Practice Questions

Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e

Q11.Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and ( 0, 0, 1 ). Let B = (1, 4, −1) and C = (2, 0, −2). Then among the statements (S1) : △ABC is an isosceles right angled triangle, and (S2) : the area of △ABC is 9√22 , (1) both are true (2) only (S2) is true (3) only (S1) is true (4) both are false

Q11.Let A = [aij] = [ log5log51288 log4log4255 ] . If Aij is the cofactor of aij, Cij = ∑2k=1 aikAjk, 1 ≤i, j ≤2, and C = [Cij], then 8|C| is equal to : (1) 288 (2) 222 (3) 242 (4) 262

Q11.Let the area enclosed between the curves |y| = 1 −x2 and x2 + y2 = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β −γ| equals. (1) 27 (2) 33 (3) 15 (4) 18

Q11.Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum is : values of 16 ((sec−1 x)2 + (cosec−1 x)2) (1) 24π2 (2) 22π2 (3) 31π2 (4) 18π2

Q11.Let the position vectors of three vertices of a triangle be 4→p + →q −3→r, −5→p + →q + 2→r and 2→p−→q+ 2→r. If the →p+→q+→r position vectors of the orthocenter and the circumcenter of the triangle are and α→p + β→q + γ→r 4 respectively, then α + 2β + 5γ is equal to : (1) 3 (2) 4 (3) 1 (4) 6 → →

Q11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then

Q11.The area of the region {(x, y) : x2 + 4x + 2 ≤y ≤|x + 2|} is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3

Q12.Let Sn = 12 + 16 + 121 + 201 + … upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is √2026 S2025 , then the absolute difference betwen 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25

Q12. (λ −1)x + (λ −4)y + λz = 5 If the system of equations λx + (λ −1)y + (λ −4)z = 7 has infinitely many solutions, then λ2 + λ is (λ + 1)x + (λ + 2)y −(λ + 2)z = 9 equal to (1) 6 (2) 10 (3) 20 (4) 12

Q12.Let f : R →R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y ∈R. If f ′(0) = 4a and f satisfies f ′′(x) −3af ′(x) −f(x) = 0, a > 0, then the area of the region R = {(x, y) ∣0 ≤y ≤f(ax), 0 ≤x ≤2} is: (1) e2 −1 (2) e2 + 1 (3) e4 + 1 (4) e4 −1

Q12.The remainder, when 7103 is divided by 23 , is equal to : (1) 6 (2) 17 (3) 9 (4) 14

Q12.Let →a = 3^i −^j + 2^k, b =→a× (^i −2^k) and→c= b × ^k. Then the projection of→c−2^j on →a is : (1) 2√14 (2) √14 (3) 3√7 (4) 2√7

Q12.Let |z1 −8 −2i| ≤1 and |z2 −2 + 6i| ≤2, z1, z2 ∈C . Then the minimum value of |z1 −z2| is : (1) 13 (2) 10 (3) 3 (4) 7

Q12.For positive integers n, if 4an = (n2 + 5n + 6) and Sn = ∑nk=1 ( ak1 ), then the value of (1) 540 (2) 675 (3) 1350 (4) 135

Q12.Let x = x(y) be the solution of the differential equation y = (x −y dxdy ) ( xy ), cos(x(2)) is equal to : (1) 1 −2(loge 2)2 (2) 1 −2 (loge 2) (3) 2 (loge 2) −1 (4) 2(loge 2)2 −1

Q12.Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A →B such that 1 ∈f( A) is equal to : (1) 151 (2) 139 (3) 163 (4) 127

Q12.The area (in sq. units) of the region {(x, y) : 0 ≤y ≤2|x| + 1, 0 ≤y ≤x2 + 1, |x| ≤3} is (1) 80 (2) 64 3 3 (3) 32 (4) 17 3 3

Q13. The number of real solution(s) of the equation x2 + 3x + 2 = min{|x −3|, |x + 2|} is : (1) 1 (2) 0 (3) 2 (4) 3

Q13.The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is (1) 36000 (2) 37000 (3) 34000 (4) 35000

Q13.Let f : R −{0} →R be a function such that f(x) −6f ( x1 ) = 3x35 −52 . If the limx→0 ( αx1 + f(x)) = β; α, β ∈R, then α + 2β is equal to (1) 5 (2) 3 (3) 4 (4) 6 n > 0, then I(9, 14) + I(10, 13) is

Q13.Let f : R −{0} →(−∞, 1) be a polynomial of degree 2, satisfying f(x)f ( x1 ) = f(x) + f ( x1 ). If f(K) = −2K , then the sum of squares of all possible values of K is : (1) 7 (2) 6 (3) 1 (4) 9 and a

Q13.The sum, of the squares of all the roots of the equation x2 + |2x −3| −4 = 0, is (1) 3(3 −√2) (2) 6(3 −√2) (3) 6(2 −√2) (4) 3(2 −√2)

Q13.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 1 cm/min. The surface area (in cm2 ) of the chocolate ball (without the ice- 4π cream layer) is : (1) 196π (2) 256π (3) 225π (4) 128π

Q13.If αx + βy = 109 is the equation of the chord of the ellipse x29 + y24 = 1 , whose mid point is ( 52 , 12 ), then α + β is equal to : (1) 58 (2) 46 (3) 37 (4) 72