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

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.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.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.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

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

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

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.The function f: R->R, f(x) = x2+2x−15 , x ∈R is x2−4x+9 (1) one-one but not onto. (2) both one-one and onto. (3) onto but not one-one. (4) neither one-one nor onto. JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper 1 ), x ≠0 x then

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

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.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. 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 f(x) = {x30 sin, x (= 0 (1) f ′′ ( π2 ) = 24−π22π (2) f ′′ ( π2 ) = 12−π22π (3) f ′′(0) = 1 (4) f ′′(0) = 0

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 𝑓: 𝑅→𝑅 be defined as 𝑎−𝑏cos2𝑥 ; 𝑥< 0 𝑥2 𝑓𝑥= 𝑥2 + 𝑐𝑥+ 2; 0 ≤𝑥≤1 2𝑥+ 1; 𝑥> 1 If 𝑓 is continuous everywhere in 𝑅 and 𝑚 is the number of points where 𝑓 is NOT differential then 𝑚 + 𝑎 + 𝑏 + 𝑐 equals: JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper (1) 1 (2) 4 (3) 3 (4) 2 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 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.Let 𝑓𝑥= 2𝑥2 + 5𝑥- 3, 𝑥∈𝑅. If 𝑚 and 𝑛 denote the number of points where 𝑓 is not continuous and not differentiable respectively, then 𝑚+ 𝑛 is equal to: (1) 5 (2) 2 (3) 0 (4) 3

Q73.Let g : R →R be a non constant twice differentiable such that g′( 21 ) = g′( 23 ). If a real valued function f is defined as f(x) = 12 [ g(x) + g(2 −x)], then (1) f ′′(x) = 0 for atleast two x in (0, 2) (2) f ′′(x) = 0 for exactly one x in (0, 1) (3) f ′′(x) = 0 for no x in (0, 1) (4) f ′( 23 ) + f ′( 21 ) = 1