AIEEE 2010 SOLVED PAPER 26 A/2 Chandra Vihar, I.P. EXTENSION, New Delhi-92 Phone : 9711343889 Email : ambikanworld@yahoo.co.in, Web : www.ambikanworld.com, www.ambikanworld.in | Enmed Test Series | Knowledge Booster | Super Mathmind Olympiad | Super Scientist Examination | MATHEMATICS 61. Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w); m p S { , n q æ ö =ç ÷ è ø| m,n, p and q are integers such that n, q ¹ 0 and qm = pn}. Then (1) R is an equivalence relation but S is not an equivalence relation (2) Neither R nor S is an equivalence relation (3) S is an equivalence relation but R is not an equivalence relation (4) R and S both are equivalence relations 62. The number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals (1) 0 (2) 1 (3) 2 (4) ¥ 63. If a and b are the roots of the equation x2 – x + 1 = 0, then a2009 + b2009 = (1) –2 (2) –1 (3) 1 (4) 2 64. Consider the system of linear equations ; x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 3 3x1 + 5x2 + 2x3 = 1 The system has (1) infinite number of solutions (2) exactly 3 solutions (3) a unique solution (4) no solution 65. There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is (1) 3 (2) 36 (3) 66 (4) 108 66. Le f : (–1, 1) ® R be a differentiable function with f(0) = – 1 and f ¢ (0) = 1. Let g(x) = [f (2f(x) + 2)]2. Then g¢(0) = (1) 4 (2) –4 (3) 0 (4) –2 67. Let f : R ® R be a positive increasing function with x f(2x) lim f(x) ®¥ = (1) 1 (2) 23 (3) 32 (4) 3 68. Let p(x) be a function defined on R such that p¢(x) = p¢(1 – x), for all x Î [0, 1], p (0) = 1 and p (1) = 41. Then 10 p(x)dx ò equals (1) 41 (2) 21 (3) 41 (4) 42 69. A person is to count 4500 currency notes. Le an denote the number of notes he counts in the nth minute. If a1 = a2 = ... = a10 = 150 and a10, a11, ... are in an AP with common difference –2, then the time taken by him to count all notes is (1) 24 minutes (2) 34 minutes (3) 125 minutes (4) 135 minutes 70. The equation of the tangent to the curve 2 4 y x x = + , that is parallel to the x-axis, is (1) y = 0 (2) y = 1 (3) y = 2 (4) y = 32 26 A/2 Chandra Vihar, I.P. EXTENSION, New Delhi-92 Phone : 9873786724, 9711343889 Email : ambikanworld@yahoo.co.in, Web : www.ambikanworld.com, www.ambikanworld.in | Enmed Test Series | Knowledge Booster | Super Mathmind Olympiad | Super Scientist Examination | ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com 71. The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = 32p is (1) 4 2 2 - (2) 4 2 2 + (3) 4 2 1 - (4) 4 2 1 + 72. Solutin of the differential equation cosxdy y(sinx y)dx,0 x 2p = - < < is (1) sec x = (tan x + c) y (2) y sec x = tan x + c (3) y tan x = sec x + c (4) tan x = (sec x + c) y 73. Let ˆ ˆ a j k = - r and ˆ ˆ ˆ c i j k = - - r . then the vector b r satisfying a b c 0 ´ + = r r r r and a.b 3 = r r (1) ˆ ˆ ˆ i j 2k - + - (2) ˆ ˆ ˆ 2i j 2k - + (3) ˆ ˆ ˆ i j 2k - - (4) ˆ ˆ ˆ i j 2k + - 74. If the vectors ˆ ˆ ˆ a i j 2k = - + r , ˆ ˆ ˆ ˆ b 2i 4j k = + + and ˆ ˆ ˆ c i j k =l + +m r are mutually orthogonal, then ( , ) l m = (1) (–3, 2) (2) (2, –3) (3) (–2, 3) (4) (3, –2) 75. If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is (1) x = 1 (2) 2x + 1 = 0 (3) x = – 1 (4) 2x – 1 = 0 76. The line L given by x y 5 b + = 1 pass through the point (13, 32). The line K is parallel to L and has the equation x y 1 c 3 + = . Then the distance between L and K is (1) 23 15 (2) 17 (3) 17 15 (4) 23 17 77. A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle q with the positive z-axis, then q equals (1) 30° (2) 45° (3) 60° (4) 75° 78. Let s be a non-empty subset of R. Consider the following statement : P : There is a rational number xÎ S such that x > 0. Which of the following statements is the negation of the statement P? (1) There is a rational number x Î S such that x < 0. (2) There is no rational number x ÎS such than x < 0. (3) Every rational number x ÎS satisfies x < 0. (4) x Î S and x < 0 Þ x is not rational. 79. Let 4 cos( ) 5 a +b = and let sin 5 ( ) , 13 a-b = where 0 , 4p £ a b £ . Then tan 2a = (1) 25 16 (2) 56 33 (3) 19 12 (4) 20 7 80. The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if (1) – 85 < m < – 35 (2) – 35 < m < 15 (3) 15 < m < 65 (4) 35 < m < 85 81. For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is (1) 52 (2) 11 2 (3) 6 (4) 13 2 82. An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is (1) 13 (2) 27 (3) 1 21 (4) 2 233 26 A/2 Chandra Vihar, I.P. EXTENSION, New Delhi-92 Phone : 9873786724, 9711343889 Email : ambikanworld@yahoo.co.in, Web : www.ambikanworld.com, www.ambikanworld.in | Enmed Test Series | Knowledge Booster | Super Mathmind Olympiad | Super Scientist Examination | ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com 83. For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is (1) There is a regular polygon with r 1 R 2 = (2) There is a regular polygon with r 1 R 2 = (3) There is a regular polygon with r 2 R 3 = (4) There is a regular polygon with r 3 R 2 = 84. The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is (1) less than 4 (2) 5 (3) 6 (4) at least 7 85. Let f : R ® R be defined by {k 2x,if x 1 2x 3,if x 1 f(x) - £- + > - = If f has a local minimum at x = – 1 , then a possible value of k is (1) 1 (2) 0 (3) 12 - (4) –1 Directions : Questions number 86 to 90 are Assertion -Reason type questions. Each of these questions contains two statements. Statement -1: (Assertion )and Statement -2 : (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice. 86. Four numbers are chosen at random (without replacement)from the set {1, 2, 3, ...20}. Statement -1: The probability that the chosen numbers when arranged in some order will form an AP is 1 85 Statement -2 : If the four chosen numbers from an AP, then the set on all possible values of common difference is ( 1, 2, 3, 4, 5) ± ± ± ± ± . (1) Statement -1 is true, Statement -2 is true ; Statement -2 is a correct explanation for Statement -1. (2) Statement -1 is true, Statement -2 is true ; Statement -2 is not a correct explanation for Statement -1 (3) Statement -1 is true, Statment -2 is false (4) Statement -1 is false, Statment -2 is true. 87. Let 10 10 10 1 J 2 j j 1 j 1 10 S j(j 1) C ,S j C = = = - = å å and 10 210 3 j j 1 S j C = =å Statement -1 : S3 = 55 × 29. Statement -2: S1 = 90 × 28 and S2 = 10 × 28 . (1) Statement -1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1. (2) Statement -1 is true, Statement -2 is true ; Statement -2 is not a correct explanation for Statement -1. (3) Statement -1 is true Statement -2 is false. (4) Statement -1 is false, Statement -2 is true 88. Statement -1 : The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5. Statement -2: The plane x -y + z = 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).4 26 A/2 Chandra Vihar, I.P. EXTENSION, New Delhi-92 Phone : 9873786724, 9711343889 Email : ambikanworld@yahoo.co.in, Web : www.ambikanworld.com, www.ambikanworld.in | Enmed Test Series | Knowledge Booster | Super Mathmind Olympiad | Super Scientist Examination | ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com ambikanworld.com (1) Statement -1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1. (2) Statement -1 is true, Statement -2 is true ; Statement -2 is not a correct explanation for Statement -1. (3) Statement -1 is true, Statement -2 is false. (4) Statement -1 is false, Statement -2 is true . 89. Let f : R ® R be a continuous function defined by x x 1 f(x) e 2e- = + Statement -1 : f(c) = 1, 3 for some c ÎR. Statement -2 : 0 < f(x) 1 2 2 £ , for all c ÎR (1) Statement -1 is true, Statement -2 is true ; Statement -2 is a correct explanation for Statement -1. (2) Statement -1 is true, Statement -2 is true ; Statement -2 is not a correct explanation for Statement -1. (3) Statement -1 is true ,Statement -2 is false. (4) Statement -1 is false, Statement -2 is true. 90. Let A be a 2 × 2 matrix with non-zero entries and let A2 = I , where I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A. Statement -1 : Tr(A) = 0. Statement -2 : |A| = 1. (1) Statement -1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1. (2) Statement -1 is true, Statement -2 is true ; Statement -2 is not a correct explanation for Statement -1. (3) Statement -1 is true Statement -2 is false. (4) Statement -1 is false, Statement -2 is true.