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

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.

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

1. Prove that âˆš5 is irrational.

Solutions: Let us assume, that âˆš5 is rational number.

i.e.Â âˆš5 = x/y (where, x and y are co-primes)

yâˆš5= x

Squaring both the sides, we get,

(yâˆš5)2 = x2

â‡’5y2 = x2â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

Thus, x2 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,

5y2 = (5k)2

â‡’y2 = 5k2

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

Clearly, x and y are not co-primes. Thus, our assumption aboutÂ âˆš5 is rational is incorrect.

Hence,Â âˆš5 is an irrational number.

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

Solutions: Let us assume 3 + 2âˆš5 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, âˆš5 is also a rational number. But this contradicts the fact that âˆš5 is irrational.

So, we conclude that 3 + 2âˆš5 is irrational.

3. Prove that the following are irrationals:

(i) 1/âˆš2

(ii) 7âˆš5

(iii) 6 + âˆš2

Solutions:

(i) 1/âˆš2

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) 7âˆš5

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 +âˆš2

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.