Practice Questions

Q74.The number of real roots of the equation 𝑒6𝑥- 𝑒4𝑥- 2𝑒3𝑥- 12𝑒2𝑥+ 𝑒𝑥+ 1 = 0 is: (1) 2 (2) 4 (3) 6 (4) 1 𝑑𝑥 is

Q74.The local maximum value of the function, f(x) = ( x2 )x2 , x > 0, e (1) 1 (2) ( √e4 ) 4 e (3) (e) 2e (4) (2√e) 1 π x x )dx is :

Q74.Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that ∫10 P(x)dx = 1 and P(x) leaves remainder 5 when it is divided by (x −2) Then the value of 9(b + c) is equal to: (1) 9 (2) 15 (3) 7 (4) 11

Q74.The value of the integral ∫ sin θ⋅sin 2θ(sin6 θ+sin41−cosθ+sin22θθ)√2 sin4 θ+3 sin2 θ+6 (1) 1 32 (2) 1 32 18 [11 −18 sin2 θ + 9 sin4 θ −2 sin6 θ] + c 18 [9 −2 sin6 θ −3 sin4 θ −6 sin2 θ] + c (3) 1 32 (4) 1 −32 18 [11 −18 cos2 θ + 9 cos4 θ −2 cos6 θ] + c 18 [9 −2 cos6 θ −3 cos4 θ −6 cos2 θ] + c

Q74.The value of lim n1 ∑nj=1 (2j−1)+4n(2j−1)+8n is equal to: n→∞ (1) 5 + loge( 32 ) (2) 2 −loge( 23 ) (3) 3 + 2 loge( 23 ) (4) 1 + 2 loge( 32 ) π dx is equal to : cos x)(sin4 x+cos4 x)

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:

Q74.The sum of possible values of x for tan−1(x + 1) + cot−1( x−11 ) = tan−1( 318 ) is: (1) −324 (2) −314 (3) −304 (4) −334

Q74.The range of a ∈R for which the function f(x) = (4a −3)(x + loge 5) + 2(a −7) cot( x2 ) sin2( x2 ), x ≠2nπ, n ∈N , has critical points, is : (1) (−3, 1) (2) [−43 , 2] (3) [1, ∞) (4) (−∞, −1] JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper

Q74.Let 𝑓 be any continuous function on 0, 2 and twice differentiable on 0, 2 . If 𝑓0 = 0, 𝑓1 = 1 and 𝑓2 = 2, then : (1) 𝑓"𝑥> 0 for all 𝑥∈0, 2 (2) 𝑓'𝑥= 0 for some 𝑥∈0, 2 (3) 𝑓"𝑥= 0 for some 𝑥∈0, 2 (4) 𝑓"𝑥= 0 for all 𝑥∈0, 2 2 𝜋𝑥

Q74.Let f : R →R be defined as f(x) = e−x sin x. If F : [0, 1] →R is a differentiable function such that F(x) = ∫x0 f(t)dt, then the value of ∫10 (F ′(x) + f(x))exdx lies in the interval (1) [ 327360 , 360329 ] (2) [ 360330 , 360331 ] (3) [ 331360 , 360334 ] (4) [ 360335 , 360336 ] dx = αe−1 + βe−12 + γ, where α, β, γ are integers and [x] denotes the greatest

Q74.The area of the region: R = {(x, y) : 5x2 ≤y ≤2x2 + 9} is (1) 9√3 square units (2) 12√3 square units (3) 11√3 square units (4) 6√3 square units

Q74.If ∫100π0 sin2x xx dx = 1+4π2απ3 π −[ π ]) e ( α is: (1) 200(1 −e−1) (2) 100(1 −e) (3) 50(e −1) (4) 150(e−1 −1)

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.Let f : R →R be a function defined as , if x < 0 ⎧ sin(a+1)x+sin2x 2x f(x) = ⎨ b , if x = 0 √x+bx3−√x , if x > 0 ⎩ bx5/2 If f is continuous at x = 0 , then the value of a + b is equal to : (1) −52 (2) −2 (3) −3 (4) −32

Q74.Let f(x) cos(2 sin(cot−1 √1−x )), (1) (1 −x)2f ′(x) + 2(f(x))2 = 0 (2) (1 + x)2f ′(x) + 2(f(x))2 = 0 (3) (1 −x)2f ′(x) −2(f(x))2 = 0 (4) (1 + x)2f ′(x) −2(f(x))2 = 0

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 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.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.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 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.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.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 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.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