Practice Questions

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.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.If the function f(x) = 2x3 −9x2 + 12a2x + 1, a > 0 has a local maximum at x = α and a local minimum at x = α2 , then α and α2 are the roots of the equation : JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper (1) x2 −6x + 8 = 0 (2) x2 + 6x + 8 = 0 (3) 8x2 + 6x −1 = 0 (4) 8x2 −6x + 1 = 0 = π6 . Then eα and e−α are the roots of the equation :

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

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

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 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 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.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 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 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.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 :

Q75.If ∫10 √3+x+√1+x1 (1) 4 (2) 10 (3) 7 (4) 8

Q75.For 0 < a < 1, the value of the integral ∫0 1 - 2𝑎cos𝑥+ 𝑎2 is : (1) 𝜋2 (2) 𝜋2 𝜋+ 𝑎2 𝜋- 𝑎2 𝜋 𝜋 (3) (4) 1 - 𝑎2 1 + 𝑎2 JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

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