Practice Questions

Q19.Let I(x) = ∫ 11 15 . If I(37) −I(24) = 4 1 − 1 b, c ∈N (x−11) 13 (x+15) 13 ( b 13 c 13 ), (1) 22 (2) 39 (3) 40 (4) 26

Q19.Let S = N ∪{0}. Define a relation R from S to R by : R = {(x, y) : loge y = x loge ( 25 ), x ∈ S, y ∈R} Then, the sum of all the elements in the range of R is equal to : (1) 10 (2) 3 9 2 (3) 5 (4) 5 2 3

Q20.Let the area of the region {(x, y) : 2y ≤x2 + 3, y + |x| ≤3, y ⩾|x −1|} be A. Then 6 A is equal to : (1) 16 (2) 12 (3) 14 (4) 18

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

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.Let E : x2 + y2 = 1, a > b and H : x2 − y2 = 1. Let the distance between the foci of E and the foci of H a2 b2 A2 B2 be 2√3. If a −A = 2, and the ratio of the eccentricities of E and H is 13 , then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7 = α × 229 , then α is equal to ______

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.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

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.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

Q20.Let z1, z2 and z3 be three complex numbers on the circle |z| = 1 with arg (z1) = −π4 , arg (z2) = 0 and arg (z3) = π4 . If |z1¯z2 + z2¯z3 + z3¯z1|2 = α + β√2, α, β ∈Z, then the value of α2 + β2 is : (1) 24 (2) 29 (3) 41 (4) 31

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

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 ______.

Q64. Choose the correct answer from the options given below : (1) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (2) (A)-(I), (B)-(III), (C)-(II), (D)-(IV) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

Q68. Choose the correct answer from the options given below : (1) (A)-(IV), (B)-(III), (C)-(II), (D)-(I) (2) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)

Q61.Let r and θ respectively be the modulus and amplitude of the complex number z = 2 −i(2 tan 5π8 ), then (r, θ) is equal to (1) (2 sec 3π8 , 3π8 ) (2) (2 sec 3π8 , 5π8 ) (3) (2 sec 5π8 , 3π8 ) (4) (2 sec 11π8 , 11π8 )

Q61.Let 𝑆= 𝑥∈𝑅: √3 + √2 𝑥+ √3 −√2 𝑥= 10. Then the number of elements in 𝑆 is: (1) 4 (2) 0 (3) 2 (4) 1

Q61.Let α, β; α > β , be the roots of the equation x2 −√2x −√3 = 0. Let Pn = αn −βn, n ∈N . Then (11√3 −10√2)P10 + (11√2 + 10)P11 −11P12 is equal to (1) 10√3P9 (2) 11√3P9 (3) 10√2P9 (4) 11√2P9

Q61.If z = 21 −2i, is such that z + 1 = αz + β(1 i), (1) −4 (2) 3 (3) 2 (4) −1

Q61.Let 𝑆 be the set of positive integral values of 𝑎 for which 𝑎𝑥2 + 2𝑎+ 1𝑥+ 9𝑎+ 4 < 0, ∀𝑥∈ℝ. Then, the number 𝑥2 - 8𝑥+ 32 of elements in 𝑆 is: (1) 1 (2) 0 (3) ∞ (4) 3

Q61.Let α, β be the roots of the equation x2 + 2√2x −1 = 0. The quadratic equation, whose roots are α4 + β4 and 1 (α6 + β6), is : 10 (1) x2 −190x + 9466 = 0 (2) x2 −180x + 9506 = 0 (3) x2 −195x + 9506 = 0 (4) x2 −195x + 9466 = 0

Q61.The number of solutions, of the equation 𝑒sin𝑥−2𝑒−sin𝑥= 2 is (1) 2 (2) more than 2 (3) 1 (4) 0

Q61.The sum of all the solutions of the equation (8)2x −16 ⋅(8)x + 48 = 0 is : (1) 1 + log8(6) (2) 1 + log6(8) (3) log8(6) (4) log8(4) –z+1 1

Q61.If S = z ∈C : |z −i| = |z + i| = |z −1|, then, n(S) is: (1) 1 (2) 0 (3) 3 (4) 2

Q61.Let 𝛼 and 𝛽 be the roots of the equation 𝑝𝑥2 + 𝑞𝑥−𝑟= 0, where 𝑝≠0. If 𝑝, 𝑞 and 𝑟 be the consecutive terms of a non-constant G.P and 1 1 3 then the value of 𝛼−𝛽2 is: 𝛼+ 𝛽= 4, (1) 80 (2) 9 9 20 (3) (4) 8 3