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# NCERT Solutions for Class 11 Maths Chapter 5-Complex Numbers and Quadratic Equations Exercise 5.3

## NCERT Solutions for Class 11 Maths Chapter 5-Complex Numbers and Quadratic Equations Exercise 5.3

Last updated date: 19th Mar 2023
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NCERT Solutions for Class 11 Maths Chapter 5-Complex Numbers and Quadratic Equations Exercise 5.3 are comprehensive and extensive solutions. They have been prepared by Vedantu’s proficient and skilled subject experts. The Maths NCERT Solutions Class 11 Chapter 5 Exercise 5.3 is accurate and has been prepared as per the CBSE guidelines. On Vedantu get the Exercise 5.3 Class 11 Maths NCERT Solutions in the form of free PDF downloads to prepare better for the examination.

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

Exercise 5.3

1. Solve the equation ${{x}^{2}}+3=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=0$, and

$c=3$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{\pm \sqrt{0-12}}{2}$

$\Rightarrow x=\frac{\pm 2\sqrt{3}\times \sqrt{-1}}{2}$

$\Rightarrow x=\pm \sqrt{3}i$.

2. Solve the equation $2{{x}^{2}}+x+1=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=2$,

$b=1$, and

$c=1$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-1\pm \sqrt{1-8}}{4}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}\times \sqrt{-1}}{4}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}i}{4}$.

3. Solve the equation ${{x}^{2}}+3x+9=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=3$, and

$c=9$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-3\pm \sqrt{9-36}}{2}$

$\Rightarrow x=\frac{-3\pm 3\sqrt{3}\times \sqrt{-1}}{2}$

$\Rightarrow x=\frac{-3\pm 3\sqrt{3}i}{2}$.

4. Solve the equation $-{{x}^{2}}+x-2=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=-1$,

$b=1$, and

$c=-2$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-1\pm \sqrt{1-8}}{-2}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}\times \sqrt{-1}}{-2}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}i}{-2}$.

5. Solve the equation ${{x}^{2}}+3x+5=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=3$, and

$c=5$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-3\pm \sqrt{9-20}}{2}$

$\Rightarrow x=\frac{-3\pm \sqrt{11}\times \sqrt{-1}}{2}$

$\Rightarrow x=\frac{-3\pm \sqrt{11}i}{2}$.

6. Solve the equation ${{x}^{2}}-x+2=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=-1$, and

$c=2$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{1\pm \sqrt{1-8}}{2}$

$\Rightarrow x=\frac{1\pm \sqrt{7}\times \sqrt{-1}}{2}$

$\Rightarrow x=\frac{1\pm \sqrt{7}i}{2}$.

7. Solve the equation $\sqrt{2}{{x}^{2}}+x+\sqrt{2}=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=\sqrt{2}$,

$b=1$, and

$c=\sqrt{2}$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-1\pm \sqrt{1-8}}{2\sqrt{2}}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}\times \sqrt{-1}}{2\sqrt{2}}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}i}{2\sqrt{2}}$.

8. Solve the equation $\sqrt{3}{{x}^{2}}-\sqrt{2}x+3\sqrt{3}=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=\sqrt{3}$,

$b=-\sqrt{2}$, and

$c=3\sqrt{3}$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{\sqrt{2}\pm \sqrt{2-36}}{2\sqrt{3}}$

$\Rightarrow x=\frac{\sqrt{2}\pm \sqrt{34}\times \sqrt{-1}}{2\sqrt{3}}$

$\Rightarrow x=\frac{\sqrt{2}\pm \sqrt{34}i}{2\sqrt{3}}$.

9. Solve the equation ${{x}^{2}}+x+\frac{1}{\sqrt{2}}=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=1$, and

$c=\frac{1}{\sqrt{2}}$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-1\pm \sqrt{1-2\sqrt{2}}}{2}$

$\Rightarrow x=\frac{-1\pm \sqrt{2\sqrt{2}-1}\times \sqrt{-1}}{2}$

$\Rightarrow x=\frac{-1\pm \left( \sqrt{2\sqrt{2}-1} \right)i}{2}$.

10. Solve the equation ${{x}^{2}}+\frac{x}{\sqrt{2}}+1=0$.

Ans: As, the given equation is a quadratic polynomial. Therefore, we can find the roots of the equation by determining the discriminant.

Hence, we have $a=1$,

$b=\frac{1}{\sqrt{2}}$, and

$c=1$.

Therefore, roots will be –

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow x=\frac{-\frac{1}{\sqrt{2}}\pm \sqrt{\frac{1}{2}-4}}{2}$

$\Rightarrow x=\frac{-1\pm \sqrt{1-8}}{2\sqrt{2}}$

$\Rightarrow x=\frac{-1\pm \sqrt{7}i}{2\sqrt{2}}$.

### NCERT Solution Class 11 Maths of Chapter 5 Exercise

 Chapter 5 - Complex Numbers and Quadratic Equations Exercises in PDF Format Exercise 5.1 14 Questions & Solutions Exercise 5.2 8 Questions & Solutions Exercise 5.3 10 Questions & Solutions Miscellaneous Exercise 20 Questions & Solutions

## Class 11 Maths NCERT Solutions Chapter 5 Exercise 5.3

Class 11 is an important year for a student. This is the class when you decide which stream of subjects you want to study. It is the stepping stone towards your future. Hence it is important that you take class 11 very seriously. It may not be your board year, but it helps you prepare the ground for the same. If your concepts are clear and strong in class 11, then learning more complex topics in class 12 will not give you sleepless nights.

NCERT Solutions for Class 11 Maths Chapter 5 deals with various significant topics and subtopics related to Complex Numbers and Quadratic Equations. Some important concepts covered in this chapter include Definition of Complex Numbers, Algebra of Complex Numbers, Argand Plane, Modulus and Argument of Complex Number, Polar Form, De-Moivre’s Theorem, Cube Root of Unity, n nth Root of Unity, Square Root of Complex Numbers, LOCI in Complex Plane, Vectorial Representation of a Complex Number, Important Properties of Complex Numbers, Quadratic Expression, Roots of Quadratic Equations, Nature of Roots, Graph of Quadratic Expression, Solution of Quadratic Inequalities, Maximum and Minimum Value of Quadratic Expression, Theory of Equations, Location of Roots, Maximum and Minimum Values of Rational Expression, Common Roots, Resolution into Two Linear Factors, Formation of A Polynomial Equation and Transformation of Equations.

Exercise 5.3 Maths Class 11 deals with various concepts related to the Algebra of Complex Numbers. Some of the important concepts covered in Class 11 Maths NCERT Solution Exercise 5.3 are Addition of Two Complex Numbers, Difference of Two Complex Numbers, Multiplication of Two Complex Numbers, Division of Two Complex Numbers, Power of I, The Square Roots of a Negative Real Number, and Identities.

### CBSE Maths Class 11 NCERT Solutions Chapter 5 Exercise 5.3

NCERT Solutions for Class 1 Maths Chapter 5 Exercise 5.3 Chapter include all possible questions from this section of the chapter. Since the NCERT Exercise follows the CBSE question pattern, solving the Class 11 Exercise 5.3 will give you a fair idea of the kind of question types that are important from an exam point of view.   Class 11 Maths Exercise 5.3 Solutions extensively cover all concepts of the chapter. This ensures that you don’t miss out on anything.

Vedantu’s Class 11 Maths NCERT Solutions Chapter 5.3 has been prepared in a step by step manner to make understanding of the exercise easier, faster and more effective. They of the finest quality and adhere to all CBSE (Central Board for Secondary Education) and NCERT (National Council of Educational Research and Training) guidelines. Vedantu’s NCERT Solutions have been prepared on the basis of proven and effective study strategies that help you develop a thorough conceptual knowledge of the subject matter quickly and increase your retention rate. These solutions are the shortest, simplest and most logical solutions to the question in the Class 10 Maths Chapter 5 Exercise 5.3. They are just what you need to ace your examinations.

On Vedantu, your entire course content is available in a single place in a well-structured manner on the basis of class, subject, and chapter. This has made learning a very time and effort saving exercise. Vedantu has the perfect resource for solid exam preparation. Vedantu’s expert teachers ensure that Vedantu’s NCERT Solutions focus on complete conceptual clarity. Vedantu gives identical importance to all subjects, all chapters, all topics and all subtopics of a chapter. Vedantu’s Maths NCERT solutions comprise of all solved examples and step by step answers to all exercise questions.

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### An Overview of the Important Topics Covered in Exercise 5.3 of Class 11 Maths NCERT Solutions

Exercise 5.3 of Class 11 Maths NCERT Solutions is mainly based on solving various quadratic equations.

Below are some key learnings from Exercise 5.3 of NCERT Solutions Class 11 Maths.

• The general form of a quadratic equation can be written as ax2 + bx + c where a, b, and c are the coefficients.

• The quadratic equations (in the real number set) have two real roots and a non-negative discriminant that is b2 – 4ac ≥ 0.

• The quadratic equations (in the complex number set) have two non-real roots and a negative discriminant value that is b2 – 4ac < 0.

This exercise consists of questions based on finding the complex roots of quadratic equations. It also includes multiple examples and problems to explain the use of complex numbers in quadratic equations. These solutions will be helpful for the students to build a deeper understanding of the concepts.