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

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The knowledgeable professors at Extramarks have produced NCERT Solutions For Class 11 Maths Chapter 5 Exercise 5.1 These Mathematics NCERT solutions aid students in finding quick, accurate, and effective solutions to problems. Additionally, Extramarks offers detailed solutions to every NCERT question, ensuring that students comprehend them and succeed in their board exams.

The CBSE Syllabus for 2022–2023 includes the chapter Complex Numbers and Quadratic Equations, which covers various important Mathematical theories and equations. Numerous practise problems are provided for each of these concepts in the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 which will aid students in understanding more complex ideas in the future. Extramarks offers remedies to all of these issues along with adequate justifications. These Extramarks NCERT solutions assist students who want to do well in their examinations.

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NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations (Ex 5.1) Exercise 5.1

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Numerous Mathematical ideas, theorems, and formulas found in the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 are essential for calculating algebraic sums. As it offers step-by-step solutions to all questions given in the exercises for this chapter, the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 are quite helpful in this regard. On Extramarks, the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 can be downloaded as a free PDF. Senior Mathematics teachers at Extramarks have crafted these answers in accordance with CBSE standards. To understand all the exercise questions better, students should download the Extramarks Learning App.

NCERT Solutions for Class 11 Maths Chapters

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The questions from Class 11 Maths Chapter 5 Exercise 5.1 are covered in the Class 11 Maths Chapter 5 Exercise 5.1 solutions. Sets, trigonometric functions, sequences and series, three-dimensional geometry, relations and functions, statistics, the binomial theorem, probability, and other topics are discussed in chapters of CBSE Class 11 Mathematics. Students can readily download the PDF versions of the solutions to these chapters and study them at their own pace.

NCERT Solutions for Class 11 Mathematics –Chapter 1 – Sets

This chapter discusses the concept of sets and how they are representedEmpty set, Finite and Infinite sets, Equal sets, Subsets, Power set, and Universal set are among the topics covered in this chapter. The chapter also teaches students how to create Venn diagrams and introduces them to the ideas of set intersection and union. This chapter also discusses the properties of a set’s complement and the difference between sets. The chapter includes six exercises and one more, giving students enough problems to work through and, as a result, better understanding the idea.

NCERT Solutions for Class 11 Mathematics – Chapter 2 Relations and Functions

The terms ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, definition of relation, graphical diagrams, domain, co-domain, and range of a relation were introduced to students In addition to the issue of relations, students study the topic of functions. Among the topics that fall under the concept of functions are the pictorial representation of a function, its domain, co-domain, and range, real-valued functions’ domain and range, and its graphs. Other topics include constants, identities, polynomials, rationales, moduli, signums, exponentials, logarithms, and the greatest integer functions. Students will have the ability to link objects from two sets together into pairs before introducing relationships between the two objects in the pair. After that, they will discover special relations, that can be considered functions.

NCERT Solutions for Class 11 Mathematics – Chapter 3 – Trigonometric Functions

The chapter covers positive and negative angles, how to measure angles in radians and degrees, and how to convert between different measures. The definition of trigonometric functions using the unit circle, the general solution of trigonometric equations, the signs, domain, and range of trigonometric functions, and their graphs are all covered in this chapter. The chapter teaches students how to describe sin (xy) and cos (xy) in terms of sin x, sin y, cos x, and cos y, along with their straightforward applications, and how to deduce the identities of sin 2x, cos 2x, tan 2x, and sin 3x, cos 3x, and tan 3x.

NCERT Solutions for Class 11 Mathematics – Chapter 4 –Principal of Mathematical Induction

The chapter on the “Principle of Mathematical Induction” discusses a variety of subjects, including how to demonstrate an induction and why it is useful by using natural numbers as the least inductive subset of real numbers. The chapter’s exercise covers issues with the Principle of Mathematical Induction, as well as some of its straightforward applications.

NCERT Solutions for Class 11 Mathematics – Chapter 5 –Complex Numbers and Quadratic Equations

Complex numbers are necessary, and their algebraic features are briefly described. Complex numbers are represented in the Argand plane and Polar form. Solving quadratic equations in the complex number system is a statement of the Fundamental Theorem of Algebra. In addition to this, students can also download the PDF notes of the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 on the Extramarks mobile application or the Extramarks website. The NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 have answers to all the questions mentioned in 5 Exercise 5.1 of the chapter

NCERT Solutions for Class 11 Mathematics – Chapter 6 –Linear Inequalities

As the name suggests, the chapter Linear Inequalities discusses the idea of linear inequality. This chapter also discusses the concepts of algebraic solutions of linear inequalities in one variable and their representation on the number line, graphical representations of linear inequalities in two variables, as well as the graphical method of solving a system of linear inequalities in two variables. This chapter has a total of 3 activities and a supplemental exercise that cover all the chapter’s subjects.

NCERT Solutions for Class 11 Mathematics – Chapter 7 –Permutations and Combinations

A combination is a grouping of objects where the order doesn’t matter, whereas a permutation is an arrangement of several elements taken at once in a specific order. The chapter covers the fundamental principles of counting, factorial n. (n! ), permutations, combinations, formula derivation, connections between formulae, and straightforward applications. There are four exercises in all, plus a bonus exercise with questions on every subject covered in the chapter. Students can find the detailed concepts of the above-mentioned chapters, along with exercises that have been curated just to fit their needs, on the Extramarks mobile application or the Extramarks website.

NCERT Solution Class 11 Maths Chapter 5 Exercise

(Include a Table for Exercises and No of Questions)

Extramarks’ subject specialists have put up the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 that are available on the Extramarks mobile application and the Extramarks website. The solutions have mostly been created to make all the Class 11 Mathematics problems that are presented in the textbooks recommended by the CBSE syllabus simpler. All 16 chapters on the syllabus are covered by these NCERT Solutions for Class 11 Maths along with NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 which are arranged methodically. Therefore, the NCERT Class 11 Maths solutions provide comprehensive learning and aid in the improvement of students’ arithmetic abilities as well as their logical and reasoning abilities. To make learning much more comfortable, the complex problems have been clarified and made simpler in these solutions.

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Complex Numbers and Quadratic Equations Exercise 5.1 – An Overview

The inability to resolve some quadratic equations is used as a motivation for the necessity of complex numbers, particularly the number 1. In this chapter, students can also learn about the polar representation of complex numbers, the Argand plane, and the algebraic features of complex numbers. The fundamental theorem of algebra is stated, as are the square root of a complex number and the solutions to quadratic equations (with real coefficients) in the complex number system. This chapter has three exercises that cover these subjects, as well as a random exercise. Complex numbers and quadratic equations are covered in NCERT’s fifth chapter. Complex numbers and quadratic equations are covered in detail in the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1. The key formulas from NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 are also included in the revision notes. NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 provide thorough explanations of concepts using examples and exercises. The topics covered in the NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 range from complex number algebra through quadratic equations. In addition to covering NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1, NCERT notes also cover chapter 5 notes.

Class 11 Maths Chapter 5 Exercise 5.1 – All Questions

The Extramarks-provided NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 were created with the careful consideration of experienced teaching professionals. The carefully constructed NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 by Extramarks were created in accordance with the syllabus and updated advice. Students are instructed to complete Class 11 Exercise 5.1 in order to improve their knowledge of complex numbers and quadratic equations and to jump-start their exam preparation. It is suggested that students download the NCERT Solution PDF for better learning, to increase their chances of performing well on the exam. The NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 are very handy when it comes to clarifying the basic understanding of the chapter and for better grades in Class 12.

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Q.1 Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

$\begin{array}{l} 1 . (5 i) (- \frac{3}{5} i) \\ 2. i^{9} + i^{19} \\ 3. i^{- 39} \\ 4. 3 (7 + i 7) + i (7 + i 7) \\ 5. (1 - i) - (- 1 + i 6) \\ 6. (\frac{1}{5} + i \frac{2}{5}) - (4 + i \frac{5}{2}) \\ 7. [(\frac{1}{3} + i \frac{7}{3}) + (4 + i \frac{1}{3})] - (- \frac{4}{3} + i) \\ 8. {(1 - i)}^{4} \\ 9 . {(\frac{1}{3} + 3 i)}^{3} \\ 10 . {(- 2 - \frac{1}{3} i)}^{3} \end{array}$

Ans.
1.

$\begin{array}{l} (5 i) (- \frac{3}{5} i) = 5 \times - \frac{3}{5} i^{2} \\ = - 3 (- 1) [∵ i^{2} = - 1] \\ = 3 + 0 i, which is in the form of a + ib. \end{array}$

$\begin{array}{l} i^{9} + i^{19} = i^{4 \times 2 + 1} + i^{4 \times 4 + 3} \\ = {(i^{4})}^{2} i + {(i^{4})}^{4} i^{3} \\ = {(1)}^{2} \times i + {(1)}^{4} (- i) [∵ i^{4} = 1 {and i}^{3} = - i] \\ = i - i \\ = 0 + 0 i, which is in the form of a + ib. \end{array}$

$i^{- 39} = \frac{1}{i^{39}}$ $\begin{array}{l} \end{array}$ $\begin{array}{l} = \frac{1}{i^{4 \times 9 + 3}} \\ = \frac{1}{{(i^{4})}^{9} . i^{3}} \\ = \frac{1}{{(1)}^{9} . (- i)} \\ = - \frac{1}{i} \times \frac{i}{i} \\ = - \frac{i}{i^{2}} \\ = - \frac{i}{- 1} \\ = 0 + i, which is in the form of a + ib. \end{array}$

4.
3(7 + i 7)+ i(7 + i 7) = 21 + 21i + 7i + 7i²
= 21 + 28i + 7(– 1)
= 21 + 28i – 7
= 14 + 28i, which is in the form of a + ib.

5.
(1 – i) – (–1 + i 6) = 1 – i + 1 – i 6
= 2 – 7i, which is in the form of a + ib.

$\begin{array}{l} (\frac{1}{5} + i \frac{2}{5}) - (4 + i \frac{5}{2}) = (\frac{1}{5} - 4) + i (\frac{2}{5} - \frac{5}{2}) \\ = - \frac{19}{5} + i (\frac{4 - 25}{10}) \\ = - \frac{19}{5} - i \frac{21}{10}, \\ which is in the form of a + ib. \end{array}$

$\begin{array}{l} [(\frac{1}{3} + i \frac{7}{3}) + (4 + i \frac{1}{3})] - (- \frac{4}{3} + i) = (\frac{1}{3} + 4 + \frac{4}{3}) + i (\frac{7}{3} + \frac{1}{3} - 1) \\ = (\frac{1 + 12 + 4}{3}) + i (\frac{7 + 1 - 3}{3}) \\ = (\frac{17}{3}) + i (\frac{5}{3}), \\ Which is in the form of a + ib . \end{array}$

$\begin{array}{l} {(1 - i)}^{4} = {{(1 - i)}^{2}}^{2} \\ = {(1 - 2 i + i^{2})}^{2} \end{array}$ $\begin{array}{l} \end{array}$ $\begin{array}{l} = {(1 - 2 i - 1)}^{2} \\ = {(- 2 i)}^{2} \\ = 4 i^{2} \\ = 4 (- 1) \\ = - 4 + i 0, \\ which is in the form of a + ib . \end{array}$

$\begin{array}{l} {(\frac{1}{3} + 3 i)}^{3} = {(\frac{1}{3})}^{3} + 3 {(\frac{1}{3})}^{2} (3 i) + 3 (\frac{1}{3}) {(3 i)}^{2} + {(3 i)}^{3} \\ [∵ {(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 {ab}^{2} + b^{3}] \\ = \frac{1}{27} + i + i^{2} 9 + i^{3} 27 \\ = \frac{1}{27} + i - 9 - i 27 \\ = (\frac{1}{27} - 9) + i (1 - 27) \\ = (\frac{1 - 243}{27}) + i (- 26) \\ = (\frac{- 242}{27}) - i 26 \\ which is in the form of a + ib . \end{array}$

10.

$\begin{array}{l} {(- 2 - \frac{1}{3} i)}^{3} = {(- 2)}^{3} + 3 {(- 2)}^{2} (- \frac{1}{3} i) + 3 (- 2) {(- \frac{1}{3} i)}^{2} + {(- \frac{1}{3} i)}^{3} \\ [∵ {(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 {ab}^{2} + b^{3}] \\ = - 8 - 4 i - \frac{2}{3} i^{2} - \frac{1}{27} i^{3} \\ = - 8 - 4 i - \frac{2}{3} (- 1) - \frac{1}{27} (- i) [∵ i^{2} = - 1 {and i}^{3} = - i] \\ = - 8 + \frac{2}{3} - 4 i + \frac{1}{27} i \\ = - \frac{22}{3} - (\frac{108 - 1}{27}) i \\ = - \frac{22}{3} - (\frac{107}{27}) i, \\ which is in the form of a + ib . \end{array}$

Q.2 Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.

11. 4 – 3i
12.

$\sqrt{5} + 3 i$

13. – i

Ans.

11.

$\begin{array}{l} Multiplicative inverse of (4 - 3i) = \frac{1}{(4 - 3i)} \times \frac{(4 + 3i)}{(4 + 3i)} \\ = \frac{(4 + 3i)}{(4^{2} - {9i}^{2})} \end{array}$

$\begin{array}{l} [∵ (a - b) (a + b) = a^{2} - b^{2}] \\ = \frac{(4 + 3i)}{(16 + 9)} \\ = \frac{4}{25} + \frac{3}{25} i \\ Thus, the multiplicative inverse of (4 - 3 i) is (\frac{4}{25} + \frac{3}{25} i) . \end{array}$

12.

$\begin{array}{l} Multiplicative inverse of (\sqrt{5} + 3 i) = \frac{1}{(\sqrt{5} + 3 i)} \times \frac{(\sqrt{5} - 3 i)}{(\sqrt{5} - 3 i)} \\ = \frac{(\sqrt{5} - 3 i)}{{{(\sqrt{5})}^{2} - {(3 i)}^{2}}} \\ = \frac{(\sqrt{5} - 3 i)}{{5 - 9 i^{2}}} \\ = \frac{(\sqrt{5} - 3 i)}{(5 + 9)} \\ = \frac{\sqrt{5}}{14} - i \frac{3}{14} \\ Thus, the multiplicative inverse of (\sqrt{5} + 3 i) is (\frac{\sqrt{5}}{14} - i \frac{3}{14}) . \end{array}$

13.

$\begin{array}{l} Multiplicative inverse of (- i) = \frac{1}{(- i)} \times \frac{i}{i} \\ = \frac{i}{- i^{2}} \\ = \frac{i}{- (- 1)} \\ = \frac{i}{1} \\ = i, \\ Thus, the multiplicative inverse of - i is i. \end{array}$

Q.3

$\begin{array}{l} Express the following expression in the form of a+ib: \\ \frac{(3+i \sqrt{5}) (3-i \sqrt{5})}{(\sqrt{3} + \sqrt{2} i) – (\sqrt{3} – \sqrt{2} i)} \end{array}$

Ans.

$\begin{array}{l} \frac{(3 + i \sqrt{5}) (3 - i \sqrt{5})}{(\sqrt{3} + \sqrt{2} i) - (\sqrt{3} - \sqrt{2} i)} = \frac{3^{2} - {(i \sqrt{5})}^{2}}{\sqrt{3} + \sqrt{2} i - \sqrt{3} + \sqrt{2} i} [\begin{array}{l} ∵ (a - b) (a + b) \\ = a^{2} - b^{2} \end{array}] \\ = \frac{(9 - i^{2} 5)}{2 \sqrt{2} i} \\ = \frac{{9 - (- 1) 5}}{2 \sqrt{2} i} \end{array}$ $\begin{array}{l} \end{array}$

$\begin{array}{l} = \frac{14}{2 \sqrt{2} i} \\ = \frac{7}{\sqrt{2} i} \times \frac{\sqrt{2} i}{\sqrt{2} i} \\ = \frac{7 \sqrt{2} i}{i^{2} (2)} \\ = \frac{7 \sqrt{2} i}{(- 1) 2} [∵ i^{2} = - 1] \\ = 0 - \frac{7 \sqrt{2} i}{2}, which in the form of a + ib. \end{array}$

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

NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations (Ex 5.1) Exercise 5.1

Access NCERT Solutions for Class 11 Mathematics Chapter – 5 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapters

NCERT Solution Class 11 Maths Chapter 5 Exercise

Complex Numbers and Quadratic Equations Exercise 5.1 – An Overview

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

NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations (Ex 5.1) Exercise 5.1

Access NCERT Solutions for Class 11 Mathematics Chapter – 5 Complex Numbers and Quadratic Equations

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Complex Numbers and Quadratic Equations Exercise 5.1 – An Overview

Class 11 Maths Chapter 5 Exercise 5.1 – All Questions

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