Practice Questions

Q72.Let 𝑓: 𝑅→𝑅 and 𝑔: 𝑅→𝑅 be defined as 𝑓𝑥= log𝑒𝑥, 𝑥> 0 and 𝑔𝑥= 𝑥, 𝑥≥0 . Then, 𝑔𝑜𝑓: 𝑅→𝑅 is: 𝑒−𝑥, 𝑥≤0 𝑒𝑥, 𝑥< 0 (1) one-one but not onto (2) neither one-one nor onto (3) onto but not one-one (4) both one-one and onto

Q72.Let f : [−1, 2] →R be given by f(x) = 2x2 + x + [x2] −[x], where [t] denotes the greatest integer less than or equal to t. The number of points, where f is not continuous, is : (1) 5 (2) 6 (3) 3 (4) 4

Q72.If the domain of the function f(x) = cos−1( 2−|x|4 ) equal to : (1) 12 (2) 9 (3) 11 (4) 8

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

Q72.Let the sum of the maximum and the minimum values of the function f(x) = 2x2+3x+82x2−3x+8 be mn , where gcd(m, n) = 1. Then m + n is equal to : (1) 195 (2) 201 (3) 217 (4) 182 2x , x < 0

Q72.Let y = loge( 1−x21+x2 ), (1) 732 (2) 746 (3) 742 (4) 736

Q72.Given that the inverse trigonometric function assumes principal values only. Let x, y be any two real numbers in [−1, 1] such that cos−1 x −sin−1 y = α, −π2 ≤α ≤π. Then, the minimum value of x2 + y2 + 2xy sin α is (1) 0 (2) -1 (3) 1 2 (4) −12 72x−9x−8x+1

Q72.Consider the function f : [ 12 , 1] →R defined by f(x) = 4√2x3 −3√2x −1. Consider the statements (I) The curve y = f(x) intersects the x-axis exactly at one point (II) The curve y = f(x) intersects the x-axis at x = cos 12π Then (1) Only (II) is correct (2) Both (I) and (II) are incorrect (3) Only (I) is correct (4) Both (I) and (II) are correct

Q72.If the function f(x) = sin 3x+α sin x−β cos 3x , x ∈R , is continuous at x = 0 , then f(0) is equal to : x3 (1) 2 (2) -2 (3) 4 (4) -4

Q72.Let the range of the function f(x) = 2+sin 3x+cos1 3x , x ∈R be [a, b]. If α and β are respectively the A.M. and the G.M. of a and b, then αβ is equal to (1) π (2) √π (3) 2 (4) √2

Q72.Suppose for a differentiable function h, h(0) = 0, h(1) = 1 and h′(0) = h′(1) = 2. If g(x) = h (ex)eh(x) , then g′(0) is equal to: (1) 5 (2) 4 (3) 8 (4) 3

Q72.If 𝑎= sin−1sin5 and 𝑏= cos−1cos5, then 𝑎2 + 𝑏2 is equal to (1) 4𝜋2 + 25 (2) 8𝜋2 −40𝜋+ 50 (3) 4𝜋2 −20𝜋+ 50 (4) 25

Q72.Let a and b be real constants such that the function 𝑓 defined by 𝑓𝑥= 𝑥2 + 3𝑥+ 𝑎, 𝑥≤1 be differentiable 𝑏𝑥+ 2, 𝑥> 1 2 on 𝑅. Then, the value of ∫-2 𝑓𝑥𝑑𝑥 equals 15 19 (1) (2) 6 6 (3) 21 (4) 17

Q72.The number of critical points of the function f(x) = (x −2)2/3(2x + 1) is (1) 1 (2) 2 (3) 0 (4) 3 6

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. 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) = ( 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.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.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.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.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.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.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 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