Practice Questions

Q86.Let A = [ 21 −11 ] . If the sum of the diagonal elements of A13 is 3n , then n is equal to_________

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

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

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 _______

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

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.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.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

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

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