Practice Questions

Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

Q2. Let f : R →R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a−1 )x + b, a ≠1. If f(x + y) = f(x) + f(y) + 1 −27 xy , then the value of 28 ∑5i=1 |f(i)| is (1) 545 (2) 715 (3) 735 (4) 675

Q21.Let A be a square matrix of order 3 such that det(A) = −2 and det(3 adj(−6 adj(3A))) = 2m+n ⋅3mn, m > n . Then 4 m + 2n is equal to _______ , then m −n is equal to _______

Q21.If ∑30r=1 r2(30Cr)230Cr−1

Q21.Let S = {x : cos−1 x = π + sin−1 x + sin−1(2x + 1)}. Then ∑x∈ S(2x −1)2 is equal to ______.

Q21.Let P be the image of the point Q(7, −2, 5) in the line L : x−12 = y+13 = 4z and R(5, p, q) be a point on Then the square of the area of △PQR is ________. x + 1 + C, where C is the

Q21.If 24 ∫ 0 4 (sin 4x − 12π + [2 sin x])dx = 2π + α, where [⋅] denotes the greatest integer function, then α is equal to _______.

Q21.Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B , and a denotes the area of △SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is -

Q21.Let the circle C touch the line x −y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 4 along the line −3x + 2y = 1. Let H be the hyperbola x2 −y2 = 1, whose one of the foci is the √13 α2 β2 centre of C and the length of the transverse axis is the diameter of C . Then 2α2 + 3β2 is equal to ______

Q22.Let f : (0, ∞) →R be a twice differentiable function. If for some a ≠0, ∫10 f(λx)dλ = af(x), f(1) = 1 and f(16) = 18 , then 16 −f ′ ( 161 ) is equal to _______.

Q22.Let a1, a2, … , a2024 be an Arithmetic Progression such that a1 + (a5 + a10 + a15 + … + a2020) + a2024 = 2233. Then a1 + a2 + a3 + … + a2024 is equal to _______ 1 2 3 , then α is equal to ________ (3x + t = 5eα ( 85 )

Q22.If for some α, β; α ≤β, α + β −8 and sec2 (tan−1 α) + cosec2 (cot−1 β) −36, then α2 + β is_______. Q23. ⎡x⎤ Let A be a 3 × 3 matrix such that X TAX = O for all nonzero 3 × 1 matrices X = y . If ⎣z ⎦ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡1 ⎤ ⎡ 0 ⎤ A 1 = 4 , A 2 = 4 , and det(adj(2(A + 1))) −2α3β5γ, α, β, γ ∈N , then α2 + β2 + γ 2 ⎣ 1⎦ ⎣ −5 ⎦ ⎣1⎦ ⎣−8 ⎦ is_____. x ≥0. Then

Q22.If ∫2x2+5x+9 dx = x√x2 + x + 1 + α√x2 + x + 1 + β loge x + 12 + √x2 + √x2+x+1 constant of integration, then α + 2β is equal to _______.

Q22.The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1

Q22.Let A = {1, 2, 3}. The number of relations on A , containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is ______ -

Q22.If the equation a(b −c)x2 + b(c −a)x + c(a −b) = 0 has equal roots, where a + c = 15 and b = 365 , then a2 + c2 is equal to

Q22.Let M denote the set of all real matrices of order 3 × 3 and let S = {−3, −2, −1, 1, 2}. Let S1 = {A = [aij] ∈M : A = AT and aij ∈ S, ∀i, j}, S2 = {A = [aij] ∈M : A = −AT and aij ∈ S, ∀i, j}, S3 = {A = [aij] ∈M : a11 + a22 + a33 = 0 and aij ∈ S, ∀i, j}. If n ( S1 ∪2 US3) = 125α, then α equals _______

Q22.The roots of the quadratic equation 3x2 −px + q = 0 are 10th and 11th terms of an arithmetic progression with common difference 32 . If the sum of the first 11 terms of this arithmetic progression is 88 , then q −2p is equal to -.

Q23.The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

Q23. If α = 1 + ∑6r=1(−3)r−1 12C2r−1 , then the distance of the point (12, √3) from the line αx −√3y + 1 = 0 is _________. be an ellipse. Ellipses E1 's are constructed such that their centres and eccentricities are

Q23.If limt→0 (∫10 5)tdx)

Q23.The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is -

Q23.If the set of all values of a, for which the equation 5x3 −15x −a = 0 has three distinct real roots, is the interval (α, β), then β −2α is equal to ______

Q23.Let →c be the projection vector of →b = λ^i + 4^k, λ > 0, on the vector →a = ^i + 2^j + 2^k. If |→a + →c| = 7, then the area of the parallelogram formed by the vectors →b and →c is ________