# NCERT solutions class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

## NCERT solutions for class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

Hello to Everyone who have come here for the the NCERT Solutions of Chapter 5 Complex Numbers class 11. (Complex Numbers and Quadratic Equations class 11)

All the Exercises (Ex 5.1 , Ex 5.2 , Ex 5.3 and Miscellaneous exercise) of Complex numbers and Quadratic Equations class 11 are very well explained and Notes are prepared by HarMohit Singh. (Subject Teacher)

Chapter 5 Complex Numbers class 11 is one of the most important chapter which is the base of your higher studies in College (B.tech, Maths Hons etc) and so you should properly understand it.

As you all know I always start from the basics and so Firstly we will understand Basics of Complex Numbers class 11 and after that we will understand the Basics of Quadratic Equations class 11.

## Basics of Complex Numbers class 11

For understanding Complex Numbers we should know What are Real Numbers? I tried to explain it below in the Notes. I f you want more better knowledge of Real Numbers Click Here.

## Chapter 5 Exercise 5.1 class 11 maths

Ch 5 maths class 11 Exercise 5.1 Complex Numbers and Quadratic Equations NCERT solutions.

## Chapter 5 Exercise 5.2 class 11 maths

Ch 5 maths class 11 Exercise 5.2 Complex Numbers and Quadratic Equations NCERT solutions.

### What is Argand Plane?

As you all know about Cartesian plane in which we plot Real numbers on Graph papers(Till class 10), Similarly Argand Plane is Plotting of Complex Numbers on Graph papers.

### Polar Representation of Complex Numbers

Here I have Explained Basics of Polar Representation of Complex Numbers class 11.

### Chapter 5 Exercise 5.2 class 11

Ch 5 ex 5.2 class 11 ncert solutions.

## Chapter 5 Exercise 5.3 class 11 maths

Ch 5 maths class 11 Exercise 5.3 Complex Numbers and Quadratic Equations NCERT solutions.

Here I have explained What are Quadratic Equations? and How to solve Quadratic Equations? Discriminant Method

### Chapter 5 Exercise 5.3 class 11

Ch 5 ex 5.3 class 11 ncert solutions.

## Chapter 5 Miscellaneous Exercise class 11 maths

Ch 5 maths class 11 Miscellaneous Exercise Complex Numbers and Quadratic Equations NCERT solutions.

NCERT solutions class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

## What are Complex Numbers?

Complex Numbers are very little extension to real numbers that is adding of Real number with Imaginary number is called Complex Number or we can say Complex numbers are sum of Real numbers and Imaginary numbers.

Complex Number = Real number + Imaginary number

For Example we have given Real number = 5 and Imaginary number = √7 i

Hence Complex number = 5 + √7 i

Another example we have given Real number = 10 and Imaginary number = 13 i

Hence Complex number = 10 + 13 i

Generally we denote Complex Number as Z

and We generally write Complex Number as

Z = a + bi

Here ‘a’ denotes Real number and ‘b’ denotes Imaginary part

## What are Imaginary Numbers ?

Numbers like √-3, √-5, √-7 which are negative numbers with root are called Imaginary numbers.

Mathematician introduced a term IOTA (i) which is equal to √-1.

IOTA (i) = √-1

By using this value of Iota we can write √-2 = √-1 × √2 = √2 i (because i = √-1)

Similarly √-5 = √5 i

√-7 = √7 i

√-8 = √8 i

√-49 = √49 i = 7 i

## Solving Complex Numbers

No We will see How can we solve Complex Numbers. Addition, Subtraction, Multiplication, Division of Complex Numbers.

### How do we add Complex Numbers?

Addition of Complex Numbers : Real part of first complex number is added to Real part of second complex number and Imaginary part of first complex number is added to Imaginary part of second complex number. Don’t worry take an Example:

Suppose we have two complex numbers

First Complex number = 5 + 3i

Second Complex number = 7 + 6i

Adding both Complex Number = (5 + 7) + (3i+6i)

Z = 12 + 9i

### How do we Subtract Complex Numbers?

Subtraction of Complex Numbers : It is same as above Addition part. Real part is subtracted from Real part and Imaginary part is subtracted from Imaginary part.

For example:

Suppose we have two complex numbers

First Complex number = (10 + 8i)

Second Complex number = (3 + 2i)

Subtracting second Complex Number from First = (10 – 3)  + (8i – 2i)

Z = (7 + 6i)

### Multiplication of Complex Numbers

Suppose we have two complex numbers

First Complex number = (10 + 8i)

Second Complex number = (3 + 2i)

Real part is multiplied with Real part and Imaginary part is multiplied with Imaginary part.

Multiply both complex Numbers = (10 × 3) + (8i × 2i)

Z = 30 + 16 i² = 30 + 16 × -1 = 30 – 16 = 14

### Division of Complex Numbers

Suppose we have two complex numbers

First Complex number = (10 + 8i)

Second Complex number = (3 + 2i)

\frac{(10 + 8i)}{(3 + 2i)}

That is it we can also leave here BUT there is a rule in Mathematics that we can never leave i (iota) in the denominator of Fraction.

So here We have to Rationalize

\frac{(10 + 8i)}{(3 + 2i)} × \frac{(3 - 2i)}{(3 - 2i)}

Now we just have to do multiplication.

= \frac{(10 × 3) + (8i × -2i)}{(3 × 3) + (2i × -2i)}

= \frac{(30 - 16 i²)}{9 - 4 i²}

Now we know value if i² = -1

= \frac{(30 - 16 × -1)}{9 - 4× -1}

= \frac{(30 + 16)}{9 + 4}

= \frac{46}{13}

NCERT solutions for class 11 Maths Chapter 2 Relations and Functions

NCERT solutions for class 11 Maths Chapter 3 Trigonometric Functions

NCERT solutions for class 11 Maths Chapter 4 Principle of Mathematical induction

NCERT solutions for class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities

NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations

NCERT solutions for class 11 Maths Chapter 8 Binomial Theorem

NCERT solutions for class 11 Maths Chapter 9 Sequences and Series

NCERT solutions for class 11 Maths Chapter 10 straight lines