Class 11 Maths NCERT Solutions Chapter 4 Complex Numbers and Quadratic Equations | Boundless Maths
Chapter 4 Class 11 Maths NCERT Solutions · Unit I · Sets and Functions

Class 11 Maths NCERT Solutions Chapter 4: Complex Numbers and Quadratic Equations

Free, step-by-step NCERT Solutions for this chapter — the definition and algebra of complex numbers, powers of i, the modulus and conjugate, and the Argand plane — solved the way CBSE awards marks, with the key definitions and formulas right on this page.

2Exercises (incl. Misc.)
28Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 4 — Overview

A complex number a + ib solves the gap left by real numbers, using i where i² = −1. This chapter covers the algebra of complex numbers (addition, subtraction, multiplication, division, and multiplicative inverse), the cyclical powers of i, why √a × √b = √(ab) fails when both a and b are negative, the modulus |z| and conjugate z̄ of a complex number, and how z = a + ib is plotted as the point (a, b) on the Argand plane.

A note on this chapter's scope

The current NCERT textbook (2025-26 reprint) covers only complex number algebra, the modulus/conjugate, and the Argand plane in this chapter — it does not include a separate exercise on solving quadratic equations with complex roots, which appeared in older editions. If your syllabus expects that section, please check your own copy of the textbook.

How the Chapter Builds

One Idea Leads to the Next

1

Complex Numbers

Defining i, the form a + ib, real and imaginary parts, and equality of two complex numbers. §4.2.

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2

Algebra of Complex Numbers

Addition, subtraction, multiplication, division and the multiplicative inverse. Exercise 4.1.

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3

Powers of i & Negative Square Roots

The 4-cycle of iⁿ, and why √a·√b ≠ √(ab) when a, b are both negative. §4.3.5–4.3.6.

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4

Modulus & Conjugate

|z| = √(a²+b²) and z̄ = a − ib, plus the identity z·z̄ = |z|². §4.4.

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5

Argand Plane

Plotting z = a + ib as the point (a, b); modulus as distance from the origin. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 4

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Complex Numbers.

Complex Numbers & Their Algebra (§4.2–4.3)

Definition

z=a+ib,\quad i^2=-1

a is the real part (Re z), b is the imaginary part (Im z). Two complex numbers are equal only if both parts match.

Addition & subtraction

(a{+}ib)\pm(c{+}id)=(a{\pm}c)+i(b{\pm}d)

Combine real parts together and imaginary parts together.

Multiplication

(a{+}ib)(c{+}id)=(ac{-}bd)+i(ad{+}bc)

Expand like binomials, then replace i² with −1 and collect terms.

Multiplicative inverse

z^{-1}=\dfrac{\bar z}{|z|^2}=\dfrac{a}{a^2{+}b^2}-i\dfrac{b}{a^2{+}b^2}

Same technique used to divide: multiply numerator and denominator by the conjugate.

Powers of i & Negative Square Roots (§4.3.5–4.3.6)

Cycle of powers of i

i^{4k}=1,\; i^{4k+1}=i,\; i^{4k+2}=-1,\; i^{4k+3}=-i

Reduce any exponent by dividing by 4 and using the remainder — works for negative exponents too.

Square root of a negative number

\sqrt{-a}=\sqrt{a}\,i \quad (a>0)

√a × √b = √(ab) fails when both a and b are negative — always convert to i-form first, then multiply.

Modulus, Conjugate & the Argand Plane (§4.4–4.5)

Modulus

|z|=\sqrt{a^2+b^2}

Always a non-negative real number — the distance of the point (a, b) from the origin in the Argand plane.

Conjugate

\bar z = a-ib

Flip only the sign of the imaginary part — geometrically, the mirror image of z across the real axis.

Key identity

z\bar z=|z|^2

Multiplying a complex number by its own conjugate always gives a real, non-negative result.

Decision Guide

Which Complex Number Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Simplify an expression with iⁿ for a large or negative nPowers of i cycleReduce n using the 4-cycle (1, i, −1, −i) rather than expanding directly (§4.3.5).
Add, subtract or multiply two complex numbersAlgebra of complex numbersTreat i as a variable, expand, then substitute i² = −1 at the end (§4.3.1–4.3.3).
Divide by a complex number, or find its inverseMultiply by the conjugateMultiplying numerator and denominator by z̄ clears i from the denominator (§4.3.4).
Take the square root of a negative number√−a = √a · iNever apply √a·√b = √(ab) directly when both are negative — convert to i-form first (§4.3.6).
Find |z| or express z̄Modulus / conjugate formulas|z| is always real and non-negative; z̄ only flips the imaginary part's sign (§4.4).
Prove an identity involving z + z̄ or z·z̄z z̄ = |z|², or write z = a+ib directlyConverting to a+ib form and comparing real/imaginary parts usually settles these quickly.
Represent or interpret a complex number geometricallyArgand planePlot a + ib as the point (a, b); |z| is its distance from the origin (§4.5).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks in this chapter.

  • Treating i² as +1 — the entire chapter rests on i² = −1; a single sign slip here cascades through the rest of a calculation.
  • Applying √a × √b = √(ab) when both a and b are negative — this identity only holds for non-negative reals (or one negative, one positive). For two negatives it gives the wrong sign; always rewrite each root as (positive number)·i first.
  • Dividing by a complex number without multiplying by its conjugate — leaving i in a denominator is not a simplified final answer; always rationalize using z̄/z̄.
  • Getting the conjugate's sign wrong — only the imaginary part flips sign; the real part of z̄ is identical to the real part of z.
  • Confusing modulus and conjugate — |z| is a single non-negative real number; z̄ is still a complex number. They answer different kinds of questions.
  • Reducing powers of i incorrectly for negative exponents — for i⁻ⁿ, first rewrite as 1/iⁿ, simplify iⁿ using the 4-cycle, then take the reciprocal, rather than guessing the pattern.
  • Dropping "+ i0" or "+ 0i" when an answer turns out purely real or purely imaginary — if a question asks for the form a + ib, state both parts explicitly, even when one of them is zero.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 4 — Choose an Exercise

4.1

Exercise 4.1

Algebra of complex numbers — expressing expressions in a + ib form, and finding the multiplicative inverse · 14 questions

Solve Exercise 4.1 →
M

Miscellaneous Exercise

Modulus, conjugate and Argand-plane proofs, plus harder simplifications, tying the whole chapter together · 14 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Complex Numbers — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 4, Complex Numbers and Quadratic Equations.

How many exercises are there in Chapter 4, Complex Numbers and Quadratic Equations?
In the current NCERT textbook, this chapter has one main exercise — 4.1 (Algebra of Complex Numbers, 14 questions) — plus a Miscellaneous Exercise of 14 questions, totalling 28 questions. Earlier editions also included a separate exercise on solving quadratic equations with complex roots; check your own copy of the textbook if that section is expected.
What is i, and why is it needed?
i is defined as the square root of −1, so that i² = −1. It is needed because the equation x² + 1 = 0 has no solution among the real numbers, since the square of every real number is non-negative. Introducing i extends the number system to complex numbers, where such equations do have solutions.
What is the difference between the modulus and the conjugate of a complex number?
For z = a + ib, the modulus |z| is the non-negative real number √(a²+b²) — it measures the distance of the point (a, b) from the origin in the Argand plane. The conjugate, written z̄, is the complex number a − ib — geometrically, it is the mirror image of z across the real axis. The modulus is always a real number; the conjugate is always a complex number.
Where can I find the official NCERT textbook for this chapter?
Complex Numbers and Quadratic Equations is Chapter 4 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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