Practice Questions

Q86.The vector →a and →b are not perpendicular and →c and →d are two vectors satisfying: →b × →c = →b × →d and →a ⋅→d = 0 . Then the vector →d is equal to →a⋅→c →b⋅→c (1) →c + (2) →b + ( →a⋅→b )→b ( →a⋅→b )→c →a⋅→c →b⋅→c (3) →c (4) →b −( →a⋅→b )→b −( →a⋅→b )→c z−3 , then λ equals and the plane x + 2y + 3z = 4 is cos−1 λ

Q70.Let f : R →R be a positive increasing function with limx→∞ f(3x)f(x) = 1. Then limx→∞ f(2x)f(x) (1) 2 (2) 3 3 2 (3) 3 (4) 1

Q75.The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0 , is (1) 5 (2) 6 (3) at least 7 (4) less than 4

Q89.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 . Statement-2: If the 85 four chosen numbers from an AP, then the set of all possible values of common difference is {±1, ±2, ±3, ±4, ±5}. (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is false Statement-2 is not the correct explanation for Statement-1 (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Q68.If P and Q are the points of intersection of the circles x2 + y2 + 3x + 7y + 2p −5 = 0 and x2 + y2 + 2x + 2y −p2 = 0, then there is a circle passing through P, Q and (1, 1) for (1) all values of p (2) all except one value of p (3) all except two values of p (4) exactly one value of p

Q75. a a + 1 a −1 a + 1 b + 1 c −1 Let a, b, c be such that b(a + c) ≠0. If −b b + 1 b −1 + a −1 b −1 c + 1 = 0, then the c c −1 c + 1 (−1)n+2a (−1)n+1b (−1)nc value of ' n ' is (1) zero (2) any even integer (3) any odd integer (4) any integer JEE Main 2009 JEE Main Previous Year Paper

Q79.Let f(x) = x|x| and g(x) = sin x. Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at x = 0 . (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-2 is not a correct explanation for Statement-1 Statement-1 (3) Statement-1 is true, Statement-2 is false (4) Statement-1 is false, Statement-2 is true

Q85.The differential equation which represents the family of curves y = c1ec2x , where c1 and c2 are arbitrary constants is (1) y′ = y2 (2) y′′ = y′y (3) yy′′ = y′ (4) yy′′ = (y′)2

Q71.Statement - 1: For every natural number n ≥2, 1 + 1 + … + 1 > √n. Statement −2 : For every √1 √2 √n natural number n ≥2, √n(n + 1) < n + 1. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q75.How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? (1) 8 ⋅6C4 ⋅7C4 (2) 6.8 ⋅7C4 (3) 6 ⋅7 ⋅8C4 (4) 7 ⋅6C4 ⋅8C4

Q82.Let p be the statement " x is an irrational number", q be the statement " y is a transcendental number", and r be the statement " x is a rational number iff y is a transcendental number". Statement −1 : r is equivalent to either q or p Statement −2 : r is equivalent to ∼(p ↔∼q). (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q87.Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A . Assume that A2 = 1. Statement -1: If A ≠1 and A ≠−1, then det A = −1. Statement −2 : If A ≠1 and A ≠−1, then tr(A) ≠0. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1

Q96.Let I = ∫10 sin√xx dx and J = ∫10 cos√xx (1) I > 32 and J > 2 (2) I < 23 and J < 2 (3) I < 32 and J > 2 (4) I > 23 and J < 2