Practice Questions

Q61.Let f(x) be a quadratic polynomial such that f(−2) +f(3) = 0. If one of the roots of f(x) = 0 is −1, then the sum of the roots of f(x) = 0 is equal to (1) 11 (2) 7 3 3 (3) 12 (4) 14 3 3

Q61.If A = ∑∞n=1 (3+(−1)n)n and B = ∑∞n=1 (3+(−1)n)n , then B is equal to (1) 11 (2) 1 9 (3) −119 (4) −113 Q62. 16 sin(20°) sin(40°) sin(80°) is equal to (1) √3 (2) 2√3 (3) 3 (4) 4√3 y2

Q61.Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 −α3033 is equal to: (1) 1 (2) α (3) 1 + α (4) 1 + 2α

Q61.Let 𝐴= 𝑥∈𝑅: 𝑥+ 1 < 2 and 𝐵= 𝑥∈𝑅: 𝑥- 1 ≥2. Then which one the following statements is NOT true? (1) 𝐴- 𝐵= -1, 1 (2) 𝐵- 𝐴= 𝑅- -3, 1 (3) 𝐴∩𝐵= ( - 3, - 1] (4) 𝐴∪𝐵= 𝑅- [1, 3 )

Q61.If the sum of the squares of the reciprocals of the roots α and β of the equation 3x2 + λx −1 = 0 is 15 , then 6(α3 + β3)2 is equal to (1) 46 (2) 36 (3) 24 (4) 18

Q61.Let A = {z ∈C : z−1z+1 < 1} and B = {z ∈C : arg( z+1z−1 ) = 2π3 }. Then A ∩B is (1) a portion of a circle centred at (0, −1√3 ) that (2) a portion of a circle centred at (0, −1√3 ) that lies in the second and third quadrants only lies in the second quadrant only (3) an empty set (4) a portion of a circle of radius 2 that lies in the √3 third quadrant only

Q61.The sum of all real roots of equation (e2x −4)(6e2x −5ex + 1) = 0 is (1) ln 4 (2) −ln 3 (3) ln 3 (4) ln 5

Q61.If 𝛼, 𝛽, 𝛾, 𝛿 are the roots of the equation 𝑥4 + 𝑥3 + 𝑥2 + 𝑥+ 1 = 0, then 𝛼2021 + 𝛽2021 + 𝛾2021 + 𝛿2021 is equal to (1) 4 (2) 1 (3) -4 (4) -1

Q61.If 𝑧≠0 be a complex number such that 𝑧- 𝑧= 2, then the maximum value of 𝑧 is (1) √2 (2) 1 (3) √2 - 1 (4) √2 + 1

Q61.The number of points of intersection |z −(4 + 3i)| = 2| and |z| + |z −4| = 6, z ∈C is (1) 1 (2) 2 (3) 3 (4) 4

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

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