Practice Questions

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.If f : R →R is given by f(x) = x + 1, then the value of lim n1 [f(0) + f( n5 ) + f( 10n ) + … . . +f( 5(n−1)n )] is: n→∞ (1) 3 (2) 5 2 2 (3) 1 (4) 7 2 2 + √x2 + x ∈R. Then which one of the

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.If Un = (1 + n2 2 n −4 n2 1 )(1 22 ) (1 n2 ) , then n→∞(Un)lim n2 is equal to (1) 16e2 (2) 4e (3) e24 (4) 16e2 dx is equal to Q75. ∫166 loge x2+loge(x2−44x+484)loge x2 (1) 5 (2) 10 (3) 8 (4) 6

Q74.If ∫cos𝑥- sin𝑥 𝑑𝑥= 𝑎sin-1sin𝑥+ cos𝑥 + 𝑐, where 𝑐 is a constant of integration, then the ordered pair 𝑎, 𝑏 is √8 - sin2𝑥 𝑏 equal to: (1) 1, - 3 (2) 3, 1 (3) -1, 3 (4) 1, 3 Q75. ∫0𝑥2 sin√𝑡𝑑𝑡 lim is equal to: 𝑥→0 𝑥3 JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper 2 (1) 0 (2) 3 (3) 3 (4) 1 2 15

Q74.Let a be a real number such that the function f(x) = ax2 + 6x −15, x ∈R is increasing in (−∞, 43 ) and decreasing in ( 34 , ∞) . Then the function g(x) = ax2 −6x + 15, x ∈R has a (1) local maximum at x = −34 (2) local minimum at x = −34 (3) local maximum at x = 34 (4) local minimum at x = 34

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.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 the integral ∫10 (1+x)(1+3x)(3+x)√xdx is: (1) π 4 (1 −√32 ) (2) π8 (1 −√36 ) (3) π 8 (1 −√32 ) (4) π4 (1 −√36 )

Q74.Let α, β, γ be the real roots of the equation, x3 + ax2 + bx + c = 0, ( a, b, c ∈R and a, b ≠0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0, γu + αv + βw = 0 has non-trivial solution, then the value of a2 is b (1) 5 (2) 3 (3) 1 (4) 0

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 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 : [0, ∞) →[0, 3] be a function defined by f(x) = {max{sin2 + cos x,t :x0>≤tπ ≤π}, x ∈[0, π] the following is true ? (1) f is continuous everywhere but not differentiable (2) f is differentiable everywhere in (0, ∞) exactly at one point in (0, ∞) (3) f is not continuous exactly at two points in (4) f is continuous everywhere but not differentiable (0, ∞) exactly at two points in (0, ∞)

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 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.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.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 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

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.Let 1 / 2 𝑥𝑛 ∀𝑛> 𝑚 and 𝑛, 𝑚∈𝑁. Consider a matrix 𝐴= where 𝐽𝑛, 𝑚= ∫0 𝑥𝑚- 1𝑑𝑥, 𝑎𝑖𝑗3 × 3 J6 + 𝑖, 3 - J𝑖+ 3, 3 , 𝑖≤𝑗 a𝑖𝑗= Then adj A-1 is : 0 , 𝑖> 𝑗. (1) (15 ) 2 × 234 (2) (15 ) 2 × 242 (3) (105 ) 2 × 236 (4) (105 ) 2 × 238

Q74.Consider function f : A →B and g : B →C(A, B, C ⊆R) such that (gof)−1 exists, then: (1) f and g both are one-one (2) f and g both are onto (3) f is one-one and g is onto (4) f is onto and g is one-one JEE Main 2021 (25 Jul Shift 2) JEE Main Previous Year Paper ∫x0 (5 + |1 −t|)dt, , then

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

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.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.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]