NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers – FREE PDF Download
NCERT Class 10 Maths Ch 1 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 10 Maths Chapter 1 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. This is where CoolGyan comes in. Our free Class 10 Real Numbers solutions will help you understand the chapter thoroughly.
NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers
1. Prove that 
is irrational.
irrational by contradiction. Let us suppose that is rational. It means that we have co-prime integers a and b (b ≠ 0) such that 
⇒ b=a
Squaring both sides, we get
… (1)
It means that 5 is factor of 
Hence, 5 is also factor of a by Theorem. … (2)
If, 5 is factor of a, it means that we can write a = 5c for some integer c.
Substituting value of a in (1),
It means that 5 is factor of 
.
Hence, 5 is also factor of b by Theorem. … (3)
From (2) and (3), we can say that 5 is factor of both a and b.
But, a and b are co-prime.
Therefore, our assumption was wrong. cannot be rational. Hence, it is irrational.
2. Prove that (3 + 2) is irrational.
Let us suppose that (3+2) is rational.
It means that we have co-prime integers a and b (b ≠ 0) such that
⇒ 
⇒ 
⇒ 
… (1)
a and b are integers.
It means L.H.S of (1) is rational but we know that is irrational. It is not possible. Therefore, our supposition is wrong. (3+2) cannot be rational.
Hence, (3+2) is irrational.
3. Prove that the following are irrationals.
(i) 
(ii) 
(iii) 
irrational by contradiction. Let us suppose that is rational.
It means we have some co-prime integers a and b (b ≠ 0) such that
12√=ab12=ab
⇒ 
… (1)
R.H.S of (1) is rational but we know that 
is irrational.
It is not possible which means our supposition is wrong.
Therefore, cannot be rational.
Hence, it is irrational.
(ii) We can prove irrational by contradiction.
Let us suppose that is rational.
It means we have some co-prime integers a and b (b ≠ 0) such that
⇒ 
… (1)
R.H.S of (1) is rational but we know that is irrational.
It is not possible which means our supposition is wrong.
Therefore, cannot be rational.
Hence, it is irrational.
(iii) We will prove irrational by contradiction.
Let us suppose that () is rational.
It means that we have co-prime integers a and b (b ≠ 0) such that
⇒ 
⇒ 
… (1)
a and b are integers.
It means L.H.S of (1) is rational but we know that is irrational. It is not possible.
Therefore, our supposition is wrong. () cannot be rational.
Hence, () is irrational.
