Practice Questions

Q75.If 𝑥 is the greatest integer ≤𝑥, then 𝜋2 ∫0 sin 2 𝑥- 𝑥[𝑥]d𝑥 is equal to : (1) 2 ( 𝜋+ 1 ) (2) 4 ( 𝜋- 1 ) (3) 2 ( 𝜋- 1 ) (4) 4 ( 𝜋+ 1 ) 𝑥2 is equal to: 𝑥𝑦2 +

Q75.If y = y(x) is the solution of the differential equation dxdy + (tan x)y = sin x, 0 ≤x ≤π3 , with y(0) = 0, then y( π4 ) is equal to (1) 1 loge 2 4 loge 2 (2) ( 2√21 ) (3) loge 2 (4) 12 loge 2

Q75.The number of real roots of the equation e4x + 2e3x −ex −6 = 0 is : (1) 0 (2) 1 (3) 4 (4) 2

Q75.The real valued function f(x) = cosec−1x , where [x] denotes the greatest integer less than or equal to x, is √x−[x] defined for all x belonging to: (1) all reals except integers (2) all non-integers except the interval [ −1, 1] (3) all integers except 0, −1, 1 (4) all reals except the Interval [−1, 1] = √1 −x, then what is the common domain of the

Q75.The function 𝑓( 𝑥) , that satisfies the condition 𝑓(𝑥) = 𝑥+ 𝜋/ 2 sin𝑥cos𝑦𝑓(𝑦)d𝑦, is : ∫0 (1) 𝑥+ 𝜋 (2) 𝑥+ ( 𝜋+ 2 ) sin𝑥 2sin𝑥 (3) 𝑥+ 2 (𝜋- 2)sin𝑥 (4) 𝑥+ ( 𝜋- 2 ) sin𝑥 3 𝜋

Q75.If f(x) = { 5x + 1, xx >≤22 (1) f(x) is not continuous at x = 2 (2) f(x) is everywhere differentiable (3) f(x) is continuous but not differentiable at x = 2 (4) f(x) is not differentiable at x = 1

Q75.Let A1 be the area of the region bounded by the curves y = sin x, y = cos x and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sin x, y = cos x, x-axis and x = π2 in the first quadrant. Then, (1) 2A1 = A2 and A1 + A2 = 1 + √2 (2) A1 : A2 = 1 : √2 and A1 + A2 = 1 (3) A1 : A2 = 1 : 2 and A1 + A2 = 1 (4) A1 = A2 and A1 + A2 = √2 . If the curve intersects the line

Q75.The integral ∫ e4 logee3x+5e3loge 2x+5e2loge x−7e2loge 2xloge x (where c is a constant of integration) (1) loge x2 + 5x −7 + c (2) 4 loge x2 + 5x −7 + c (3) 1 4 loge x2 + 5x −7 + c (4) loge √x2 + 5x −7 + c π

Q75.The value of ∫π/2−π/2 cos21+3xx (1) π2 (2) π4 (3) 2π (4) 4π

Q75.The value of the definite integral ∫ −π4 4 (1+ex (1) −π2 (2) 2√2π (3) −π4 (4) √2π

Q75.Let a be a positive real number such that ∫a0 ex−[x]dx = 10e −9 where, [x] is the greatest integer less than or equal to x. Then, a is equal to: (1) 10 −loge(1 + e) (2) 10 + loge 2 (3) 10 + loge 3 (4) 10 + loge(1 + e) −x + √1 +

Q75.Let f be a twice differentiable function defined on R such that f(0) = 1, f ′(0) = 2 and f ′(x) ≠0 for all f(x) f ′(x) x ∈R. If = 0, for all x ∈R, then the value of f(1) lies in the interval f ′(x) f ′′(x) JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (1) (9, 12) (2) (3, 6) (3) (0, 3) (4) (6, 9)

Q75.Let y = y(x) be the solution of the differential equation cosec2 xdy + 2dx = (1 + y cos 2x) cosec2 xdx, with y( π4 ) = 0. Then, the value of (y(0) + 1)2 is equal to: (1) e1/2 (2) e−1/2 (3) e−1 (4) e →

Q75.The value of ∫ −π2 2 ( 1+sin21+πsin (1) π (2) 5π 2 2 (3) 3π (4) 3π 2 4 dx = αe−1 + β, where α, β ∈R, 5α + 6β = 0, and [x] denotes the

Q75.Let g(t) = ∫π/2−π/2(cos π4 t + f(x))dx, where f(x) = loge(x 1), following is correct? (1) g(1) = g(0) (2) √2 g(1) = g(0) (3) g(1) = √2 g(0) (4) g(1) + g(0) = 0

Q75.The inverse of y = 5log x is: (1) x = 5log y (2) x = ylog 5 log y (3) y = x 1 1 log 5 (4) x = 5

Q75.Let g(x) = ∫x0 f(t)dt, where f is continuous function in [0, 3] such that 31 ≤f(t) ≤1 for all t ∈[0, 1] and 0 ≤f(t) ≤12 for all t ∈(1, 3]. The largest possible interval in which g(3) lies is : (1) [−1, −12 ] (2) [−32 , −1] (3) [ 31 , 2] (4) [1, 3]

Q75.The value of the definite integral ∫𝜋/5𝜋/2424 1 + 3√tan2𝑥 𝜋 𝜋 (1) (2) 3 6 𝜋 𝜋 (3) (4) 12 18

Q75.If the integral ∫100 [sinex−[x]2πx] integer less than or equal to x, then the value of α + β + γ is equal to: (1) 0 (2) 20 (3) 25 (4) 10

Q75.The value of lim n1 ∑2n−1r=0 n2+4r2n2 is: n→∞ (1) 1 tan−1(2) (2) tan−1(4) 2 (3) 1 2 tan−1(4) (4) 41 tan−1(4) JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper 2 dx is:

Q75.Let f : (a, b) →R be twice differentiable function such that f(x) = ∫xa g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g′(x) = 0 has at least : (1) twelve roots in (a, b) (2) five roots in (a, b) (3) seven roots in (a, b) (4) three roots in (a, b)

Q75.The area of the region bounded by the parabola (y −2)2 = (x −1), the tangent to it at the point whose ordinate is 3 and the x -axis, is: (1) 4 (2) 6 (3) 9 (4) 10 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper

Q75.The value of ∫1−1 x2e[x3]dx, where [t] denotes the greatest integer ≤t, is : (1) e+1 (2) e−1 3 3e (3) 1 (4) e+1 3e 3e then this

Q75.If y = y(x) is the solution of the differential equation, dxdy + 2y tan x = sin x, y( π3 ) = 0, then the maximum value of the function y(x) over R is equal to : (1) 8 (2) 21 (3) −154 (4) 18

Q76.Let C1 be the curve obtained by the solution of differential equation 2xy dxdy = y2 −x2, x > 0 . Let the curve C2 be the solution of x2−y22xy = dxdy . If both the curves pass through (1, 1), then the area (in sq. units) enclosed by the curves C1 and C2 is equal to : (1) π −1 (2) π2 −1 (3) π + 1 (4) π4 + 1 → → = 3 and