Practice Questions

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 2 and 6 are the roots of the equation ax2 + bx + 1 = 0, then the quadratic equation, whose roots are 2a+b1 and 1 , is : 6a+b (1) 2x2 + 11x + 12 = 0 (2) x2 + 8x + 12 = 0 (3) 4x2 + 14x + 12 = 0 (4) x2 + 10x + 16 = 0

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 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.If 𝛼, 𝛽 are the roots of the equation, x2 - x - 1 = 0 and Sn = 2023𝛼n + 2024𝛽n, then (1) 2 S12 = S11 + S10 (2) S12 = S11 + S10 (3) 2 S11 = S12 + S10 (4) S11 = S10 + S12 4! ! 5! ( ) (

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.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 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.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

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.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 1 + 1 + … + 1 = m and 1⋅21 + 2⋅31 + … + 99⋅1001 = n , then the point (m, n) lies on the √1+√2 √2+√3 √99+√100 line (1) 11(x −1) −100(y −2) = 0 (2) 11x −100y = 0 (3) 11(x −2) −100(y −1) = 0 (4) 11(x −1) −100y = 0

Q62.The value of 1×22+2×32+…+100×(101)2 is 12×2+22×3+….+1002×101 (1) 32 (2) 31 31 30 (3) 306 (4) 305 305 301 JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

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.Let 0 ≤r ≤n. If n+1Cr+1 : nCr : n−1Cr−1 = 55 : 35 : 21, then 2n + 5r is equal to: JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper (1) 50 (2) 62 (3) 55 (4) 60

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

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 𝛼= and 𝛽= 3! )4!.! Then : 4! 5! ( ( ) ) (1) 𝛼∈N and 𝛽∉N (2) 𝛼∉N and 𝛽∈N (3) 𝛼∈N and 𝛽∈N (4) 𝛼∉N and 𝛽∉N

Q62.The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is (1) 48 (2) 56 (3) 24 (4) 16

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

Q63.Let a, ar, ar2 , be an infinite G.P. If ∑∞n=0 arn = 57 and ∑∞n=0 a3r3n = 9747, then a + 18r is equal to (1) 46 (2) 38 (3) 31 (4) 27 is

Q63.In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70 and the 3 product of the third and fifth terms is 49 . Then the sum of the 4th , 6th and 8th terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78

Q63.If 𝑛 is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then 𝑛 is equal to: (1) 47 (2) 53 (3) 51 (4) 43