Practice Questions

Q60.Following tetrapeptide can be represented as JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper ( F, L, D, Y, I, Q, P are one letter codes for amino acids) (1) FIQY (2) FLDY (3) YQLF (4) PLDY

Q61.The equation e4x + 8e3x + 13e2x −8ex + 1 = 0, x ∈R has : (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative

Q61.Let the complex number 𝑧= 𝑥+ 𝑖𝑦 be such that is purely imaginary. If 𝑥+ 𝑦2 = 0, then 𝑦4 + 𝑦2 - 𝑦 is 2𝑧+ 𝑖 equal to (1) 2 (2) 3 3 2 3 4 (3) (4) 4 3

Q61.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: is a real number }. Then which of the following is NOT correct? 4𝑧+ 2𝑖 (1) 𝑦+ 𝑥2 + 𝑦2 ≠- 1 (2) (𝑥, 𝑦) = 0, - 1 4 2 (3) 𝑥= 0 (4) 𝑦∈- ∞, - 1 ∪-1 ∞ 2 2,

Q61.The sum of all the roots of the equation 𝑥2 - 8𝑥+ 15 - 2𝑥+ 7 = 0 is (1) 9 - √3 (2) 9 + √3 (3) 11 - √3 (4) 11 + √3

Q61.The number of integral values of k, for which one root of the equation 2x2 −8x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) 2 (2) 0 (3) 1 (4) 3

Q61.Let a ≠b be two non-zero real numbers. Then the number of elements in the set X = {z ∈C : Re(az2 + bz) = a and Re(bz2 + az) = b} is equal to (1) 0 (2) 1 (3) 3 (4) 2

Q61.Let 𝑝, 𝑞∈ℝ and (1 - √3𝑖) 200 = 2199 (𝑝+ 𝑖𝑞), 𝑖= √-1. Then, 𝑝+ 𝑞+ 𝑞2 and 𝑝- 𝑞+ 𝑞2 are roots of the equation. (1) 𝑥2 + 4𝑥- 1 = 0 (2) 𝑥2 - 4𝑥+ 1 = 0 (3) 𝑥2 + 4𝑥+ 1 = 0 (4) 𝑥2 - 4𝑥- 1 = 0

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

Q61.The number of real solutions of the equation 3(x2 + x21 ) −2(x + x1 ) + 5 = 0 , is (1) 4 (2) 0 (3) 3 (4) 2 2π 2π 3 1+sin 9 +i cos 9

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.The number of real roots of the equation x|x| −5|x + 2| + 6 = 0 , is (1) 5 (2) 4 (3) 6 (4) 3 ¯ ¯

Q61.Let λ ≠0 be a real number. Let α, β be the roots of the equation 14x2 −31x + 3λ = 0 and α, γ be the roots of the equation 35x2 −53x + 4λ = 0. Then 3αβ and 4αγ are the roots of the equation : (1) 7x2 + 245x −250 = 0 (2) 7x2 −245x + 250 = 0 (3) 49x2 −245x + 250 = 0 (4) 49x2 + 245x + 250 = 0

Q61.The number of integral solution 𝑥 of 7 ≥0 is log𝑥+ 2𝑥- 3 2 (1) 7 (2) 8 (3) 6 (4) 5

Q61.Let a ∈R and let α, β be the roots of the equation x2 + 60 41 x + a = 0. If α4 + β4 = −30, then the product of all possible values of a is _____ .

Q61.Let 𝛼, 𝛽 be the roots of the equation 𝑥2 - √2𝑥+ 2 = 0 Then 𝛼14 + 𝛽14 is equal to (1) -64 (2) -64√2 (3) -128 (4) -128√2

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.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 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.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.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.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.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.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 _____.