Practice Questions

Q61.Let α, β be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0 . Then α8 + β8 is equal to JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper (1) 10 (2) 100 (3) 50 (4) 160

Q61.Let 𝑆𝑛 be the sum of the first 𝑛 terms of an arithmetic progression. If 𝑆3𝑛= 3𝑆2𝑛, then the value of 𝑆4𝑛 is : 𝑆2𝑛 JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper (1) 6 (2) 4 (3) 2 (4) 8

Q61.The sum of the roots of the equation, 𝑥+ 1 - 2log23 + 2𝑥+ 2log410 - 2-𝑥= 0, is : (1) log214 (2) log212 (3) log213 (4) log211

Q61.The value of 4 + 1 1 is: 5+ 1 4+ 1 5+ 4+……∞ (1) 2 + 52 √30 (2) 2 + √54 √30 (3) 4 + 4 √30 (4) 5 + 25 √30 √5

Q61.If for x ∈(0, π2 ), log10 sin x + log10 cos x = −1 and log10(sin x + cos x) = 12 (log10 n −1), n > 0 , then the value of n is equal to : (1) 20 (2) 12 (3) 9 (4) 16

Q61.If α and β are the distinct roots of the equation x2 + (3) 1 1 4 x + 3 2 = 0 , then the value of α96(α12 −1) + β96(β12 −1) is equal to: (1) 56 × 325 (2) 56 × 324 (3) 52 × 324 (4) 28 × 325

Q61.A natural number has prime factorization given by n = 2x3y5z , where y and z are such that y + z = 5 and y−1 + z−1 = 56 , y > z. Then the number of odd divisors of n, including 1 , is: (1) 12 (2) 6 (3) 11 (4) 6x

Q61.If x2 + 9y2 −4x + 3 = 0, x, y ∈R, then x and y respectively lie in the intervals (1) [−13 , 13 ] and [−13 , 13 ] (2) [1, 3] and [−13 , 13 ] (3) [−13 , 13 ] and [1, 3] (4) [1, 3] and [1, 3]

Q61.The least value of |z| where z is complex number which satisfies the inequality ||z|+1| loge 2) e( (|z|+3)(|z|−1) ≥log√2 5√7 + 9i , i = √−1, is equal to : (1) 3 (2) √5 (3) 2 (4) 8

Q61.The number of real solutions of the equation, x2 −|x| −12 = 0 is: (1) 2 (2) 3 (3) 1 (4) 4

Q61.The sum of 10 terms of the series 3 + 5 + 7 + … is : 12×22 22×32 32×42 (1) 143 (2) 99 144 100 (3) 1 (4) 120121

Q61.Let S1, S2 and S3 be three sets defined as : z −1 S1 = ≤√2}, {z ∈C S2 = {z ∈C : Re((1 −i)z) ≥1} and S3 = {z ∈C : Im(z) ≤1}. Then, the set S1 ∩S2 ∩S3 (1) is a singleton (2) has exactly two elements (3) has infinitely many elements (4) has exactly three elements

Q61.The value of 3 + 1 1 is equal to 4+ 1 3+ 1 4+ 3+…∞ (1) 1. 5 + √3 (2) 2 + √3 (3) 3 + 2√3 (4) 4 + √3 ¯¯

Q61.Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be ( 103 , 73 ). If α, β are the roots of the equation ax2 + bx + 1 = 0, then the value of α2 + β2 −αβ is: (1) −71256 (2) 25669 (3) 256 71 (4) −69256

Q61.The equation arg( z+1z−1 ) = π4 represents a circle with: (1) centre at (0, 0) and radius √2 (2) centre at (0, 1) and radius 2 (3) centre at (0, −1) and radius √2 (4) centre at (0, 1) and radius √2 22

Q61.Let 𝑝 and 𝑞 be two positive numbers such that 𝑝+ 𝑞= 2 and 𝑝4 + 𝑞4 = 272. Then 𝑝 and 𝑞 are roots of the equation: (1) 𝑥2 - 2𝑥+ 2 = 0 (2) 𝑥2 - 2𝑥+ 8 = 0 (3) 𝑥2 - 2𝑥+ 136 = 0 (4) 𝑥2 - 2𝑥+ 16 = 0

Q61.The integer k, for which the inequality x2 −2(3k −1)x + 8k2 −7 > 0 is valid for every x in R is: (1) 4 (2) 2 (3) 3 (4) 0 JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper ¯¯

Q62.The area of the triangle with vertices P(z), Q(iz) and R(z + iz) is (1) 1 (2) 12 z 2 (3) 1 (4) 1 z + iz 2 2 2

Q62.The sum of the series 1 + 2 + + … + 2100 when x = 2 is: x+1 x2+1 x4+1 x2100+1 (1) 1 − 2101 (2) 1 + 2101 4101−1 4101−1 (3) 1 + 2100 (4) 1 − 2100 4101−1 4201−1

Q62.Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3r2, then r2 −d is equal to : (1) 7 −√3 (2) 7 + 3√3 (3) 7 −7√3 (4) 7 + √3

Q62.A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = √5 inches and ∠PCB = tan−1(2). The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is : (1) tan−1( 43 ) (2) tan−1( 21 ) (3) tan−1( 43 ) (4) tan−1(1)

Q62.The sum of all those terms which are rational numbers in the expansion of 1 1 12 3 + 3 4 (2 ) is: (1) 89 (2) 27 (3) 35 (4) 43 , then the

Q62.If 0 < x < 1 and y = 21 x2 + 32 x3 + 43 x4 + … … , then the value of e1+y at x = 21 is: (1) 1 e2 (2) 2e 2 (3) 2e2 (4) 21 √e

Q62.If S = {z ∈C : z+2iz−i ∈R}, then (1) S is a circle in the complex plane (2) S contains exactly two elements (3) S contains only one element (4) S is a straight line in the complex plane

Q62.If sum of the first 21 terms of the series log91/2 x + log91/3 x + log91/4 x + … . . where x > 0 is 504, then x is equal to (1) 243 (2) 9 (3) 7 (4) 81