JEE Main Maths · Complex Numbers PYQ
JEE Main Complex Numbers Previous Year Questions (2002–2026)
Complex Numbers is a P1 Algebra foundation chapter — ~5% JEE Main weightage with 1–2 questions per session. Mastery here supports Quadratic Equations (complex roots), Trigonometry (De Moivre's) and Coordinate Geometry (Argand plane). High-yield chapter for dedicated Algebra students.
Complex Numbers PYQs from 2002 to 2026, tagged by sub-topic (modulus/argument, De Moivre's, nth roots of unity, geometry). Every solution shows z = a + bi decomposition and trigonometric form conversions.
Complex Numbers at a Glance
Try 5 Real Complex Numbers PYQs
5 real JEE Main questions on Complex Numbers, straight from past papers. Try them, then check the answer.
- Q1JEE Main 2026 (Online) 24th January Morning Shift
The number of the real solutions of the equation: $x|x+3|+|x-1|-2$${}=0$ is
3
2
5
4
- Q2JEE Main 2025 (Online) 4th April Evening Shift
Let the product of $\omega_1$${}=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2$${}=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be $\alpha+i \beta$, $i$${}=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $\mathrm{p}+\mathrm{q}$ is equal to :
- 130
- 150
- 160
- 140
- Q3JEE Main 2024 (Online) 29th January Evening Shift
Let the set $C$${}=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$. Then $\sum\limits_{(x, y) \in C}(x+y)$ is equal to _________.
- Q4JEE Main 2023 (Online) 8th April Evening Shift
Let $A$${}=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in A is :
- $3 \pi$
- $\pi$
- $2 \pi$
- $4 \pi$
- Q5JEE Main 2022 (Online) 26th July Morning Shift
The number of distinct real roots of the equation $x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1$${}=0$ is ______________.
Practise the full Complex Numbers set
Every past-paper question of this chapter, with solutions, timing and your accuracy tracked.
Key Sub-Topics & What's Tested
Complex Number Basics
z = a + bi, real/imaginary parts, conjugate z̄ = a - bi, addition, multiplication, division.
Modulus & Argument
|z| = √(a² + b²), arg(z) ∈ (-π, π], polar form z = r(cos θ + i sin θ), relationship |z₁ z₂| = |z₁||z₂|.
Argand Plane
Geometric representation of complex numbers, rotation by θ = multiplication by cis θ, distance between two complex numbers.
De Moivre's Theorem
(cos θ + i sin θ)^n = cos nθ + i sin nθ, applications to trigonometric identities, nth roots of complex number.
Nth Roots of Unity
Solutions of zⁿ = 1: ω = e^(2πi/n), cube roots of unity (1, ω, ω²), properties 1 + ω + ω² = 0, ω³ = 1.
Geometric Interpretation
|z - a| = r is circle, |z - a| + |z - b| = 2k is ellipse (for 2k > |a - b|), arg((z-a)/(z-b)) = const is arc.
Complex Conjugate Properties
z + z̄ = 2 Re(z), z - z̄ = 2i Im(z), z·z̄ = |z|², conjugate of sum = sum of conjugates.
Triangle Inequality
||z₁| - |z₂|| ≤ |z₁ ± z₂| ≤ |z₁| + |z₂|, equality conditions, applications in range problems.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Modulus / Argument Calculation | 25% | Find |1 + i|^n where n = 10. |
| Nth Roots of Unity Problem | 20% | If ω is cube root of unity, evaluate 1 + ω + ω² and 1 + ω² + ω⁴. |
| Geometric Locus | 15% | |z - 2| + |z + 2| = 10 represents which conic? |
| De Moivre's Application | 15% | Evaluate (1 + i)^8 using De Moivre's theorem. |
| Complex Equation Solving | 15% | Find all z satisfying z² + |z| = 0. |
| Conjugate / Triangle Inequality | 10% | Prove that |z₁ + z₂|² + |z₁ - z₂|² = 2(|z₁|² + |z₂|²). |
How to Solve Complex Numbers PYQs
- 1Convert to polar form for De Moivre. z = r·cis θ where cis θ = cos θ + i sin θ. Powers and roots become rotations in polar form — much faster than Cartesian.
- 2Memorise cube roots of unity properties. 1 + ω + ω² = 0, ω³ = 1, ω·ω̄ = 1, ω̄ = ω². These identities appear frequently across PYQs.
- 3For geometric locus, interpret |z - a| and arg. |z - a| = r → circle radius r centre a. |z - a| - |z - b| = const → hyperbola. arg((z-a)/(z-b)) = const → arc.
- 4Use conjugate for real/imaginary extraction. Re(z) = (z + z̄)/2, Im(z) = (z - z̄)/(2i). Useful when separating real and imaginary parts in equations.
- 5Nth roots of unity lie on unit circle. They are evenly spaced around |z| = 1. Sum of all n roots = 0 for n ≥ 2. Key property.
Common Mistakes That Cost Marks
- Wrong sign for conjugate. z̄ = a - bi (flips sign of imaginary part only). Not -a - bi.
- Missing periodicity of argument. arg(z) is defined modulo 2π. Principal value is in (-π, π]. Ignoring this gives ambiguous answers.
- Applying De Moivre's to r^n incorrectly. (r·cis θ)^n = rⁿ·cis(nθ). Both magnitude AND angle change.
- Using ω² as ω instead of conjugate. For cube roots, ω² = ω̄ (complex conjugate). Confusing these breaks identity calculations.
- Wrong triangle inequality direction. |z₁ + z₂| ≤ |z₁| + |z₂| (sum). |z₁ - z₂| ≥ ||z₁| - |z₂|| (difference). Don't confuse.
Related JEE Main Practice
Frequently asked questions
How many Complex Numbers PYQs should I solve?
Target 60–80 PYQs across 2010–2026. Given the chapter's role as foundation for multiple downstream topics, consistent practice has compound returns.
Is De Moivre's theorem tested every year?
Yes — at least one PYQ per paper uses De Moivre's theorem directly or implicitly (via polar form of complex powers/roots). Mastering it is non-negotiable.
What's the most-tested Complex Numbers pattern?
Cube roots of unity identities (1 + ω + ω² = 0, ω³ = 1) — appears in ~5 PYQs per 10-year window in different disguises.
Do I need hyperbolic identities for Complex Numbers?
No — JEE Main stays within standard trigonometric (cos, sin) and exponential complex identities. Hyperbolic functions are beyond the syllabus here.
How should I approach geometric locus PYQs?
Let z = x + iy. Translate condition to x, y equation. Recognise standard conic form (circle, ellipse, parabola, hyperbola). Many PYQs are disguised conic-section problems.
Does Complex Numbers connect with Quadratic Equations?
Yes — quadratic equations with negative discriminant have complex roots. PYQs test this link via problems asking for product/sum of complex roots.
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