Practice Questions

Q73.If the function f(x) = , x ≠0 √2−√1+cos x is continuous at x = 0, then the value of a2 is equal to { a loge 2 loge 3 , x = 0 (1) 968 (2) 1152 (3) 746 (4) 1250

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

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 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 ∫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 the function 𝑓: −∞, −1 →𝑎, 𝑏 defined by 𝑓𝑥= 𝑒𝑥3 −3𝑥+ 1 is one-one and onto, then the distance of the point 𝑃2𝑏+ 4, 𝑎+ 2 from the line 𝑥+ 𝑒−3𝑦= 4 is: (1) 2√1 + 𝑒6 (2) 4√1 + 𝑒6 (3) 3√1 + 𝑒6 (4) √1 + 𝑒6 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q73.The function f : N −{1} →N; defined by f(n) = the highest prime factor of n, is : (1) both one-one and onto (2) one-one only (3) onto only (4) neither one-one nor onto JEE Main 2024 (27 Jan Shift 1) JEE Main Previous Year Paper Q74. , x < 3 ⎧ a(7x−12−x2)b|x2−7x+12| Consider the function f(x) = sin(x−3) ,where [x] denotes the greatest integer less than or equal x−[x] ⎨ 2 , x > 3 ⎩ b , x = 3 to x . If S denotes the set of all ordered pairs (a, b) such that f(x) is continuous at x = 3, then the number of elements in S is : (1) 2 (2) Infinitely many (3) 4 (4) 1 dx = a + b√2 + c√3, where a, b, c are rational numbers, then 2a + 3 b −4c is equal to :

Q73.Let 𝑓: 𝑅- {0} →𝑅 be a function satisfying 𝑓 𝑥 𝑓( 𝑥) for all 𝑥, 𝑦, 𝑓( 𝑦) ≠0. If 𝑓' (1) = 2024, then 𝑦= 𝑓( 𝑦) (1) 𝑥𝑓'𝑥- 2024𝑓𝑥= 0 (2) 𝑥𝑓'𝑥+ 2024𝑓𝑥= 0 (3) 𝑥' (𝑥) + 𝑓(𝑥) = 2024 (4) 𝑥𝑓' (𝑥) - 2023𝑓(𝑥) = 0

Q73.If y(θ) = cos 3θ+42 coscosθ+cos2θ+52θcos θ+2 , then at θ = π2 , y′′ + y′ + y is equal to : (1) 21 (2) 1 (3) 2 (4) 32 20

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