NCERT Solutions for Class 11 Maths Chapter 5: Complex Numbers and Quadratic Equations - Exercise 5.1

Chapter 5.1

1. Express the given complex number in the form $a+ib  :  (5i)\left( -  \dfrac{3}{5}i \right)$ 

Ans:

$\Rightarrow $ $(5i)\left( \dfrac{-3}{5}i \right)=-5\times \dfrac{3}{5}\times i\times i$

$=-3{{i}^{2}}$

$=-3(-1)$       $\left[ {{i}^{2}}=-1 \right]$

$=3$

2. Express the given complex number in the form $a+ib:{{i}^{9}}+{{i}^{19}}$

Ans:

$\Rightarrow $${{i}^{9}}+{{i}^{19}}={{i}^{4\times 2+1}}+{{i}^{4\times 4+3}}$

$={{\left( {{i}^{4}} \right)}^{2}}\cdot i+{{\left( {{i}^{4}} \right)}^{4}}\cdot {{i}^{3}}$

$=1\times i+1\times (-i)\quad \left[ {{i}^{4}}=1,{{i}^{3}}=-i \right]$

$=i+(-i)$

$=0$

3. Express the given complex number in the form $a  +ib:{{i}^{-39}}$

Ans:

$\Rightarrow $${{i}^{  -  39}}$$=  {{i}^{-  4  \times   9  -  3}}  =  {{\left( {{i}^{4}} \right)}^{-  9}}.{{i}^{-  3}}$

$=  {{\left( 1 \right)}^{-9}}.{{i}^{-  3}}    \left[ {{i}^{4}}  =  1 \right]$

$=  \dfrac{1}{{{i}^{3}}}  =  \dfrac{1}{-  i}      \left[ {{i}^{3}}  =  -  i \right]$

$=  \dfrac{-  1}{i}  \times   \dfrac{i}{i}  $$=  \dfrac{-  i}{{{i}^{2}}}  =  \dfrac{-  i}{-  1}  =  i      \left[ {{i}^{2}}  =  -  1 \right]$

4. Express the given complex number in the form a  $a  +  ib:$$3\left( 7  +  i7 \right)  +  i\left( 7  +  i7 \right)$

Ans:

$\Rightarrow $$3\left( 7  +  i7 \right)  +  i\left( 7  +  i7 \right)  =  21  +  21i  +7i  +7{{i}^{2}}$

$=  21  +  28i  +7\times \left( -  1 \right)        \left[ \because   {{i}^{2}}  =  -  1 \right]$

$=  14  +  28i$

5. Express the given complex number in the form\[a  +  ib  :  \left( 1  -  i \right)  -  \left( -  1  +  i6 \right)\].

Ans:

\[\Rightarrow \left( 1  -  i \right)  -  \left( -  1  +  i6 \right)  \]

\[=  1  -  i  +  1  -  6i\]

\[=2  -  7i\]

6. Express the given complex number in the form$a  +  ib:$$\left( \dfrac{1}{5}  +  i\dfrac{2}{5} \right)  -  \left( 4  +  i\dfrac{5}{2} \right)$.

Ans:

$\Rightarrow   \left( \dfrac{1}{5}  +  i\dfrac{2}{5} \right)  -  \left( 4  +  i\dfrac{5}{2} \right)  $

$=  \dfrac{1}{5}  +  \dfrac{2}{5}i  -  4  -  \dfrac{5}{2}i$

$=  \left( \dfrac{1}{5}  -  4 \right)  +  \left( \dfrac{2}{5}  -  \dfrac{5}{2} \right)i$

$=  \left( \dfrac{1  -  \left( 4  \times   5 \right)}{5} \right)  +  \left( \dfrac{\left( 2  \times   2 \right)  -  \left( 5  \times   5 \right)}{5  \times   2} \right)i$

$=  -  \left( \dfrac{1  -  20}{5} \right)  +  \left( \dfrac{4  -  25}{10} \right)i$

$=  -  \dfrac{19}{5}  +  \left( \dfrac{-  21}{10} \right)i$

$=  -  \dfrac{19}{5}  -  \dfrac{21}{10}i$

7. Express the given complex number in the form$a  +  ib:$$\left[ \left( \dfrac{1}{3}  +  i\dfrac{7}{3} \right)  +  \left( 4  +  i\dfrac{1}{3} \right)  -  \left( -  \dfrac{4}{3}  +  i \right) \right]$.

Ans:

$\Rightarrow $$\left[ \left( \dfrac{1}{3}  +  i\dfrac{7}{3} \right)  +  \left( 4  +  i\dfrac{1}{3} \right)  -  \left( -  \dfrac{4}{3}  +  i \right) \right]$

$\dfrac{1}{3}  +  \dfrac{7}{3}i  +  4  +  \dfrac{1}{3}i  +  \dfrac{4}{3}  -   i$

$=\left( \dfrac{1}{3}  +  4  +  \dfrac{4}{3} \right)  +  \left( \dfrac{7}{3}  +  \dfrac{1}{3}  -  1 \right)i$

$=  \left( \dfrac{1  +  \left( 4  \times   3 \right)  +  4}{3} \right)  +  \left( \dfrac{7  +  1  -  \left( 1  \times   3 \right)}{3} \right)i$

$=  \left( \dfrac{1  +  12  +  4}{3} \right)  +  \left( \dfrac{7  +  1  -  3}{3} \right)i$

$a  +  ib:$\[=  \dfrac{17}{3}  +  i\dfrac{5}{3}\]

8. Express the given complex number in the form $a  +  ib:$${{\left( 1  -  i \right)}^{4}}$

Ans:

$\Rightarrow $\[{{\left( 1  -  i \right)}^{4}}  =  {{\left[ {{\left( 1  -  i \right)}^{2}} \right]}^{2}}\]

\[=  {{\left[ {{1}^{2}}  +  {{i}^{2}}  -2i \right]}^{2}}\]

\[{{\left( 1  -  i \right)}^{4}}  =  {{\left[ {{\left( 1  -  i \right)}^{2}} \right]}^{2}}\]

\[={{\left[ 1  -  1  -  2i \right]}^{2}}\]

\[={{\left( 2i \right)}^{2}}\]

\[=4{{i}^{2}}\]

\[=-4        \left[ {{i}^{2}}  =  -  1 \right]\]

\[a  +  ib  = -4  +  0i \]

9. Express the given complex number in the form $a  +  ib:$ ${{\left( \dfrac{1}{3}  +  3i \right)}^{3}}$.

Ans:

$\Rightarrow $${{\left( \dfrac{1}{3}  +  3i \right)}^{3}}  =  {{\left( \dfrac{1}{3} \right)}^{3}}  +  {{\left( 3i \right)}^{3}}  +  3\left( \dfrac{1}{3} \right)\left( 3i \right)\left( \dfrac{1}{3}  +  3i \right)$

$=  \dfrac{1}{27}  +  27{{i}^{3}}  +  3i\left( \dfrac{1}{3}  +  3i \right)$

$=  \dfrac{1}{27}  +  27\left( -  i \right)  +  i  +  9{{i}^{2}}        \left[ {{i}^{3}}  =  -  i \right]$

$=  \dfrac{1}{27}  -  27i  +  i  +  9{{i}^{2}}                \left[ {{i}^{2}}  =  -  1 \right]$

$=  \left( \dfrac{1}{27}  -  9  \right)  +  i\left( -  27  +  1 \right)$

$a  +  ib$$=  \dfrac{-  242}{27}  -  26i$

10. Express the given complex number in the form $a  +  ib:$${{\left( -  2  -  \dfrac{1}{3}i \right)}^{3}}$.

Ans:

$\Rightarrow $${{\left( -2  -  \dfrac{1}{3}i \right)}^{3}}  =  {{\left( -  1 \right)}^{3}}\left( 2  +  \dfrac{  1  }{3}i \right)^3$

$=  -  \left[ {{2}^{3}}  +  {{\left( \dfrac{i}{3} \right)}^{3}}  +  3\left( 2 \right)\left( \dfrac{i}{3} \right)\left( 2 +  \dfrac{i}{3} \right) \right]$

$=  -  \left[ 8  +  \dfrac{{{i}^{3}}}{27}  +  2i\left( 2  +\dfrac{i}{3} \right) \right]        \left[ {{i}^{3}}  =  -  i \right]$

$= - \left[ 8  -  \dfrac{i}{27}  +  4i   -  \dfrac{2{{i}^{2}}}{3} \right]            \left[ {{i}^{2}}  =  -  1 \right]$

$=  -  \left[ \dfrac{22}{3}  +  \dfrac{107i}{27} \right]$

$a  +  ib$$=  -\dfrac{22}{3}  -  \dfrac{107}{27}i$

11. Find the multiplicative inverse of the complex number \[4  -  3i\text{ }.\]

Ans:

Let \[z  =  4  -  3i\text{ }\]

Then,

\[\overline{z}  =  4  +  3i  \]and \[\left| z \right|  =  {{4}^{2}}  +  {{\left( -  3 \right)}^{2}}  =  16  +  9  =  25\]

The multiplicative inverse of \[4  -  3i\text{ }\]is given by 

$\Rightarrow $${{z}^{-  1}}  =  \dfrac{\overline{z}}{{{\left| z \right|}^{2}}}$

\[=  \dfrac{4  +  3i}{25}  \]

\[=  \dfrac{4}{25}  +  \dfrac{3}{25}i\]

12. Find the multiplicative inverse of the complex number$\sqrt{5}  +  3i$.

Ans:

Let $z  =  \sqrt{5}  +  3i$

Then, $\overline{z}  =  \sqrt{5}  -  3i$

$\left| z \right|  =  {{\left( \sqrt{5} \right)}^{2}}  +{{\left( 3 \right)}^{2}}  =  5  +  9  =  14$

The multiplicative inverse of the complex number $\sqrt{5}  +  3i$ is given by 

${{z}^{-  1}}  =  \dfrac{\overline{z}}{{{\left| z \right|}^{2}}}$

$ =   \dfrac{\sqrt{5}  -  3i}{14}   $

$=  \dfrac{\sqrt{5}}{14}  -  \dfrac{3}{14}i$

13. Find the multiplicative inverse of the complex number $-  i  .$

Ans:

Let $z  =  -  i  $

Then, $\overline{z}  =  i$ 

${{\left| z \right|}^{2}} =   1$

The multiplicative inverse of the complex number $-  i  $

$\Rightarrow $${{z}^{-  1}}  =  \dfrac{\overline{z}}{{{\left| z \right|}^{2}}}$

$  =  \dfrac{i}{1}  $

$=   i$

14. Express the following expression in the form of $a  +  ib  .$

$\dfrac{\left( 3  +  i\sqrt{5} \right)\left( 3  -  i\sqrt{5} \right)}{\left( \sqrt{3}  +  \sqrt{2 }i \right)  -  \left( \sqrt{3}  -  i \sqrt{2} \right)}$

Ans:

$\Rightarrow $$\dfrac{\left( 3  +  i\sqrt{5} \right)\left( 3  -  i\sqrt{5} \right)}{\left( \sqrt{3}  +  \sqrt{2 }i \right)  -  \left( \sqrt{3}  -  i \sqrt{2} \right)}$

$=  \dfrac{\left( {{3}^{2}}  -  {{\left( i\sqrt{5} \right)}^{2}} \right)}{\left( \sqrt{3}  +  \sqrt{2 }i \right)  -  \left( \sqrt{3}  -  i \sqrt{2} \right)}  $            $\left[ \left( a  +  b \right)\left( a  -  b \right)  =  {{a}^{2}}  -  {{b}^{2}} \right]$

$=\,\dfrac{\left( 9  -  5{{i}^{2}} \right)}{2\sqrt{2}i}$

$=  \dfrac{9  -  5\left( -  1 \right)}{2\sqrt{2}i}          \left[ {{i}^{2}}  =  -  1 \right]  $

$=  \dfrac{14i}{2\sqrt{2}{{i}^{2}}}$

$=  \dfrac{14i}{2\sqrt{2}\left( -  1 \right)}$

$=  \dfrac{-  7i}{\sqrt{2}}\times   \dfrac{\sqrt{2}}{\sqrt{2}}$

$=  \dfrac{-  7\sqrt{2}i}{2}$

NCERT Solutions for Class 11 Maths Chapters

• Chapter 1 - Sets

• Chapter 2 - Relations and Functions

• Chapter 3 - Trigonometric Functions

• Chapter 4 - Principle of Mathematical Induction

• Chapter 5 - Complex Numbers and Quadratic Equations

• Chapter 6 - Linear Inequalities

• Chapter 7 - Permutations and Combinations

• Chapter 8 - Binomial Theorem

• Chapter 9 - Sequences and Series

• Chapter 10 - Straight Lines

• Chapter 11 - Conic Sections

• Chapter 12 - Introduction to Three Dimensional Geometry

• Chapter 13 - Limits and Derivatives

• Chapter 14 - Mathematical Reasoning

• Chapter 15 - Statistics

• Chapter 16 - Probability

NCERT Solution Class 11 Maths of Chapter 5 Exercise

Complex Numbers and Quadratic Equations Exercise 5.1– An Overview

1. Complex Numbers

A complex number is defined as the sum of a real number and an imaginary number. A complex number is expressed in the form of a + ib and is usually denoted by z. Here, we should note that both a and b are real numbers.

2. Algebra of Complex Numbers

In this section, the following subtopics are covered. 

• Addition of two complex numbers.

• Difference between two complex numbers.

• Multiplication of two complex numbers.

• Division of two complex numbers.

• Power of i.

• The square roots of a negative real number.

• Identities.

3. The Modulus and the Conjugate of a Complex Number

• The modulus of a complex number is the distance between the origin and the point on the argand plane that represents the complex number z.

• By taking the same real part of the complex number and changing the imaginary part of the complex number to its additive inverse, the conjugate of the complex number is formed. Conjugate complex numbers are those in which the sum and product of two complex numbers are both real numbers.

Class 11 Maths Chapter 5 Exercise 5.1 – All Questions

Equip yourself with the necessary learning tools to ace your revision game and perform exceptionally well in your upcoming exams. To begin with, download our PDF version of NCERT Solution of Mathematics for Class 11 Chapter 5 and figure out the best methods of approaching chapter-based problems.

There are a total of 14 questions in Exercise 5.1, each of which intends to test students’ knowledge about the different concepts discussed in this particular section. 

Here’s how our NCERT Solution of Mathematics for Class 11 Chapter 5 will help you solve these questions of Class 11 Maths Chapter 5 Exercise 5.1 –

• Complex Numbers Class 11 – Question 1 to 9

The basic concepts of both complex numbers and quadratic equations will help students to solve these types of problems with confidence. As per directions, students have to express the given complex numbers in the form of a + ib. 

A fair idea of the techniques that are required to express a given complex number in the format of summation of a Natural Number and Complex Number will allow you to solve these problems accurately. Find the best shortcut techniques for solving these problems with the help of our solutions for Class 11 Ex. 5.1. 

• Class 11 Maths Ex. 5.1 – Question 10 to 13

These particular sets of problems direct students to figure out the multiplicative inverse of the given complex numbers. In this case, you need a quick revision before solving these problems - access our Complex Numbers and Quadratic Equations Exercise 5.1 solutions that are available online. The step by step explanations that accompany each problem offers you a better understanding of the methods used and its following outcome. 

With regular practice of NCERT Solution of Mathematics for Class 11 Chapter 5, you will be able to approach similar problems in exams in no time. Download our solutions now and make your process of exam preparation fun and easy.  

• Exercise 5.1 Class 11 Maths – Question 14

Do you know how to integrate the form (a+b)(a−b)=a2−b2 to solve problems based on the summation of natural numbers and complex numbers?

Learn about the steps and gain valuable insights into the same by going through detailed explanations and prompt solutions that are available at our learning portal. It will help you clear all your doubts and queries related to the techniques used to solve such problems. It further offers a fair idea on how to develop effective shortcut techniques to solve them in no time and with negligible mistakes. 

Download the PDF version on our NCERT Solution of Mathematics for Class 11 Chapter 5 and get all your doubts and queries cleared. In case, you still require a little help with the chapter and would appreciate the help of an expert teaching professional – refer to our expert-approved latest study tools. You may also enrol in our free Master Class and better a helpful insight into how to tackle challenging questions related to this chapter with more confidence and convenience. 

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