Practice Questions

Q73.Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement p ⇒(q ∨r) is: (1) (p ∨q) ⇒r (2) (p ⇒q) ∨(p ⇒r) (3) (p ⇒∼q) ∧(p ⇒r) (4) (p ⇒q) ∧(p ⇒∼r)

Q73.The statement ∼(p ↔~q) is (1) A tautology (2) A fallacy (3) Equivalent to p ↔q (4) Equivalent to ~p ↔q

Q73.If limx→2 tan(x−2{x2+k+2x−2k}x2−4x+4 JEE Main 2014 (11 Apr Online) JEE Main Previous Year Paper (1) 0 (2) 1 (3) 2 (4) 3

Q73.Let x , M and σ2 be respectively the mean, mode and variance of n observations x1, x2, . ..., xn and di = −xi −a, i = 1, 2, . ..., n, where a is any number. Statement I: Variance of d1, d2, . . . , dn is σ2 . ¯Statement II: Mean and mode of d1, d2, . . . . , dn are −x −a and −M −a, respectively. (1) Statement I and Statement II are both true (2) Statement I and Statement II are both false (3) Statement I is true and Statement II is false (4) Statement I is false and Statement II is true

Q73.If OB is the semi-minor axis of an ellipse, F1 and F2 are its focii and the angle between F1B and F2B is a right angle, then the square of the eccentricity of the ellipse is (1) 1 (2) 1 4 √2 (3) 1 (4) 1 2 2√2

Q74.Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB −BA is (1) skew symmetric (2) I or − I, where I is an identity matrix (3) symmetric (4) neither symmetric nor skew symmetric

Q74.Let ¯X and M.D. be the mean and the mean deviation about ¯X of n observations xi, i = 1, 2,n. If each of the observations is increased by 5 , then the new mean and the mean deviation about the new mean, respectively, are : – (1) ¯X , M.D. (2) X + 5, M.D. – (3) ¯X , M.D. +5 (4) X + 5, M.D. +5 JEE Main 2014 (12 Apr Online) JEE Main Previous Year Paper

Q74.If f(x) is continuous and f( 29 ) = 29 , then lim f( 1−cosx2 3x ) equals to x→0 (1) 8 (2) 0 9 (3) 2 (4) 9 9 2

Q74.The variance of the first 50 even natural numbers is : JEE Main 2014 (06 Apr) JEE Main Previous Year Paper (1) 437 (2) 4374 (3) 833 (4) 833 4

Q74.The proposition ∼(p∨∼q)∨∼(p ∨q) is logically equivalent to: (1) p (2) q (3) ∼p (4) ∼q

Q75.A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45°. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to 30°. Then the speed (in m/s) of the bird is (1) 20√2 (2) 20(√3 −1) (3) 40(√2 −1) (4) 40(√3 −√2)

Q75. r 2r −1 3r −2 If Δr = n2 n −1 a , then the value of ∑n−1r=1 Δr 2 1 n(n −1) (n −1)2 12 (n −1)(3n + 4) (1) Is independent of both a and n (2) Depends only on a (3) Depends only on n (4) Depends both on a and n

Q75.Two ships A and B are sailing straight away from a fixed point O along routes such that ∠AOB is always 120∘ . At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr ): (1) 260 (2) 260 √37 37 (3) 80 (4) 80 √37 37

Q75.A relation on the set A = {x : |x| < 3, x ∈Z}, where Z is the set of integers is defined by R = {(x, y) : y = |x|, x ≠−1}. Then the number of elements in the power set of R is: (1) 32 (2) 16 (3) 8 (4) 64

Q75.The contrapositive of the statement "I go to school if it does not rain" is (1) If it rains, I go to school. (2) If it rains, I do not go to school. (3) If I go to school, it rains. (4) If I do not go to school, it rains.

Q76.The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be α. After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to β. Then the height (in metres) of the tower is: (1) 2 sin α sin β (2) sin α sin β sin(β−α) cos(β−α) (3) 2 sin(β−α) (4) cos(β−α) sin α sin β sin α sin β

Q76.In a set of 2n distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations : (1) Increases by 2 . (2) Increases by 1 . (3) Decreases by 2 . (4) Decreases by 1 .

Q76.If X = {4n −3n −1 : n ∈N} and Y = {9(n −1) : n ∈N}, where N is the set of natural numbers, then X ∪Y is equal to (1) X (2) Y (3) N (4) Y −X

Q76. y 1 2 x 6 If A = and B = ⎡x⎤ be such that AB = , then: [3 −1 2 ] [8 ] 1 ⎣ ⎦ (1) y = 2x (2) y = −2x (3) y = x (4) y = −x

Q76.The principal value of tan−1(cot 43π4 ) is (1) π 4 (2) −π4 (3) 3π 4 (4) −3π4

Q77.Let P be the relation defined on the set of all real numbers such that P = {(a, b) : sec2 a −tan2 b = 1}. Then, P is (1) reflexive and symmetric but not transitive (2) symmetric and transitive but not reflexive (3) reflexive and transitive but not symmetric (4) an equivalence relation

Q77.If A is a 3 × 3 non-singular matrix such that AA′ = A′A and B = A−1A′, then BB′ equals, where X ′ denotes the transpose of the matrix X . (1) B−1 (2) (B−1)′ (3) I + B (4) I Q78. 3 1 + f(1) 1 + f(2) If α, β ≠0, f(n) = αn + βn and 1 + f(1) 1 + f(2) 1 + f(3) = K(1 −α)2(1 −β)2(α −β)2 , then K is 1 + f(2) 1 + f(3) 1 + f(4) equal to (1) 1 (2) −1 (3) αβ (4) αβ1

Q77.The function f(x) = |sin 4x| + |cos 2x|, is a periodic function with a fundamental period (1) π (2) 2π (3) π (4) π 4 2 f is

Q77.If a2 b2 c2 ⎞ ∣(a + λ)2 (b + λ)2 (c + λ2) (a −λ)2 (b −λ2) (−λ2 ⎠ a2 b2 c2 = kλ a b c , λ ≠0 1 1 1 then k is equal to: (1) 4λabc (2) −4λabc (3) 4λ2 (4) −4λ2 Q78. 1 cos θ 1 If f(θ) = −sin θ 1 −cos θ and A and B are respectively the maximum and the minimum values of −1 sin θ 1 f(θ), then (A, B) is equal to: (1) (3, −1) (2) (4, 2 −√2) (3) (2 + √2, 2 −√2) (4) (2 + √2, −1)

Q77.Let A(2, 3, 5), B(−1, 3, 2) and C(λ, 5, μ) be the vertices of a △ABC. If the median through A is equally inclined to the coordinate axes, then: (1) 5λ −8μ = 0 (2) 8λ −5μ = 0 (3) 10λ −7μ = 0 (4) 7λ −10μ = 0