Practice Questions

Q73.Let ∫2−tan3+tan xx dx = 12 (αx + loge |β sin x + γ cos x|) + C , where C is the constant of integration. Then α + βγ is equal to : (1) 7 (2) 4 (3) 1 (4) 3

Q73.If loge y = 3 sin−1 x, then (1 −x2)y′′ −xy′ at x = 12 is equal to (1) 3eπ/6 (2) 9eπ/2 (3) 3eπ/2 (4) 9eπ/6 y ≥0, y(0) = 0. Then at x = 2, y′′ + y + 1 is equal to

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

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 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 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.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.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.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 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 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 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.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.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

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

Q75.Let 𝑓, 𝑔: 0, ∞→𝑅 be two functions defined by 𝑓𝑥= 𝑥𝑡−𝑡2𝑒−𝑡2𝑑𝑡 and 𝑔𝑥= 𝑥2 𝑡 12𝑒−𝑡2𝑑𝑡. Then the ∫−𝑥 ∫0 value of 9𝑓√log𝑒9 + 𝑔√log𝑒9 is equal to (1) 6 (2) 9 (3) 8 (4) 10

Q75.If the area of the region {(x, y) : x2a ≤y ≤1x , 1 ≤x ≤2, 0 < a < 1} is (loge 2) −17 then the value of 7a −3 is equal to: (1) 0 (2) 2 (3) -1 (4) 1 dy

Q75.The area (in square units) of the region bounded by the parabola y2 = 4(x −2) and the line y = 2x −8. (1) 8 (2) 9 (3) 6 (4) 7

Q75.The area enclosed between the curves y = x|x| and y = x −|x| is : (1) 4 (2) 1 3 (3) 2 (4) 8 3 3

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

Q75.If the value of the integral ∫1−1 cos1+3xαx (1) π (2) π 3 6 (3) π (4) π 4 2