Practice Questions

Q74.The shortest distance between the line x −y = 1 and the curve x2 = 2y is: (1) 1 (2) 1 2 √2 (3) 1 (4) 0 2√2 dx, x > 0, is equal to

Q74.The value of ∑100n=1 ∫nn−1 ex−[x]dx , where [x] is the greatest integer ≤x, is: (1) 100(e −1) (2) 100e (3) 100(1 −e) (4) 100(1 + e) dx is:

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

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

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 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.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 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.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 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 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

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 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.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.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 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 ∫ −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

Q76.Let a vector αˆi + βˆj be obtained by rotating the vector √3ˆi +ˆj by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices (α, β), (0, β) and (0, 0) is equal to (1) 1 (2) 1 2 (3) 1 (4) 2√2 √2

Q76. y sin x 1 dy ⎡ ⎤ Let y = y(x) satisfies the equation dx −|A| = 0, for all x > 0, where A = 0 −1 1 . If y(π) = π + 2, ⎣ 2 0 x1 ⎦ then the value of y( π2 ) is: (1) π 2 + π4 (2) π2 −1π (3) 3π 2 −1π (4) π2 −4π −−−−−

Q76.Let y = y(x) be the solution of the differential equation cos sin x + cos x + = + y sin sin x + cos x + 0 ≤x ≤π2 , y(0) = 0. Then, y( π3 ) is x(3 3)dy (1 x(3 3))dx, equal to: JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper 2 loge( 2√3+1011 ) (1) 2 loge( 2√3+96 ) (2) 2 loge( 3√3−84 ) (3) 2 loge( √3+72 ) (4)

Q76.Let us consider a curve, y = f(x) passing through the point (−2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf ′(x) = x2. Then (1) x3 −3xf(x) −4 = 0 (2) x2 + 2xf(x) −12 = 0 (3) x3 + xf(x) + 12 = 0 (4) x2 + 2xf(x) + 4 = 0

Q76.The integral ∫ 1 dx is equal to : (where C is a constant of integration) 4√(x−1)3(x+2)5 (1) 5 1 4 + C 4 3 ( x−1x+2 ) 4 + C (2) 34 ( x+2x−1 ) (3) 4 x−1 54 (4) 3 x+2 14 3 ( x+2 ) + C 4 ( x−1 ) + C JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper