Complex Numbers

24 solved problems from JEE previous year papers.

Complex Numbers

Let |z₁ − 8 − 2i| ≤ 1 and |z₂ − 2 + 6i| ≤ 2, z₁, z₂ ∈ C. Then the Minimum Value of |z₁ − z₂| Is

JEE Main 2025 PYQ: z1 and z2 satisfy |z1-8-2i|≤1 and |z2-2+6i|≤2. Find the minimum value of |z1-z2| by interpreting the…
Complex Numbers

Let z₁ and z₂ Be Two Complex Numbers Such That z₁ + z₂ = 5 and z₁³ + z₂³ = 20 + 15i. Then |z₁⁴ + z₂⁴| Equals

JEE Main 2024 PYQ: given z1+z2=5 and z1³+z2³=20+15i, find |z1⁴+z2⁴| by working through z1z2, then z1²+z2², using the standard…
Complex Numbers

Let |z−i|/|2z+i| = 1/3, z ∈ C, Be the Equation of a Circle With Centre at C. If the Area of the Triangle Whose Vertices Are (0,0), C and (α, 0) Is 11 Square Units, Then α² Equals

JEE Main 2025 PYQ: |z-i|/|2z+i|=1/3 traces a circle with center C. Given the triangle (0,0), C, (α,0) has area 11, find α² using…
Complex Numbers

Two Numbers k₁ and k₂ Are Randomly Chosen From the Set of Natural Numbers. Then the Probability That the Value of i^k₁ + i^k₂ (Where i = √−1) Is Non-Zero Is Equal To

JEE Main 2025 PYQ: k1 and k2 are chosen randomly from the natural numbers — find the probability that i^k1+i^k2 is non-zero,…
Complex Numbers

|z/(z+i)|=2 Represents a Circle — Find Distance of Centre From D(1,5)

The locus |z/(z+i)|=2 in the complex plane is a circle with centre P. Find the distance of P from the point D(1,5) by converting…
Complex Numbers

Let |(z̄ − i)/(2z̄ + i)| = 1/3, z ∈ C, Be the Equation of a Circle With Centre C. If the Area of the Triangle Whose Vertices Are (0, 0), C and (α, 0) Is 11 Square Units, Then α² Equals

JEE Main 2025 PYQ: find alpha squared given that |(z-bar – i)/(2z-bar + i)| = 1/3 is a circle with centre C, and the…
Complex Numbers

If Re((z−1)/(2z+i)) = 1, Where z = x + iy, Then the Point (x, y) Lies On

Find the locus of (x,y) given Re((z-1)/(2z+i))=1 for z=x+iy, by rationalising the fraction and simplifying its real part into a…
Complex Numbers

Let z₁, z₂ and z₃ Be Three Complex Numbers on the Circle |z| = 1 With arg(z₁) = −π/4, arg(z₂) = 0 and arg(z₃) = π/4. If |z₁z̄₂ + z₂z̄₃ + z₃z̄₁|² = α + β√2, α, β ∈ Z, Then α² + β² Is

Given z1,z2,z3 on |z|=1 with arg(z1)=−π/4, arg(z2)=0, arg(z3)=π/4, find α²+β² where |z1z̄2+z2z̄3+z3z̄1|²=α+β√2.
Complex Numbers

The Sum of All Possible Values of θ ∈ [−π, 2π] for Which (1 + i cos θ)/(1 − 2i cos θ) Is Purely Imaginary Is Equal To

Find the sum of all possible values of θ ∈ [-π, 2π] for which (1+i cosθ)/(1-2i cosθ) is purely imaginary. Multiple-choice: 3π,…
Complex Numbers

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−5i|} Is

Find the sum of the squares of the modulus of complex numbers z=a+ib (a,b integers) satisfying |z−1|≤1 and |z−5|≤|z−5i|, by…
Complex Numbers

Let α, β Be the Roots of the Equation x² − x + 2 = 0 With Im(α) > Im(β). Then α⁶ + α⁴ + β⁴ − 5α² Is Equal To

Given α, β are the roots of x²−x+2=0 with Im(α)>Im(β), find α⁶+α⁴+β⁴−5α² by repeatedly reducing powers using α²=α−2.
Complex Numbers

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

Given |z+2|=1 and Im((z+1)/(z+2))=1/5, find |Re(z+2)| by extracting y directly from the imaginary-part condition and using the…
Complex Numbers

If z Is a Complex Number Such That |z| ≥ 2, Then the Minimum Value of |z + 1/2| Is

Given |z|≥2, find the minimum value of |z+1/2| by writing z in polar form and minimising the distance from z to the point −1/2.
Complex Numbers

Let z Be a Complex Number Such That the Real Part of (z − 2i)/(z + 2i) Is Zero. Then the Maximum Value of |z − (6 + 8i)| Is Equal To

Given the real part of (z−2i)/(z+2i) is zero, show z traces a circle of radius 2 about the origin, then find the maximum distance…
Complex Numbers

The Number of Complex Numbers z Satisfying |z| = 1 and |z/z̄ + z̄/z| = 1 Is

Find how many complex numbers z satisfy both |z|=1 and |z/z̄+z̄/z|=1 simultaneously, by reducing the condition to x²−y²=±1/2 on…
Complex Numbers

If S = {z ∈ C : |z − i| = |z + i| = |z − 1|}, Then n(S) Is

Find the number of complex numbers z satisfying |z−i|=|z+i|=|z−1| simultaneously, by recognising z as the circumcenter of the…
Complex Numbers

If the Centre and Radius of the Circle |(z−2)/(z−3)| = 2 Are Respectively (α, β) and γ, Then 3(α + β + γ) Is Equal To

The locus |(z−2)/(z−3)|=2 in the complex plane is a circle with center (α,β) and radius γ. Find 3(α+β+γ) by converting to…
Complex Numbers

If z₁, z₂ Are Two Distinct Complex Numbers Such That |z₁−2z₂|/|1/2−z₁z̄₂| = 2, Then

If z1, z2 are distinct complex numbers with |z1−2z2|/|1/2−z1z̄2|=2, determine which circle condition z1 and z2 must satisfy, by…
Complex Numbers

Roots of 2z²−3z−2i=0 — Find 16·Re(E)·Im(E) for a Power-Sum Ratio

If α, β are roots of 2z²−3z−2i=0, find 16·Re(E)·Im(E) where E=(α^19+β^19+α^11+β^11)/(α^15+β^15), using a Newton’s-identity…
Complex Numbers

Let α and β Be Two Roots of the Equation x² + 2x + 2 = 0. Then α¹⁵ + β¹⁵ Is Equal To

If α, β are the roots of x²+2x+2=0, find α¹⁵+β¹⁵ by converting the complex roots to polar form and using De Moivre’s…
Complex Numbers

|z|≤1: Minimum Value of |z+1/2(3+4i)|

If z is a complex number with |z|≤1, find the minimum value of |z+1/2(3+4i)| by interpreting it as the distance from a fixed…
Complex Numbers

α,β Roots of x²−x+1=0 — Find α¹⁰¹+β¹⁰⁷

If α,β are the distinct complex roots of x²−x+1=0, find α¹⁰¹+β¹⁰⁷ by recognising the roots as negatives of cube roots of unity.
Complex Numbers

Let A = {z ∈ C : |(z+1)/(z−1)| < 1} and B = {z ∈ C : arg((z−1)/(z+1)) = 2π/3}. Then A ∩ B Is

Let A: |(z+1)/(z−1)|
Complex Numbers

Complex Roots of x²-(3-2i)x-(2i-2)=0, Find αγ+βδ

A quadratic with complex coefficients and complex roots is easier factorised directly than approached through Vieta’s…