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

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

Q76.The area (in sq. units) of the region, given by the set 𝑥, 𝑦∈𝑅× 𝑅∣𝑥≥0, 2𝑥2 ≤𝑦≤4 - 2𝑥 is : JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper 8 17 (1) (2) 3 3 (3) 13 (4) 7 3 3

Q76.Which of the following statement is correct for the function g(α) for α ∈R such that π 3 sinα x dx g(α) = ∫ π 6 cosα x+sinα x (1) g(α) is a strictly increasing function (2) g(α) has an inflection point at α = −12 (3) g(α) is a strictly decreasing function (4) g(α) is an even function

Q76.If 𝑦d𝑦 𝜙𝑦2 d𝑥= 𝑥2 𝑦2 , 𝑥> 0, 𝜙> 0, and 𝑦( 1 ) = - 1, then 𝜙𝑦24 𝜙' 𝑥2 (1) 2𝜙1 (2) 𝜙1 (3) 4𝜙2 (4) 4𝜙1 𝑑𝑦 2𝑥𝑦+ 2𝑦· 2𝑥

Q76.The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of 2 k bacteria is 2000 after hours, then ( logek 2 ) is equal to: 6 ) loge( 5 (1) 8 (2) 4 (3) 16 (4) 2 is equal to: × × ×

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)