NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1 Complex Numbers and Quadratic Equations - PDF Download

NCERT solutions class 11 maths chapter 4 complex numbers and quadratic equations exercise 4.1 is based on the concepts of algebra, modulus and conjugate of complex numbers. Algebra is the study of complex numbers and the operations that can be performed on them. By solving this exercise, you will be able to acquire the basic knowledge of working with complex numbers in a concise manner.

Class 11 maths chapter 4 exercise 4.1 NCERT solutions consists of a comprehensive set of questions intended to provide a comprehensive understanding of the topic. This exercise consists of 14 questions that are well-suited for building a basic foundation for advanced algebra. Ex 4.1 class 11 maths solutions are designed by the expert faculty of eSaral for students to score good marks in exams. NCERT solutions class 11 maths ch 4 ex 4.1 is also made available here in PDF format to prepare for examination. You can download these free PDFs from the official website of eSaral. 

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Topics Covered in Exercise 4.1 Class 11 Mathematics Questions

Ex 4.1 class 11 maths chapter 4 is all about topics like complex numbers, algebra of complex numbers and identities. Our subject experts of mathematics have provided detailed solutions for each topic here. You can check them below.

1.  Complex Numbers - A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number
   
2. Algebra of Complex Numbers 

In this section, we will explore the complex number algebra. 

Addition of two complex numbers

Let x=a+ib and y=c+id

Therefore x+y= (a+c) +i(b+d)

The addition of complex numbers satisfy the following properties:

1. The closure law The sum of two complex numbers is a complex number
2. The commutative law For any two complex numbers x+y = y+x
3. The associative law For any three complex numbers (x+y)+z=x+(y+z)
4. The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number x, x + 0 = x.
5. The existence of additive inverse To every complex number x = a + ib, we have the complex number – a + i(– b) (denoted as – x), called the additive inverse or negative of x. We observe that x + (–x) = 0 (the additive identity).

Difference of two complex numbers

            Let x=a+ib and y=c+id

            Therefore x-y= x+(-y)

Multiplication of two complex numbers

           Let x=a+ib and y=c+id

           Therefore xy=ac-bd+i(ad+bc)

The multiplication of complex numbers possesses the following properties, which we state without proof.

1. The closure law The product of two complex numbers is a complex number
2. The commutative law For any two complex numbers xy=yx
3. The associative law For any three complex numbers (xy)z=x(yz)
4. The distributive law For any three complex numbers x(y+z) = xy+xz and (x+y)z=xz+yz
5. The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z.

Division of two complex numbers 

Let x=a+ib and y=c+id

Therefore x/y=x.(1/y)

Power of i     

In general, for any integer k, $i^{4 k}=1, i^{4 k+1}=i, i^{4 k+2}=-1, i^{4 k+3}=-i$

The square roots of a negative real number   

i²=-1, and (-1)²=i²=-1

Identities

we can prove the following identities:

(i) $\left(z_1-z_2\right)^2=z_1^2-2 z_1 z_2+z_2^2$

(ii) $\left(z_1+z_2\right)^3=z_1^3+3 z_1^2 z_2+3 z_1 z_2^2+z_2^3$

(iii) $\left(z_1-z_2\right)^3=z_1^3-3 z_1^2 z_2+3 z_1 z_2^2-z_2^3$

(iv) $z_1^2-z_2^2=\left(z_1+z_2\right)\left(z_1-z_2\right)$

3. The Modulus and the Conjugate of a Complex Number

  If  x = a + ib then  | x | = √a²+b²

Properties

(I) |x y|=|x||y|

(II) |x/y|=|x|/|y|

(III) (xy)'=x'y'

(IV) (x±y)’=x’±y’

(V) (x/y)’=x’/y’

4. Argand Plane and Polar Representation

The plane having a complex number assigned to each of its points is called the complex plane or the Argand plane.

In the argand plane, the modulus of complex number x + iy  =√(x²+y²) is the distance from the origin to the point.

The representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are, respectively, the points P (x, y) and Q (x, – y).

Tips for Solving Exercise 4.1 Class 11 Chapter 4 Complex Numbers and Quadratic Equations

In this exercise, there are a number of questions that are based on the calculation of multiplicative inverse for complex numbers. eSaral's experts of mathematics have provided some really helpful tips to solve ex 4.1 questions. Here, you can find them below.

1. Students need to understand the core concepts of algebra of two complex numbers to solve exercise 4.1.
2. To solve questions in NCERT solutions you must understand the essential definitions, terms, concepts, formulas, and other material.
3. Having an in-depth understanding of each topic will make sure you get the most out of this chapter.

Importance of Solving Ex 4.1 Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Ex 4.1 class 11 maths chapter 4 NCERT solutions provides questions and examples related to topics mentioned in exercise 4.1 to solve questions and find answers with ease. Here, we have combined some important benefits of solving questions of ex 4.1 class 11 maths chapter 4 complex numbers and quadratic equations.

1. You must try to solve all the questions provided in ex 4.1. Attempting all the questions will equip you with the crucial algebraic skills for enhancing your confidence and potential.
2. You can build up your maths problem-solving skills over time by practicing the problems in NCERT solutions class 11 maths chapter 4 ex 4.1 step by step.
3. NCERT solutions class 11 maths chapter 4 ex 4.1 helps students remember key formulas and concepts so they can revise them quickly.
4. By solving questions of NCERT solutions class 11 maths chapter 4 ex 4.1 will help you to solve questions asked in the final exam.

Frequently Asked Questions

Question 1. What are complex numbers according to NCERT solutions class 11 maths chapter 4 ex 4.1?

Answer 1. A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.

Question 2. What is the multiplicative identity in complex numbers?

Answer 2. If you have a complex number as z then multiplicative identity is expressed as z(1/z)=1.