Practice Questions

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 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 [t] denote the greatest integer less than or equal to t. Let f : [0, ∞) →R be a function defined by f(x) = [ x2 + 3] −[√x]. Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then ∑a∈S a 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 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.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.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 = [ 21 −11 ] . If the sum of the diagonal elements of A13 is 3n , then n is equal to_________

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.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.Let A = {(x, y) : 2x + 3y = 23, x, y ∈N} and B = {x : (x, y) ∈A}. Then the number of one-one functions from A to B is equal 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 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

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

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

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 _____.