Practice Questions

Q80.If f(x) = xex(1−x), x ∈R, then f(x) is (1) decreasing on [−1/2, 1] (2) decreasing on R (3) increasing on [−1/2, 1] (4) increasing on R

Q81.If x + |y| = 2y, then y as a function of x, at x = 0 is (1) differentiable but not continuous (2) continuous but not differentiable (3) continuous as well as differentiable (4) neither continuous nor differentiable

Q81.If a metallic circular plate of radius 50 cm is heated so that its radius increases at the rate of 1 mm per hour, then the rate at which, the area of the plate increases (in cm2/ hour) is (1) 5π (2) 10π (3) 100π (4) 50π

Q81.Let a, b ∈R be such that the function f given by f(x) = ln |x| + bx2 + ax, x ≠0 has extreme values at x = −1 and x = 2. Statement 1: f has local maximum at x = −1 and at x = 2. Statement 2: a = 12 and b = −14 (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; statement (4) Statement 1 is true, statement 2 is false 2 is not a correct explanation for statement 1

Q81.The weight W of a certain stock of fish is given by W = nw, where n is the size of stock and w is the average weight of a fish. If n and w change with time t as n = 2t2 + 3 and w = t2 −t + 2, then the rate of change of W with respect to t at t = 1 is (1) 1 (2) 8 (3) 13 (4) 5 JEE Main 2012 (19 May Online) JEE Main Previous Year Paper

Q81.The integral of x2−x w.r.t. x is x3−x2+x−1 (1) 1 2 log (x2 + 1 + c) (2) 12 log x2 −1 + c (3) log (x2 + 1 + c) (4) log x2 −1 + c

Q82. f(x) = ∫ dx is a polynomial of degree sin6 x (1) 5 in cot x (2) 5 in tan x (3) 3 in tan x (4) 3 in cot x

Q82.If dx d G(x) = etanx x , x ∈(0, π/2), then ∫1/21/4 x2 ⋅etan(πx2)dx is equal to (1) G(π/4) −G(π/16) (2) 2[G(π/4) −G(π/16)] (3) π[G(1/2) −G(1/4)] (4) G(1/√2) −G(1/2)

Q82.If a circular iron sheet of radius 30 cm is heated such that its area increases at the uniform rate of 6πcm2/hr, then the rate (in mm/hr ) at which the radius of the circular sheet increases is (1) 1.0 (2) 0.1 (3) 1.1 (4) 2.0

Q82.If the integral ∫ tan5 tanx−2x dx = x + a ln | sin x −2 cos x| + k, then a is equal to JEE Main 2012 (Offline) JEE Main Previous Year Paper (1) −1 (2) −2 (3) 1 (4) 2 dt, then g(x + π) equals

Q82.If f(x) = ∫( x2+sin21+x2 x ) sec2 xdx and f(0) = 0 , then f(1) equals (1) tan 1 −π4 (2) tan 1 + 1 (3) π 4 (4) 1 −π4

Q83.If g(x) = ∫x0 cos 4t (1) g(x) (2) g(x) + g(π) g(π) (3) g(x) −g(π) (4) None of these

Q83.The value of the integral ∫0.90 [x −2[x]]dx, where [.] denotes the greatest integer function is (1) 0.9 (2) 1.8 (3) −0.9 (4) 0

Q83.The area enclosed by the curves y = x2, y = x3 , x = 0 and x = p, where p > 1 , is 1/6 . The p equals (1) 8/3 (2) 16/3 (3) 2 (4) 4/3

Q83.If [x] is the greatest integer ≤x, then the value of the integral ∫0.9−0.9 ([x2] + log ( 2−x2+x ))dx is (1) 0.486 (2) 0.243 (3) 1.8 (4) 0

Q83.Let f(x) be an indefinite integral of cos3 x. Statement 1: f(x) is a periodic function of period π. Statement 2: cos3 x is a periodic function. (1) Statement 1 is true, Statement 2 is false. (2) Both the Statements are true, but Statement 2 is not the correct explanation of Statement 1. (3) Both the Statements are true, and Statement 2 is correct explanation of Statement 1. (4) Statement 1 is false, Statement 2 is true.

Q84.The area of the region bounded by the curve y = x3 , and the lines, y = 8 , and x = 0 , is (1) 8 (2) 12 (3) 10 (4) 16

Q84.If a straight line y −x = 2 divides the region x2 + y2 ≤4 into two parts, then the ratio of the area of the smaller part to the area of the greater part is (1) 3π −8 : π + 8 (2) π −3 : 3π + 3 (3) 3π −4 : π + 4 (4) π −2 : 3π + 2 d2y

Q84.The area bounded by the parabola y2 = 4x and the line 2x −3y + 4 = 0, in square unit, is (1) 2 (2) 1 5 3 (3) 1 (4) 1 2 JEE Main 2012 (26 May Online) JEE Main Previous Year Paper = x is

Q84.If ∫xe tf(t)dt = sin x −x cos x −x22 , for all x ∈R −{0}, then the value of f ( π6 ) is (1) 1/2 (2) 1 (3) 0 (4) −1/2

Q84.The area bounded between the parabolas x2 = 4y and x2 = 9y, and the straight line y = 2 is (1) 20√2 (2) 10√2 3 (3) 20√2 (4) 10√2 3

Q85.The population p(t) at time t of a certain mouse species satisfies the differential equation dp(t)dt = 0.5 p(t) −450. If p(0) = 850 , then the time at which the population becomes zero is (1) 2 ln 18 (2) ln 9 (3) 1 2 ln 18 (4) ln 18

Q85.The general solution of the differential equation dx dy + x2 y = x2 is (1) y = cx−3 −x24 (2) y = cx3 −x24 (3) y = cx2 + x35 (4) y = cx−2 + x35

Q85.Statement 1: The degrees of the differential equations dy + y2 = x and + y = sin x are equal. Statement dx dx2 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined. (1) Statement 1 is true, Statement 2 is true, (2) Statement 1 is false, Statement 2 is true. Statement 2 is not a correct explanation of Statement 1. (3) Statement 1 is true, Statement 2 is false. (4) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

Q85.The integrating factor of the differential equation (x2 −1 dxdy + 2)xy (1) 1 (2) x2 −1 x2−1 (3) x2−1 (4) x x x2−1