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# NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.3) Exercise 1.3

Last updated date: 20th Mar 2023
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Mathematics is an essential subject of the Class 10 board exam. The syllabus of Class 10 Mathematics has 15 different chapters on various. The Central Board of Secondary Education (CBSE) has included many topics of Mathematics such as Algebra, Geometry, Trigonometry, Statistics, etc. Chapter 1 of Class 10 Mathematics is all about real numbers. This chapter is a vital part of Arithmetic. The students should study and practice this chapter sincerely. There are many exercises and a practice set of this chapter. Here, we are discussing NCERT Solution for Class 10 Maths chapter 1 exercise 1.3.  The students will be benefited by the questions of chapter 1 ex 1.3 Class 10 maths. You can also download NCERT Solutions for Class 10 Maths to help you revise the complete syllabus and score more marks in your examination. Also, NCERT Solution Class 10 Science is available for free on Vedantu.

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## Maths Syllabus of CBSE Class 10

1. Prove that $\sqrt{5}$ is irrational.

Ans: We have to prove that $\sqrt{5}$ is irrational.

We will use a contradiction method to prove it.

Let $\sqrt{5}$ is a rational number of the form $\dfrac{a}{b}$, where $b\ne 0$ and $a$ and $b$ are coprime i.e. $a$ and $b$ have only $1$ as a common factor.

Let $\sqrt{5}=\dfrac{a}{b}$

Now, squaring both sides, we get

${{\left( \sqrt{5} \right)}^{2}}={{\left( \dfrac{a}{b} \right)}^{2}}$

$\Rightarrow 5=\dfrac{{{a}^{2}}}{{{b}^{2}}}$

$\Rightarrow {{a}^{2}}=5{{b}^{2}}$ …….(1)

If ${{a}^{2}}$ is divisible by $5$ then $a$ is also divisible by $5$.

Let $a=5k$, where, $k$ is any integer.

Again squaring both sides, we get

$\Rightarrow {{a}^{2}}={{\left( 5k \right)}^{2}}$

Substitute the value in eq. (1), we get

$\Rightarrow {{\left( 5k \right)}^{2}}=5{{b}^{2}}$

$\Rightarrow {{b}^{2}}=5{{k}^{2}}$ …..(2)

If ${{b}^{2}}$ is divisible by $5$ then $b$ is also divisible by $5$.

From, eq. (1) and (2), we can conclude that $a$ and $b$ have $5$ as a common factor.

Therefore, we can say that $\sqrt{5}$ is irrational.

Hence proved.

2. Prove that $3+2\sqrt{5}$ is irrational.

Ans: We have to prove that $3+2\sqrt{5}$ is irrational.

We will use a contradiction method to prove it.

Let $3+2\sqrt{5}$ is a rational number of the form $\dfrac{a}{b}$, where $b\ne 0$ and $a$ and $b$ are coprime i.e. $a$ and $b$ have only $1$ as a common factor.

Let $3+2\sqrt{5}=\dfrac{a}{b}$

$\Rightarrow 2\sqrt{5}=\dfrac{a}{b}-3$

$\Rightarrow \sqrt{5}=\dfrac{1}{2}\left( \dfrac{a}{b}-3 \right)$ ……..(1)

From eq. (1) we can say that $\dfrac{1}{2}\left( \dfrac{a}{b}-3 \right)$ is rational so $\sqrt{5}$ must be rational.

But this contradicts the fact that $\sqrt{5}$ is irrational. Hence the assumption is false.

Therefore, we can say that $3+2\sqrt{5}$ is irrational.

Hence proved.

3. Prove that following are irrationals:

(i) $\dfrac{1}{\sqrt{2}}$

Ans: We have to prove that $\dfrac{1}{\sqrt{2}}$ is irrational.

We will use a contradiction method to prove it.

Let $\dfrac{1}{\sqrt{2}}$ is a rational number of the form $\dfrac{a}{b}$, where $b\ne 0$ and $a$ and $b$ are coprime i.e. $a$ and $b$ have only $1$ as a common factor.

Let $\dfrac{1}{\sqrt{2}}=\dfrac{a}{b}$

$\Rightarrow \sqrt{2}=\dfrac{b}{a}$ ………..(1)

From eq. (1) we can say that $\dfrac{b}{a}$ is rational so $\sqrt{2}$ must be rational.

But this contradicts the fact that $\sqrt{2}$ is irrational. Hence the assumption is false.

Therefore, we can say that $\dfrac{1}{\sqrt{2}}$ is irrational.

Hence proved.

(ii) $7\sqrt{5}$

Ans: We have to prove that $7\sqrt{5}$ is irrational.

We will use a contradiction method to prove it.

Let $7\sqrt{5}$ is a rational number of the form $\dfrac{a}{b}$, where $b\ne 0$ and $a$ and $b$ are coprime i.e. $a$ and $b$ have only $1$ as a common factor.

Let $7\sqrt{5}=\dfrac{a}{b}$

$\Rightarrow \sqrt{5}=\dfrac{a}{7b}$ ………..(1)

From eq. (1) we can say that $\dfrac{a}{7b}$ is rational so $\sqrt{5}$ must be rational.

But this contradicts the fact that $\sqrt{5}$ is irrational. Hence the assumption is false.

Therefore, we can say that $7\sqrt{5}$ is irrational.

Hence proved.

(iii) $6+\sqrt{2}$

Ans: We have to prove that $6+\sqrt{2}$ is irrational.

We will use a contradiction method to prove it.

Let $6+\sqrt{2}$ is a rational number of the form $\dfrac{a}{b}$, where $b\ne 0$ and $a$ and $b$ are coprime i.e. $a$ and $b$ have only $1$ as a common factor.

Let $6+\sqrt{2}=\dfrac{a}{b}$

$\Rightarrow \sqrt{2}=\dfrac{a}{b}-6$ ………..(1)

From eq. (1) we can say that $\dfrac{a}{b}-6$ is rational so $\sqrt{2}$ must be rational.

But this contradicts the fact that $\sqrt{2}$ is irrational. Hence the assumption is false.

Therefore, we can say that $6+\sqrt{2}$ is irrational.

Hence proved.

### Maths Syllabus of CBSE Class 10

The Mathematics syllabus of Class 10 contains some crucial part of Mathematics such as Geometry, Algebra, Arithmetic, Trigonometry, Statistics, etc. There are 15 chapters in Class 10 in Mathematics. The chapters are real numbers, polynomials, pair of linear equations in two variables, Arithmetic progressions, quadratic equations, triangles, introduction to Trigonometry, coordinate Geometry, applications of Trigonometry, constructions, circles, the area related to circles, surface areas and volumes, probability, Statistics. There are chapters from different parts of Mathematics.

### CBSE Class 10 Mathematics Chapter 1

Central Board of Secondary Education (CBSE) has included some vital concepts of Arithmetic in the Class 10 Mathematics syllabus. Chapter 1 of Class 10 Mathematics is one of them. This chapter contains all about real numbers. Real numbers are a strong conceptual topic of Mathematics. From this chapter, the students will get to know the definition, properties and classification of the real number. This chapter is also an essential part of the mathematical number system. In this chapter, some examples and property proof is included for the students. By solving those examples, the students can learn the real number system easily. The property proofs will be beneficial for them. The students should practise the exercises of the textbook. They will get a good topic insight from Class 10 maths ex 1.3 NCERT solutions and it will be efficient for your practice.

### NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3

Class 10 maths ex 1.3 is an important practice set for the students. This exercise contains some testing problems of real numbers. By practising those problems, the students will get a clear understanding of the properties and classification of real numbers. The examples are much efficient to clear the conceptual part. The solutions of this exercise are provided by NCERT. Class 10 maths ex 1.3 NCERT solutions are available online for free. The students can download it for their convenience.

### Necessity of NCERT Solutions Class 10 Maths Ch 1 Ex 1.3

Practising exercises of individual Mathematics chapters is essential for the Class 10 board exam. The students should practise the exercises of specific chapters sincerely. If they get solutions for those exercises after solving, that will be beneficial for the students. That is why Class 10 maths ex 1.3 NCERT solutions are necessary. It increases the confidence of the students.

### Irrational Numbers

Any number which cannot be represented in the form of p/q (where p and q are integers and q≠0.) is an irrational number. Examples √2,π, e, and so on.

Interesting Results of Number Theory

• If a number p (a prime number) divides a2, then p divides a. Example: 3 divides 62 i.e 36, which implies that 3 divides 6.

• The addition or subtraction of a rational and an irrational number always results in an irrational number.

• The multiplication value of a non-zero rational number and an irrational number is always irrational.

• The quotient obtained when division operation is performed between a non-zero rational number and an irrational number is always irrational.

• √m is irrational when ‘m’ is a prime. For example, 11 is a prime number and √11 is irrational which can be proved by the method of “Proof by contradiction”.

In a contradiction method, we start with an assumption which is contrary to what we are required to prove. Using a series of logical deductions from this assumption of contradiction, we will reach a mathematical inconsistency (error) – which enables us to conclude that our assumption of contradiction was incorrect.

Let’s suppose, we are going to prove √2 is an irrational number through the method of contradiction. Firstly, we assume the contradiction that √2 is a rational number and So it can be written in the form of a/b, where a and b are two co-prime numbers and n ≠ 0. By observing, we will find that there exist no coprime integers a and b for √2, so our assumption was wrong.

### Solved Examples

1. Prove that, √5 is an international number.

Solution:

Suppose, √5 is rational. Therefore, there are two integers a and b, where a/b=√5.

Suppose, the common factor of a and b is other than 1. So, we can assume them as co-prime numbers by dividing by the common factor.

a = b√5

Or,  5b2 = a2

Therefore, a2 is divisible by 5 as well as a is divisible by 5.

Let, a=5k, where k is an integer

Now, (5k)2=5b2

Or, b2 = 5k2

So, b2 and b both are divisible by 5.

Hence, we can say that a and b have common factor 5, which contradicts the co-prime fact.

Therefore, √5 cannot be expressed in p/q form. √5 is an irrational number (proved).

2. 3+2√5 is irrational - prove this.

Solution:

Suppose, 3+2√5 is a rational number.

That means, there are two integers called a and b such that

a/b=3+2√5

Or, a/b - 3 = 2√5

Or, √5 = 1/2 (a/b-3)

As a and b are integers and 1/2 (a/b - 3) will also be rational. So, √5 is rational.

This fact contradicts that √5 is irrational.

Hence, our consumption is false and √5 is an irrational number.

## FAQs on NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers

1. Why Practising Chapter 1 Class 10 Maths Exercise 1.3 is Important?

Class 10 board exam is the first board exam of the students. The students read and work hard to score well in their first board exam. The Mathematics syllabus of Class 10 has diversity in the chapters. The students have to read all the different chapters and topics with the same importance. The students should practise the chapters after completing the entire syllabus. There are several exercises and practise sets for each chapter of Class 10 Mathematics. Chapters 1 Class 10 maths exercise 1.3 is one of them. Chapter 1 Class 10 maths exercise 1.3 help the students to get a strong concept of this chapter and in practising.

Mathematics is a practice-based subject. The more the students will practise, the more they will get strong on the topics. They should read and complete their Mathematics syllabus. After that, they should practise as much as they can. There are individual exercises and practise sets for all the chapters of Class 10 Mathematics. For chapter 1 the exercises are ex 1.1, ex 1.2 and ex 1.3 Class 10 maths. They can download all the exercises of chapter 1 from different educational websites. Chapter 1 ex 1.3 Class 10 maths pdf is available online for free. The students can download the pdf or click the direct download link and get it.

3. How many questions are there in Exercise 1.3 of 10th Maths?

Chapter 1 in the syllabus for Class 10 Mathematics is “Real Numbers”. The chapter consists of four exercises in total. Exercise 1.3 of this chapter contains three questions in total.  Although most of these problems are short, their solutions are long. The exercise includes questions that require students to prove the given roots as irrational. These questions and the examples related to them are important and are often asked in the exam as well.

4. What is the main concept to learn exercise 1.3 of 10th Maths?

The 1st chapter in Class 10 Maths includes everything about real numbers which is an important topic in the Class 10 syllabus. Exercise 1.3 of this chapter includes questions of the proof of the irrationality of roots. Examples 9, 10, and 11 are also based on questions in this exercise. Solving these questions might seem difficult to understand at first. Students can take the help of NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1.

5. Do I need to practice all the questions provided in Class 10 Maths Chapter 1 NCERT Solutions?

Class 10 Board Question Papers are framed directly based on the Class 10 Maths NCERT. Hence, it is important to practice all the questions that are provided in the NCERT Solution for Class 10 Maths Chapter 1. Any of them can be asked in the exams and it is not possible to know which ones they might be. Ignoring any questions or examples can lead to a loss of marks.

6. Where can I find NCERT Solutions for Class 10 Maths Exercise 1.3?

Students who need help understanding the method of solving the questions given in Exercise 1.3 can take the help of the NCERT Solutions for Class 10 Maths available on Vedantu. These solutions have been framed by experts of the subject in a language easy for students to understand and be able to grasp all the concepts taught in this exercise. You can find these solutions on the page NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 at free of cost. The PDFs of these NCERT Solutions are available on the Vedantu app for free.

7. How many exercises and questions are covered in NCERT Solutions for Class 10 Maths Chapter 1?

Chapter 1 in Class 10 Maths is known as Real Numbers. There are a total of four exercises covered in the NCERT Solutions for this chapter. The following is a list of total questions provided in each exercise:

• Exercise 1.1 - Five Questions

• Exercise 1.2 - Seven Questions

• Exercise 1.3 - Three Questions

• Exercise 1.4 - Three Questions

Students can find step-by-step NCERT Solutions for all four exercises on the page NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1.