Practice Questions

Q85.Let f be a differentiable function in the interval (0, ∞) such that f(1) = 1 and limt→x t2f(x)−x2f(t)t−x = 1 each x > 0 . Then 2f(2) + 3f(3) is equal to _______

Q85.Let 𝐴= 1, 2, 3, 4 and 𝑅= ( 1, 2 ) , ( 2, 3 ) , ( 1, 4 ) be a relation on 𝐴. Let 𝑆 be the equivalence relation on 𝐴 such that 𝑅⊂𝑆 and the number of elements in 𝑆 is 𝑛. Then, the minimum value of 𝑛 is _______ 4𝑥

Q85.Consider two circles 𝐶1: 𝑥2 + 𝑦2 = 25 and 𝐶2: ( 𝑥- 𝛼) 2 + 𝑦2 = 16, where 𝛼∈( 5, 9 ) . Let the angle between . If the the two radii (one to each circle) drawn from one of the intersection points of C1and C2 be sin-1√638 length of common chord of 𝐶1 and 𝐶2 is 𝛽, then the value of ( 𝛼𝛽) 2 equals _________. JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper

Q85.The value of limx→0 2 ( 1−cos x√cos 2x3√cosx2 3x……10√cos 10x )

Q85.If α = limx→0+ e√tan x−e√x and β = limx→0(1 + sin x) 1 ( √tan x−√x ) 2 cot x are the roots of the quadratic equation ax2 + bx −√e = 0, then 12 loge(a + b) is equal to__________

Q85.Consider the function f : R →R defined by f(x) = 2x . If the composition of √1+9x2 f, (f ∘f ∘f ∘⋯∘f) (x) = 210x , then the value of √3α + 1 is equal to ______ √1+9αx2 10 times 

Q85.The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If 𝜇 and 𝜎2 denote the mean and variance of the correct observations respectively, then 15𝜇+ 𝜇2 + 𝜎2 is equal to _________. JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

Q85.Let L1, L2 be the lines passing through the point P(0, 1) and touching the parabola 9x2 + 12x + 18y −14 = 0. Let Q and R be the points on the lines L1 and L2 such that the △PQR is an isosceles triangle with base QR. If the slopes of the lines QR are m1 and m2 , then 16 (m21 + m22) is equal to _______

Q85.Let 𝐴= 1, 2, 3, . ...100 . Let 𝑅 be a relation on 𝐴 defined by 𝑥, 𝑦∈𝑅 if and only if 2𝑥= 3𝑦. Let 𝑅1 be a symmetric relation on 𝐴 such that 𝑅⊂𝑅1 and the number of elements in 𝑅1 is 𝑛. Then the minimum value of 𝑛 is _______.

Q85.Let the slope of the line 45x + 5y + 3 = 0 be 27r1 + 9r22 for some r1, r2 ∈R. Then 8t2 is equal to ______. lim 3 3r2x x→3(∫x 2 −r2x2−r1x3−3x dt)

Q85.If the points of intersection of two distinct conics x2 + y2 = 4b and x216 + y2b2 then 3√3 times the area of the rectangle formed by the intersection points is _______.

Q85.Let f(x) = x3 + x2f ′(1) + xf ′′(2) + f ′′′(3), x ∈R. Then f ′(10) is equal to + x −y, ∀x, y ∈(0, ∞). Then

Q85.Let 𝑥 denote the fractional part of 𝑥 and 𝑓𝑥= cos−11 −𝑥2sin−11 −𝑥 , 𝑥≠0. If 𝐿 and 𝑅 respectively denotes the 𝑥−𝑥3 32 left hand limit and the right hand limit of 𝑓𝑥 at 𝑥= 0, then 𝜋2𝐿2 + 𝑅2 is equal to __________.

Q85.A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.

Q86.Let A be a 2 × 2 real matrix and I be the identity matrix of order 2 . If the roots of the equation |A - xI | = 0 be -1 and 3, then the sum of the diagonal elements of the matrix A2 is _____. π

Q86.The number of elements in the set 𝑆= 𝑥, 𝑦, 𝑧: 𝑥, 𝑦, 𝑧∈𝑍, 𝑥+ 2𝑦+ 3𝑧= 42, 𝑥, 𝑦, 𝑧≥0 equals ________

Q86.If the variance 𝜎2 of the data xi 0 1 5 6 10 12 17 is 𝑘 then the value of 𝑘 is ______ {where . fi 3 2 3 2 6 3 3 denotes the greatest integer funciton}

Q86.Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax + by + c = 0 be concurrent at the point P and Q(α, β) be a point such that the system of equations x + y + z = 6, 2x + 5y + αz = β and x + 2 y + 3 z = 4, has infinitely many solutions. Then (PQ)2 is equal to _______. JEE Main 2024 (29 Jan Shift 2) JEE Main Previous Year Paper r be differentiable in (−∞, 0) ∪(0, ∞) and f(1) = 1. Then r2−x2 −r3e }

Q86.If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, α, β, 60 where α > β are 56 and 66. 2 respectively, then α2 + β2 is equal to JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q86. X α 1 0 −3 Let the mean and the standard deviation of the probability distribution be μ and σ, P(X) 31 K 16 41 respectively. If σ −μ = 2, then σ + μ is equal to________ JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

Q86.Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2 sin−1 x + 3 cos−1 x = 2π5 , is _______

Q86.Let A = {1, 2, 3, … . 7} and let P(A) denote the power set of A . If the number of functions f : A →P(A) such that a ∈f(a), ∀a ∈A is mn, m and n ∈N and m is least, then m + n is equal to ______. 1 , |x| ≥2 |x|

Q86.Let αβγ = 45; α, β, γ ∈R. If x(α, 1, 2) + y(1, β, 2) + z(2, 3, γ) = (0, 0, 0) for some x, y, z ∈R, xyz ≠0, then 6α + 4β + γ is equal to _______

Q86.Let f : R →R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = −1, f(3) = 2 and f(4) = −2. Then, the minimum number of zeros of (3f ′f ′′ + ff ′′′)(x) is _______

Q86.Let A be a non-singular matrix of order 3 . If det(3 adj(2 adj((det A)A))) = 3−13 ⋅2−10 and det(3 adj(2 A)) = 2m ⋅3n , then |3 m + 2n| is equal to