Practice Questions

Q21.If →a and →b makes an angle cos−1 ( 9 ) with each other, then |→a + →b| = √2|→a −→b| value of n is ____

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.Let S1 = {z ∈C : |z| ≤5}, S2 = {z ( z+1−√3i1−√3i ) ≥0} area of the region S1 ∩S2 ∩S3 is : (1) 125π (2) 125π 12 4 (3) 125π (4) 125π 24 6 Q62.60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is : (1) JBBOH (2) OBBJH (3) OBBHJ (4) HBBJO

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

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

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.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.If z1, z2 are two distinct complex number such that z1−2z21 = 2, then 2 −z1¯z2 (1) z1 lies on a circle of radius 21 and z2 lies on a (2) both z1 and z2 lie on the same circle. both z1 and circle of radius 1 . z2 lie on the same circle. (3) either z1 lies on a circle of radius 21 or z2 lies on (4) either z1 lies on a circle of radius 1 or z2 lies on a a circle of radius 1 . circle of radius 1 . 2

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

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 be the distinct roots of the equation x2 −(t2 −5t + 6)x + 1 = 0, t ∈R and an = αn + βn . Then the minimum value of a2023+a2025 is a2024 (1) −1/4 (2) −1/4 (3) −1/2 (4) 1/4

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

Q62.The number of common terms in the progressions 4, 9, 14, 19, … …, up to 25th term and 3, 6, 9, 12,.... up to 37th term is : (1) 9 (2) 5 (3) 7 (4) 8 n

Q62.If 𝑧 is a complex number such that 𝑧≤1, then the minimum value of 𝑧+ 1 + 4𝑖 is: 23 5 (1) 2 (2) 2 3 (3) (4) 3 2

Q62.Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to (1) 18 (2) 16 (3) 12 (4) 15

Q62.Let 𝑆= 𝑧∈𝐶: 𝑧−1 = 1 and √2 −1𝑧+ ¯𝑧- 𝑖𝑧- ¯𝑧= 2√2. Let 𝑧1, 𝑧2 ∈𝑆 be such that 𝑧1 = max𝑧∈𝑠𝑧 and 2 𝑧2 = min𝑧∈𝑠𝑧. Then √2𝑧1 −𝑧2 equals: (1) 1 (2) 4 (3) 3 (4) 2

Q62.Let α and β be the sum and the product of all the non-zero solutions of the equation (¯z)2 + |z| = 0, z ∈ C. Then 4 (α2 + β2) is equal to : (1) 6 (2) 8 (3) 2 (4) 4

Q62.For 0 < 𝑐< 𝑏< 𝑎, let ( 𝑎+ 𝑏– 2𝑐) 𝑥2 + ( 𝑏+ 𝑐– 2𝑎) 𝑥+ ( 𝑐+ 𝑎– 2𝑏) = 0 and 𝛼≠1 be one of its root. Then, among the two statements (I) If 𝛼∈-1, 0, then 𝑏 cannot be the geometric mean of 𝑎 and 𝑐. (II) If 𝛼∈0, 1, then 𝑏 may be the geometric mean of 𝑎 and 𝑐. (1) Both (I) and (II) are true (2) Neither (I) nor (II) is true (3) Only (II) is true (4) Only (I) is true 1 2 3

Q62.Let z be a complex number such that the real part of z−2i is zero. Then, the maximum value of |z −(6 + 8i)| z+2i is equal to (1) 12 (2) 10 (3) 8 (4) ∞

Q62.Let z be a complex number such that |z + 2| = 1 and Im ( z+2 ) = 5 . Then the value of |Re(z + 2)| is (1) 2√6 (2) 24 5 5 (3) 1+√6 (4) √6 5 5

Q62.The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175

Q62.If the sum of the series 1 + 1 + … + 1 is equal to 5 , then 50 d is equal to : 1⋅(1+d) (1+d)(1+2 d) (1+9 d)(1+10 d) (1) 10 (2) 5 (3) 15 (4) 20

Q62.Let 𝑧1 and 𝑧2 be two complex number such that 𝑧1 + 𝑧2 = 5 and 𝑧13 + 𝑧23 = 20 + 15𝑖. Then 𝑧14 + 𝑧24 equals- (1) 30√3 (2) 75 (3) 15√15 (4) 25√3