Practice Questions

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

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 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.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.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 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.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.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 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 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.Suppose that the number of terms in an A.P. is 2k, k ∈N . If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to : (1) 6 (2) 5 (3) 8 (4) 4 y+2

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.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

Q13.The area of the region, inside the circle (x −2√3)2 + y2 = 12 and outside the parabola y2 = 2√3x is : (1) 3π + 8 (2) 6π −16 (3) 3π −8 (4) 6π −8

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.Let L1 : x−11 = y−2−1 = z−12 and L2 : x+1−1 = y−22 = 1z be two lines. Let L3 be a line passing through the point (α, β, γ) and be perpendicular to both L1 and L2 . If L3 intersects L1 , then |5α −11β −8γ| equals : (1) 20 (2) 18 (3) 25 (4) 16

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

Q14.Let the foci of a hyperbola be (1, 14) and (1, −12). If it passes through the point (1, 6), then the length of its latus-rectum is : (1) 24 (2) 25 5 6 (3) 144 (4) 288 5 5 is equal to :

Q14.Let M and m respectively be the maximum and the minimum values of 1 + sin2 x cos2 x 4 sin 4x f(x) = sin2 x 1 + cos2 x 4 sin 4x , x ∈R Then M 4 −m4 is equal to : sin2 x cos2 x 1 + 4 sin 4x (1) 1280 (2) 1295 (3) 1215 (4) 1040

Q14. The function f : (−∞, ∞) →(−∞, 1), defined by f(x) = 2x−2−x2x+2−x is : (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto

Q14. IfI(m, n) = ∫10 xm−1(1 −x)n−1dx, m, (1) I(19, 27) (2) I(9, 1) (3) I(1, 13) (4) I(9, 13)

Q14.If A and B are the points of intersection of the circle x2 + y2 −8x = 0 and the hyperbola x29 −y24 = 1 point P moves on the line 2x −3y + 4 = 0, then the centroid of △PAB lies on the line : (1) x + 9y = 36 (2) 4x −9y = 12 (3) 6x −9y = 20 (4) 9x −9y = 32