Chapter 1 Complex Numbers Class 10 Maths — full chapter solved. All exercises (1.1–1.4) and Review Exercise 1 with step-by-step solutions. Free PDF, Al Huda Science Academy.

Chapter 1: Complex Numbers — Class 10 Maths Full Solved (New Book)

By Sir Usman Adal | Al Huda Science Academy | Board Exam Preparation Series

This is the complete, fully solved guide to Chapter 1: Complex Numbers for Class 10 Mathematics (New Book). Every exercise — 1.1, 1.2, 1.3, 1.4, and the Review Exercise — is solved step by step below. Click any exercise to open its full solution, or scroll down for a quick summary of everything the chapter covers.

📄 Want everything in one file? Download the complete Chapter 1 PDF (all exercises combined) at the bottom of this page.

📚 Chapter 1 — Table of Contents

Click any exercise below to jump straight to its full solved solution:

Exercise 1.1 — Powers of i, Simplification, Finding x and y Simplifying powers of i (like i⁵, i¹⁶, i⁻¹⁹), writing expressions in terms of i, and solving for unknowns by comparing real and imaginary parts.
Exercise 1.2 — Operations on Complex Numbers, Inverses Addition, subtraction, multiplication, division of complex numbers, additive & multiplicative inverses, and verifying algebraic properties (commutative, associative).
Exercise 1.3 — Modulus and Conjugate Finding the modulus of a complex number, verifying conjugate identities (sum, product, quotient), and evaluating real/imaginary parts of products.
Exercise 1.4 — Real/Imaginary Parts, Simultaneous Equations Finding real and imaginary parts of powers and inverses of complex numbers, and solving simultaneous linear equations with complex coefficients.
Review Exercise 1 — Mixed Practice A full review combining conjugate identities, modulus, division, and simultaneous equations from across the entire chapter — ideal for exam revision.

What is a Complex Number? (Quick Recap)

A complex number is written in the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1. Complex numbers let us work with square roots of negative numbers, which have no solution within real numbers alone.

Chapter 1 — Key Formulas at a Glance

Powers of i (cycle of 4): i⁰ = 1, i¹ = i, i² = −1, i³ = −i, i⁴ = 1

Conjugate: if z = a + bi, then z̄ = a − bi

Modulus: |z| = √(a² + b²)

Multiplicative inverse: z⁻¹ = (a − bi)/(a² + b²)

Key identity: z · z̄ = |z|²

Conjugate distributes over operations:

conjugate(z₁ + z₂) = conjugate(z₁) + conjugate(z₂)
conjugate(z₁ · z₂) = conjugate(z₁) · conjugate(z₂)
conjugate(z₁ / z₂) = conjugate(z₁) / conjugate(z₂)

Division rule: always multiply numerator and denominator by the conjugate of the denominator.

How to Use This Chapter for Exam Preparation

Start with Exercise 1.1 to build comfort with powers of i — almost every later exercise depends on this.
Work through 1.2 and 1.3 together, since they build the operations (add/subtract/multiply/divide) and properties (modulus, conjugate) you’ll use constantly.
Exercise 1.4 is the most exam-heavy — simultaneous equations with complex coefficients show up often in board papers.
Finish with the Review Exercise a few days before your test as a timed mixed-practice run.

Frequently Asked Questions

How many exercises are in Chapter 1 Complex Numbers, Class 10 (New Book)?

The chapter has five exercises (1.1 to 1.4& review exercise) plus one Review Exercise at the end, all covered on this page.

What is the most important concept in Chapter 1 Complex Numbers?

Understanding that i² = −1 and knowing the 4-cycle of powers of i is the foundation — nearly every question in the chapter builds on it.

Is Chapter 1 Complex Numbers hard for Class 10 students?

It’s mostly algebraic manipulation once the rules for i are clear. The main difficulty students face is sign errors when i² = −1 is substituted, so careful step-by-step practice (as in the exercises above) is the best way to build confidence.