Practice Questions

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

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.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.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

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

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

Q79.Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p′i(x) and p′′i(x) be the first and second order derivatives of pi(x) respectively. Let, p1(x) p′1(x) p′′1x( A(x) = ⎡ p2(x) p′2(x) p′′2( ⎤ ⎞ p3(x) p′3(x) p′′3(x ⎣ ⎦ ⎠ and B(x) = [A(x)]TA(x). Then determinant of B(x) : (1) is a polynomial of degree 6 in x. (2) is a polynomial of degree 3 in x. (3) is a polynomial of degree 2 in x. (4) does not depend on x.

Q79.If a, b, c are non - zero real numbers and if the system of equations (a −1)x = y + z JEE Main 2014 (09 Apr Online) JEE Main Previous Year Paper (b −1)y = x + z (c −1)z = x + y has a non - trivial solution, then ab + bc + ca equals : (1) −1 (2) a + b + c (3) abc (4) 1 is equal to :

Q79.If g is the inverse of a function f and f ′(x) = 1 , then g′(x) is equal to 1+x5 (1) 1 (2) 1 + {g(x)}5 1+{g(x)}5 (3) 1 + x5 (4) 5x4

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.If f(x) = x2 −x + 5, x > 21 , and g(x) is its inverse function, then g′(7) equals: (1) −13 (2) 131 (3) 3 1 (4) −113 x , ≠0 1 ) x and g(x) = xf(x) Statement I: f is x

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.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 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.If the volume of a spherical ball is increasing at the rate of 4π cc / sec then the rate of increase of its radius (in cm / sec), when the volume is 288π cc is (1) 1 (2) 1 9 6 (3) 1 (4) 1 24 36

Q81.Let f(x) = x|x|, g(x) = sin x and h(x) = (g ∘f)(x). Then (1) h(x) is not differentiable at x = 0. (2) h(x) is differentiable at x = 0, but h′(x) is not continuous at x = 0 (3) h′(x) is continuous at x = 0 but it is not (4) h′(x) is differentiable at x = 0 differentiable at x = 0

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 non-zero real numbers b and c are such that min f(x) > max g(x), where f(x) = x2 + 2bx + 2c2 and g(x) = −x2 −2cx + b2, (x ∈R); then cb lies in the interval , (1) (√2, ∞) (2) [ 12 1 ) √2 , √2] (3) (0, 12 ) (4) [ √21

Q82.The slope of the line touching both the parabolas y2 = 4x and x2 = −32y is (1) 1 (2) 2 8 3 (3) 1 (4) 3 2 2 x dx, is equal to

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

Q82.Let f and g be two differentiable functions on R such that f ′(x) > 0 and g′(x) < 0 for all x ∈R. Then for all x : (1) f(g(x)) > f(g(x −1)) (2) f(g(x)) > f(g(x + 1)) (3) g(f(x)) > g(f(x −1)) (4) g(f(x)) < g(f(x + 1)) JEE Main 2014 (12 Apr Online) JEE Main Previous Year Paper

Q82.For the curve y = 3 sin θ cos θ, x = eθ sin θ, 0 ≤θ ≤π, the tangent is parallel to x-axis when θ is: (1) 3π (2) π 4 2 (3) π (4) π 4 6