# NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

**
NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations:
**
In the earlier classes you have studied the quadratic equations. You must have come across some equations like x
^{
2
}
+2=0, x
^{
2
}
=-2, for which there is no real solution. How to solve these quadratic equations? In
**
**
NCERT solutions for class 11 maths chapter 5 complex numbers and quadratic equations, you will learn to solve equations like x
^{
2
}
+2=0. This chapter will introduce you to a new term called i (iota),
. Using this you will solve the quadratic equation
with
. This chapter is useful not only in solving quadratic equations but also in solving the alternating current circuits and in vector analysis. In CBSE NCERT solutions for class 11 maths chapter 5 complex numbers and quadratic equations, you will learn to solve quadratic equations that have imaginary roots. In this chapter, there are 32 questions in 3 exercises. All the questions are explained in solutions of NCERT for class 11 maths chapter 5 complex numbers and quadratic equations in a detailed manner. It will be very useful for you to understand the concepts.
Check all
**
NCERT solutions
**
from class 6 to 12 to learn CBSE science and maths. There are three exercises and a miscellaneous exercise in this chapter which are explained below.

##
**
The main topics of the NCERT Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are
**

5.1 Introduction

5.2 Complex Numbers

5.3 Algebra of Complex Numbers

5.4 The Modulus and the Conjugate of a Complex Number

5.5 Argand Plane and Polar Representation

5.6 Quadratic Equations

##
**
The NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations is given below:
**

##
**
Solutions of NCERT for class 11 maths chapter 5
Complex Numbers and Quadratic Equations-Exercise: 5.1
**

**
Question:1
**
Express each of the complex number in the form
.

**
Answer:
**

On solving

we will get

Now, in the form of
we can write it as

**
Question:2
**
Express each of the complex number in the form
.

**
Answer:
**

We know that

Now, we will reduce
into

Now, in the form of
we can write it as

Therefore, the answer is

**
Question:3
**
Express each of the complex number in the form a+ib.

**
Answer:
**

We know that

Now, we will reduce
into

Now, in the form of
we can write it as

Therefore, the answer is

**
Question:4
**
Express each of the complex number in the form a+ib.

**
Answer:
**

Given problem is

Now, we will reduce it into

Therefore, the answer is

**
Question:5
**
Express each of the complex number in the form
.

**
Answer:
**

Given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:6
**
Express each of the complex number in the form
.

**
Answer:
**

Given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:7
**
Express each of the complex number in the form
.

**
Answer:
**

Given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:8
**
Express each of the complex number in the form
.

**
Answer:
**

The given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:9
**
Express each of the complex number in the form
.

**
Answer:
**

Given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:10
**
Express each of the complex number in the form
.

**
Answer:
**

Given problem is

**
**

Now, we will reduce it into

**
**

Therefore, the answer is

**
Question:11
**
Find the multiplicative inverse of each of the complex numbers.

**
Answer:
**

Let
**
**

Then,

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

**
Question:12
**
Find the multiplicative inverse of each of the complex numbers.

**
Answer:
**

Let
**
**

Then,

**
**

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

**
Question:13
**
Find the multiplicative inverse of each of the complex numbers.

**
Answer:
**

Let
**
**

Then,

**
**

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

**
Question:14
**
Express the following expression in the form of

**
Answer:
**

Given problem is

Now, we will reduce it into

Therefore, answer is

##
**
Solutions of NCERT for class 11 maths chapter 5
Complex Numbers and Quadratic Equations-Exercise: 5.2
**

**
Question:1
**
Find the modulus and the arguments of each of the complex numbers.

**
Answer:
**

Given the problem is

**
**

Now, let

Square and add both the sides

Therefore, the modulus is 2

Now,

Since, both the values of
is negative and we know that they are negative in III quadrant

Therefore,

Argument =

Therefore, the argument is

**
Question:2
**
Find the modulus and the arguments of each of the complex numbers.

**
Answer:
**

Given the problem is

Now, let

Square and add both the sides

Therefore, the modulus is 2

Now,

Since values of
is negative and value
is positive and we know that this is the case in II quadrant

Therefore,

Argument =

Therefore, the argument is

**
Question:3
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given problem is

Now, let

Square and add both the sides

Therefore, the modulus is

Now,

Since values of
is negative and value
is positive and we know that this is the case in the IV quadrant

Therefore,

Therefore,

Therefore, the required polar form is

**
Question:4
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given the problem is

Now, let

Square and add both the sides

Therefore, the modulus is

Now,

Since values of
is negative and value
is positive and we know that this is the case in II quadrant

Therefore,

Therefore,

Therefore, the required polar form is

**
Question:5
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given problem is

Now, let

Square and add both the sides

&nbsnbsp;

Therefore, the modulus is

Now,

Since values of both
and
is negative and we know that this is the case in III quadrant

Therefore,

Therefore,

Therefore, the required polar form is

**
Question:6
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given problem is

Now, let

Square and add both the sides

Therefore, the modulus is
**
3
**

Now,

Since values of
is negative and
is Positive and we know that this is the case in II quadrant

Therefore,

Therefore,

Therefore, the required polar form is

**
Question:7
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given problem is

Now, let

Square and add both the sides

Therefore, the modulus is
**
2
**

Now,

Since values of Both
and
is Positive and we know that this is the case in I quadrant

Therefore,

Therefore,

Therefore, the required polar form is

**
Question:8
**
Convert each of the complex numbers in the polar form:

**
Answer:
**

Given problem is

Now, let

Square and add both the sides

Therefore, the modulus is
**
1
**

Now,

Since values of Both
and
is Positive and we know that this is the case in I quadrant

Therefore,

Therefore,

Therefore, the required polar form is

##
**
CBSE NCERT solutions for class 11 maths chapter 5
Complex Numbers and Quadratic Equations-Exercise: 5.3
**

**
Question:1
**
Solve each of the following equations:
**
**

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation is given by the formula

In this case value of
**
a = 1 , b = 0 and c = 3
**

Therefore,

Therefore, the solutions of requires equation are

**
Question:2
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case value of
**
a = 2 , b = 1 and c = 1
**

Therefore,

Therefore, the solutions of requires equation are

**
Question:3
**
Solve each of the following equations:
**
**

**
Answer:
**

Given equation is

**
**

Now, we know that the roots of the quadratic equation are given by the formula

In this case value of
**
a = 1 , b = 3 and c = 9
**

Therefore,

Therefore, the solutions of requires equation are

**
Question:4
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation is given by the formula

In this case value of
**
a = -1 , b = 1 and c = -2
**

Therefore,

Therefore, the solutions of equation are

**
Question:5
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case value of
**
a = 1 , b = 3 and c = 5
**

Therefore,

Therefore, the solutions of the equation are

**
Question:6
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case value of
**
a = 1 , b = -1 and c = 2
**

Therefore,

Therefore, the solutions of equation are

**
Question:7
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation is given by the formula

In this case the value of

Therefore,

Therefore, the solutions of the equation are

**
Question:8
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of the equation are

**
Question:9
**
Solve each of the following equations:
**
**

**
Answer:
**

Given equation is

**
**

Now, we know that the roots of the quadratic equation is given by the formula

In this case the value of

Therefore,

Therefore, the solutions of the equation are

**
Question:10
**
Solve each of the following equations:

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of the equation are

##
**
NCERT solutions for class 11 maths chapter 5
Complex Numbers and Quadratic Equations-Miscellaneous Exercise
**

**
Question:1
**
Evaluate
.

**
Answer:
**

The given problem is

Now, we will reduce it into

Now,

Therefore, answer is

**
Question:2
**
For any two complex numbers
and
, prove that

**
Answer:
**

Let two complex numbers are

Now,

**
Hence proved
**

**
Question:3
**
Reduce
to the standard form.

**
Answer:
**

Given problem is

Now, we will reduce it into

Now, multiply numerator an denominator by

Therefore, answer is

**
Question:4
**
If
, prove that

**
Answer:
**

the given problem is

Now, multiply the numerator and denominator by

Now, square both the sides

On comparing the real and imaginary part, we obtain

Now,

**
Hence proved
**

**
Question:5(i)
**
Convert the following in the polar form:

**
Answer:
**

Let

Now, multiply the numerator and denominator by

Now,

let

On squaring both and then add

Now,

Since the value of
is negative and
is positive this is the case in II quadrant

Therefore,

Therefore, the required polar form is

**
Question:5(ii)
**
Convert the following in the polar form:

**
Answer:
**

Let

Now, multiply the numerator and denominator by

Now,

let

On squaring both and then add

Now,

Since the value of
is negative and
is positive this is the case in II quadrant

Therefore,

Therefore, the required polar form is

**
Question:6
**
Solve each of the equation:
**
**

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

**
Question:7
**
Solve each of the equation:
**
**

**
Answer:
**

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

**
Question:8
**
Solve each of the equation:
**
.
**

**
Answer:
**

Given equation is

**
**

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

**
Question:9
**
Solve each of the equation:
**
**

**
Answer:
**

Given equation is

**
**

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

**
Question:10
**
If
, find
.

**
Answer:
**

It is given that

Then,

Now, multiply the numerator and denominator by

Now,

Therefore, the value of

is

**
Question:11
**
If
, prove that
.

**
Answer:
**

It is given that

Now, we will reduce it into

On comparing real and imaginary part. we will get

Now,

**
Hence proved
**

**
Question:12(ii)
**
Let
Find

**
Answer:
**

It is given that

Therefore,

NOw,

Now,

Therefore,

Therefore, the answer is
**
0
**

**
Question:13
**
Find the modulus and argument of the complex number
.

**
Answer:
**

Let

Now, multiply the numerator and denominator by

Therefore,

Square and add both the sides

Therefore, the modulus is

Now,

Since the value of
is negative and the value of
is positive and we know that it is the case in II quadrant

Therefore,

Argument

Therefore, Argument and modulus are
respectively

**
Question:14
**
Find the real numbers x andy if
is the conjugate of
.

**
Answer:
**

Let

Therefore,

Now, it is given that

Compare (i) and (ii) we will get

On comparing real and imaginary part. we will get

On solving these we will get

Therefore, the value of x and y are 3 and -3 respectively

**
Question:15
**
Find the modulus of
.

**
Answer:
**

Let

Now, we will reduce it into

Now,

square and add both the sides. we will get,

Therefore, modulus of

is
**
2
**

**
Question:16
**
If
, then show that

**
Answer:
**

it is given that

Now, expand the Left-hand side

On comparing real and imaginary part. we will get,

Now,

**
Hence proved
**

**
Question:17
**
If
and
are different complex numbers with
, then find
.

**
Answer:
**

Let

and

It is given that

and

Now,

Therefore, value of
is
**
1
**

**
Question:18
**
Find the number of non-zero integral solutions of the equation
.

**
Answer:
**

Given problem is

Now,

**
x = 0
**
is the only possible solution to the given problem

Therefore, there are
**
0
**
number of non-zero integral solutions of the equation

**
Question:19
**
If
then show that

**
Answer:
**

It is given that

Now, take mod on both sides

Square both the sides. we will get

**
Hence proved
**

**
Question:20
**
If
then find the least positive integral value of
.

**
Answer:
**

Let

Now, multiply both numerator and denominator by

We will get,

We know that

Therefore, the least positive integral value of
is
**
4
**

##
**
NCERT solutions for class 11 mathematics
**

##
**
NCERT solutions for class 11- Subject wise
**

##
**
Some important point to remember:
**

## As mentioned in the first paragraph and

and any number can be represented as a complex number of the form a+ ib where a is the real part and b is the imaginary part, for example, 1=1+0i. A complex number a+ib in the X-Y plane is represented as follows

Where is

So a complex number of the form a+ ib can be represented as and the above representation is known as the polar form of a complex number. The polar form of the complex number makes the problem very easy to solve. There are many problems in the CBSE NCERT solutions for class 11 maths chapter 5 complex numbers and quadratic equations which are explained using the polar form of the complex number and some are solved using 2-D geometry.

So, NCERT solutions for class 11 maths chapter 5 complex numbers and quadratic equations can make learning easier for you so that you can score well.

**
Happy Reading !!!
**