Practice Questions

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

Q72.If 𝑓𝑥= 4𝑥+ 3 , 𝑥≠2 and ( 𝑓𝑜𝑓) ( 𝑥) = 𝑔( 𝑥) , where 𝑔: 𝑅- 2 →𝑅- 2 then ( 𝑔𝑜𝑔𝑜𝑔) ( 4 ) is equal to 6𝑥- 4 3 3 3, 19 19 (1) - (2) 20 20 (3) -4 (4) 4 Q73. 𝑔𝑥, 𝑥≤0 Let 𝑔𝑥 be a linear function and 𝑓𝑥= 1 , is continuous at 𝑥= 0. If 𝑓'1 = 𝑓−1, then the value of 1 + 𝑥 𝑥, 𝑥> 0 2 + 𝑥 𝑔3 is JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper 1 4 1 4 (1) 3log𝑒 1 (2) 3log𝑒 9 + 1 9𝑒 3 4 4 (3) log𝑒 9 −1 (4) log𝑒 1 9𝑒 3 𝑥𝑦𝑥- 1𝑥- 2

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

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.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.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 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 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(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 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→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.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 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.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.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