle of : a. 300 b. 450 c. 600 d. 750
• 35. 33 Problem The value of 1 1 1 is : x x 2 x x 2 x x 2 (2  2 ) (3  3 ) (5  5 ) (2x  2 x )2 (3x  3 x )2 (5x  5 x )2 a. 0 b. 30x c. 30-x d. 1
• 36. 34 Problem 4 1 0 0 The rank of the matrix   is :  3 0 1 0 6 0 2 0   a. 4 b. 3 c. 2 d. 1
• 37. 35 Problem The values of a for which the system of equations ax + y + z = 0, x – ay + z = 0, x + y + z = 0 possesses non-zero solutions, are given by : a. 1, 2 b. 1, -1 c. 0 d. none of these
• 38. 36 Problem If A is skew symmetric matrix of order n and C is a column matrix of order n x 1, then CTAC is : a. An identity matrix of order n b. An identity matrix of order 1 c. A zero matrix of order 1 d. None of these
• 39. 37 Problem y z xz xy If yz zx yx = kxyz, then the value of k is : xy zx xy a. 2 b. 4 c. 6 d. 8
• 40. 38 Problem Let f(x) be a polynomial function of the second degree. If f(1) = f(1) and a1, a2, a3 are in A.P., then f’(a1), f’(a2),f’(a3) are in : a. A.P. b. G.P. c. H.P. d. None of these
• 41. 39 Problem The curve given by x + y = exy has a tangent parallel to the y-axis at the point : a. (0, 1) b. (1, 0) c. (1, 1) d. (- 1, -1)
• 42. 40 Problem Let f(x) = 1 + 2x2 + 22 x4 + …+ 210x20. Then f(x) has : a. More than one minimum b. Exactly one minimum c. At least one maximum d. None of these
• 43. 41 Problem 4x 2  1 The interval in which the function f (x)  is decreasing, is : x a. (-1, -1) b. (1, 1) c. (-1, 1) d. [- 1, 1]
• 44. 42 Problem A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume if its height h and radius r are related by : a. 2 h = r b. h = 4 r c. h = 2r d. h = r
• 45. 43 Problem If f(x) = x2 – 2x + 4 on [1, 5], then the value of a constant c such that is : a. 0 b. 1 c. 2 d. 3
• 46. 44 Problem 1  cos 4 x The value of k for which the function  , x 0 is continuous f (x)   8x 2  k x 0 at x = 0, is :  a. k = 0 b. k = 1 c. k = - 1 d. none of these
• 47. 45 Problem If f(x) = cos x cos 2x cos 4x cos 8x cos 16x, then   is : f   4 a. 2 b. 0 1 c. 2 d. 3 2
• 48. 46 Problem Let f (x)  x 2dx and f(0) = 0. Then f(1) is :  2 (1  x )(1  1  x )2 a. log (1 + 2 ) b.  log(1  2)  4  c. log(1  2)  4 d. none of these
• 49. 47 Problem 2 The value of 1 [f {g(x)}]1 {g(x)}g’(x)dx, where g(1) = g (2), is equal to : a. 1 b. 2 c. 0 d. none of these
• 50. 48 Problem x 1 If  0 f (t )dt  x   x t f (t ) dt , then the value of f(1) is : 1 a. 2 b. 0 c. 1 d. - 1 2
• 51. 49 Problem a na If f(x) = f(a + x) and  f (x)dx  k, then  f (x) dx is equal to : 0 0 a. n k b. (n - 1) k c. (n + 1)k d. 0
• 52. 50 Problem If  b x3dx  0 and if  b x2dx 2 then the values of a and b are respectively  a a 3 : a. 1, 1 b. -1, -1 c. 1, -1 d. - 1, 1
• 53. 51 Problem A vector has components 2a and 1 with respect to a rectangular Cartesian system. The axes are rotated through an angle about the origin in the anticlockwise direction. If the vector has components a + 1 and 1 with respect to the new system, then the values of a are : a. 1, - 1/3 b. 0 c. - 1 , 1/3 d. 1, -1
• 54. 52 Problem   Let i j ˆ  j ˆ a  ˆ  ˆ  k , c  ˆ  k , If b is a vector satisfying      a x b  c and a  b  3, then b is : 1 ˆ ˆ a. (5i  2ˆ  2k) j 3 1 ˆ ˆ b. (5i  2ˆ  2k ) j 3 ˆ j ˆ 3i  ˆ  k c. d. none of these
• 55. 53 Problem      Let OA  a, OB  10a  2b and OC  b , where O, A and C are non-collinear points. Let p denote the area of the quadrilateral OABC and q denote the area of the p parallelogram with OA and OC as adjacent sides. Then is equal to : q a. 4 b. 6   |a  b| c.  2| a|   |a  b| d.  2| a|
• 56. 54 Problem The value of x so that the four points A = {0, 2, 0}, B = (1, x, 0), C = (1, 2, 0) and D = (1, 2, 1) are coplanar, is : a. 0 b. 1 c. 2 d. 3
• 57. 55 Problem Constant forces i j ˆ i j ˆ j ˆ P1  ˆ  ˆ  k , P2  ˆ  ˆ  k and P3  ˆ  k act on a particle at a point a. The work done when the particle is displaced from the point A to B where i j ˆ i j ˆ A  4ˆ  ˆ  k and B  6ˆ  ˆ  3k is : a. 3 b. 9 c. 20 d. none of these
• 58. 56 Problem dy The solution of the differential equation x  y  x cos x  sin x, dx  , is : given that y = 1, when x = x  2 a. y = sin x – cos x b. y = cos x c. y = sin x d. y = sin x + cos x
• 59. 57 Problem From a point on the ground at a distance 70 feet from the foot of a vertical wall, a ball is thrown at an angle of 450 which just clears the top of the wall and afterwards strikes the ground at a distance 30 feet on the other side of the wall. The height of the wall is : a. 20 feet b. 21 feet c. 10 feet d. 105 feet
• 60. 58 Problem Three coplanar forces acting on a particle are in equilibrium. The angle between the first and the second is 600 and that between the second and the third is 1500. The ratio of the magnitude of the forces are : a. 1 : 1 : 3 b. 1 : 3: 1 c. 3: 1:1 d. 3: 3: 1
• 61. 59 Problem A particle having simultaneous velocities 3 m/s, 5 m/s and 7 m/s, is at rest. The angle between the first two velocities is : a. 300 b. 450 c. 600 d. 900
• 62. 60 Problem A cyclist is beginning to move with an acceleration of 1 m/s2 and a boy, who is 40 m behind the cyclist, starts running at 9 m/s to meet him. The boy will be able to meet the cyclist after : a. 6 sec b. 8 sec c. 9 sec d. 10 sec
• 63. 61 Problem Two bodies slide from rest down two smooth inclined planes commencing at the same point and terminating in the same horizontal plane. The ratio of the velocities attained if inclinations to the horizontal of the planes are 300 and 600 respectively, is : a. 3 :1 b. 2 : 3 c. 1 : 1 d. 1 : 2
• 64. 62 Problem A die is thrown 2n + 1 times. The probability that faces with even numbers show odd number of times, is : 2n  1 a. 4n  3 n b. 2n c. n 1 2n  1 d. none of these
• 65. 63 Problem The probability that exactly one of the independent events A and B occurs, is equal to : a. P (A) + P (B) + 2P (A B) b. P (A) + P (B) – P (A B) c. P(A’) + P (B’) = 2P (A’ B’) d. None of these
• 66. 64 Problem A bag contains 30 tickets, numbered from 1 to 30. Five tickets are drawn at random and arranged in the ascending order. The probability that the third number is 20, is : 20 a. C2 x 10C2 30 C5 19 C2 x 10C2 b. 30 C5 19 c. C2 x 11C2 30 C5 d. none of these
• 67. 65 Problem The probability that at least one of the events A and B occur is 0.6 If A and B occur simultaneously with probability 0.2, then P( A)  P(B) is : a. 0.4 b. 0.8 c. 1.2 d. 1.4
• 68. 66 Problem The relation of “congruence modulo” is :  a. Reflexive only b. Symmetric only c. Transitive only d. An equivalence relation
• 69. 67 Problem If flow values of switches x1, x2 and x3- are respectively 0, 0 and 1, then the flow value of the circuit s = (x’1.x’2.x3) + (x1.x’2.x’3) + (x’1.x2.x’3) is : a. 0 b. 1 c. 2 d. none of these
• 70. 68 Problem In a Boolean Algebra a v (a’ b) is equal to : a. a v b b. a  b c. a’ d. b’
• 71. 69 Problem The range of the function f (x)  ( x  1)(3  x ) is : a. [- 1, 1] b. (- 1, 1) c. (- 3, 3) d. (- 3, 1)
• 72. 70 Problem The coefficients of x in the quadratic equation x2 + bx + c = 0 was taken as 17 in place of 13, its roots were found to be – 2 and –15. The correct roots of the original equation are : a. - 10, - 3 b. - 9, - 4 c. - 8, - 5 d. - 7, - 6
• 73. FOR SOLUTIONS VISIT WWW.VASISTA.NET