Practice Questions

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.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.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

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.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)

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.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.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

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.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 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.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

Q78.If B is a 3 × 3 matrix such that B2 = 0, then det. [(I + B)50 −50B] is equal to : (1) 1 (2) 2 (3) 3 (4) 50

Q78.Let A be a 3 × 3 matrix such that 1 2 3 0 0 1 A ⎡ 0 2 3⎤ = ⎡1 0 0 ⎤ 0 1 1 0 1 0 ⎣ ⎦ ⎣ ⎦ Then A−1 is: (1) 3 1 2 (2) 3 2 1 ⎡3 0 2 ⎤ ⎡ 3 2 0⎤ 1 0 1 1 1 0 ⎣ ⎦ ⎣ ⎦ (3) 0 1 3 (4) 1 2 3 ⎡0 2 3 ⎤ ⎡ 0 1 1⎤ 1 1 1 0 2 3 ⎣ ⎦ ⎣ ⎦

Q78.Let f : R →R be defined by f(x) = |x|−1|x|+1 , then (1) one-one but not onto (2) neither one-one nor onto (3) both one-one and onto (4) onto but not one-one Q79. √2+cosx−1 , x ≠π If the function f(x) = (π−x)2 is continuous at x = π, then k equals { k, x = π (1) 14 (2) 0 (3) 2 (4) 12 JEE Main 2014 (19 Apr Online) JEE Main Previous Year Paper

Q79.Statement I: The equation (sin−1 x)3+ (cos−1 x)3 −aπ3 = 0 has a solution for all a ≥ 321 . Statement II: For π 2 any x ∈R, sin−1 x + cos−1 x = 2 and 0 ≤(sin−1 x −π4 ) ≤9π216 (1) Both statements I and II are true. (2) Both statements I and II are false. (3) Statement I is true and statement II is false. (4) Statement I is false and statement II is true.

Q80.Let f : R →R be a function such that |f(x)| ≤x2, for all x ∈R. Then, at x = 0, f is (1) differentiable but not continuous (2) neither continuous nor differentiable (3) continuous as well as differentiable (4) continuous but not differentiable

Q80.If y = enx , then dx2d2y . d2xdy2 (1) ne−nx (2) −ne−nx (3) nenx (4) 1 x ∈R, then the equation f(x) = 0 has :

Q80.Let f be an odd function defined on the set of real numbers such that for x ≥0, f(x) = 3 sin x + 4 cos x. Then f(x) at x = −11π6 is equal to: JEE Main 2014 (11 Apr Online) JEE Main Previous Year Paper (1) 3 2 + 2√3 (2) −32 + 2√3 (3) 2 3 −2√3 (4) −32 −2√3

Q80.If f& g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 & f(1) = 6, then for some c∈]0, 1[ (1) f ′(c) = g′(c) (2) f ′(c) = 2g′(c) (3) 2f ′(c) = g′(c) (4) 2f ′(c) = 3g′(c)

Q81.If x = −1 and x = 2 are extreme points of f(x) = α log|x| + βx2 + x, then (1) α = 2, β = −12 (2) α = 2, β = 12 (3) α = −6, β = 12 (4) α = −6, β = −12

Q81.Let f, g : R →R be two functions defined by f(x) = {x0,sin ( , = 0 a continuous function at x = 0. Statement II: g is a differentiable function at x = 0. (1) Both statement I and II are false. (2) Both statement I and II are true. (3) Statement I is true, statement II is false. (4) Statement I is false, statement II is true.

Q81.If f(x) = ( 35 )x + ( 45 )x −1, (1) No solution (2) More than two solutions (3) One solution (4) Two solutions

Q82.If the Rolle's theorem holds for the function f(x) = 2x3 + ax2 + bx in the interval [−1, 1] for the point c = 21 , then the value of 2a + b is: (1) -1 (2) 2 (3) 1 (4) -2 Q83. ∫ sin8 x−cos8 x dx is equal to (1−2 sin2 x cos2 x) (1) −12 sin 2x + c (2) −sin2 x + c (3) −12 sin x + c (4) 21 sin 2x + c Q84. 21 dx equals The integral ∫ ln(1+2x)1+4x2 0 (1) π 4 ln 2 (2) 16π ln 2 (3) π 8 ln 2 (4) 32π ln 2