Practice Questions

Q61.Let α, β, γ be the three roots of the equation x3 + bx + c = 0 if βγ = 1 = −α then b3 + 2c3 −3α3 −6β3 −8γ 3 is equal to (1) 155 (2) 21 8 (3) 169 (4) 19 8

Q61.Let S = {α : log2(92α−4 + 13) −log2( 25 ⋅32α−4 + 1) = 2}. Then the maximum value of β for which the equation x2 −2(∑α∈s α) 2x + ∑a∈s (α + 1)2β = 0 has real roots, is _____ .

Q61.Let 𝑥2 - 4 𝑥2 - 4 𝑆= 𝑥: 𝑥∈ℝ and √3 + √2 + √3 - √2 = 10. Then 𝑛𝑆 is equal to (1) 2 (2) 4 (3) 6 (4) 0 𝑧- 2

Q62.The number of ways of selecting two numbers a and b, a ∈{2, 4, 6, … … , 100} and b ∈{1, 3, 5, … … , 99} such that 2 is the remainder when a + b is divided by 23 is (1) 186 (2) 54 (3) 108 (4) 268 JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q62.For a ∈C, let A = {z ∈C :Re (a + z) >Im (a + z)} and B = {z ∈C :Re (a + z) <Im (a + z)} . Then among the two statements: (S1) : If Re (a), Im (a) > 0, then the set A contains all the real numbers (S2) : If Re (a), Im (a) < 0, then the set B contains all the real numbers, (1) Only (S2) is true (2) only (S1) is true (3) Both are true (4) Both are false z2+8iz−15 : α −1311 i ∈S, α ∈R −{0}, then 242α2 is equal to

Q62.If the set {Re ( 2−3z+5zz−z+zz ) : z ∈C, Re z = 3} is equal to the interval (α, β], then 24(β −α) is equal to (1) 36 (2) 27 (3) 30 (4) 42

Q62.Let a, b be two real numbers such that ab < 0 . If the complex number 1+aib+i is of unit modulus and a + ib lies on the circle |z −1| = |2z| , then a possible value of 1+[a]4b , where [t] is greatest integer function, is : (1) 0 (2) −1 (3) 1 (4) 21

Q62.Let z be a complex number such that z−2iz+i = 2, z ≠−i. Then z lies on the circle of radius 2 and centre (1) (2, 0) (2) (0, 2) (3) (0, 0) (4) (0, −2)

Q62.The value of ( 1+sin 2π9 −i cos 2π9 ) is (1) −1 (2) 1 2 (1 −i√3) 2 (1 −i√3) (3) −1 + i) 2 (√3 −i) (4) 12 (√3

Q62.The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

Q62.If the center and radius of the circle = 2 are respectively 𝛼, 𝛽 and 𝛾, then 3𝛼+ 𝛽+ 𝛾 is equal to 𝑧- 3 (1) 11 (2) 9 (3) 10 (4) 12

Q62.Let A = {θ ∈(0, 2π) : 1+2i1−i sinsinθθ is purely imaginary} Then the sum of the elements is in A is (1) 4π (2) 3π (3) π (4) 2π

Q62.Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is (1) 1120 (2) 3360 (3) 1680 (4) 560 1

Q62.For two non-zero complex number z1 and z2 , if Re (z1z2) = 0 and Re (z1 + z2) = 0, then which of the following are possible? (A) Im (z1) > 0 and Im (z2) > 0 (B) Im (z1) < 0 and Im (z2) > 0 (C) Im (z1) > 0 and Im (z2) < 0 (D) Im (z1) < 0 and Im (z2) < 0 Choose the correct answer from the options given below: (1) B and D (2) B and C (3) A and B (4) A and C

Q62.If for z = α + iβ, |z + 2| = z + 4(1 + i), then α + β and αβ are the roots of the equation (1) x2 + 3x −4 = 0 (2) x2 + 7x + 12 = 0 (3) x2 + x −12 = 0 (4) x2 + 2x −3 = 0

Q62.Let C be the circle in the complex plane with centre z0 = 12 (1 + 3i) and radius r = 1. Let z1 = 1 + i and the complex number z2 be outside circle C such that |z1 −z0||z2 −z0| = 1 . If z0, z1 and z2 are collinear, then the smaller value of |z2|2 is equal to (1) 5 (2) 7 2 2 (3) 13 (4) 3 2 2

Q62.The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + . . . is (1) 3520 (2) 3450 (3) 3250 (4) 3420

Q62.The complex number z = πi−1 π is equal to: cos 3 +i sin 3 (1) √2i(cos 5π12 −i sin 5π12 ) (2) cos 12π −i sin 12π (3) √2(cos 12π + i sin 12π ) (4) √2(cos 5π12 + i sin 5π12 )

Q62.For all 𝑧∈𝐶 on the curve 𝐶1: | 𝑧| = 4, let the locus of the point z + 1 be the curve 𝐶2. Then z (1) the curves C1 and C2intersect at 4 points (2) the curves 𝐶1 lies inside 𝐶2 (3) the curves 𝐶1 and 𝐶2 intersect at 2 points (4) the curves 𝐶2 lies inside 𝐶1

Q62.For α, β, z ∈C and λ > 1 , if √λ −1 is the radius of the circle |z −α|2 + |z −β|2 = 2λ, then |α −β| is equal to _____.

Q62.Let 𝑆= 𝑧∈ℂ: ¯𝑧= 𝑖𝑧2 + Re ( ¯𝑧) . Then ∑𝑧∈𝑆| 𝑧| 2 is equal to (1) 5 (2) 4 2 (3) 7 (4) 3 2

Q62.For three positive integers 𝑝, 𝑞, 𝑟, 𝑥𝑝𝑞2 = 𝑦𝑞𝑟= 𝑧𝑝2𝑟 and 𝑟= 𝑝𝑞+ 1 such that 1 3, 3log𝑦𝑥, 3 log𝑧𝑦, 7log𝑥𝑧 are in A.P. with common difference 2. The 𝑟- 𝑝- 𝑞 is equal to (1) 2 (2) 6 (3) 12 (4) -6

Q62.Let the first term a and the common ratio 𝑟 of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to (1) 241 (2) 231 (3) 210 (4) 220 1 13 1 13

Q62.If 𝑎𝑛= 4𝑛2 - 16𝑛+ 15, then 𝑎1 + 𝑎2 + … . + 𝑎25 is equal to: (1) 51 (2) 49 144 138 50 52 (3) (4) 141 147 1 15

Q62.Let α = 8 −14i, A = {z ∈C : z2−(¯z)2−112iαz−α¯z = 1} and B = {z ∈C : |z + 3i| = 4} Then, ∑z∈A∩B(Re z −Imz) is equal to ________