Practice Questions

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

Q62.If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to: (1) 364 (2) 240 (3) 333 (4) 360

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.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.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.If α, β ∈R are such that 1 −2i (here i2 = −1) is a root of z2 + αz + β = 0, then (α −β) is equal to: (1) −7 (2) 7 (3) −3 (4) 3

Q62.If the equation a z 2 + αz + αz + d = 0 represents a circle where a, d are real constants then which of the following condition is correct? (1) |α|2 −ad ≠0 (2) |α|2 −ad > 0 and a ∈R −{0} (3) |α|2 −ad ≥0 and a ∈R (4) α = 0, a, d ∈R+

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.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.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 C be the set of all complex numbers. Let S1 = {z ∈C : |z −2| ≤1} and 2 ¯S2 = {z ∈C : z(1 + i) + z(1 −i) ≥4}. Then, the maximum value of z −52 for z ∈S1 ∩S2 is equal to : (1) 3+2√2 (2) 5+2√2 4 2 (3) 3+2√2 (4) 5+2√2 2 4

Q62.The number of solutions of the equation 32tan2𝑥+ 32sec2𝑥= 81, 0 ≤𝑥≤ 𝜋 is : 4 (1) 0 (2) 2 (3) 1 (4) 3 JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper 𝑧- 𝑖

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.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.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.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

Q62.A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is: (1) 1050 (2) 1625 (3) 575 (4) 560

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 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.If 𝑏 is very small as compared to the value of 𝑎, so that the cube and other higher powers of 𝑏 can be neglected 𝑎 in the identity 1 1 1 1 … . + 𝛼𝑛+ 𝛽𝑛2 + 𝛾𝑛3 𝑎- 𝑏+ 𝑎- 2𝑏+ 𝑎- 3𝑏+ 𝑎- 𝑛𝑏= then the value of 𝛾 is : (1) 𝑎2 + 𝑏 (2) 𝑎+ 𝑏 3𝑎3 3𝑎2 (3) 𝑏2 (4) 𝑎+ 𝑏2 3𝑎3 3𝑎3

Q62.Consider a rectangle ABCD having 5, 6, 7, 9 points in the interior of the line segments AB, BC, CD, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β −α) is equal to (1) 795 (2) 1173 (3) 1890 (4) 717