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 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 ______

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

Q21.If ∑30r=1 r2(30Cr)230Cr−1

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

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

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.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 -.

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

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.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 y = y(x) is the solution of the differential equation, ( 2 ), −2 ≤x ≤2, y(2) = 4 , then y2(0) is equal to √4 −x2 dxdy = ((sin−1 x 2 x π2−8 ( 2 )) −y) sin−1

Q23.If limt→0 (∫10 5)tdx)

Q23.Let y = y(x) be the solution of the differential equation 2 cos x dxdy = sin 2x −4y sin x, x ∈(0, π2 ). If y ( π3 ) = 0, then y′ ( π4 ) + y ( π4 ) 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 ________