JEE Maths›Complex Numbers & Quadratic Equations›Question Types
Top Question Types

Complex Numbers & Quadratic Equations - How It Appears in JEE

12 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

01

Modulus/Argument Manipulation

Pattern

Given conditions on |z| or arg(z), find a value or simplify

How to recognize

Question mentions |z−a|, arg(z/w), or asks for |z|²

→ Convert to x+iy form or use geometric interpretation on Argand plane. Use |z|² = z·z̄ for algebraic approach.
02

Locus-Based Questions

Pattern

Find the locus of z satisfying a given condition

How to recognize

Question says 'locus of z' or 'set of points z such that'

→ Put z = x + iy, substitute, simplify to standard curve equation. Or use geometric properties directly.
03

Conjugate & Division Problems

Pattern

Simplify expressions involving z̄, division of complex numbers

How to recognize

Expression has z̄, z/w, or 'real/imaginary part of'

→ Multiply by conjugate of denominator. Use z·z̄ = |z|². Separate real and imaginary parts.
04

De Moivre's Theorem Applications

Pattern

Find zⁿ, expand (cosθ + i sinθ)ⁿ, or find nth roots

How to recognize

Powers of complex numbers, 'find all roots', or trigonometric identity derivation

→ Convert to polar form → apply De Moivre's → convert back. For roots, use zₖ = r^(1/n) · e^(i(θ+2πk)/n).
05

Roots of Unity

Pattern

Problems involving ω, cube roots, nth roots of unity

How to recognize

Mentions ω, '1+ω+ω²', 'primitive root', or symmetric expressions

→ Use 1+ω+ω²=0 and ω³=1. For nth roots, sum=0 and product=(-1)^(n+1).
06

Geometry on Argand Plane

Pattern

Find area, distance, angle, or geometric properties using complex numbers

How to recognize

Triangle/polygon formed by complex numbers, rotation, midpoint, centroid

→ Use |z₁−z₂| for distance, arg((z₃−z₁)/(z₂−z₁)) for angle, rotation by multiplying by e^(iθ).
07

Triangle Inequality Applications

Pattern

Find maximum/minimum of |z₁ + z₂| or |f(z)| given constraints

How to recognize

Min/max of modulus, 'range of |z|', or optimization on complex plane

→ Apply ||z₁|−|z₂|| ≤ |z₁±z₂| ≤ |z₁|+|z₂|. Draw on Argand plane for geometric insight.
08

Quadratic Equations with Complex Roots

Pattern

Find roots, form equations, or use properties of complex roots

How to recognize

Quadratic with Δ<0, 'complex roots α,β', sum/product of roots

→ Use conjugate pair property. Sum = -b/a, product = c/a. New equation from roots: x² - (sum)x + (product) = 0.
09

Purely Real / Purely Imaginary Conditions

Pattern

Find conditions for z or f(z) to be purely real or purely imaginary

How to recognize

'z is purely real', 'imaginary part = 0', 'lies on real/imaginary axis'

→ z is real ⟺ z = z̄. z is purely imaginary ⟺ z = −z̄ (and z≠0). Set Im(z)=0 or Re(z)=0.
10

Apollonius Circle / Section Formula

Pattern

|z−a| = k|z−b| type problems

How to recognize

Ratio of distances to two fixed points, |z−a|/|z−b| = constant

→ If k=1 → perpendicular bisector (line). If k≠1 → Apollonius circle. Square both sides and substitute z=x+iy.
11

Rotation Problems

Pattern

One complex number obtained by rotating another

How to recognize

'z₂ is obtained by rotating z₁ by angle θ', perpendicular complex numbers

→ z₂ − c = (z₁ − c)·e^(iθ) where c is center of rotation. For 90° rotation, multiply by i.
12

Complex Number Equations

Pattern

Solve z² = w, zⁿ = w, or systems of complex equations

How to recognize

Solve for z, 'find all z satisfying', polynomial equations in z

→ Convert to polar form. Use De Moivre for zⁿ = w. For z² = a+bi, let z=x+iy and compare real/imaginary parts.