Practice Questions

Q73.Suppose f(x) = (2x+2−x) tan x√tan−1(x2−x+1) . Then the value of f ′(0) is equal to (7x2+3x+1)3 (1) π (2) 0 (3) √π (4) π2 π + = 4 ( π + a) −2, then the value of a is

Q73.Let 𝑓𝑥= 2𝑥2 + 5𝑥- 3, 𝑥∈𝑅. If 𝑚 and 𝑛 denote the number of points where 𝑓 is not continuous and not differentiable respectively, then 𝑚+ 𝑛 is equal to: (1) 5 (2) 2 (3) 0 (4) 3

Q73.Let g(x) = 3f x + f(3 - x) and f" (x) > 0 for all x ∈( 0, 3 ) . If g is decreasing in ( 0, α ) and increasing in 3 ( α, 3 ) , then 8α is (1) 24 (2) 0 (3) 18 (4) 20

Q73.Let I(x) = ∫ dx. If I(0) = 3, then I ( 12π ) is equal to sin2 x(1−cot x)2 JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper (1) 2√3 (2) √3 (3) 3√3 (4) 6√3 n ∈N, satisfies 147I20 = 148I21 is

Q73.Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS . Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)2 is equal to : (1) 72 (2) 60 (3) 64 (4) 80

Q74.Consider the function 𝑓: 0, ∞→𝑅 defined by 𝑓𝑥= 𝑒−log𝑒𝑥. If 𝑚 and 𝑛 be respectively the number of points at which 𝑓 is not continuous and 𝑓 is not differentiable, then 𝑚+ 𝑛 is (1) 0 (2) 3 (3) 1 (4) 2

Q74.Let 𝑓𝑥= 𝑥+ 32𝑥- 23, 𝑥∈[ - 4, 4]. If 𝑀 and 𝑚 are the maximum and minimum values of 𝑓, respectively in [ - 4, 4], then the value of 𝑀- 𝑚 is : (1) 600 (2) 392 (3) 608 (4) 108

Q74.For the function f(x) = sin x + 3x −2π (x2 + x), where x ∈[0, π2 ], consider the following two statements : (I) f is increasing in (0, π2 ) . (II) f ′ is decreasing in (0, π2 ) . Between the above two statements, (1) only (II) is true. (2) only (I) is true. (3) neither (I) nor (II) is true. (4) both (I) and (II) are true dy is :

Q74.The value of 1 1 2𝑥3 −3𝑥2 −𝑥+ 1 3𝑑𝑥 is equal to: ∫0 (1) 0 (2) 1 (3) 2 (4) -1 𝜋 Q75. 3 If ∫ cos4𝑥𝑑𝑥= 𝑎𝜋+ 𝑏√3, where 𝑎 and 𝑏 are rational numbers, then 9𝑎+ 8𝑏 is equal to: 0 (1) 2 (2) 1 3 (3) 3 (4) 2

Q74.The value of n→∞∑nlim k=1 (n2+k2)(n2+3k2)n3 is : (1) (2√3+3)π (2) 13π 24 8(4√3+3) (3) 13(2√3−3)π (4) π 8 8(2√3+3)

Q74.The integral ∫ x8 - x2dx 1 is equal to : x12 + 3x6 + 1tan-1x3 + x3 (1) 1 13 (2) 1 12 logtan-1x3 + x3 + C logetan-1x3 + x3 + C 1 1 3 + + C (3) logetan-1x3 + x3 + C (4) logetan-1x3 x3 𝜋 𝑑𝑥

Q74.The value of k ∈N for which the integral In = ∫10 (1 −xk) ndx, (1) 14 (2) 8 (3) 10 (4) 7

Q74.Let f(x) = x5 + 2ex/4 for all x ∈R. Consider a function g(x) such that (g ∘f)(x) = x for all x ∈R. Then the value of 8g′(2) is : (1) 2 (2) 8 (3) 4 (4) 16 is equal to :

Q74.Let ∫x0 √1 −(y′(t))2dt = ∫x0 y(t)dt, 0 ≤x ≤3, (1) 1 (2) 2 (3) √2 (4) 1/2 is

Q74.Let ∫logeα 4 √ex−1dx (1) x2 + 2x −8 = 0 (2) x2 −2x −8 = 0 (3) 2x2 −5x + 2 = 0 (4) 2x2 −5x −2 = 0

Q74.Let β(m, n) = ∫10 xm−1(1 −x)n−1 dx, m, n > 0 . If ∫10 (1 −x10) dx = a × β(b, c), then 100(a + b + c) equals____ (1) 1021 (2) 2120 (3) 2012 (4) 1120 JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

Q74.If 5𝑓𝑥+ 4𝑓 𝑥= 𝑥2 −2, ∀𝑥≠0 and 𝑦= 9𝑥2𝑓𝑥, then 𝑦 is strictly increasing in: (1) 0, 1 ∪1 ∞ (2) −1 0 ∪1 ∞ √5 √5, √5, √5, (3) −1 0 ∪0, 1 (4) −∞, 1 ∪0, 1 √5, √5 √5 √5 𝜋 Q75. 4 𝑥𝑑𝑥 The value of the integral ∫ equals: 0 sin42𝑥+ cos42𝑥 (1) √2𝜋2 (2) √2𝜋2 8 16 (3) √2𝜋2 (4) √2𝜋2 32 64

Q74.The function f(x) = x , x ∈R −{−2, 8} x2−6x−16 (1) decreases in (−2, 8) and increases in (2) decreases in (−∞, −2) ∪(−2, 8) ∪(8, ∞) (−∞, −2) ∪(8, ∞) (3) decreases in (−∞, −2) and increases in (8, ∞) (4) increases in (−∞, −2) ∪(−2, 8) ∪(8, ∞) sin 2 x+cos 2 x dx = A√cos θ tan x −sin θ + B√cos θ −sin θ cot x + C, where C is the integration

Q74.Let f(x) = 3√x −2 + √4 −x be a real valued function. If α and β are respectively the minimum and the maximum values of f , then α2 + 2β2 is equal to (1) 42 (2) 38 (3) 24 (4) 44 dx is π2 . Then, a value of α is

Q74.If ∫ dx = 121 tan−1(3 tan x)+ constant, then the maximum value of a sin x + b cos x, is : a2 sin2 x+b2 cos2 x (1) √40 (2) √41 (3) √39 (4) √42

Q74.The parabola y2 = 4x divides the area of the circle x2 + y2 = 5 in two parts. The area of the smaller part is equal to: (1) 1 3 + 5 sin−1 ( √52 ) (2) 31 + √5 sin−1 ( √52 ) (3) 3 2 + 5 sin−1 ( √52 ) (4) 32 + √5 sin−1 ( √52 )

Q74.If the value of the integral ∫ −π2 2 ( x21+πxcos x 1+sin2 x π 1+e(sin x)2023 )dx (1) 3 (2) −32 (3) 2 (4) 32 JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q74.The area of the region 𝑥, 𝑦: 𝑦2 ≤4𝑥, 𝑥< 4, > 0, 𝑥≠3 is 𝑥- 3𝑥- 4 (1) 16 (2) 64 3 3 8 32 (3) (4) 3 3

Q74.The interval in which the function f(x) = xx, x > 0, is strictly increasing is (1) (0, 1e ] (2) (0, ∞) (3) [ 1e , ∞)]V (4) [ e21 , 1) cos2 x sin2 x dx is equal toQ75. ∫π/40 x+sin3 (cos3 x)2 (1) 1/6 (2) 1/3 (3) 1/12 (4) 1/9

Q75.The area of the region in the first quadrant inside the circle x2 + y2 = 8 and outside the parabola y2 = 2x is equal to : (1) π 2 −13 (2) π −13 (3) π 2 −23 (4) π −23