Practice Questions

Q61.If z = x + iy, xy ≠0, satisfies the equation z2 + iz = 0, then z2 is equal to : (1) 9 (2) 1 (3) 4 (4) 14

Q61.The sum of all possible values of θ ∈[−π, 2π], for which 1−2i1+i coscosθθ is purely imaginary, is equal (1) 3π (2) 2π (3) 5π (4) 4π

Q61.If 𝑧 is a complex number, then the number of common roots of the equation 𝑧1985 + 𝑧100 + 1 = 0 and 𝑧3 + 2𝑧2 + 2𝑧+ 1 = 0, is equal to : (1) 1 (2) 2 (3) 0 (4) 3

Q61.Consider the following two statements : Statement I : For any two non-zero complex numbers z1, z2 , (|z1| + |z2|) z1 + z2 ≤2 (|z1| + |z2|), and |z1| |z2| Statement II : If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that |y−z| a = |z−x|b = |x−y|c , then y−za2 + z−xb2 + x−yc2 = 1. Between the above two statements, (1) Statement I is correct but Statement II is (2) both Statement I and Statement II are correct. incorrect. (3) both Statement I and Statement II are incorrect. (4) Statement I is incorrect but Statement II is correct.

Q61.If 𝛼, 𝛽 are the roots of the equation, x2 - x - 1 = 0 and Sn = 2023𝛼n + 2024𝛽n, then (1) 2 S12 = S11 + S10 (2) S12 = S11 + S10 (3) 2 S11 = S12 + S10 (4) S11 = S10 + S12 4! ! 5! ( ) (

Q61.The area (in sq. units) of the region S = {z ∈C : |z −1| ≤2; (z + ¯z) + i(z −¯z) ≤2, Im(z) ≥0} is (1) 7π (2) 7π 3 4 (3) 17π (4) 3π 8 2

Q61.If 2 and 6 are the roots of the equation ax2 + bx + 1 = 0, then the quadratic equation, whose roots are 2a+b1 and 1 , is : 6a+b (1) 2x2 + 11x + 12 = 0 (2) x2 + 8x + 12 = 0 (3) 4x2 + 14x + 12 = 0 (4) x2 + 10x + 16 = 0

Q62.Let Sa denote the sum of first n terms an arithmetic progression. If S20 = 790 and S10 = 145, then S15−S5 is : JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper (1) 395 (2) 390 (3) 405 (4) 410

Q62.Let 𝛼= and 𝛽= 3! )4!.! Then : 4! 5! ( ( ) ) (1) 𝛼∈N and 𝛽∉N (2) 𝛼∉N and 𝛽∈N (3) 𝛼∈N and 𝛽∈N (4) 𝛼∉N and 𝛽∉N

Q81.There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

Q81.Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 −17x2 −12x + 9 = 0 and (4 + x21) (4 + x22) (4 + x23) (4 + x24) = 12516 m. Then the value of m is

Q81.Let the complex numbers α and lie on the circles z - z02 = 4 and z - z02 = 16 respectively, where z0 = 1 + i. ¯α Then, the value of 100 | α| 2 is__________.

Q81.The lines 𝐿1, 𝐿2, . .. , 𝐿20 are distinct. For 𝑛= 1, 2, 3, . .. , 10 all the lines 𝐿2𝑛−1 are parallel to each other and all the lines 𝐿2𝑛 pass through a given point 𝑃. The maximum number of points of intersection of pairs of lines from the set 𝐿1, 𝐿2, . .. , 𝐿20 is equal to:

Q81.Let α, β be roots of x2 + √2x −8 = 0. If Un = αn + βn , then U10+√2U9 is equal to______ 2U8

Q81.Let a = 1 + 2C23! + 3C24! + 4C25! + … 1! + 2! + 3! + … Then 2b is equal to a2

Q81.Let the set C = {(x, y) ∣x2 −2y = 2023, x, y ∈N}. Then ∑(x,y)∈C(x y)

Q81.The number of distinct real roots of the equation |x + 1||x + 3| −4|x + 2| + 5 = 0, is JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper

Q81.Let α, β ∈ be roots of equation x2 −70x + λ = 0, where λ2 , λ3 ∉ . If λ assumes the minimum possible value, (√α−1+√β−1)(λ+35) then is equal to : |α−β|

Q81.The number of ways of getting a sum 16 on throwing a dice four times is______

Q81.The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is ______.

Q81.If α satisfies the equation x2 + x + 1 = 0 and (1 + α)7 = A + Bα + Cα2, A, B, C ≥0 , then 5(3 A −2 B −C) is equal to

Q81.Let 𝑎, 𝑏, 𝑐 be the length of three sides of a triangle satisfying the condition 𝑎2 + 𝑏2𝑥2 −2𝑏𝑎+ 𝑐 𝑥+ 𝑏2 + 𝑐2 = 0. If the set of all possible values of 𝑥 is in the interval 𝛼, 𝛽, then 12𝛼2 + 𝛽2 is equal to _______.

Q82.An arithmetic progression is written in the following way The sum of all the terms of the 10th row is_______

Q82.Let α, β be the roots of the equation x2 −√6x + 3 = 0 such that Im (α) >Im (β). Let a, b be integers not + i = √−1. Then n + a + b is divisible by 3 and n be a natural number such that α99β + α98 = 3n(a ib), equal to ___________.

Q82.If 1 + √3−√2 a + loge ( ab ), where a and b are + 49−20√6180 + … upto ∞= 2 + 2√3 + 5−2√618 + 9√3−11√236√3 (√b 1) integers with gcd(a, b) = 1, then 11a + 18 b is equal to ______