Practice Questions

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.The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to (1) 205 (2) 615 (3) 510 (4) 430 JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper

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 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 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.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.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

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

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

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

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

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

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 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.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.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

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