Practice Questions

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

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

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

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.The value of the integral ∫1−1 loge(√1 x)dx is equal to: (1) 2 1 loge 2 + π4 −32 (2) 2 loge 2 + π4 −1 (3) loge 2 + π2 −1 (4) 2 loge 2 + π2 −12

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.If In = ∫ π2 cotn xdx, then 4 (1) I2 + I4, (I3 + I5)2, I4 + I6 are in G. P. (2) I2 + I4, I3 + I5, I4 + I6 are in A. P. (3) 1 , 1 , 1 are in A. P. (4) 1 , 1 , 1 are in G. P. I2+I4 I3+I5 I4+I6 I2+I4 I3+I5 I4+I6 is equal to lim n1 + (n+1)2n + (n+2)2n + … + (2n−1)2n ]

Q76.Let slope of the tangent line to a curve at any point P(x, y) be given by xy2+yx x + 2y = 4 at x = −2, then the value of y, for which the point (3, y) lies on the curve, is : (1) −43 (2) 3518 (3) −1819 (4) −1811 −−

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

Q76.If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is x2−4x+y+8x−2 , curve also passes through the point: (1) (5, 4) (2) (4, 4) (3) (4, 5) (4) (5, 5)

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.If the area of the bounded region R = {(x, y) : max{0, loge x} ≤y ≤2x, 21 ≤x ≤2} is, α(loge 2)−1 + β(loge 2) + γ then the value of (α + β −2γ)2 is equal to: (1) 8 (2) 2 (3) 4 (4) 1 = 3x + 4y, with y(0) = 0. If

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.The area (in sq. units) of the part of the circle 𝑥2 + 𝑦2 = 36, which is outside the parabola 𝑦2 = 9𝑥, is equal to (1) 12𝜋+ 3√3 (2) 24𝜋+ 3√3 (3) 24𝜋- 3√3 (4) 12𝜋- 3√3

Q76.If the value of the integral ∫50 x+[x]ex−[x] greatest integer less than or equal to x; then the value of (α + β)2 is equal to : (1) 25 (2) 100 (3) 36 (4) 16