PrepWiser › Learn › Mathematics › Complex NumbersComplex Numbers — JEE Main PYQs & Practice Questions Complex Numbers is a Mathematics chapter in the JEE Main syllabus. This page brings together 169 previous-year and practice questions on Complex Numbers, organised into 6 topics: Algebra of Complex Numbers, Modulus and Argument, Polar and Euler Form, De Moivre's Theorem, Cube Roots of Unity and Geometry in Argand Plane. Most questions come with a step-by-step solution you can work through free in PrepWiser.
Download Complex Numbers JEE Main PYQs as a free PDF
Revise this chapter: Revision notes · Formula sheet
Chapter weightage : how many questions each chapter had per shift in the 2024–26 papers.
Algebra of Complex Numbers If the set {Re((z-z̄+z z̄)/(2-3 z+5 z̄)): z ∈ ℂ, Re(z)=3} is equal to the interval (α… The imaginary part of ( 3 + 2√(- 54) )^(1/2) - ( 3 - 2√(- 54) )^(1/2) can be : Let z₁,z₂∈ ℂ be the distinct solutions of the equation z²+4z-(1+12i)=0. Then |z₁|²+|z₂|²… For a ∈ ℂ, let A={z ∈ ℂ: Re(a+z̄) > Im(ā+z)} and B={z ∈ ℂ: Re(a+z̄)<Im(ā+z)}. Then… Let A={θ ∈(0,2 π): (1+2 i sin θ)/(1-i sin θ) is purely imaginary }. Then the sum of the… If z=1/2-2 i is such that |z+1|=α z+β(1+i), i=√-1 and α, β ∈ ℝ, then α+β is equal to Let S={z:3≤ |2z-3(1+i)|≤ 7} be a set of complex numbers. Then Min_(z∈ S)|(z+1/2(5+3i))|… Let α and β be the roots of the equation x² + (2i - 1) = 0. Then, the value of |α⁸ + β⁸|… If α, β ∈ R are such that 1 - 2i (here i² = -1) is a root of z² + αz + β = 0, then (α -… Let x and y be real numbers such that 50(2x/(1+3i)-y/(1-2i))=31+17i,i=√-1. Then the… If the set R={(a, b): a+5 b=42, a, b ∈ ℕ} has m elements and Σn=1ᵐ(1-i^(n !))=x+i y… Sum of squares of modulus of all the complex numbers z satisfying z̅ = iz² + z² - z is… Let S={z∈ ℂ:z²+√6iz-3=0}. Then Σ_(z∈ S)z⁸ is equal to : Let z=a+i b, b ≠ 0 be complex numbers satisfying z²=z̄ · 2^(1-|z|). Then the least value… If α + iβ and γ + iδ are the roots of x² - (3 - 2i)x - (2i - 2) = 0, i = √-1, then α γ +… Let a,b∈ ℂ. Let α,β be the roots of the equation x²+ax+b=0. If β-α=√11 and β²-α²=3i√11… Let S={z∈ ℂ:|(z-6i)/(z-2i)|=1 and |(z-8+2i)/(z+2i)|=3/5}. Then Σ_(z∈ s)|z|² is equal to : The real part of the complex number ((1 + 2i)⁸ . (1 - 2i)²)/((3 + 2i) . (4 - 6i)̅) is… Let z ∈ C be such that (z²+3 i)/(z-2+i)=2+3 i. Then the sum of all possible values of z²… Let α and β be the sum and the product of all the non-zero solutions of the equation… Modulus and Argument If z=x+i y satisfies |z|-2=0 and |z-i|-|z+5 i|=0, then : If the equation, x² + bx + 45 = 0 (b ∈ R) has conjugate complex roots and they satisfy… Let the complex number z = x + iy be such that (2z - 3i)/(2z + i) is purely imaginary… The modulus of 3 - 4i is: If z is a complex number such that |z| ≤ 1, then the minimum value of |z+1/2(3+4 i)| is : If z and ω are two complex numbers such that | zω | = 1 and arg (z) - arg (ω ) = 3π/2… If z ≠ 0 be a complex number such that |z-1/z|=2, then the maximum value of |z| is : If α denotes the number of solutions of |1-i|ˣ=2ˣ and β=(|z|/(arg (z))), where… Let z and ω be two complex numbers such that ω = zz̅ - 2z + 2,| (z + i)/(z - 3i) | = 1… Consider the following two statements : Statement I: For any two non-zero complex… Let r and θ respectively be the modulus and amplitude of the complex number z=2-i(2 tan… Among the statements (S1) : The set {z ∈ ℂ-{-i}:|z|=1 and (z-i)/(z+i) is purely real }… The least value of |z| where z is complex number which satisfies the inequality exp (… If for z=α+i β,|z+2|=z+4(1+i), then α+β and α β are the roots of the equation : Let z₁, z₂ and z₃ be three complex numbers on the circle |z|=1 with arg (z₁)=-π/4, arg… For z ∈ ℂ if the minimum value of (|z-3 √2|+|z-p √2 i|) is 5 √2, then a value Question… Let z be complex number such that | (z - i)/(z + 2i) | = 1 and |z| = 5/2. Then the value… If |z₁ z₂| = |z₁| |z₂| for all complex numbers z₁, z₂. If z = 4 + 3i, then |z|² = x. Let a complex number z, |z| ≠ 1, satisfy log _(1/√2)( (|z| + 11)/(|z| - 1)² ) ≤ 2. Then… Polar and Euler Form De Moivre's Theorem The value of ( (- 1 + i√3)/(1 - i) )³⁰ is : Let i = √(- 1). If ( - 1 + i√3 )²¹/(1 - i)²⁴ + ( 1 + i√3 )²¹/(1 + i)²⁴ = k, and n =… If ( √3 + i )¹⁰⁰ = 2⁹⁹(p + iq), then p and q are roots of the equation : Let p,q∈ℝ and ( 1 - √3 i )²⁰⁰ = 2¹⁹⁹(p + iq),i = √(- 1) then p+q+q² and p-q+q² are roots… If z=√3/2+i/2,i=√-1, then (z²⁰¹-i)⁸ is equal to If z=2+3 i, then z⁵+(z̄)⁵ is equal to : Let α, β be the roots of the equation x²-√6 x+3=0 such that Im(α)>Im(β). Let a, b be… Let α, β be the roots of the equation x²-√2 x+2=0. Then α¹⁴+β¹⁴ is equal to The sum of 162ᵗʰ power of the roots of the equation x³ - 2x² + 2x - 1 = 0 is ________. If ( (1 + i)/(1 - i) )^(m/2) = ( (1 + i)/(1 - i) )^(n/3) = 1, (m, n ∈ N) then the… The value of ( (1 + sin 2π/9 + i cos 2π/9)/(1 + sin 2π/9 - i cos 2π/9) )³ is Cube Roots of Unity If α, β, γ, δ are the roots of the equation x⁴+x³+x²+x+1=0, then α²⁰²¹+β²⁰²¹+γ²⁰²¹+δ²⁰²¹… If a and b are real numbers such that ( 2 + α )⁴ = a + bα where α = (- 1 + i√3)/2 then a… Let α be a root of the equation 1 + x² + x⁴ = 0. Then, the value of α¹⁰¹¹ + α²⁰²² -… If x²+x+1=0, then the value of (x+1/x)⁴+(x²+1/x²)⁴+(x³+1/x³)⁴+… +(x²⁵+1/x²⁵)⁴ is: If α satisfies the equation x²+x+1=0 and (1+α)⁷=A+B α+C α², A, B, C ≥ 0, then 5(3 A-2… If z² + z + 1 = 0, z ∈ C, then | Σₙ₌₁¹⁵ ( zⁿ + ( - 1)ⁿ1/zⁿ )² | is equal to _________. Let integers a, b ∈[-3,3] be such that a+b ≠ 0. Then the number of all possible ordered… Let α be a solution of x² + x + 1 = 0, and for some a and b in R, [4, a, b] [1, 16, 13… If z is a complex number, then the number of common roots of the equations… Let α=(-1+i√3)/2 and β=(-1-i√3)/2,i=√-1. If… Let z = (1 - i√3)/2, i = √(- 1). Then the value of 21 + ( z + 1/z )³ + ( z² + 1/z² )³ +… Let α = (- 1 + i√3)/2. If a = ( 1 + α )Σₖ₌₀¹⁰⁰ α ²ᵏ and b = Σₖ₌₀¹⁰⁰ α ³ᵏ, then a and b… If α is a root of the equation x²+x+1=0 and Σk=1ⁿ(αᵏ+1/αᵏ)²=20, then n is equal to… If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x³) + x g(x³) is… Geometry in Argand Plane For all z ∈ C on the curve C₁:|z|=4, let the locus of the point z+1/z be the curve C₂… Let C be the set of all complex numbers. Let S₁ = { z ∈ C||z - 3 - 2i|² = 8} S₂ = { z ∈… Let a circle C in complex plane pass through the points z₁ = 3 + 4i, z₂ = 4 + 3i and z₃… If the four complex numbers z,z̅ ,z̅ - 2 Re ( z̅ ) and z-2Re(z) represent the vertices… If S = { z ∈ C:(z - i)/(z + 2i) ∈ R }, then : If S={z ∈ C:|z-i|=|z+i|=|z-1|}, then, n(S) is : Let C be the set of all complex numbers. Let S₁ = z∈C : |z - 2| ≤ 1 and S₂ = z∈C : z(1 +… Let z₁=2+3i and z₂=3+4i. The set S = { z ∈ ℂ:| z - z₁ |² - | z - z₂ |² = | z₁ - z₂ |² }… Let P={z ∈ ℂ:|z+2-3 i| ≤ 1} and Q={z ∈ ℂ: z(1+i)+z̄(1-i) ≤-8}. Let in P ∩ Q, |z-3+2 i|… If Re ( (z - 1)/(2z + i) ) = 1, where z = x + iy, then the point (x, y) lies on a : Let the curve z(1+i)+z̄(1-i)=4, z ∈ C, divide the region |z-3| ≤ 1 into two parts of… Let the minimum value v₀ of v=|z|²+|z-3|²+|z-6 i|², z ∈ ℂ is attained at z=z₀. Then |2… If for the complex numbers z satisfying | z - 2 - 2i | ≤ 1, the maximum value of | 3iz +… The sum of the square of the modulus of the elements in the set {z=a+ib: a, b ∈ Z, z ∈… Let S₁={z₁ ∈ C:|z₁-3|=1/2} and S₂={z₂ ∈ C:|z₂-| z₂+1||=|z₂+| z₂-1||}. Then, for z₁ ∈ S₁… Let A = { z ∈ C:1 ≤ |z - (1 + i)| ≤ 2} and B = { z ∈ A:|z - (1 - i)| = 1}. Then, B : Let S₁={z ∈ C:|z| ≤ 5}, S₂={z ∈ C: Im((z+1-√3 i)/(1-√3 i)) ≥ 0} and S₃={z ∈ C: Re(z) ≥… Let S={z=x+i y:|z-1+i| ≥|z|,|z|<2,|z+i|=|z-1|}. Then the set of all values of x, for… The equation of a circle is Re(z²) + 2(Im(z))² + 2Re(z) = 0, where z = x + iy. A line… The equation arg ( (z - 1)/(z + 1) ) = π/4 represents a circle with : More Mathematics chapters Frequently asked questions How many Complex Numbers questions are there for JEE Main? PrepWiser has 169 Complex Numbers questions for JEE Main, grouped into 6 topics such as Algebra of Complex Numbers, Modulus and Argument and Polar and Euler Form.
Are these Complex Numbers questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Complex Numbers questions? Open any Complex Numbers question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.