Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations NCERT Solutions | Full Guide

Class 11 Maths Chapter 4 – Complex Numbers and Quadratic Equations

Introduction

This chapter introduces a new type of number called complex numbers. These numbers help solve equations that do not have real solutions.

You will also learn how to solve quadratic equations in a better and more complete way using formulas and roots.

Complex Numbers

A complex number is written as:

$$z = a + ib$$

Where
a = real part
b = imaginary part
i = √(-1)

Properties of i

• i² = -1

• i³ = -i

• i⁴ = 1

Operations on Complex Numbers

Let z₁ = a + ib and z₂ = c + id

• Addition: (a+c) + i(b+d)

• Subtraction: (a-c) + i(b-d)

• Multiplication: (ac - bd) + i(ad + bc)

Conjugate of Complex Number

If z = a + ib

Then conjugate is

z̄ = a - ib

Modulus of Complex Number

|z| = √(a² + b²)

Argand Plane

• Real part plotted on x-axis

• Imaginary part plotted on y-axis

Quadratic Equation

Standard form:

Discriminant

D = b² - 4ac

• D > 0 → real and distinct roots

• D = 0 → equal roots

• D < 0 → complex roots

Solved NCERT Examples (Step by Step)

Example 1

Find i⁵

Solution:
i⁴ = 1
i⁵ = i⁴ × i = i

Example 2

Add (3 + 2i) and (1 - 4i)

Solution:
= (3+1) + (2i - 4i)
= 4 - 2i

Example 3

Multiply (2 + 3i)(1 + i)

Solution:
= 2 + 2i + 3i + 3i²
= 2 + 5i - 3
= -1 + 5i

Example 4

Find modulus of 3 + 4i

Solution:
|z| = √(3² + 4²)
= √(9 + 16) = 5

Example 5

Find conjugate of 5 - 2i

Solution:
= 5 + 2i

Example 6

Solve x² - 5x + 6 = 0

Solution:
(x-2)(x-3)=0
x = 2, 3

Example 7

Solve x² + 4 = 0

Solution:
x² = -4
x = ±2i

Example 8

Find discriminant of x² + 2x + 5

Solution:
D = 2² - 4×1×5 = 4 - 20 = -16

Example 9

Solve using formula: x² - 2x + 5 = 0

Solution:
x = [2 ± √(-16)]/2
= 1 ± 2i

Example 10

Find product of roots of x² + 7x + 10

Solution:
Product = c/a = 10

Practice Focus (From NCERT Exercises)

• Simplifying powers of i

• Operations on complex numbers

• Finding modulus and conjugate

• Solving quadratic equations

• Using discriminant

15 FAQs 

1. What is a complex number?

A number of the form a + ib where i = √(-1).

2. What is i?

Imaginary unit defined as √(-1).

3. Value of i²?

-1.

4. What is conjugate?

Changing sign of imaginary part.

5. Modulus formula?

|z| = √(a² + b²).

6. What is Argand plane?

Graphical representation of complex numbers.

7. Standard quadratic form?

ax² + bx + c = 0.

8. Quadratic formula?

(-b ± √(b² - 4ac)) / 2a.

9. What is discriminant?

b² - 4ac.

10. When are roots equal?

When D = 0.

11. When are roots complex?

When D < 0.

12. Sum of roots?

-b/a.

13. Product of roots?

c/a.

14. What is real part?

The value a in a + ib.

15. What is imaginary part?

The value b in a + ib.