AMU - Mathematics - 2001

• 1. AMU PAST PAPERSMATHEMATICS- UNSOLVED PAPER - 2001

2. SECTION I Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them iscorrect. Indicate you choice of the correct answer for each part in your answer-book bywriting the letter (a), (b), (c) or (d) whichever is appropriate 3. 01 Problem The area of the triangle formed by the complex number z, iz, z + iz in the argand diagram is :1 a. | z |22 b. | z |2 c. 2 | z |2 d. none of these 4. 02 Problem The points with position vectors 60 3, 40 8, ai 52are collinear ifijij j : a. a = 40 b. a = 20 c. a = - 40 d. a = 30 5. 03 Problem Which of the following is correct ? a.A BAA(A B)A A b. c. (AB ) AA d. (A B)A B 6. 04 Problem The locus of the point of intersection of lines x cos y sin a and x sin y cos b ( is a variable) : a. 2(x2 + y2) = a2 + b2 b. x2 y2 = a2 b2 c. x2 + y2 = a2 + b2 d. none of these 7. 05 Problem If l m m l 3 m3 n3 is :, then 3p p r p q3 r3 a. 1 b. lmn c. - 1 d. 0 8. 06 Problem The solution of 3x 2 2 = 2x 1 are : a. (2, 4) b. (1, 4) c. (3, 4) d. (1, 3) 9. 07 Problem The two circles x2 + y2 2x 2y 7 = 0 and (x2 + y2) 8x + 29y = 0 : a. Touch externally b. Touch internally c. Cut each orthogonally d. Do not cut each other 10. 08 Problem In a circle with centre O, AB and CD are two diameter perpendicular to each other. The length of the Chord AC is : a. 2AB 1 b. 2AB c. AB d. AB 11. 09 Problem The eccentricity of the conic x2 4y = 1 is : 3 a.22 b. 3 c. 2 55 d. 2 12. 10 Problemx2 y2 the value of m for which y = mx + 6 is a tangent to the hyperbola1 is 100 49 : a. 201717 b.20 3 c. 2020 d.3 13. 11 Problem the equation of an ellipse with one vertex at the point (3, 1) the nearer focus 2 at the point (1, 1) and e is : 3 a.(x 3)2 (y 1)2 13620(x 3)2(y 1)2 b.1 2036 (x 3)2(y1)2 c.136 20 (x 3)2 (y1)2 d.136 20 14. 12 Problem The line x y = 1 touches the hyperbola 3x2 4y2 = 12 at the point : a. (- 4, 1) b. (2, -3) c. (4, 3) d. (- 4, - 3) 15. 13 Problem A rational number lying between 3 and 5 is : a. 2.132 b. 1.413 c. 3.126 d. 2.561 16. 14 Problem One root of the equation 5x2 + 13x + m = 0 is reciprocal of the other if m equals : a. 0 b. 51 c. 6 d. 6 17. 15 Problem If the equation x2 + px + q = 0 has roots u and v where p,q are non-zero constants. Then :1 1 a. qx2 + px + 1 = has rootsandu b. (x - p) (x + q) = 0 has roots u2 and v2 c. x2 + p2x + q2 = 0 has roots u2 and v2 u u d. x2 + qx + p = 0 has rootsand u 18. 16 Problem The number of proper subset of {a, b, c} is : a. 3 b. 8 c. 6 d. 7 19. 17 Problem Fifteen coupons are numbered 1, 2, 3, ., 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is : 6 a.916 63 b.5 4 8 c.15 53 d.5 20. 18 Problem If A and B are square matrix of the same order such that AB = A and BA = B, then A and B are both : a. Singular b. Non-singular c. Idempotent d. Involutory 21. 19 Problema c b a2 c2 b2 Ifthen :b b a andb2 b2 a2c a c c2 a2 c2 a. = a2b2c2 b. = a2b2c2 c.= abc d. none of these 22. 20 Problem If p + q + r = 0 = a + b + c, then the value of the determinant pa qb rc is : qc ra pb rb pc qa a. 0 b. pa + qb + rc c. 1 d. none of these 23. 21 Problem If two angles of on triangle are 720 and 800 respectively and that of another triangle are 280 and 720 respectively, then the triangle are : a. Similar b. Congruent c. Obtuse angle d. Equal in area 24. 22 Problem x The range of f ( x)is : (1 x 2 )1 a. ,121, b. 2 1 1, c.2 2, d. 25. 23 Problem the period of the function cos 3x is : a. b. 2 c. 3 d. 4 26. 24 Problem A drawer contains 5 brown socks and 4 blue socks well mixed. A man reaches the drawer and pulls out 2 socks at random, What is the probability that they match ? a. 294 b. 95 c. 95 d. 8 27. 25 Problem11 the coefficient of x-3 in expansion of 1is :xx a. - 33 b. - 330 c. 330 d. 33 28. 26 Problem If are the roots of ax2 + bx + c = 0 and if , then a. a(b2 4ac) = 4c b. b2 4ac = a c. a(b2 + 4ac) d. b2 + 4ac = a 29. 27 Problem In 9th term of the series 1 + 5 + 13 + 29 .. is given by : a. 505 b. 515 c. 525 d. 171 30. 28 Problem xsin x If then f (x), lim f (x) : x cos2 x x a. 0 b. c. 1 d. none of these 31. 29 Problem x Let f(x) be defined for all x > 0 and by continuous. Let f(x) satisfy f = f(x) f(y) y for all x, y and f(o) = 1, Then :1f as x 0 a. x b. f(x) is bounded c. xf (x) 0 as x0 d. f(x) = in x 32. 30 Problem If m and n are any two odd numbers with n < m. The largest integer which divide all possible numbers of the type m2 = n2, is : a. 9 b. 8 c. 4 d. 6 33. 31 Problem If a < 0, then f(x) = eax + e-ax is decreasing for : a. x > 0 b. x < 0 c. x > 1 d. x < 1 34. 32 Problem the function f(x) = x2 log x is the interval [1, e] has : a. a point of maximum and minimum b. a point of maximum only c. no point of maximum and minimum [1, e] d. no point of maximum and minimum 35. 33 Problem The area of the circle exceeds the area of regular polygon of n sides of equal perimeter in the ratio of : a. cot :n n b. tan n:n c. cos n:n : d. sin n n 36. 34 Problem 1 If a, b, c an three unit vectors such that b is not parallel to c and a x(b x c ) b, 2 then the angle betweena and c: a.3 b. 2 c.6 d.4 37. 35 Problem If a and b are two non-collinear vectors and is a vector coplanar with a and b then : a. r a b b. r a b c. r manb d. r ab 38. 36 Problem If x [0, 5], then what is the probability that x 2 3x + 27 = 0 ? a. 451 b.52 c. 53 d. 5 39. 37 Problemb If f(a + b - x) = f(x), then xf (x) dx is equal to :aa b b a. xf (b x )dx2 aa b b b. f (x)dx2 aa b b c. f ( x )dx2 a d. none of these 40. 38 Problem If 1xthen :dx tan a b,1 sin x2 a. a ,b R4 b. 5a,bR 4 c. a ,b R4 d. none of these 41. 39 Problem 24 If A and B have change of solving problem in physics correctly are5 and 11 respectively. If the probability of their matching a common is . If the probability of1 their answer is 105 correct is :4 a.514 b. 2541 c. 2340 d. 41 42. 40 Problem A letter is known to have come either from LONDON or LEBANON on the postmark only the two consecutive letter ON are legible. The probability that it come from LONDON is : a. 121715 b. 1713 c. 1714 d. 17 43. 41 Problembc ca ab In a ABC ,is equal to :r1 r2 r3 a. 1 b. r1r2r3 c. 0 d. abc 44. 42 Problem 45 The value of cos , cos .cos .cos :3 777 a. 16 1 b. 1611 c. 1616 d. 11 45. 43 Problem The circles x2 + y2 10x + 16 = 0 and x2 + y2 = r2 intersect each other in two distinct if : a. r < 2 b. r > 8 c. 2 < r < 8 d. 2 r8 46. 44 Problem If two even A and B one such that P (A /B) = P(A), then a. B is impossible b. Vector only c. B is implied by A d. None of these 47. 45 Problem The line y = mx + 1 is tangent to the parabola y2 = 4x if m equal to : a. 1 b. 2 c. 3 d. 4 48. 46 Problem The angle between the pair of tagents drawn to the ellipse 3x2 + 2y2 = 5 from the point (1, 2) is :6 a. tan 151 12 b. tan 51 12 c. tan 51 12 d. tan 5 49. 47 Problem the radius of the circular section of the sphere by the plane r .( i j k) 3 3is a. 3 b. 4 c. 5 d. none of these 50. 48 Problem2c The integral|x c | dx equal :0 a. Cc b. 2 c. 4c d. 2c 51. 49 Problem tan2 A = 2 tan2 B + 1, then cos 2A equals : a. sin2 B b. - sin2 B c. tan2 B d. tan B 52. 50 Problem The value of 3 cosec 200 sec200 is equal to : 1. 2 2. 1 3. 4 4. -4 53. 51 Problem There are 6 positive and 8 negative numbers. Four numbers are chosen at random without replacement and multiplied the probability that the product is a positive numbers : a. 0.504 b. 0.503 c. 0.502 d. 0.501 54. 52 Problem If the line of regression of Y and x and x on y are respectively y = kx + 9 and x = 4y + 5, then : a. 0 k 41 b. 0 k41 c. k4 d. none of these 55. 53 Problem The correlation coefficient between the variable xi and yi positive, then the curve passing through (xi yi) : a. Collinear b. Straight line c. Slopes downwards through (xi, yi) d. Rising upwards to the right 56. 54 Problem The mean of n items is X . If the first item is increased by 1, second by 2 and so on, the new mean is : a. xX2 b. X +x n 1 c. X 2 d. none of these 57. 55 Problem AB is vertical pole. The end A is on the level ground. C is the middle point of AB. P is a point on the level ground. The portion BC subtends an angle B at P. If AP = n AB, then tan B is equal to : n a. 2n2 1 n b.n2 1 n c.2n1 d. none of these 58. 56 Problem If | k | = 5 and 003600 , then the number of different solutions of3cos 4 sin k is : a. Zero b. Two c. One d. Infinite 59. 57 Problem1 If y 2x 1for real x, then the least value of y is : 2x 1 a. 1 b. 2 c. - 1 d. - 2 60. 58 Problem If f(x) = (x - 1) (x - 3) (x - 4) (x - 6) + 19 for all real value of x is : a. Positive b. Negative c. Zero d. None of these 61. 59 Problem3 The equation of the line with gradient which 2is concurrent with the lines 4x + 3y 7 = 0 and 8x + 5y 1 = 0 is : a. 3x + 2y 2 = 0 b. 3x + 2y 63 = 0 c. 2y 3x 2 = 0 d. none of these 62. 60 Problem If X. A0, X. B 0 and X.C 0 for some non-zero vector X , then [A B C] is equal to : a. 2 b. 1 c. 0 d. - 2 63. 61 Problem The system of simultaneous equations kx + 2y z = 1, (k - 1)y 2z = 2 and (k + 2)z = 3 have a solution if k equals : a. 0 b. - 2 c. 1 d. - 1 64. 62 Problem A single letter is selected from the word ALIGARH MUSLIM UNIVERISTY the Probability that it is A vowel is : a. 11.16! 25! b. 160!161.10 c. 25!16.10! d.13! 65. 63 Problem1 2 The principal value of sin sinis : 32 a.32 b. -3 c. 34 d. 3 66. 64 Problem The imaginary part of log 1 ei is : a. b. - c. 2 d. - 2 67. 65 Problem If the second term of G.P. is 1 and the sun of its infinte number is 4 then its first term is : a. 3 b. 2 c. 1 d. 4 68. 66 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. ay = x + 1 d. y ax 1 = 0 69. 67 Problem In a binomial distribution, the mean is 4 and variances is 3. Then its mode is : a. 4 b. 5 c. 6 d. 7 70. 68 Problem If be a complex number and arg ( z ) then :4 a. Re(z) = Im(z) only b. Re(z) = Im(z) > 0 c. Re(z2) = Im(22) d. None of these 71. 69 Problem The number of number that can be formed with the digits 1, 2, 3, 4, 2, 1. so that odd digits always occupy the odd planes is : a. 36 b. 72 c. 144 d. 18 72. 70 Problem 1 1 1 If the product of the roots of the equation is 0, then sum x a x b x c of its root is :2bc a.b c2bc b. b cb c c. 2bcb c d. 2bc 73. 71 Problem The domain of f(x) = cos-1 (3x - 1) is : a.20, 3 b.2 2, 3 3 c.20, 3 d. none of these 74. 72 Problem If 2x2 + 4xy y2 12x 6y + 15 = 0, where x and y are real, then : a. x lie between 1 and 2 b. y lie between 1 and 2 c. x and y real numbers d. none of these 75. 73 Problem 1 3 3 If one end of a diameter of the sphere x2 + y2 + z2 2x 2z 2 = 0 is then 1 , , 2 2 2 other end is :1 1 1 a. 1,,2 2 21 3 3 b. 12,2,2 c. 1 1 11 ,,2 2 2 d. none of these 76. 74 Problem The function, f(x) = log (1 + ax) log (1 - bx) not defined at x = 0. The value which should be assigned to at x = 0 so that it is continuous at x = 0 is : a. a + b b. a b c. a d. b 77. 75 Problem The rectangular garden is of 20 x5 metre and a circular garden is of radius 5 metre, the ratio of areas of rectangular garden top circular garden is : a. 15 : b. 10 : c. 4 : d. 20 : 78. 76 Problem nC + nCr 1 + nCr 2 is equal to :r a. n + 1C r b. nC r+1 c. n - 1C r+1 d. none of these 79. 77 Problem The greatest possible number of points of intersection of 8 straight lines and 4 circles is : a. 32 b. 64 c. 76 d. 104 80. 78 Problem The value of a and b such that x(1 a cos x) b sin x are :lim1x 0 x3 a. (- 1, 1) b. (- 2, 2) c. (-3, 3) d. none of these 81. 79 Problem The length of the sub tangent to the curve x y 3 at the point (4, 1) is : a. 21 b.2 c. 3 d. 4 82. 80 Problem The angel between the planes 2x y + z = 6 and x + y + 2z = 7 is : a.6 b.4 c. 3 d. 2 83. 81 Problem A boat goes 70 km in 10 hours along the stream and returns back the same distance in 14 hours. Find the speed of boat : a. 10 km/hr b. 6 km/hr c. 7 km/hr d. 5 km/hr 84. 82 Problem 20 men complete one-third of a pieces of work in 20 days. How many more men should be employed to finish the rest of the work in 25 more days : a. 10 b. 12 c. 15 d. 20 85. 83 Problem A point on the curve y = x2 which is closets to the line 2x y 4 = 0 is : a. (0, 0) b. (1, 1) c. (2, 4) d. (3, 9) 86. 84 Problem The angles of pentagon an in A.P. one of the angles in degree must be :b a. 54 b. 72 c. 90 d. 108 87. 85 Problem If the function f(x) = x3 6x2 + ax + b defined on [1, 3], satisfies the Rolles theorem for2 3 1 , them : c3 a. a = 1, b = 6 b. a = -1, b = 6 c. a = 11, b R d. none of these 88. 86 Problem If y 1 / m [x 1 x 2 ] , then (1 + x2)y2 + xy1 is equal to : a. m2y b. my2 c. m2y2 d. none of these 89. 87 Problem If f : RR is given as f(x) = | x | and A = {x R/X > 0}then f-1 (A) equal to a. R b. R- {0} c. d. A 90. 88 Problem Four faces of a regular tetrahedron are to be painted with different colors the number of way in which this can be down is: a. 1 b. 2! c. 3! d. 4! 91. 89 Problem 21 a a If the (r + 1)th term in the expansion of 3contains a and b to b 3 b one and the same power, then the value of r is : a. 9 b. 10 c. 8 d. 6 92. 90 Problem the equation xy = 1 represents : a. straight line b. pair of straight line c. an ellipse d. a hyperbola 93. 91 Problem The vertex of the parabola (x - 4)2 + 2y = 9 is : a. (2, 8) b. (7, 2) 94, c.2 94, d.2 94. 92 Problem The value of 2.357 is :2355 a.10012355 b.9992355 c. 1111 d. none of these 95. 93 Problem After interesting x Ams between 2 and 38, the sum of the resulting progression is 200. The value of x is : a. 10 b. 8 c. 9 d. none of these 96. 94 Problem(2 x) (y 2) The equation 3, if x = -2, then y equal to : 2 2 a. - 2 b. - 7 c. 7 d. 5 97. 95 Problem The graph of a linear equation our the set of real is : a. A point b. Circle c. A curve d. Straight line 98. 96 Problem If x, y, z are real and distinct, then x2 + 4y2 + 9z2 6yz 3zx 2xy is always : a. Non-negative b. Non-positive c. Zero d. None of these 99. 97 Problem If each of third order determinant of value is multiplied by 4, then value of the new determinant is : a. b. 21 c. 64 d. 128 100. 98 Problem The number of generators of an infinite cyclic group is : a. 1 b. 2 c. infinite d. none of these 101. 99 Problem The radius of a sphere initially at zero increases at the rate of 5 cm/sec. Then its volume after 1 sec. Is increasing at the rate of : a. 50 b. 5 c. 500 d. none of these 102. 100 Problem 1 (x x 3 )1 / 3The value of dx is : 1/3x4a. 6b. 3c. 2d. 2/3 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET