NCERT solutions Class 10 Maths Chapter 1 Real Numbers Exercise 1.3 can be downloaded from here. These solutions for questions given in NCERT are prepared by our subject experts. Thatâ€™s why these are a helpful resource for students appearing for Class 10 first term exams. These experts review this NCERT Maths Solution for Class 10 â€“ in two ways â€“ chapter and exercise-wise, so that they would be useful for students in solving the problems easily.

They prefer to make these NCERT Solutions for Class 10 in such a way that they should be easily understandable by the students. Exercise 1.3 is the third exercise of Chapter- 1, Real Numbers in Class 10. The important topic discussed here is revisiting irrational numbers. It is compulsory to have a basic understanding of rational and irrational numbers before solving Exercise 1.3. Students can also practise how to solve a problem by following the NCERT guidelines. The solutions provided here follow the same.

### Download PDF of NCERT Solutions for Class 10 Maths Chapter 1- Real Number Exercise 1.3

### Access Answers of Maths NCERT Class 10 Chapter 1 â€“ Real Number Exercise 1.3

**1. Prove that âˆš**5 **is irrational.**

**Solutions: **Let us assume, that **âˆš**5 is rational number.

i.e.Â **âˆš**5 = x/y (where, x and y are co-primes)

y**âˆš**5= x

Squaring both the sides, we get,

(y**âˆš**5)^{2} = x^{2}

â‡’5y^{2} = x^{2}â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

Thus, x^{2} is divisible by 5, so x is also divisible by 5.

Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get,

5y^{2} = (5k)^{2}

â‡’y^{2} = 5k^{2}

is divisible by 5 it means y is divisible by 5.

Clearly, x and y are not co-primes. Thus, our assumption aboutÂ **âˆš**5 is rational is incorrect.

Hence,Â **âˆš**5 is an irrational number.

**2. Prove that 3 + 2âˆš5 + is irrational.**

**Solutions: **Let us assume 3 + 2**âˆš**5 is rational.

Then we can find co-prime x and y (y â‰ 0) such that 3 + 2âˆš5 = x/y

Rearranging, we get,

Since, x and y are integers, thus,

is a rational number.

Therefore, **âˆš**5 is also a rational number. But this contradicts the fact that **âˆš**5 is irrational.

So, we conclude that 3 + 2**âˆš**5 is irrational.

**3. Prove that the following are irrationals: **

**(i) 1/âˆš2 **

**(ii) 7âˆš5**

**(iii) 6 + **âˆš**2**

**Solutions: **

**(i) 1/**âˆš**2**

Let us assume 1/âˆš2 is rational.

Then we can find co-prime x and y (y â‰ 0) such that 1/âˆš2 = x/y

Rearranging, we get,

âˆš2 = y/x

Since, x and y are integers, thus, âˆš2 is a rational number, which contradicts the fact that âˆš2 is irrational.

Hence, we can conclude that 1/âˆš2 is irrational.

**(ii) 7**âˆš**5**

Let us assume 7âˆš5 is a rational number.

Then we can find co-prime a and b (b â‰ 0) such that 7âˆš5 = x/y

Rearranging, we get,

âˆš5 = x/7y

Since, x and y are integers, thus, âˆš5 is a rational number, which contradicts the fact that âˆš5 is irrational.

Hence, we can conclude that 7âˆš5 is irrational.

**(iii) 6 +**âˆš**2**

Let us assume 6 +âˆš2 is a rational number.

Then we can find co-primes x and y (y â‰ 0) such that 6 +âˆš2 = x/yâ‹…

Rearranging, we get,

âˆš2 = (x/y) â€“ 6

Since, x and y are integers, thus (x/y) â€“ 6 is a rational number and therefore, âˆš2 is rational. This contradicts the fact that âˆš2 is an irrational number.

Hence, we can conclude that 6 +âˆš2 is irrational.

Exercise 1.1 Solutions 5 Question ( 4 long, 1 short)

Exercise 1.2 Solutions 14 Question ( 4 long, 3 short)

Exercise 1.4 Solutions 3 Question ( 3 short)

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

Exercise 1.3 of NCERT solutions for class 10 maths chapter 1 Real Numbers is the third exercise of Chapter 1 of Class 10 Maths. Real Numbers is introduced in Class 9 and is discussed more in detail in Class 10. It is crucial to have a fair knowledge of the topic â€“ irrational numbers to understand these solutions. The exercise discusses how to prove that root p is irrational.

- Revisiting Irrational Numbers â€“ It includes 3 questions based on the theorem where question no â€“ 3 has 3 roots to be proved as irrational.

### Key Features of NCERT Solutions for Class 10 Maths Chapter 1- Real Number Exercise 1.3

- These NCERT SolutionsÂ help you solve and revise all questions of Exercise 1.3.
- After going through the stepwise solutions given by our subject expert teachers, you will be able to score more marks.
- These solutions are prepared by following NCERT Guidelines.
- It helps in scoring well in maths in exams.
- It contains all the important questions from the examination point of view.