Practice Questions

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.Suppose a1, a2, … , an, … be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a15 < 120 , then the sum of the JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper first ten terms of the progression is equal to (1) 290 (2) 380 (3) 460 (4) 510

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

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 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.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.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.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.If the minimum value of 𝑓𝑥= 5𝑥2 + 𝛼 𝑥> 0, is 14, then the value of 𝛼 is equal to 2 𝑥5, (1) 32 (2) 64 (3) 128 (4) 256 2

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.If x = ∑∞n=0 an, y = ∑∞n=0 bn, z = ∑∞n=0 cn , where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc ≠0, then (1) x, y, z are in A.P. (2) x, y, z are in G.P. (3) x 1 , 1y , 1z are in A.P. (4) x1 + 1y + 1z = 1 −(a + b + c)

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.If (20−a)(40−a) 1 + (40−a)(60−a)1 + … … + (180−a)(200−a)1 = 2561 , then the maximum value of a is (1) 198 (2) 202 (3) 212 (4) 218

Q62.Let {an}∞n=0 be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 −an + 1 for all n ≥0 . Then, ∑∞n=2 an7n is equal to (1) 6 (2) 7 343 216 (3) 8 (4) 49 343 216 5 10

Q62.Let x, y > 0 . If x3y2 = 215 , then the least value of 3x + 2y is JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper (1) 30 (2) 32 (3) 36 (4) 40

Q62.For 𝑛∈𝑁, let 𝑆𝑛= 𝑧∈𝐶: 𝑧- 3 + 2𝑖= 𝑛 and 𝑇𝑛= 𝑧∈𝐶: 𝑧- 2 + 3𝑖= 1 Then the number of elements in the 4 𝑛. set 𝑛∈𝑁: 𝑆𝑛∩𝑇𝑛= 𝜙 is (1) 0 (2) 2 (3) 3 (4) 4

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

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

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

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 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.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 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 = 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.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