Practice Questions

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.Consider the matrices : A = [ 23 −5m ], B = [ 20m ] and X = [ xy ] . Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0 ), be the interval (a, b). Then 8 ∫ba |A|dm 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.If 𝑦= √𝑥+ 1𝑥2 −√𝑥 1 then 96𝑦'𝜋 is equal to: 𝑥√𝑥+ 𝑥+ √𝑥+ 153cos2𝑥−5cos3𝑥, 6 𝑥

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 [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.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 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.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 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 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.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.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 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 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

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.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.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 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.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 ______.