Practice Questions

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.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

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.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

Q83.The integral ∫(1 + x −1x )ex+ 1 JEE Main 2014 (06 Apr) JEE Main Previous Year Paper (1) (x + 1)ex+ x1 + c (2) −xex+ x1 + c (3) (x −1)ex+ x1 + c (4) xex+ x1 + c π x 2 −4 sin x2 dx equals

Q83. sin2 x cos2 x The integral dx is equal to: ∫ 2 (sin3 x + cos3 x) 1 1 (1) + c (2) − + x (1+cot3 x 3(1+tan3 c) (3) sin3 x + c (4) − cos3 x + c (1+cos3 x 3(1+sin3 x

Q83.The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = √3 is: (1) 3 4 √3π (2) 83 √3π (3) 4π (4) 2π > 0) is equal to:

Q84.The integral ∫x cos−1 ( 1+x21−x2 )dx(x (1) −x + (1 + x2) tan−1 x + c (2) x −(1 + x2) cot−1 x + c (3) −x + (1 + x2) cot−1 x + c (4) x −(1 + x2) tan−1 x + c

Q84.Let, the function F be defined as F(x) = ∫x1 ett dt, x > 0, then the value of the integral ∫x1 t+aet dt, where a > 0, is (1) ea[F(x) −F(1 + a)] (2) e−a[F(x + a) −F(a)] (3) ea[F(x + a) −F(1 + a)] (4) e−a[F(x + a) −F(1 + a)]

Q84.The integral ∫ √1 + 4 sin2 0 (1) 4√3 −4 (2) 4√3 −4 −π3 (3) π −4 (4) 2π3 −4 −4√3

Q84.If [ ] denotes the greatest integer function, then the integral ∫π0 [cos xdx is equal to: (1) π (2) 0 2 (3) −1 (4) −π2

Q85.If for a continuous function f(x), ∫t−π(f(x) + xdx) = π2 −t2 , for all t ≥−π, then f (−π3 ) is equal to: (1) π (2) π2 (3) π (4) π 3 6

Q85.The area of the region (in square units ) above the x-axis bounded by the curve y = tan x, 0 ≤x ≤π2 and the tangent to the curve at x = π4 is (1) 2 1 (log 2 −12 ) (2) 12 (1 + log 2) (3) 1 2 (1 −log 2) (4) 12 (log 2 + 12 )

Q85.Let A = {(x, y) : y2 ≤4x, y −2x ≥−4}. The area of the region A in square units is (1) 10 (2) 8 (3) 9 (4) 11

Q85.If for n ≥1, Pn = ∫e1 (log xn)dx, then P10 −90P8 is equal to: (1) −9 (2) 10e (3) −9e (4) 10 Φ, is given by

Q86.If the differential equation representing the family of all circles touching x-axis at the origin is (x2 −y2) dxdy = g(x)y, then g(x) equals (1) 1 x2 (2) 2x 2 (3) 1 x (4) 2x2 2 → → →

Q86.The general solution of the differential equation, sin 2x −y = 0, is : dx ( dy −√tan x) (1) y√tan x = x + c (2) y√cot x = tan x + c (3) y√tan x = cot x + c (4) y√cot x = x + c

Q86.If the general solution of the differential equation y′ = xy + Φ ( xy ), for some function y ln |cx| = x, where c is an arbitrary constant, then Φ(2) is equal to: (1) 4 (2) 1 4 (3) −4 (4) −14 →

Q86.Let the population of rabbits surviving at a time t be governed by the differential equation dp(t) . If p(0) = 100, then p(t) equals dt = 12 {p(t) −400} (1) 600 − 500 e 2t (2) 400 −300 e −t2 (3) 400 − 300 et/2 (4) 300 −200 e −t2 2 → → → → a b then λ is equal to

Q86.If dxdy + ytan x=sin 2x and y(0) = 1, then y(π) is equal to (1) −1 (2) 5 (3) 1 (4) −5 → → →

Q87.If x = 3ˆi −6ˆj −ˆk , y = ˆi + 4ˆj −3ˆk and →z= 3ˆi −4ˆj −12ˆk, then the magnitude of the projection of x ×→y on →zis (1) 14 (2) 12 (3) 15 (4) 10

Q87.If ×→b →b ×→c c = λ [→a ×→a] [ c] (1) 0 (2) 1 (3) 2 (4) 3 y−3

Q87.If →a = 2, b = 3 and 2→a− b = 5, then 2→a+ b equals : (1) 5 (2) 7 (3) 17 (4) 1 y−2

Q87.If |→c|2 = 60 and →c × (^i + 2^j + 5^k) = 0, then a value of →c ⋅(−7^i + 2^j + 3^k) is: (1) 4√2 (2) 12 (3) 24 (4) 12√2 y−2

Q88.A symmetrical form of the line of intersection of the planes x = ay + b and z = cy + d is (1) x−b a = y−11 = z−dc (2) x−b−aa = y−11 = z−d−cc (3) x−a b = y−01 = z−cd (4) x−b−ab = y−10 = z−d−cd