Practice Questions

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 A = [ 21 −11 ] . If the sum of the diagonal elements of A13 is 3n , then n is equal to_________

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 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 𝑓: 0, ∞→𝑅 and 𝐹𝑥= ∫ 𝑡𝑓𝑡𝑑𝑡. If 𝐹𝑥2 = 𝑥4 + 𝑥5, then 12 𝑓𝑟2 is equal to: ∑𝑟= 1 0

Q86.Let a, b, c ∈N and a < b < c. Let the mean, the mean deviation about the mean and the variance of the 5 observations 9, 25, a, b, c be 18,4 and 1365 , respectively. Then 2a + b −c is equal to__________

Q86.From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is σ2 , then 96σ2 is equal to ______

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.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 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 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 𝑓: ℝ→ℝ be a function defined by 𝑓𝑥= 𝑓1 −𝑎 and 𝑀= ∫ 𝑥sin4𝑥1 −𝑥𝑑𝑥, 4𝑥+ 2 𝑓𝑎 𝑓1 −𝑎 𝑁= 𝛼𝑀= 𝛽𝑁, 𝛼, 𝛽∈ℕ, then the least value of 𝛼2 + 𝛽2 is equal to ______ ∫ sin4𝑥1 −𝑥𝑑𝑥; 𝑎≠12. If 𝑓𝑎 𝑥

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

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

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

Q87.Let fx = x 1 - t where g is a continuous odd function. If 2 fx + x2cosx dx = π 2 - α, then α is equal ∫0 gtloge 1 + tdt, ∫-π 1 + ex α 2 to _____.

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.If ∫cosec5 xdx = α cot x cosec x (cossc2 x + 32 ) + β logϵ tan x2 + C where α, β ∈R and C is the constant of integration, then the value of 8(α + β) equals _______

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.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.If a function f satisfies f( m + n) = f( m) + f(n) for all m, n ∈N and f(1) = 1, then the largest natural number λ such that ∑2022k=1 f(λ + k) ≤(2022)2 is equal to _________ JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper Q88. ⎧ ( 78 ) tantan 7x8x , 0 < π x < 2 a −8, x = π2 Let f : (0, π) →R be a function given by f(x) = ⎨ b | tan π x < π ⎩ (1 + | cot x|) x|, 2 < where a, b ∈Z. If f is continuous at x = π2 , then a2 + b2 is equal to

Q87.If the range of f(θ) = sin4 θ+3 cos2 θ , θ ∈R is [α, β] , then the sum of the infinite G.P., whose first term is 64 and sin4 θ+cos2 θ the common ratio is α , is equal to________ β