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Edited By Ramraj Saini | Updated on Sep 21, 2023 11:22 PM IST

**NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations **are discussed here. These NCERT solutions are created by expert team at Careers360 keeping in mind of latest syllabus of CBSE 2023-24. 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 this class 11 maths chapter 5 question answer, you will learn to solve equations like x^{2 }+ 2 = 0.

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This Story also Contains

- Complex Numbers and Quadratic Equations Class 11 Questions And Answers
- Complex Numbers and Quadratic Equations Class 11 Questions And Answers PDF Free Download
- Complex Numbers and Quadratic Equations Class 11 Solutions - Important Formulae
- Complex Numbers and Quadratic Equations Class 11 NCERT Solutions (Intext Questions and Exercise)
- Highlights Of NCERT Class 11 Chapter 5 Complex Numbers And Quadratic Equations
- Complex Numbers And Quadratic Equations Exercise Wise Solutions
- NCERT Solutions For Class 11 Mathematics - Chapter Wise
- Benefits of NCERT Class 11 Maths ch 5 Question Answer
- NCERT Solutions For Class 11 - Subject wise
- NCERT Books and NCERT Syllabus

The CBSE Syllabus for 2023 includes the chapter Complex Numbers and Quadratic Equations, which comprises significant mathematical theorems and formulae. The NCERT textbook provides ample practice problems to cover all these concepts, facilitating students' comprehension of advanced concepts in the future. To aid in this process, Careers360 offers detailed solutions for all complex numbers class 11 problems in the textbook. These complex numbers and quadratic equations class 11 solutions are particularly beneficial for students who aim to pass their exams with last-minute preparations. However, the primary focus of NCERT solutions for class 11 is to master concepts and develop a deeper understanding of the subject.

**Complex Numbers:**

A complex number is expressed as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.

Imaginary Numbers: The square root of a negative real number is called an imaginary number, represented as √-1 = i.

Equality of Complex Number: Two complex numbers z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} are equal if x_{1} = x_{2} and y_{1} = y_{2}.

**Algebra of Complex Numbers:**

Addition: (z_{1 }+ z_{2}) = (x_{1} + x_{2}) + i(y_{1} + y_{2})

Subtraction: (z_{1} - z_{2}) = (x_{1} - x_{2}) + i(y_{1} - y_{2})

Multiplication: (z_{1} * z_{2}) = (x_{1}x_{2} - y_{1}y_{2}) + i(x_{1}y_{2} + x_{2}y_{1})

Division: (z_{1} / z_{2}) = [(x_{1}x_{2} + y_{1}y_{2}) + i(x_{2}y_{1} - x_{1}y_{2})] / (x_{2}^{2} + y_{2}^{2}), where z_{2} ≠ 0.

Conjugate of Complex Number: The conjugate of a complex number z = x + iy is represented as z¯ = x - iy.

Modulus of a Complex Number: |z| = √(x^{2} + y^{2})

Argument of a Complex Number: The angle made by the line joining the point z to the origin, with the positive X-axis in an anti-clockwise sense is called the argument (arg) of the complex number.

**Principal Value of Argument:**

When x > 0 and y > 0 ⇒ arg(z) = θ

When x < 0 and y > 0 ⇒ arg(z) = π - θ

When x < 0 and y < 0 ⇒ arg(z) = -(π - θ)

When x > 0 and y < 0 ⇒ arg(z) = -θ

Polar Form of a Complex Number: z = |z| (cosθ + isinθ), where θ = arg(z).

The general polar form of z is z = |z| [cos(2nπ + θ) + isin(2nπ + θ)], where n is an integer.

Free download **NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations **for CBSE Exam.

**Class 11 maths chapter 5 NCERT solutions - 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

**Class 11 maths chapter 5 ncert solutions - 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

**Class 11 maths chapter 5 ncert solutions - 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

**Complex numbers and quadratic equations class 11 solutions - 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 **

The NCERT Solutions for Chapter 5 - Complex Numbers and Quadratic Equations in Class 11 Maths consists of 3 exercises and a miscellaneous exercise to provide enough practice problems for the students to comprehend all the concepts. The PDF of NCERT Solutions for Class 11 Maths includes detailed explanations of the following topics and sub-topics:

- 5.1 Introduction

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Download EBookThis section of ch 5 maths class 11 covers quadratic equations with no real solutions and their solutions using complex numbers. The solution to the quadratic equation ax^2 + bx + c = 0 has been derived when the discriminant D = b^2 - 4ac is less than 0.

- 5.2 Complex Numbers It defines complex numbers and explains the real and imaginary parts of complex numbers with examples.
- 5.3 Algebra of Complex Numbers This section covers the basic BODMAS operations on complex numbers i.e:

5.3.1 Addition of two complex numbers

5.3.2 Difference of two complex numbers

5.3.3 Multiplication of two complex numbers

5.3.4 Division of two complex number

5.3.5 Power of i

5.3.6 The square roots of a negative real number

5.3.7 Identities

After completing these exercises of complex no class 11 , students will have a better understanding of the fundamental BODMAS operations on complex numbers, as well as their properties, the power of i, square roots of negative real numbers, and complex number identitie

- 5.4 The Modulus and the Conjugate of a Complex Number This section provides detailed explanations of the modulus and conjugate of a complex number with solved examples.
- 5.5 Argand Plane and Polar Representation This section explains the complex plane or Argand plane and polar representation of complex numbers, including how to write ordered pairs for complex numbers.

- A complex number is a number expressed in the form of a + ib where a and b are real numbers. The real part of the complex number is denoted by "a" while the imaginary part is denoted by "b".
- If z1 = a + ib and z2 = c + id are two complex numbers, then their addition is (a + c) + i(b + d) and their product is (ac - bd) + i(ad + bc).
- For any non-zero complex number z = a + ib (where a and b are not equal to zero), there exists a unique complex number called the multiplicative inverse of z, denoted by 1/z or z
^{(-1)}. - For any integer k, i
^{(4k)}equals 1, i^{(4k+1)}equals i, i^{(4k+2)}equals -1, and i^{(4k+3)}equals -i. - The polar form of the complex number z = x + iy is represented as r(cosθ + i sinθ), where r is the modulus or absolute value of the complex number, and θ is the argument or angle made by the complex number with the positive x-axis.
- A polynomial equation of degree n has exactly n roots, which may or may not be complex.

- NCERT Solutions for Exercise 5.1
- NCERT Solutions for Exercise 5.2
- NCERT Solutions for Exercise 5.3
- NCERT Solutions for Class 11 Maths Chapter 5 Miscellaneous Exercise

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 | Permutation and Combinations |

chapter-8 | Binomial Theorem |

chapter-9 | Sequences and Series |

chapter-10 | Straight Lines |

chapter-11 | Conic Section |

chapter-12 | Introduction to Three Dimensional Geometry |

chapter-13 | Limits and Derivatives |

chapter-14 | Mathematical Reasoning |

chapter-15 | Statistics |

chapter-16 | Probability |

**Clarity of concepts:** The NCERT Solutions for maths chapter 5 class 11 provide clear explanations and examples that help students to understand the concepts easily. The solutions are designed to make the learning process easier and more enjoyable.

**Practice:** The class 11 complex numbers and quadratic equations NCERT solutions come with a wide range of practice problems that help students to practice and master the concepts covered in the chapter. The more problems they solve, the better they become at the topic.

**Exam preparation:** NCERT Solutions for ch 5 maths class 11 are designed to help students prepare for their exams. The solutions provide a comprehensive overview of the chapter, including all the important topics and subtopics.

NCERT solutions for class 11 biology |

NCERT solutions for class 11 maths |

NCERT solutions for class 11 chemistry |

NCERT solutions for class 11 physics |

**Complex Numbers And Quadratic Equation Class 11 Chapter-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.

So a complex number of the form a+ ib can be represented as

and

and the above representation is known as the polar form of a complex number. T he polar form of the complex number makes the problem very easy to solve. There are many problems in the 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 !!! **

1. What are important topics of the chapter Complex Numbers and Quadratic Equations ?

Class 11 maths NCERT solutions chapter 5 includes topics such as Complex numbers, algebra of complex numbers, modulus and the conjugate of a complex number, argand plane and polar representation, and quadratic equations are the important topics in this chapter. to get command in these topics students can practice problems from complex numbers class 11 pdf.

2. How does the NCERT solutions are helpful ?

NCERT solutions are highly beneficial for students as they provide well-structured and comprehensive solutions to the textbook questions. These solutions are designed by subject matter experts and cover all the important topics and concepts in a clear and concise manner. Students can practice complex numbers class 11 solutions to get good hold on the concepts of chapter 5 class 11.

3. What are the most difficult chapters in the class 11 maths ?

Most of the students consider permutation and combination, trigonometry as the most difficult chapters in class 11 maths but with the rigorous practice students can get command on them also.

4. How many chapters are there in CBSE class 11 maths ?

There are 16 chapters starting from set to probability in the CBSE class 11 maths.

5. Does CBSE provides the solutions of NCERT class 11 maths ?

No, CBSE doesn’t provided NCERT solutions for any class or subject.

6. Where can I find the complete solutions of NCERT for class 11 maths ?

Here you will get the detailed NCERT solutions for class 11 maths by clicking on the link.

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