Practice Questions

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 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

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 𝐴 be a 3 × 3 matrix and det𝐴= 2. If 𝑛= det𝑎𝑑𝑗𝑎𝑑𝑗.⏟ ... 𝑎𝑑𝑗𝐴 , then the remainder when 𝑛 is divided by 9 2024 −times is equal to __________. 𝜋 Q87. 120 𝑥2sin𝑥cos𝑥 ∫ is equal to ______. 𝜋3 0 sin4𝑥+ cos4𝑥𝑑𝑥

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 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.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.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 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.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.Let for a differentiable function f : (0, ∞) →R, f(x) −f(y) ≥loge( xy ) ∑20n=1 f ′( n21 ) is equal to

Q87.If ∫ 0 4 1+sinsin2x xcos x dx = 1a loge ( 3a ) + b√3π , where

Q87.If the function f(x) = is differentiable on R, then 48(a + b) is equal to _______. { ax2 + 2b, |x| < 2 dx, where t denotes the greatest integer less than or equal to t, is _____. x+1

Q87.Let the area of the region {(x, y) : x −2y + 4 ≥0, x + 2y2 ≥0, x + 4y2 ≤8, y ≥0} be mn , where m and n are coprime numbers. Then m + n is equal to ______.

Q87.Let the maximum and minimum values of −x2 −12 2 + (x −7)2, x ∈R be M and m , (√8x −4) respectively. Then M2 −m2 is equal to _________ π

Q87.Let 𝐴= 1, 2, 3, . ..20 . Let 𝑅1 and 𝑅2 two relation on 𝐴 such that 𝑅1 = {𝑎, 𝑏: 𝑏 is divisible by 𝑎} 𝑅2 = {𝑎, 𝑏: 𝑎 is an integral multiple of 𝑏} Then, number of elements in 𝑅1 −𝑅2 is equal to __________. 𝛼𝜋+ 𝛽log𝑒3 + 2√2, where 𝛼, 𝛽 are integers, then 𝛼2 + 𝛽2 equals __________

Q87.Let f(x) = √limr→x{ 2r2[(f(r))2−f(x)f(r)] f(r) the value of ae , such that f(a) = 0, is equal to ______. π

Q87.Let [t] denote the largest integer less than or equal to t. If + = a + b√2 −√3 −√5 + c√6 −√7, where a, b, c ∈Z, then a + b + c is equal ∫30 ([x2] [ x22 ])dx to_______

Q87.Let f(x) = 2x −x2, x ∈R. If m and n are respectively the number of points at which the curves y = f(x) and y = f ′(x) intersects the x−axis, then the value of m + n is

Q87.The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _______.

Q87.The number of distinct real roots of the equation |x||x + 2| −5|x + 1| −1 = 0 is_______

Q87.Let A be the region enclosed by the parabola y2 = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the region A is________ + C, where C is the constant of integration, then the value of

Q87.For n ∈N , if cot−1 3 + cot−1 4 + cot−1 5 + cot−1 n = π4 , then n is equal to_____ ∫1 (1−x7)kdx 0

Q87.Let 𝑆= −1, ∞ and 𝑓: 𝑆→ℝ be defined as 𝑓𝑥= ∫ 𝑒𝑡−1112𝑡−15𝑡−27𝑡−3122𝑡−1061𝑑𝑡. Let 𝑝= Sum −1 of square of the values of 𝑥, where 𝑓𝑥 attains local maxima on 𝑆. and 𝑞= Sum of the values of 𝑥, where 𝑓𝑥 attains local minima on 𝑆. Then, the value of 𝑝2 + 2𝑞 is ________ 𝜋 1 Q88. 2 11 5 If the integral 525 ∫ sin2𝑥 cos 2 𝑥1 + cos 2𝑥 2𝑑𝑥 is equal to 𝑛√2 −64, then 𝑛 is equal to ________ 0 → → → → →

Q87.Three points 𝑂0, 0, 𝑃𝑎, 𝑎2, 𝑄−𝑏, 𝑏2, 𝑎> 0, 𝑏> 0, are on the parabola 𝑦= 𝑥2. Let 𝑆1 be the area of the region bounded by the line 𝑃𝑄 and the parabola, and 𝑆2 be the area of the triangle 𝑂𝑃𝑄. If the minimum value 𝑆1 𝑚 of is 𝑛, gcd𝑚, 𝑛= 1, then 𝑚+ 𝑛 is equal to: 𝑆2 JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper