Practice Questions

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

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. (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.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 |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

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.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.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.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. 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.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

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.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 R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: (1) 10 (2) 7 (3) 8 (4) 9

Q14.The number of complex numbers z , satisfying |z| = 1 and z¯z + ¯zz = 1, is : (1) 4 (2) 8 (3) 10 (4) 6 Q15. ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡4⎤ ⎡0⎤ ⎡2 ⎤ ⎡1 ⎤ Let A = [aij] be 3 × 3 matrix such that A 1 = 0 , A 1 = 1 and A 1 = 0 , then a23 equals : ⎣ 0 ⎦ ⎣ 1 ⎦ ⎣3⎦ ⎣0⎦ ⎣2 ⎦ ⎣0 ⎦ (1) -1 (2) 2 (3) 1 (4) 0 2 x sin 2 dx equals : 3 3

Q14.The perpendicular distance, of the line x−1 2 = −1 = z+32 from the point P(2, −10, 1), is : (1) 6 (2) 5√2 (3) 4√3 (4) 3√5

Q14.Let Tr be the rth term of an A.P. If for some m, Tm = 251 , T25 = 201 , and 20 ∑25r=1 Tr = 13, then 5 m ∑2r=mm Tr is equal to (1) 98 (2) 126 (3) 142 (4) 112

Q15.Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of x is (1) 28/75 (2) 18/25 (3) 26/75 (4) 14/25 x > 0 and f(2) = 3. Then f(6) is equal to

Q15.Let ABC be a triangle formed by the lines 7x −6y + 3 = 0, x + 2y −31 = 0 and 9x −2y −19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y −53 = 0. Then h2 + k2 + hk is equal to: (1) 47 (2) 37 (3) 36 (4) 40 is:

Q15. x + y + 2z = 6 If the system of linear equations : 2x + 3y + az = a + 1 where a, b ∈R, has infinitely many solutions, then −x −3y + bz = 2 b 7a + 3b is equal to : (1) 16 (2) 12 (3) 22 (4) 9 = 0, y ∈(−π2 , π2 ) with

Q15. A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and B wins if he throws a sum of 8 before A throws a sum of 5 . The probability, that A wins if A makes the first throw, is (1) 8 (2) 9 17 19 (3) 9 (4) 8 17 19

Q15.Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C , whose mid-point is (1, 2), is : (1) √3 (2) 2√2 (3) 2√3 (4) 4√2

Q15.If f(x) = ∫ 1 dx, f(0) = −6, then f(1) is equal to : x1/4(1+x1/4) (1) 4 (loge 2 −2) (2) 2 −loge2 2 (3) loge 2 + 2 (4) 4 (loge 2 + 2)

Q16.A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ2 denote the mean and variance of X , then the value of 64 (μ + σ2) is : (1) 51 (2) 64 (3) 32 (4) 48