Practice Questions

Q72.A variable line L passes through the point (3, 5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is : (1) 30 (2) 25 (3) 40 (4) 35

Q73. x2 ⎧ 1−cos where α, β ∈R. If f is continuous at Let f : R →R be a function given by f(x) = ⎨ α, x = 0, β√1−cos x ⎩ x , x > 0 x = 0, then α2 + β2 is equal to : (1) 3 (2) 12 (3) 48 (4) 6 JEE Main 2024 (04 Apr Shift 1) JEE Main Previous Year Paper

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 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.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 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.The function f(x) = 2x + 3x 23 , x ∈R, has (1) exactly one point of local minima and no point of (2) exactly one point of local maxima and no point local maxima of local minima (3) exactly one point of local maxima and exactly (4) exactly two points of local maxima and exactly one point of local minima one point of local minima

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) = ( x1 ) 2x; x > 0 attains the maximum value at x = 1e then : (1) eπ < πe (2) eπ > πe (3) (2e)π > π(2e) (4) e2π < (2π)e 1

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.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

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

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

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

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