Maths · Class 11 · Complex Numbers and Quadratic Equations

Complex Numbers for JEE

Everything you need for this topic on one page: the ideas, the formulas, the traps, and free lectures that explain it well.

Last updated: 26 October 2026

Direct answer

Algebra of complex numbers, modulus and argument, polar and Euler forms, cube roots of unity, De Moivre's theorem and loci in the Argand plane.

Key ideas

Complex Numbers: what you must understand

|z|² = z z̄
arg(z₁z₂) = arg z₁ + arg z₂
Cube roots of unity: 1 + ω + ω² = 0, ω³ = 1
|z − z₀| = r is a circle

Formula sheet

Must-know formulas for Complex Numbers

z = r(cosθ + i sinθ) = re^(iθ)
De Moivre: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ
Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|
|z − z₁| = |z − z₂| is the perpendicular bisector

One-page formula sheet

Printable Complex Numbers formula sheet: formulas, key ideas and traps on one page. Save it as a PDF for offline revision.

Avoid

Common mistakes students make

Principal argument range (−π, π]
Forgetting the quadrant when finding arg
Treating i² as +1 in long expansions

For the exam

How this topic is asked in JEE

Appears frequently. Locus questions are common. Convert to x + iy when stuck.

NCERT map

Where Complex Numbers is in NCERT

Chapter names follow the current rationalised NCERT textbooks. Read these sections before solving PYQs.

Class	Book	Chapter	Sections JEE asks from
Class 11	Mathematics	Complex Numbers and Quadratic Equations	Algebra of Complex Numbers; Modulus and Conjugate of a Complex Number; Argand Plane and Polar Representation; Quadratic Equations

JEE Advanced level

A JEE Advanced-level example on Complex Numbers

Question. Let z be a non-zero complex number satisfying |z - 1| + |z + 1| = 4. If arg(z) = π/3, find the value of |z|².

Show solution

The equation |z - 1| + |z + 1| = 4 represents an ellipse with foci at (1, 0) and (-1, 0). Here 2a = 4 so a = 2, and 2c = 2 so c = 1. This gives b² = a² - c² = 3.
The ellipse equation is x²/4 + y²/3 = 1.
Since arg(z) = π/3, we can write z = r(cos π/3 + i sin π/3) = r/2 + i(r√3)/2 for some r > 0. So x = r/2 and y = (r√3)/2.
Substituting into the ellipse equation: (r/2)²/4 + ((r√3)/2)²/3 = 1. This simplifies to r²/16 + r²/4 = 1, giving 5r²/16 = 1.
Solving gives r² = 16/5. Since |z| = r, we get |z|² = 16/5.

Answer: 16/5

This example was solved twice independently and both solutions agree.

Free lectures

Best free lectures for Complex Numbers

Full-chapter free lectures picked from YouTube search results by views, from established JEE channels (mostly in Hinglish). Use the subtopic list to jump straight to the part you need (timestamps come from each video's own chapter list). Watch one, then solve PYQs the same day.

Complex Numbers in 1 Shot - All Concepts, Tricks & PYQs Covered | Class 11 | JEE Main & Advanced — JEE Wallah, 5:57:13
Jump to a subtopic (20)
- 0:00 Introduction
- 4:51 Iota and Its Powers
- 27:09 Complex Numbers
- 37:24 Algebra of Complex Numbers
- 53:21 Equality of Complex Numbers
- 1:30:13 Modulus, Argument, Conjugate of Complex Number
- 1:42:53 BREAK 1
- 1:54:43 How To Find Argument of a Complex Number?
- 2:33:47 Mixed Properties of Modulus And Conjugate
- 2:58:55 Important Result
- 3:06:38 Square Root of a Complex Number
- 3:25:29 Three Forms of a Complex Numbers
- 3:33:30 BREAK 2
- 3:56:05 Problems on Forms of Complex Numbers
- 4:08:17 Euler's Identity and Important Relation
- 4:16:15 Demoivre's Theorem
- 4:24:07 Applications of DMT
- 4:40:50 Nth Roots of Unity
- 5:13:00 Cube Roots of Unity
- 5:26:41 Important Observations
Complex Numbers FULL CHAPTER | Class 11th Maths | Arjuna JEE — Arjuna JEE, 4:36:23
Jump to a subtopic (9)
- 00:00 Introduction
- 34:09 Algebra of Complex Numbers
- 59:07 Conjugate of a Complex Numbers (2)
- 01:11:05 Modulus of a Complex Numbers (z)
- 01:34:48 Argument of a Complex Numbers
- 02:09:38 Demoivre's Theorem
- 02:20:34 Finding Cube Roots of Unity Using D.M.T
- 02:41:59 Roots of Unity
- 04:35:11 Thank You Bachoo!!

Practise

Where to practise for free

10 JEE Main 2026 PYQs on Complex Numbers, with solutions.
Read the chapter in NCERT textbooks (official, free), then solve its NCERT Exemplar (official, free) questions.
Chapter-wise past questions: JEE Main and Advanced PYQs and question bank.
Timed practice tests: NTA Abhyas practice tests (official, free) and our free mock tests.
See it move: interactive simulation.

FAQs

Frequently asked questions

Is Complex Numbers important for JEE Main 2027?

Appears frequently. It is part of the official 'Complex Numbers and Quadratic Equations' unit in the JEE Main 2027 syllabus.

What are the most important formulas in Complex Numbers?

z = r(cosθ + i sinθ) = re^(iθ); De Moivre: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ; Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|.

How should I study Complex Numbers for JEE?

Understand the key ideas from NCERT or one full lecture, learn the formula sheet, then solve 25–30 past JEE questions on the topic. Locus questions are common. Convert to x + iy when stuck.