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 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 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.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.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.If the domain of the function 𝑓𝑥= √𝑥2 −25 + + 2𝑥−15 is −∞, 𝛼∪𝛽, ∞, then 𝛼2 + 𝛽3 is equal to: 4 −𝑥2 log10𝑥2 (1) 140 (2) 175 (3) 150 (4) 125

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.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.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.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.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

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.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

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.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

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 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.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.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 ∫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 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.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 :