Practice Questions

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.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.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.The number of pairs 𝑎, 𝑏 of real numbers, such that whenever 𝛼 is a root of the equation 𝑥2 + 𝑎𝑥+ 𝑏= 0, 𝛼2 - 2 is also a root of this equation, is : (1) 6 (2) 8 (3) 4 (4) 2

Q61.Let n denote the number of solutions of the equation z2 + 3z = 0, where z is a complex number. Then the value of ∑∞k=0 nk1 is equal to (1) 1 (2) 34 (3) 32 (4) 2

Q61.Let α and β be the roots of x2 −6x −2 = 0. If an = αn −βn for n ⩾1, then the value of a10−2a83a9 is: (1) 1 (2) 3 (3) 2 (4) 4

Q61.Let a complex number be w = 1 −√3i . Let another complex number z be such that |zw| = 1 and arg(z) −arg(w) = π2 . Then the area of the triangle (in sq. units) with vertices origin, z and w is equal to (1) 4 (2) 12 (3) 1 (4) 2 4

Q61.The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is: (1) 77 (2) 82 (3) 42 (4) 35

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.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.Let α = max{82 sin 3x ⋅44 cos 3x} and β = min sin 3x ⋅44 cos 3x}. If 8x2 + bx + c = 0 is a quadratic equation x∈R x∈R{82 whose roots are α1/5 and β1/5, then the value of c −b is equal to : (1) 42 (2) 47 (3) 43 (4) 50 JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper

Q61.If (√3 + + + √3 = 0 1)x + √3 = 0 (2) x2 + (√3 1)x (1) x2 −(√3 = 0 = 0 (4) x2 −(√3 −1)x −√3 (3) x2 + (√3 −1)x −√3

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 ¯¯

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

Q62.The sum of the infinite series 1 + 32 + 327 + 1233 + 1734 + 2235 + … … is equal to: (1) 94 (2) 154 (3) 114 (4) 134

Q62.Let the lines (2 −i)z = (2 + i)z and (2 + i)z + (i −2)z −4i = 0, (here i2 = −1) be normal to a circle C . If ¯the line iz + z + 1 + i = 0 is tangent to this circle C , then its radius is : (1) 3 (2) 3√2 √2 (3) 3 (4) 1 2√2 2√2

Q62.Let C be the set of all complex numbers. Let S1 = {z ∈C |z–3–2i|2 = 8}, S2 = z ∈C| Re(z) ≥5 and ¯S3 = {z ∈C| |z–z| ≥8}. Then the number of elements in S1 ∩S2 ∩S3 is equal to (1) 1 (2) 0 (3) 2 (4) Infinite b ≠0, are equal, then the value of b is equal

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 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

Q62.Let a complex number z, |z| ≠1, satisfy log 1 |z|+11 ≤2 . Then, the largest value of |z| is equal to √2 ( (|z|−1)2 ) _________. (1) 8 (2) 7 (3) 6 (4) 5

Q62.The sum of the series ∑∞n=1 n2+6n+10(2n+1)! is equal to (1) 41 8 e + 198 e−1 + 10 (2) 418 e + 198 e−1 −10 (3) −418 e + 198 e−1 −10 (4) 418 e −198 e−1 −10 + + …

Q62.Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 −S6 is equal to: (1) 1862 (2) 1842 (3) 1852 (4) 1872

Q62.Let 𝑃1, 𝑃2 … , 𝑃15 be 15 points on a circle. The number of distinct triangles formed by points 𝑃𝑖, 𝑃𝑗, 𝑃𝑘 such that 𝑖+ 𝑗+ 𝑘≠15, is : (1) 455 (2) 419 (3) 12 (4) 443

Q62.If n ⩾2 is a positive integer, then the sum of the series n+1C2 + 2(2C2 + 3C2 + 4C2 + … + nC2) is (1) n(n−1)(2n+1) (2) n(n+1)(2n+1) 6 6 (3) n(n+1)2(n+2) (4) n(2n+1)(3n+1) 12 6