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JEE Complex Number PYQs with Solutions

551 previous year questions from Complex Number (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

551 questionsMathematics2006–2026Answers + solutions

About Complex Number PYQs

Complex Number has 551 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (181) and JEE Advanced (39), spanning 2006 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 17
  • 2025: 21
  • 2024: 24
  • 2023: 27
  • 2022: 30
  • 2021: 31
  • 2020: 18
  • 2019: 18
  • 2018: 7
  • 2017: 4
  • 2016: 3
  • 2015: 1

Complex Number questions with solutions

1

The number of values of z ∈ ℂ , satisfying the equations |z-(4+8i)|= 10 and |z-(3+5i)|+|z-(5+11i)|=4 5 , is:

JEE Main 2026 Solution View → 2

Let S = \ z ∈ ℂ : z^2 + 6 \,iz - 3 = 0\ . Then ∑ _ z ∈ S z^8 is equal to :

JEE Main 2026 Solution View → 3

Let the set of all values of k ∈ ℝ such that the equation z( z + 2 + i) + k(2 + 3i) = 0 , z ∈ ℂ , has at least one solution, be the interval [α, β] . Then 9(α + β) is equal to:

JEE Main 2026 Solution View → 4

Let z_1, z_2 ∈ ℂ be the distinct solutions of the equation z^2 + 4z - (1 + 12i) = 0 . Then |z_1|^2 + |z_2|^2 is equal to :

JEE Main 2026 Solution View → 5

Let S=\ z ∈ ℂ: z^2+4z+16=0\ . Then ∑_ z ∈ S |z+ 3 i|^2 is equal to:

JEE Main 2026 Solution View → 6

Let z be a complex number such that |z+2| = |z-2| and ((z+3)/(z-i)) = (π)/(4) . Then |z|^2 is equal to:

JEE Main 2026 Solution View → 7

Let the circles C_1 : |z| = r and C_2 : |z - 3 - 4i| = 5 , z ∈ ℂ , be such that C_2 lies within C_1 . If z_1 moves on C_1 , z_2 moves on C_2 and |z_1 - z_2| = 2 , then |z_1 - z_2|

JEE Main 2026 Solution View → 8

Let x and y be real numbers such that 50((2x)/(1+3i) - (y)/(1-2i)) = 31 + 17i , i = -1 . Then the value of 10(x - 3y) is :

JEE Main 2026 Solution View → 9

Let A=\ z ∈ ℂ:|z-2| 4\ and B=\ z ∈ ℂ:|z-2|+|z+2|=5\ . Then the max \ |z_ 1 -z_ 2 |: z_ 1 ∈ ~A. and .z_ 2 ∈ ~B\ is :

JEE Main 2026 Solution View → 10

Let z be a complex number such that |z-6|=5 and |z+2-6 i|=5 . Then the value of z^ 3 +3 z^ 2 -15 z+141 is equal to

JEE Main 2026 Solution View → 11

Let z=(1+i)(1+2 i)(1+3 i) …(1+n i) , where i= -1 . If |z|^ 2 =44200 , then n is equal to \_\_\_\_

JEE Main 2026 Solution View → 12

If z= 3 2 +(i)/(2), i= -1 , then (z^ 201 -i)^ 8 is equal to

JEE Main 2026 Solution View → 13

Let S=\ z: 3 |2 z-3(1+i)| 7\ be a set of complex numbers. Then _ z ∈ S |(z+(1)/(2)(5+3 i))| is equal to :

JEE Main 2026 Solution View → 14

Let S=\ z ∈ ℂ: 4 z^ 2 + z =0\ . Then ∑_ z ∈ S |z|^ 2 is equal to:

JEE Main 2026 Solution View → 15

Let α= -1+i 3 2 and β= -1-i 3 2 , i= -1 . If (7-7 α+9 β)^ 20 +(9+7 α-7 β)^ 20 +(-7+9 α+7 β)^ 20 +(14+7 α+7 β)^ 20 =m^ 10 , then m is \_\_\_\_\_

JEE Main 2026 Solution View → 16

Let z be the complex number satisfying |z-5| ≤ 3 and having maximum positive principal argument. Then 34| 5 z-12 5 i z+16 |^ 2 is equal to :

JEE Main 2026 Solution View → 17

If x^ 2 +x+1=0 , then the value of (x+(1)/(x))^ 4 +(x^ 2 + 1 x^ 2 )^ 4 + (x^ 3 + 1 x^ 3 )^ 4 +…+ (x^ 25 + 1 x^ 25 )^ 4 is:

JEE Main 2026 Solution View → 18

Let ℝ denote the set of all real numbers. Let z_1=1+2 i and z_2=3 i be two complex numbers, where i= -1 . Let S=\ (x, y) ∈ ℝ × ℝ:|x+i y-z_1|=2|x+i y-z_2|\ . Then which of the

JEE Advanced 2025 Solution View → 19

For a non-zero complex number z , let (z) denote the principal argument of z , with -π < (z) ≤ π . Let ω be the cube root of unity for which 0 < (ω) <π . Let α= (∑_ n=1 ^ 2025

JEE Advanced 2025 Solution View → 20

Let A= \ θ ∈[0,2 π]: 1+10 Re((2 θ+i θ)/( θ-3 i θ))=0\ . Then ∑_ θ ∈ A θ^2 is equal to

JEE Main 2025 Solution View → 21

If the locus of z ∈ C , such that Re( z-1 2 z+i )+Re( z -1 2 z -i )=2 is a circle of radius r and center (a, b) then (15 a b)/(r^2) is equal to :

JEE Main 2025 Solution View → 22

Among the statements (S1) : The set \ z ∈ ℂ-\ -i\ :|z|=1. and z-i z+i is purely real contains exactly two elements, and (S2) : The set \ z ∈ ℂ-\ -1\ :|z|=1. and z-1 z+1 is purely

JEE Main 2025 Solution View → 23

Let the product of ω_1=(8+i) θ+(7+4 i) θ and ω_2=(1+8 i) θ+(4+7 i) θ be α+i β , i= -1 . Let p and q be the maximum and the minimum values of α+β respectively.

JEE Main 2025 Solution View → 24

If α is a root of the equation x^2+x+1=0 and ∑_ k=1 ^ n (α^ k + 1 α^ k )^2=20 , then n is equal to

JEE Main 2025 Solution View → 25

Let A=\ z ∈ C:|z-2-i|=3\ , B=\ z ∈ C: Re(z-i z)=2\ and S=A ∩ B . Then ∑_ z ∈ S |z|^2 is equal to ________ .

JEE Main 2025 Solution View → 26

If z_1, z_2, z_3 ∈ C are the vertices of an equilateral triangle, whose centroid is z_0 , then ∑_ k=1 ^3(z_ k -z_0)^2 is equal to

JEE Main 2025 Solution View → 27

Let z ∈ C be such that (z^2+3 i)/(z-2+i)=2+3 i . Then the sum of all possible values of z^2 is

JEE Main 2025 Solution View → 28

Let z be a complex number such that |z|=1 . If 2+k^2 z k+ z =kz, k ∈ R , then the maximum distance of k+ik^2 from the circle |z-(1+2 i)|=1 is:

JEE Main 2025 Solution View → 29

Let integers a, b ∈[-3,3] be such that a+b ≠ 0 . Then the number of all possible ordered pairs (a, b), for which | z-a z+b |=1 and | array ccc z+1 & ω & ω^2 ω & z+ω^2 & 1 ω^2 & 1

JEE Main 2025 Solution View → 30

Let \(|z_1-8-2 i| ≤ 1\) and \(|z_2-2+6 i| ≤ 2, z_1, z_2 ∈ C\). Then the minimum value of \(|z_1-z_2|\) is :

JEE Main 2025 Solution View → 31

If α+i β and γ+i δ are the roots of x^2-(3-2 i) x-(2 i-2)=0, i= -1 , then α γ+β δ is equal to :

JEE Main 2025 Solution View → 32

Let O be the origin, the point A be z_1= 3 +2 2 i , the point B(z_2) be such that 3 |z_2|=|z_1| and (z_2)= (z_1)+(π)/(6) . Then

JEE Main 2025 Solution View → 33

If α and β are the roots of the equation 2 z^2-3 z-2 i=0 , where i= -1 , then 16 · Re( α^ 19 +β^ 19 +α^ 11 +β^ 11 α^ 15 +β^ 15 ) · lm( α^ 19 +β^ 19 +α^ 11 +β^ 11 α^ 15 +β^ 15 ) is

JEE Main 2025 Solution View → 34

The number of complex numbers z , satisfying |z|=1 and | z z + z z |=1 , is :

JEE Main 2025 Solution View → 35

Let α, β be the roots of the equation x^2-a x-b=0 with Im(α) Im(β) . Let P_n=α^n-β^n . If P_3=-5 7 i, P_4=-3 7 i, P_5=11 7 i and P_6=45 7 i , then |α^4+β^4| is equal to .

JEE Main 2025 Solution View → 36

Let | z -i 2 z +i |=(1)/(3), z ∈ C , be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points (0,0), C and (α, 0) is 11 square

JEE Main 2025 Solution View → 37

Let the curve z(1+i)+ z (1-i)=4, z ∈ C , divide the region |z-3| ≤ 1 into two parts of areas α and β . Then |α-β| equals :

JEE Main 2025 Solution View → 38

Let z_1, z_2 and z_3 be three complex numbers on the circle |z|=1 with (z_1)=(-π)/(4), (z_2)=0 and (z_3)=(π)/(4) . If |z_1 z _2+z_2 z _3+z_3 z _1|^2=α+β 2 , α, β ∈ Z , then the

JEE Main 2025 Solution View → 39

Let S=\ a+b 2 : a, b ∈ ℤ\ , T_1=\ (-1+ 2 )^n: n ∈ ℕ\ , and T_2=\ (1+ 2 )^n: n ∈ ℕ\ . Then which of the following statements is (are) TRUE?

JEE Advanced 2024 Solution View → 40

Let z be a complex number such that the real part of (z-2 i)/(z+2 i) is zero. Then, the maximum value of |z-(6+8 i)| is equal to

JEE Main 2024 Solution View → 41

The sum of the square of the modulus of the elements in the set \ z=a+ib: a, b ∈ Z, z ∈ C,|z-1| ≤ 1,|z-5| ≤|z-5 i|\ is ________

JEE Main 2024 Solution View → 42

The sum of all possible values of θ ∈[-π, 2 π] , for which (1+i θ)/(1-2 i θ) is purely imaginary, is equal

JEE Main 2024 Solution View → 43

Let z be a complex number such that |z+2|=1 and Im((z+1)/(z+2))=(1)/(5) . Then the value of |Re( z+2 )| is

JEE Main 2024 Solution View → 44

If z_1, z_2 are two distinct complex number such that | z_1-2 z_2 (1)/(2)-z_1 z _2 |=2 , then

JEE Main 2024 Solution View → 45

Let S_1=\ z ∈ C:|z| ≤ 5\ , S_2=\ z ∈ C: Im( z+1- 3 i 1- 3 i ) ≥ 0\ and S_3=\ z ∈ C: Re(z) ≥ 0\ . Then the area of the region S_1 ∩ S_2 ∩ S_3 is :

JEE Main 2024 Solution View → 46

Consider the following two statements : Statement I : For any two non-zero complex numbers z_1, z_2 , (|z_1|+|z_2|)|(z_1)/(|z_1|)+(z_2)/(|z_2|)| ≤ 2(|z_1|+|z_2|) , and Statement

JEE Main 2024 Solution View → 47

The area (in sq. units) of the region S=\ z ∈ ℂ:|z-1| ≤ 2 ;(z+ z )+i(z- z ) ≤ 2, Im(z) ≥ 0\ is

JEE Main 2024 Solution View → 48

Let α and β be the sum and the product of all the non-zero solutions of the equation ( z )^2+|z|=0 , z ∈ C. Then 4(α^2+β^2) is equal to :

JEE Main 2024 Solution View → 49

If z is a complex number such that |z| ≤ 1 , then the minimum value of |z + (1)/(2)(3 + 4i) | is:

JEE Main 2024 Solution View → 50

Let P = \ z ∈ ℂ: |z + 2 - 3i| ≤ 1\ and Q = \ z ∈ ℂ: z(1 + i) + ¯ z (1 - i) ≤ - 8\ . Let in P ∩ Q, |z - 3 + 2i| be maximum and minimum at z _ 1 and z _ 2 respectively. If | z _ 1 |

JEE Main 2024 Solution View → 51

Let S = \ z ∈ C: |z - 1| = 1and( 2 - 1)(z + ¯ z ) - i(z - ¯ z ) = 2 2 \ . Let z _ 1 , z _ 2 ∈ S be such that | z _ 1 | = max _ z ∈ s |z| and | z _ 2 | = min _ z ∈ s |z| . Then | 2

JEE Main 2024 Solution View → 52

Let z _ 1 and z _ 2 be two complex number such that z _ 1 + z _ 2 = 5 and z _ 1 ^ 3 + z _ 2 ^ 3 = 20 + 15i . Then | z _ 1 ^ 4 + z _ 2 ^ 4 | equals-

JEE Main 2024 Solution View → 53

If α denotes the number of solutions of |1 - i| ^ x = 2 ^ x and β = ((|z|)/(arg (z) )) , where z = (π)/(4) (1 + i) ^ 4 ( 1 - π · i π + i + π - i 1 + π · i ), i = - 1 , then the

JEE Main 2024 Solution View → 54

If z is a complex number, then the number of common roots of the equation z ^ 1985 + z ^ 100 + 1 = 0 and z ^ 3 + 2 z ^ 2 + 2z + 1 = 0 , is equal to :

JEE Main 2024 Solution View → 55

If z = x + iy, xy ≠ 0 , satisfies the equation z ^ 2 + i ¯ z = 0 , then | z ^ 2 | is equal to :

JEE Main 2024 Solution View → 56

Let r and θ respectively be the modulus and amplitude of the complex number z = 2 - i(2tan(5π)/(8)) , then ( r, θ ) is equal to

JEE Main 2024 Solution View → 57

Let α, β be the roots of the equation x ^ 2 - 6 x + 3 = 0 such that Im ( α ) > Im ( β ) . Let a, b be integers not divisible by 3 and n be a natural number such that α ^ 99 β + α

JEE Main 2024 Solution View → 58

If z = (1)/(2) - 2i , is such that | z + 1 | = + β ( 1 + i ) , i = - 1 and α, β ∈ R, then α + β is equal to

JEE Main 2024 Solution View → 59

Let α, β be the roots of the equation x ^ 2 - x + 2 = 0 with Im ( α ) > Im ( β ) . Then α ^ 6 + α ^ 4 + β ^ 4 - 5 α ^ 2 is equal to

JEE Main 2024 Solution View → 60

Let the complex numbers α and 1 ¯ α lie on the circles |z - z _ 0 | ^ 2 = 4 and |z - z _ 0 | ^ 2 = 16 respectively, where z _ 0 = 1 + i . Then, the value of 100 | α | ^ 2

JEE Main 2024 Solution View → 61

If S = z ∈ C: | z - i | = | z + i | = | z - 1 | , then, n ( S ) is:

JEE Main 2024 Solution View → 62

If α satisfies the equation x ^ 2 + x + 1 = 0 and ( 1 + α ) ^ 7 = A + Bα + C α ^ 2 , A, B, C ≥ 0 , then 5 ( 3A - 2B - C ) is equal to

JEE Main 2024 Solution View → 63

Let A _ 1 , A _ 2 , A _ 3 , … , A _ 8 be the vertices of a regular octagon that lie on a circle of radius 2 . Let P be a point on the circle and let P A _ i denote the distance

JEE Advanced 2023 Solution View → 64

Let A = \ (1967 + 1686isinθ)/(7 - 3icosθ): θ ∈ R\ . If A contains exactly one positive integer n , then the value of n is

JEE Advanced 2023 Solution View → 65

Let z be a complex number satisfying |z| ^ 3 + 2 z ^ 2 + 4 ¯ z - 8 = 0 , where ¯ z denotes the complex conjugate of z . Let the imaginary part of z be nonzero. Match each entry in

JEE Advanced 2023 Solution View → 66

If the set \ Re ( z - ¯ z + z ¯ z 2 - 3z + 5 ¯ z ): z ∈ ℂ, Re z = 3\ is equal to the interval ( α, β ] , then 24 (β - α) is equal to

JEE Main 2023 Solution View → 67

Let S = \ z ∈ ℂ: ¯ z = i( z ^ 2 + Re ( ¯ z ))\ . Then ∑ _ z ∈ S | z | ^ 2 is equal to

JEE Main 2023 Solution View → 68

Let w = z ¯ z + k _ 1 z + k _ 2 iz + λ ( 1 + i ) , k _ 1 , k _ 2 ∈ ℝ . . Let Re ( w ) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y

JEE Main 2023 Solution View → 69

Let C be the circle in the complex plane with centre z _ 0 = (1)/(2)(1 + 3i) and radius r = 1 . Let z _ 1 = 1 + i and the complex number z _ 2 be outside circle C such that | z _

JEE Main 2023 Solution View → 70

For a ∈ ℂ , let A = z ∈ ℂ: Re ( a + ¯ z ) > Im ( ¯ a + z ) and B = z ∈ ℂ: Re ( a + ¯ z ) ( S1 ) : If Re ( a ) , Im ( a ) > 0 , then the set A contains all the real numbers ( S2 )

JEE Main 2023 Solution View → 71

Let S = \ z ∈ ℂ - i, 2i : z ^ 2 + 8iz - 15 z ^ 2 - 3iz - 2 ∈ ℝ\ . α - (13)/(11)i ∈ S, α ∈ ℝ - 0 , then 242 α ^ 2 is equal to

JEE Main 2023 Solution View → 72

Let w _ 1 be the point obtained by the rotation of z _ 1 = 5 + 4i about the origin through a right angle in the anticlockwise direction, and w _ 2 be the point obtained by the

JEE Main 2023 Solution View → 73

Let S = \ z = x + iy: (2z - 3i)/(4z + 2i)) is a real number . Then which of the following is NOT correct?

JEE Main 2023 Solution View → 74

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

JEE Main 2023 Solution View → 75

Let A = \ θ ∈ (0, 2π): (1 + 2isinθ)/(1 - isinθ)ispurelyimaginary\ Then the sum of the elements is in A is

JEE Main 2023 Solution View → 76

If for z = α + iβ, |z + 2| = z + 4 ( 1 + i ) , then α + β and αβ are the roots of the equation

JEE Main 2023 Solution View → 77

Let a ≠ b be two non-zero real numbers. Then the number of elements in the set X = \ z ∈ C: Re(a z ^ 2 + bz) = aandRe(b z ^ 2 + az) = b\ is equal to

JEE Main 2023 Solution View → 78

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 _____.

JEE Main 2023 Solution View → 79

Let a, b be two real numbers such that ab < 0 . If the complex number (1 + ai)/(b + i) is of unit modulus and a + ib lies on the circle |z - 1| = |2z| , then a possible value of

JEE Main 2023 Solution View → 80

If the center and radius of the circle |(z - 2)/(z - 3)| = 2 are respectively (α, β) and γ , then 3(α + β + γ) is equal to

JEE Main 2023 Solution View → 81

The complex number z = (i - 1)/(cos(π)/(3) + isin(π)/(3)) is equal to:

JEE Main 2023 Solution View → 82

For all z ∈ C on the curve C _ 1 : | z | = 4 , let the locus of the point z + (1)/(z) be the curve C _ 2 . Then

JEE Main 2023 Solution View → 83

Let z = 1 + i and z _ 1 = 1 + i ¯ z ¯ z ( 1 - z ) + (1)/(z) · Then (12)/(π) arg( z _ 1 ) is equal to

JEE Main 2023 Solution View → 84

Let α = 8 - 14i, A = \ z ∈ ℂ: - ¯ α ¯ z z ^ 2 - ( ¯ z ) ^ 2 - 112i = 1\ and B = \ z ∈ ℂ: |z + 3i| = 4\ Then, ∑ _ z ∈ A ∩ B ( Re z - Imz ) is equal to ________

JEE Main 2023 Solution View → 85

For two non-zero complex number z _ 1 and z _ 2 , if Re ( z _ 1 z _ 2 ) = 0 and Re ( z _ 1 + z _ 2 ) = 0 , then which of the following are possible? (A) Im ( z _ 1 ) > 0 and Im (

JEE Main 2023 Solution View → 86

Let z be a complex number such that |(z - 2i)/(z + i)| = 2, z ≠ - i . Then z lies on the circle of radius 2 and centre

JEE Main 2023 Solution View → 87

Let z _ 1 = 2 + 3i and z _ 2 = 3 + 4i . The set S = \ z ∈ C: |z - z _ 1 | ^ 2 - |z - z _ 2 | ^ 2 = | z _ 1 - z _ 2 | ^ 2 \ represents a

JEE Main 2023 Solution View → 88

The value of ((1 + sin(2π)/(9) + icos(2π)/(9))/(1 + sin(2π)/(9) - icos(2π)/(9))) ^ 3 is

JEE Main 2023 Solution View → 89

Let p, q ∈ ℝ and ( 1 - 3 i ) ^ 200 = 2 ^ 199 ( p + iq ) , i = - 1 . Then, p + q + q ^ 2 and p - q + q ^ 2 are roots of the equation.

JEE Main 2023 Solution View → 90

Let ¯ z denote the complex conjugate of a complex number z . If z is a non-zero complex number for which both real and imaginary parts of ( ¯ z ) ^ 2 + 1 z ^ 2 are integers, then

JEE Advanced 2022 Solution View → 91

Let z be a complex number with non-zero imaginary part. If 2 + 3z + 4 z ^ 2 2 - 3z + 4 z ^ 2 is a real number, then the value of |z| ^ 2 is ______.

JEE Advanced 2022 Solution View → 92

Let ¯ z denote the complex conjugate of a complex number z and let i = - 1 . In the set of complex numbers, the number of distinct roots of the equation ¯ z - z ^ 2 = i( ¯ z + z ^

JEE Advanced 2022 Solution View → 93

If z ≠ 0 be a complex number such that |z - (1)/(z)| = 2 , then the maximum value of |z| is

JEE Main 2022 Solution View → 94

Let S = \ z = x + iy: |z - 1 + i| ≥ |z|, |z| < 2, |z + i| = |z - 1|\ . Then the set of all values of x , for which w = 2x + iy ∈ S for some y ∈ ℝ , is

JEE Main 2022 Solution View → 95

If z = 2 + 3i , then z ^ 5 + ( ¯ z ) ^ 5 is equal to:

JEE Main 2022 Solution View → 96

Let z = a + ib, b ≠ 0 be complex numbers satisfying z ^ 2 = ¯ z · 2 ^ 1 - |z| . Then the least value of n ∈ N , such that z ^ n = (z + 1) ^ n , is equal to _____ .

JEE Main 2022 Solution View → 97

Let S _ 1 = \ z _ 1 ∈ C: | z _ 1 - 3| = (1)/(2)\ and S _ 2 = \ z _ 2 ∈ C: | z _ 2 - | z _ 2 + 1|| = | z _ 2 + | z _ 2 - 1||\ . Then, for z _ 1 ∈ S _ 1 and z _ 2 ∈ S _ 2 , the

JEE Main 2022 Solution View → 98

Let S be the set of all (α, β), π Then ∑ _ (α, β) ∈ S (i Z _ αβ + 1 i ¯ Z αβ ) is equal to

JEE Main 2022 Solution View → 99

Let the minimum value v _ 0 of v = |z| ^ 2 + |z - 3| ^ 2 + |z - 6i| ^ 2 , z ∈ ℂ is attained at z = z _ 0 . Then |2 z _ 0 ^ 2 - ¯ z 03 + 3| ^ 2 + v _ 0 ^ 2 is equal to

JEE Main 2022 Solution View → 100

Let S = \ z ∈ ℂ: z ^ 2 + ¯ z = 0\ . Then ∑ _ z ∈ S (Re (z) + Im (z)) is equal to _______.

JEE Main 2022 Solution View → 101

If z = x + iy satisfies |z| - 2 = 0 and |z - i| - |z + 5i| = 0 , then

JEE Main 2022 Solution View → 102

Let O be the origin and A be the point z _ 1 = 1 + 2i . If B is the point z _ 2 , Re ( z _ 2 ) < 0 , such that OAB is a right angled isosceles triangle with OB as hypotenuse, then

JEE Main 2022 Solution View → 103

For z ∈ ℂ if the minimum value of (|z - 3 2 | + |z - p 2 i|) is 5 2 , then a value of p is _______.

JEE Main 2022 Solution View → 104

If α, β, γ, δ are the roots of the equation x ^ 4 + x ^ 3 + x ^ 2 + x + 1 = 0 , then α ^ 2021 + β ^ 2021 + γ ^ 2021 + δ ^ 2021 is equal to

JEE Main 2022 Solution View → 105

For n ∈ N , let S _ n = \ z ∈ C: |z - 3 + 2i| = (n)/(4)\ and T _ n = \ z ∈ C: |z - 2 + 3i| = (1)/(n)\ . Then the number of elements in the set \ n ∈ N: S _ n ∩ T _ n = ϕ\ is

JEE Main 2022 Solution View → 106

Let (z) represent the principal argument of the complex number z . The, |z| = 3 and arg(z - 1) - arg(z + 1) = (π)/(4) intersect:

JEE Main 2022 Solution View → 107

Let α and β be the roots of the equation x ^ 2 + (2i - 1) = 0 . Then, the value of | α ^ 8 + β ^ 8 | is equal to

JEE Main 2022 Solution View → 108

Let S = \ z ∈ C: |z - 2| ≤ 1, z(1 + i) + ¯ z (1 - i) ≤ 2\ . Let |z - 4i| attains minimum and maximum values, respectively, at z _ 1 ∈ S and z _ 2 ∈ S . If 5( | z _ 1 | ^ 2 + | z _

JEE Main 2022 Solution View → 109

Sum of squares of modulus of all the complex numbers z satisfying ¯ z = i z ^ 2 + z ^ 2 - z is equal to

JEE Main 2022 Solution View → 110

The number of elements in the set z = a + ib ∈ ℂ: a, b ∈ ℤ and 1 < |z - 3 + 2i| < 4 is _____.

JEE Main 2022 Solution View → 111

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

JEE Main 2022 Solution View → 112

Let for some real numbers α and β, a = α - iβ . If the system of equations 4ix + (1 + i)y = 0 and 8(cos(2π)/(3) + isin(2π)/(3))x + ¯ a y = 0 has more than one solution then

JEE Main 2022 Solution View → 113

The area of the polygon, whose vertices are the non-real roots of the equation ¯ z = i z ^ 2 is

JEE Main 2022 Solution View → 114

If z ^ 2 + z + 1 = 0, z ∈ C , then | ∑ _ n = 1 ^ 15 ( z ^ n + ( - 1 ) ^ a 1 z ^ n ) ^ 2 | is equal to _____.

JEE Main 2022 Solution View → 115

Let A = \ z ∈ C: |(z + 1)/(z - 1)| < 1\ and B = \ z ∈ C: arg((z - 1)/(z + 1)) = (2π)/(3)\ . Then A ∩ B is

JEE Main 2022 Solution View → 116

Let z _ 1 and z _ 2 be two complex numbers such that ¯ z 1 = i ¯ z 2 and arg( z _ 1 ¯ z 2 ) = π , then the argument of z _ 1 is

JEE Main 2022 Solution View → 117

Let a circle C in complex plane pass through the points z _ 1 = 3 + 4i, z _ 2 = 4 + 3i and z _ 3 = 5i . If z(≠ z _ 1 ) is a point on C such that the line through z and z _ 1 is

JEE Main 2022 Solution View → 118

Let S = z ∈ ℂ: |z - 3| ≤ 1 and z(4 + 3i) + ¯ z (4 - 3i) ≤ 24 . If α + iβ is the point in S which is closest to 4i , then 25(α + β) is equal to ______.

JEE Main 2022 Solution View → 119

Let A = \ z ∈ C: 1 ⩽ |z - (1 + i)| ⩽ 2\ and B = \ z ∈ A: |z - (1 - i)| = 1\ . Then, B

JEE Main 2022 Solution View → 120

Let θ _ 1 , θ _ 2 , … , θ _ 10 be positive valued angles (in radian) such that θ _ 1 + θ _ 2 + ⋯ + θ _ 10 = 2π . Define the complex numbers z _ 1 = e ^ i θ _ 1 , z _ k = z _ k - 1

JEE Advanced 2021 Solution View → 121

For any complex number w = c + id , let arg(w) ∈ ( - π, π ] , where i = - 1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg((z + α)/(z +

JEE Advanced 2021 Solution View → 122

If for the complex numbers z satisfying | z - 2 - 2i | ≤ 1 , the maximum value of | 3iz + 6 | is attained at a + ib , then a + b is equal to _____ .

JEE Main 2021 Solution View → 123

If z is a complex number such that (z - i)/(z - 1) is purely imaginary, then the minimum value of | z - ( 3 + 3i ) | is :

JEE Main 2021 Solution View → 124

A point z moves in the complex plane such that arg((z - 2)/(z + 2)) = (π)/(4), then the minimum value of | z - 9 2 - 2i | ^ 2 is equal to

JEE Main 2021 Solution View → 125

Let z _ 1 and z _ 2 be two complex numbers such that arg( z _ 1 - z _ 2 ) = (π)/(4) and z _ 1 , z _ 2 satisfy the equation |z - 3| = Re ( z ) . Then the imaginary part z _ 1 + z _

JEE Main 2021 Solution View → 126

If S = \ z ∈ C: (z - i)/(z + 2i) ∈ R\ , then

JEE Main 2021 Solution View → 127

If ( 3 + i ) ^ 100 = 2 ^ 99 ( p + iq ) , then p and q are roots of the equation :

JEE Main 2021 Solution View → 128

The least positive integer n such that ( 2i ) ^ n ( 1 - i ) ^ n - 2 , i = - 1 , is a positive integer, is ______.

JEE Main 2021 Solution View → 129

The equation arg((z - 1)/(z + 1)) = (π)/(4) represents a circle with:

JEE Main 2021 Solution View → 130

Let z = 1 - i 3 2 , i = - 1 . Then the value of 21 + (z + (1)/(z)) ^ 3 + ( z ^ 2 + 1 z ^ 2 ) ^ 3 + ( z ^ 3 + 1 z ^ 3 ) ^ 3 + … + ( z ^ 21 + 1 z ^ 21 ) ^ 3 is______.

JEE Main 2021 Solution View → 131

Let ℂ be the set of all complex numbers. Let S _ 1 = \ z ∈ ℂ: |z - 2| ≤ 1\ and S _ 2 = \ z ∈ ℂ: z(1 + i) + ¯ z (1 - i) ≥ 4\ . Then, the maximum value of |z - (5)/(2)| ^ 2 for z ∈

JEE Main 2021 Solution View → 132

If the real part of the complex number z = (3 + 2icosθ)/(1 - 3icosθ), θ ∈ (0, (π)/(2)) is zero, then the value of ^ 2 3θ + ^ 2 θ is equal to ______.

JEE Main 2021 Solution View → 133

Let C be the set of all complex numbers. Let S _ 1 = \ z ∈ C | |z – 3 – 2i| ^ 2 = 8\ , S _ 2 = z ∈ C | Re(z) ≥ 5 and S _ 3 = \ z ∈ C | |z – ¯ z | ≥ 8\ . Then the number of

JEE Main 2021 Solution View → 134

The equation of a circle is Re ( z ^ 2 ) + 2 (Im(z)) ^ 2 + 2 Re (z) = 0, where z = x + iy . A line which passes through the centre of the given circle and the vertex of the

JEE Main 2021 Solution View → 135

Let n denote the number of solutions of the equation z ^ 2 + 3 ¯ z = 0, where z is a complex number. Then the value of ∑ _ k = 0 ^ ∞ 1 n ^ k is equal to

JEE Main 2021 Solution View → 136

If the real part of the complex number (1 - cosθ + 2isinθ) ^ - 1 is (1)/(5) for θ ∈ ( 0, π ) , then the value of the integral ∫ _ 0 ^ θ sinxdx is equal to:

JEE Main 2021 Solution View → 137

If z and ω are two complex numbers such that |zω| = 1 and arg(z) - arg(ω) = (3π)/(2) , then arg( 1 - 2 ¯ z ω 1 + 3 ¯ z ω ) is: (Here arg (z) denotes the principal argument of

JEE Main 2021 Solution View → 138

Let a complex number be w = 1 - 3 i . Let another complex number z be such that | zw | = 1 and arg(z) - arg(w) = (π)/(2) . Then the area of the triangle (in sq. units) with

JEE Main 2021 Solution View → 139

If f ( x ) and g ( x ) are two polynomials such that the polynomial P ( x ) = f( x ^ 3 ) + xg( x ^ 3 ) is divisible by x ^ 2 + x + 1 , then P ( 1 ) is equal to ___ .

JEE Main 2021 Solution View → 140

If the equation a | z | ^ 2 + ¯ ¯ α z + α ¯ z + d = 0 represents a circle where a, d are real constants then which of the following condition is correct?

JEE Main 2021 Solution View → 141

Let z _ 1 , z _ 2 be the roots of the equation z ^ 2 + az + 12 = 0 and z _ 1 , z _ 2 form an equilateral triangle with origin. Then, the value of |a| is

JEE Main 2021 Solution View → 142

Let S _ 1 , S _ 2 and S _ 3 be three sets defined as S _ 1 = z ∈ ℂ: | z - 1 | ≤ 2 , S _ 2 = z ∈ ℂ: Re ( ( 1 - i ) z ) ≥ 1 and S _ 3 = z ∈ ℂ: Im ( z ) ≤ 1 . Then, the set S _ 1 ∩ S

JEE Main 2021 Solution View → 143

The area of the triangle with vertices P ( z ) , Q ( iz ) and R ( z + iz ) is

JEE Main 2021 Solution View → 144

The least value of |z| where z is complex number which satisfies the inequality e ^ (((|z| + 3)(|z| - 1))/(||z| + 1|) _ e 2) ≥ _ 2 |5 7 + 9i| , i = - 1 , is equal to :

JEE Main 2021 Solution View → 145

Let a complex number z, |z| ≠ 1 , satisfy log 1 2 ( |z| + 11 (|z| - 1) ^ 2 ) ≤ 2 . Then, the largest value of |z| is equal to _________.

JEE Main 2021 Solution View → 146

Let z and w be two complex numbers such that w = z ¯ z - 2z + 2, |(z + i)/(z - 3i)| = 1 and Re (w) has minimum value. Then, the minimum value of n ∈ N for which w ^ n is real, is

JEE Main 2021 Solution View → 147

Let z be those complex numbers which satisfy |z + 5| ≤ 4 and z(1 + i) + ¯ z (1 - i) ⩾ - 10, i = - 1 . If the maximum value of | z + 1 | ^ 2 is α + β 2 , then the value of (α + β)

JEE Main 2021 Solution View → 148

Let the lines (2 - i)z = (2 + i) ¯ z and (2 + i)z + (i - 2) ¯ z - 4i = 0, (here i ^ 2 = - 1 ) be normal to a circle C . If the line iz + ¯ z + 1 + i = 0 is tangent to this circle

JEE Main 2021 Solution View → 149

Let i = - 1 . If (- 1 + i 3 ) ^ 21 (1 - i) ^ 24 + (1 + i 3 ) ^ 21 (1 + i) ^ 24 = k, and n = [|k|] be the greatest integral part of |k| . Then ∑ _ j = 0 ^ n + 5 (j + 5) ^ 2 - ∑ _ j

JEE Main 2021 Solution View → 150

If the least and the largest real values of α, for which the equation z + α|z - 1| + 2i = 0 (z ∈ Candi = - 1 ) has a solution, are p and q respectively; then 4( p ^ 2 + q ^ 2 ) is

JEE Main 2021 Solution View → 151

For a complex number z , let Re (z) denote the real part of z . Let S be the set of all complex numbers z satisfying z ^ 4 - | z | ^ 4 = 4i z ^ 2 , where i = - 1 . Then the

JEE Advanced 2020 Solution View → 152

Let S be the set of all complex numbers z satisfying | z ^ 2 + z + 1| = 1 . Then which of the following statements is/are TRUE?

JEE Advanced 2020 Solution View → 153

Let z = x + iy be a non-zero complex number such that z ^ 2 = i |z| ^ 2 , where i = - 1 , then z lies on the :

JEE Main 2020 Solution View → 154

The region represented by z = x + iy ∈ C: | z | - Re ( z ) ≤ 1 is also given by the inequality

JEE Main 2020 Solution View → 155

The value of ( - 1 + i 3 1 - i ) ^ 30 is :

JEE Main 2020 Solution View → 156

If the four complex numbers z, ¯ z , ¯ z - 2 Re ( ¯ z ) and z - 2 Re(z) represent the vertices of a square of side 4 units in the Argand plane, then |z| is equal to :

JEE Main 2020 Solution View → 157

If a and b are real numbers such that (2 + α) ^ 4 = a + bα , where α = - 1 + i 3 2 , then a + b is equal to:

JEE Main 2020 Solution View → 158

Let u = (2z + i)/(z - ki), z = x + iy and k > 0 . If the curve represented by Re ( u ) + Im ( u ) = 1 intersects the y -axis at points P and Q where PQ = 5 then the value of k is

JEE Main 2020 Solution View → 159

If z _ 1 , z _ 2 are complex numbers such that Re ( z _ 1 ) = | z _ 1 - 1| and Re ( z _ 2 ) = | z _ 2 - 1| and arg( z _ 1 - z _ 2 ) = (π)/(6) , then Im( z _ 1 + z _ 2 ) is equal

JEE Main 2020 Solution View → 160

If ((1 + i)/(1 - i))(m)/(2) = ((1 + i)/(i - 1))(n)/(3) = 1, (m, n ∈ N) then the greatest common divisor of the least values of m and n is

JEE Main 2020 Solution View → 161

The imaginary part of (3 + 2 - 54 )(1)/(2) - (3 - 2 - 54 )(1)/(2) ,can be

JEE Main 2020 Solution View → 162

The value of ((1 + sin(2π)/(9) + icos(2π)/(9))/(1 + sin(2π)/(9) - icos(2π)/(9))) ^ 3 is

JEE Main 2020 Solution View → 163

If z is a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be

JEE Main 2020 Solution View → 164

Let z be a complex number such that |(z - i)/(z + 2i)| = 1 and |z| = (5)/(2) . Then, the value of |z + 3i| is

JEE Main 2020 Solution View → 165

Let α = - 1 + i 3 2 . If a = (1 + α) ∑ _ k = 0 ^ 100 α ^ 2k and b = ∑ _ k = 0 ^ 100 α ^ 3k , then a and b , are the roots of the quadratic equation.

JEE Main 2020 Solution View → 166

If the equation x ^ 2 + bx + 45 = 0, b ∈ R has conjugate complex roots and they satisfy |z + 1| = 2 10 , then

JEE Main 2020 Solution View → 167

If (3 + isinθ)/(4 - icosθ), θ ∈ [0 ,2 π], is a real number, then an argument of sinθ + icosθ is

JEE Main 2020 Solution View → 168

If Re((z - 1)/(2z + i)) = 1, where z = x + iy, then the point (x, y) lies on a

JEE Main 2020 Solution View → 169

Let S be the set of all complex numbers z satisfying |z - 2 + i| ≥ 5 . If the complex number z _ 0 is such that 1 | z _ 0 - 1| is the maximum of the set \ (1)/(|z - 1|): z ∈ S\

JEE Advanced 2019 Solution View → 170

Let ω ≠ 1 be a cube root of unity. Then the minimum of the set |a + bω + c ω ^ 2 | ^ 2 : a, b, c distinct non – zero integers equals ________

JEE Advanced 2019 Solution View → 171

Let z ∈ C with Im ( z ) = 10 and it satisfies (2z - n)/(2z + n) = 2i - 1 for some natural number n . Then

JEE Main 2019 Solution View → 172

The equation | z - i | = | z - 1 | , i = - 1 , represents:

JEE Main 2019 Solution View → 173

If z and ω are two complex numbers such that |zω| = 1 and arg(z) - arg( ω ) = (π)/(2) , then:

JEE Main 2019 Solution View → 174

If a > 0 and z = ( 1 + i ) ^ 2 a - i , has magnitude (2)/(5) , then - z is equal to:

JEE Main 2019 Solution View → 175

Let z ∈ C be such that |z| < 1 . If ω = (5 + 3z)/(5(1 - z)), then:

JEE Main 2019 Solution View → 176

All the points in the set S = \ (α + i)/(α - i), α ∈ R\ , i = - 1 lie on a

JEE Main 2019 Solution View → 177

If z = 3 2 + (i)/(2)(i = - 1 ), then (1 + iz + z ^ 5 + i z ^ 8 ) ^ 9 is equal to:

JEE Main 2019 Solution View → 178

If α and β be the roots of the equation x ^ 2 - 2x + 2 = 0, then the least value of n for which ((α)/(β)) ^ n = 1 is

JEE Main 2019 Solution View → 179

Let z _ 1 and z _ 2 be two complex numbers satisfying | z _ 1 | = 9 and | z _ 2 - 3 - 4i | = 4 . Then the minimum value of | z _ 1 - z _ 2 | is :

JEE Main 2019 Solution View → 180

If (z - α)/(z + α)(α ∈ R) is a purely imaginary number and |z| = 2 , then a value of α is :

JEE Main 2019 Solution View → 181

Let z be a complex number such that |z|+z=3+i ( where i= -1 ) Then |z| is equal to :

JEE Main 2019 Solution View → 182

Let (-2-(1)/(3) i)^ 3 =(x+i y)/(27)(i= -1 ), where x and y are real numbers then y-x equals

JEE Main 2019 Solution View → 183

Let z = ( 3 2 + (i)/(2)) ^ 5 + ( 3 2 - (i)/(2)) ^ 5 . If R(z) and I(z) respectively denote the real and imaginary parts of z, then

JEE Main 2019 Solution View → 184

Let z _ 1 and z _ 2 be any two non-zero complex numbers such that 3| z _ 1 | = 4| z _ 2 | . If z = 3 z _ 1 2 z _ 2 + 2 z _ 2 3 z _ 1 then maximum value of |z| is Note: In actual

JEE Main 2019 Solution View → 185

Let z _ 0 be a root of quadratic equation, x ^ 2 + x + 1 = 0 . If z = 3 + 6i z _ 0 ^ 81 - 3i z _ 0 ^ 93 , then arg (z) is equal to:

JEE Main 2019 Solution View → 186

Let A = \ θ ∈ (- (π)/(2), π): (3 + 2isinθ)/(1 - 2isinθ)ispurelyimaginary\ . Then the sum of the elements in A is:

JEE Main 2019 Solution View → 187

Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy(x, y ∈ ℝ, i = - 1 ) of the equation sz + t - z + r = 0 , where - z = x - iy . Then, which of the

JEE Advanced 2018 Solution View → 188

For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < arg(z) ≤ π then, which of the following statement(s) is (are) FALSE?

JEE Advanced 2018 Solution View → 189

The least positive integer n for which ( 1 + i 3 1 - i 3 ) ^ n = 1 is

JEE Main 2018 Solution View → 190

If |z-3+2 i| ≤ 4 then the difference between the greatest value and the least value of |z| is

JEE Main 2018 Solution View → 191

The set of all α ∈ R , for which w=(1+(1-8 α) z)/(1-z) is a purely imaginary number, for all z ∈ C satisfying |z|=1 and Re z ≠ 1 , is

JEE Main 2018 Solution View → 192

The set of all α ∈ R , for which w = (1 + (1 - 8α)z)/(1 - z) is a purely imaginary number, for all z ∈ C satisfying |z| = 1 and Re(z) ≠ 1 , is :

JEE Main 2018 Solution View → 193

If α, β ∈ C are the distinct roots of the equation x ^ 2 - x + 1 = 0, then α ^ 101 + β ^ 107 is equal to

JEE Main 2018 Solution View → 194

Let a, b, x and y be real numbers such that a - b = 1 and y ≠ 0 . If the complex number z = x + iy satisfies Im((az + b)/(z + 1)) = y, then which of the following is (are)

JEE Advanced 2017 Solution View → 195

The equation Im((iz - 2)/(z - i)) + 1 = 0, z ∈ C, z ≠ i represents a part of a circle having radius equal to :

JEE Main 2017 Solution View → 196

Let z ∈ C, the set of complex numbers. Then the equation, 2|z + 3i| - |z - i| = 0 represents:

JEE Main 2017 Solution View → 197

Let ω be a complex number such that 2ω + 1 = z where z = - 3 . If | cases 1 & 1 & 1 1 & - ω ^ 2 - 1 & ω ^ 2 1 & ω ^ 2 & ω ^ 7 cases | = 3k, Then k can be equal to:

JEE Main 2017 Solution View → 198

Let a, b ∈ ℝ and a ^ 2 + b ^ 2 ≠ 0 . Suppose S = \ z ∈ ℂ: z = (1)/(a + ibt), t ∈ ℝ, t ≠ 0\ , where i = - 1 . If z = x + iy and z ∈ S, then (x, y) lies on

JEE Advanced 2016 Solution View → 199

Let z = 1 + ai , be a complex number, a > 0 , such that z ^ 3 is a real number. Then, the sum 1 + z + z ^ 2 + … . + z ^ 11 is equal to :

JEE Main 2016 Solution View → 200

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there 2 2 units in the south-west wards direction. Then its new

JEE Main 2016 Solution View →
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Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Binomial Theorem (550)Quadratic Equation (517)Determinants (508)

Complex Number in other exams

NDA Complex Number (46)BITSAT Complex Number (30)