Practice Questions

Q62.The sum ∑21n=1 (4n−1)(4n+3)3 is equal to (1) 7 (2) 7 87 29 (3) 14 (4) 21 87 29

Q62.If 𝑧= 𝑥+ 𝑖𝑦 satisfies 𝑧- 2 = 0 and 𝑧- 𝑖- 𝑧+ 5𝑖= 0, then (1) 𝑥+ 2𝑦- 4 = 0 (2) 𝑥2 + 𝑦- 4 = 0 (3) 𝑥+ 2𝑦+ 4 = 0 (4) 𝑥2 - 𝑦+ 3 = 0 Q63. ∑𝑖,𝑛 𝑗= 0 𝑛𝐶𝑖 𝑛𝐶𝑗 is equal to 𝑖≠𝑗 (1) 22𝑛- 2𝑛𝐶𝑛 (2) 22𝑛- 1 - 2𝑛- 1𝐶𝑛- 1 1 1 2𝑛- (3) 22𝑛- 2 2𝑛𝐶𝑛 (4) 2𝑛- + 1𝐶𝑛

Q62.Let α, β be the roots of the equation x2 −√2x + √6 = 0 and 1 + 1, 1 + 1 be the roots of the equation α2 β2 x2 + ax + b = 0 . Then the roots of the equation x2 −(a + b −2)x + (a + b + 2) = 0 are : (1) non-real complex numbers (2) real and both negative (3) real and both positive (4) real and exactly one of them is positive

Q63.If the constant term in the expansion of (3x3 −2x2 + x5 ) is 2k. l, where l is an odd integer, then the value of k is equal to (1) 6 (2) 7 (3) 8 (4) 9

Q63.If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is (1) 21 (2) 22 (3) 23 (4) 24 − , x ≠0 is

Q63.Let S = {θ ∈[0, 2π] : 82 sin2 θ + 82 cos2 θ = 16} . Then to: (1) 0 (2) −2 (3) −4 (4) 12

Q63.Let 𝑎𝑛𝑛=∞ 0 be a sequence such that 𝑎0 = 𝑎1 = 0 and 𝑎𝑛+ 2 = 3𝑎𝑛+ 1 - 2𝑎𝑛+ 1, ∀𝑛≥0. Then 𝑎25𝑎23 - 2𝑎25𝑎22 - 2𝑎23𝑎24 + 4𝑎22𝑎24 is equal to (1) 483 (2) 528 (3) 575 (4) 624 Q64. ∑𝑟=20 1 𝑟2 + 1𝑟! is equal to (1) 22! - 21! (2) 22! - 221! (3) 21! - 220! (4) 21! - 20!

Q63.Let 𝑧1 and 𝑧2 be two complex numbers such that ¯𝑧1 = 𝑖¯𝑧2 and arg = 𝜋, then the argument of 𝑧1 is ¯𝑧2 (1) arg 𝑧2 = π (2) arg 𝑧2 = - 3π 4 4 π 3π (3) arg 𝑧1 = 4 (4) arg 𝑧1 = - 4

Q63.Let the sum of an infinite G. P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be 98 . Then the sum of the first 21 terms of an AP, whose first term is 10ar, nth term is an and the 25 common difference is 10 ar2 , is equal to (1) 21a11 (2) 22a11 (3) 15a16 (4) 14a16

Q63.Let S = 2 + 76 + 1272 + 2073 + 3074 + … . . then 4S is equal to JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper (1) ( 27 ) 2 (2) ( 73 ) 3 (3) 3 7 (4) ( 37 ) 4

Q63.Consider the sequence 𝑎1, 𝑎2, 𝑎3, … … such that 𝑎1 = 1, 𝑎2 = 2 and 𝑎𝑛+ 2 = + 𝑎𝑛 for 𝑛= 1, 2, 3, … 𝑎𝑛+ 1 1 1 1 1 𝑎1 + 𝑎2 𝑎2 + 𝑎3 𝑎3 + 𝑎4 𝑎30 + 𝑎31 If · · … = 2𝛼61𝐶31 then 𝛼 is equal to 𝑎3 𝑎4 𝑎5 𝑎32 (1) -30 (2) -31 (3) -60 (4) -61

Q63.The number of solutions of the equation cos(x + π3 ) cos( π3 −x) = 14 cos2 2x, x ∈[−3π, 3π] is: (1) 8 (2) 5 (3) 6 (4) 7

Q63.Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k1, k2), then k1 + k2 is equal to (1) 2 (2) 47 (3) 2 (4) 4 7

Q63.The sum of the infinite series 1 + 65 + 1262 + 2263 + 3564 + 5165 + 7066 + … is equal to: (1) 425 (2) 429 216 216 (3) 288 (4) 280 125 125

Q63.The remainder when (11)1011 + (1011)11 is divided by 9 is _____ . (1) 1 (2) 8 (3) 6 (4) 4

Q63.Let the tangents at two points A and B on the circle x2 + y2 −4x + 3 = 0 meet at origin O(0, 0). Then the area of the triangle of OAB is (1) 3√3 (2) 3√3 2 4 (3) 3 (4) 3 2√3 4√3

Q63.If m is the slope of a common tangent to the curves x2 16 + 9 = 1 and x2 + y2 = 12 , then 12m2 is equal to JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper (1) 6 (2) 9 (3) 10 (4) 12

Q63.If {ai}ni=1 , where n is an even integer, is an arithmetic progression with common difference 1 , and n ∑ni=1 ai = 192, ∑ i=12 a2i = 120 , then n is equal to (1) 18 (2) 36 (3) 96 (4) 48 JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper

Q63.Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of ΔPQR is (1) 25 (2) 25√3 4√3 2 (3) 25 (4) 25 √3 2√3

Q63.If ∑31k=1(31Ck)(31Ck−1) −∑30k=1(30Ck)(30Ck−1) = (30!)(31!)α(60!) , where (1) 1411 (2) 1320 (3) 1615 (4) 1855 + y2 −2x −4y = 0 intersect at

Q63.The value of cos( 2π7 ) + cos( 4π7 ) + cos( 6π7 ) is equal to (1) −1 (2) −12 (3) −13 (4) −14

Q63.The remainder when (2021)2022 + (2022)2021 is divided by 7 is (1) 0 (2) 1 (3) 2 (4) 6

Q63.The number of solutions of cos𝑥= sin𝑥, such that -4𝜋≤𝑥≤4𝜋 is (1) 4 (2) 6 (3) 8 (4) 12

Q63.Let a circle 𝐶 touch the lines 𝐿1: 4𝑥- 3𝑦+ 𝐾1 = 0 and 𝐿2: 4𝑥- 3𝑦+ 𝐾2 = 0, 𝐾1, 𝐾2 ∈𝑅. If a line passing through the centre of the circle 𝐶 intersects 𝐿1 at -1, 2 and 𝐿2 at 3, - 6, then the equation of the circle 𝐶 is (1) 𝑥- 12 + 𝑦- 22 = 4 (2) 𝑥- 12 + 𝑦+ 22 = 16 (3) 𝑥+ 12 + 𝑦- 22 = 4 (4) 𝑥- 12 + 𝑦- 22 = 16

Q64.Let S = {θ ∈(0, π2 ) : ∑9m=1 sec(θ + (m −1) π6 ) sec(θ + mπ6 ) = −8√3 }. Then (1) S = { 12π } (2) S = { 2π3 } (3) ∑θ∈S θ = π2 (4) ∑θ∈S θ = 3π4