Practice Questions

Q62.Let 𝑎, 𝑏∈𝑅 be such that the equation 𝑎𝑥2 - 2𝑏𝑥+ 15 = 0 has repeated root 𝛼 and if 𝛼 and 𝛽 are the roots of the equation 𝑥2 - 2𝑏𝑥+ 21 = 0, then 𝛼2 + 𝛽2 is equal to: (1) 37 (2) 58 (3) 68 (4) 92 𝑧1

Q62.Let S be the set of all (α, β), π < α, β < 2π, for which the complex number 1+2i1−i sinsinαα is purely imaginary and β 1+i cos is purely real. Let Zαβ = sin 2α + i cos 2β, (α, β) ∈S . β 1−2i cos 1 +¯ Then ∑(α,β)∈S(iZαβ iZ αβ ) is equal to (1) 3 (2) 3i (3) 1 (4) 2 −i

Q62.Consider two G.Ps. 2, 22, 23, … and 4, 42, 43, … of 60 and n terms respectively. If the geometric mean of all 225 the 60 + n terms is (2) 8 , then ∑nk=1 k(n −k) is equal to: (1) 560 (2) 1540 (3) 1330 (4) 2600 n(S) + ∑θ∈S(sec( π4 + 2θ) cosec ( π4 + 2θ)) is equal

Q62.If + + … + = then the remainder when 𝐾 is divided by 6 is 2 · 310 22 · 39 210 · 3 210 · 310, (1) 2 (2) 3 (3) 4 (4) 5

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.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: 𝑧- 1 + 𝑖≥𝑧, 𝑧< 2, 𝑧+ 𝑖= 𝑧- 1. Then the set of all values of 𝑥, for which 𝑤= 2𝑥+ 𝑖𝑦∈𝑆 for some 𝑦∈ℝ, is 1 1 1 (2) - (1) -√2, 4 2√2 √2, (3) -√2, 1 (4) - 1 1 2 √2, 2√2

Q62.Let A1, A2, A3, … … be an increasing geometric progression of positive real numbers. If A1 A3 A5 A7 = 12961 and A2 + A4 = 367 , then, the value of A6 + A8 + A10 is equal to (1) 43 (2) 33 (3) 37 (4) 48 JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper α ∈R, then the value of 16α is equal to

Q62.Let for some real numbers α and β, a = α −iβ . If the system of equations 4ix + (1 + i)y = 0 and ¯8(cos 2π3 + i sin 2π3 )x + ay = 0 has more than one solution then αβ is equal to (1) 2 −√3 (2) 2 + √3 (3) −2 + √3 (4) −2 −√3

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

Q62.The remainder when (2021)2023 is divided by 7 is JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper (1) 2 (2) 3 (3) 4 (4) 5

Q62.Let A = {z ∈C : 1 ⩽|z −(1 + i)| ⩽2} and B = {z ∈A : |z −(1 −i)| = 1} . Then, B (1) is an empty set (2) contains exactly two elements (3) contains exactly three elements (4) is an infinite set

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

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

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