Practice Questions

Q19.The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8 , is (1) 4608 (2) 5720 (3) 5719 (4) 4607

Q19.Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to : (1) √14 (2) 3√7 (3) 2√14 (4) 5√7

Q20.If the area of the region {(x, y) : −1 ≤x ≤1, 0 ≤y ≤a + e|x| −e−x, a > 0} is e2+8e+1e , then the value of is : (1) 8 (2) 7 (3) 5 (4) 6

Q20.If π 2 ≤x ≤3π4 , then cos−1 ( 1213 cos x + 135 sin x) is equal to (1) x −tan−1 43 (2) x + tan−1 45 (3) x −tan−1 125 (4) x + tan−1 125

Q20.If α > β > γ > 0, then the expression cot−1 {β (α−β) } + cot−1 {γ (β−γ) } + cot−1 {α (γ−α) } equal to : (1) π (2) 0 (3) π 2 −(α + β + γ) (4) 3π L.

Q20.If sin x + sin2 x = 1, x ∈(0, π2 ), then (cos12 x + tan12 x) + 3 (cos10 x + tan10 x + cos8 x + tan8 x) + (cos6 x + tan6 x) is equal to : (1) 4 (2) 1 (3) 3 (4) 2 π

Q21.If ∑30r=1 r2(30Cr)230Cr−1

Q21.Let S = {p1, p2 … . , p10} be the set of first ten prime numbers. Let A = S ∪P , where P is the set of all possible products of distinct elements of S . Then the number of all ordered pairs ( x, y ), x ∈S , y ∈A , such that x divides y, is ______.

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

Q21.The variance of the numbers 8, 21, 34, 47, … , 320 is

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.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.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 ∫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.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.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.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 ________

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.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.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 ______

Q24.The interior angles of a polygon with n sides, are in an A.P. with common difference 6∘ . If the largest interior angle of the polygon is 219∘ , then n is equal to . Then limx→0 (x−f(x))ex−ef(x) is equal to

Q24.Let E1 : x29 + y24 = 1 same as that of E1 , and the length of minor axis of Ei is the length of major axis of Ei+1(i ≥1). If Ai is the area of the ellipse Ei , then π5 (∑∞i=1 Ai), is equal to → → →

Q24.The focus of the parabola y2 = 4x + 16 is the centre of the circle C of radius 5 . If the values of λ, for which C passes through the point of intersection of the lines 3x −y = 0 and x + λy = 4, are λ1 and λ2, λ1 < λ2 , then 12λ1 + 29λ2 is equal to