# 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 x2+2=0, x2=-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 x2+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.

Exercise:5.1

Exercise:5.2

Exercise:5.3

Miscellaneous Exercise

## 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

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

On solving

we will get

Now, in the form of      we can write it as

We know that
Now, we will reduce      into

Now, in the form of    we can write it as

We know that
Now, we will reduce      into

Now, in the form of    we can write it as

Given problem is

Now, we will reduce it into

Given problem is

Now, we will reduce it into

Given problem is

Now, we will reduce it into

Given problem is

Now, we will reduce it into

The given problem is

Now, we will reduce it into

Given problem is

Now, we will reduce it into

Given problem is

Now, we will reduce it into

Let
Then,

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

Let
Then,

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

Let
Then,

And

Now, the multiplicative inverse is given by

Therefore, the multiplicative inverse is

Given problem is

Now, we will reduce it into

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

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

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

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

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

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

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

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

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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     .

The given problem is

Now, we will reduce it into

Now,

Let two complex numbers are

Now,

Hence proved

Question:3 Reduce      to the standard form.

Given problem is

Now, we will reduce it into

Now, multiply numerator an denominator by

Question:4 If       ,   prove that

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:

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:

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:

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:

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: .

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:

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    .

It is given that

Then,

Now, multiply the numerator and denominator  by

Now,

Therefore, the value of

is

Question:11 If    , prove that  .

It is given that

Now, we will reduce it into

On comparing real and imaginary part. we will get

Now,

Hence proved

Question:12(i) Let     Find

It is given that

Now,

And

Now,

Now,

Question:12(ii) Let   Find

It is given that

Therefore,

NOw,

Now,

Therefore,

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

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   .

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

it is given that

Now, expand the Left-hand side

On comparing real and imaginary part. we will get,

Now,

Hence proved

Let
and
It is given that

and

Now,

Therefore, value of      is  1

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

It is given that

Now, take  mod on both sides

Square both the sides. we will get

Hence proved

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

 chapter-1 NCERT solutions for class 11 maths chapter 1 Sets chapter-2 Solutions of NCERT for class 11 chapter 2 Relations and Functions chapter-3 CBSE NCERT solutions for class 11 chapter 3 Trigonometric Functions chapter-4 NCERT solutions for class 11 chapter 4 Principle of Mathematical Induction chapter-5 NCERT solutions for class 11 maths chapter 5 Complex Numbers and Quadratic equations chapter-6 CBSE NCERT solutions for class 11 maths chapter 6 Linear Inequalities chapter-7 NCERT solutions for class 11 maths chapter 7 Permutation and Combinations chapter-8 Solutions of NCERT for class 11 maths chapter 8 Binomial Theorem chapter-9 CBSE NCERT solutions for class 11 maths chapter 9 Sequences and Series chapter-10 NCERT solutions for class 11 maths chapter 10 Straight Lines chapter-11 Solutions of NCERT for class 11 maths chapter 11 Conic Section chapter-12 CBSE NCERT solutions for class 11 maths chapter 12 Introduction to Three Dimensional Geometry chapter-13 NCERT solutions for class 11 maths chapter 13 Limits and Derivatives chapter-14 Solutions of NCERT for class 11 maths chapter 14 Mathematical Reasoning chapter-15 CBSE NCERT solutions for class 11 maths chapter 15 Statistics chapter-16 NCERT solutions for class 11 maths chapter 16 Probability

## NCERT solutions for class 11- Subject wise

 Solutions of NCERT for class 11 biology CBSE NCERT solutions for class 11 maths NCERT solutions for class 11 chemistry Solutions of NCERT for Class 11 physics

## 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.