##
**Class 10, NCERT
(CBSE) solution of Mathematics **

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**Chapter 1, Real
Numbers **

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**NCERT solution for
Math Textbook Exercise 1.3**

(Page 14)

**Q 1: Prove that √5 is irrational.**

__Solution__:

Let us assume

*√5*is a rational number.
Then, we can
find two positive integers

*a*,*b*(*b*≠ 0) such thatSuppose,

*a*and

*b*have a common factor other than 1. Then we can divide them by the common factor, and assume that

*a*and

*b*are co-prime.

or,

*a = √5b*
or

*, a*^{2}= 5b^{2}
Therefore

*, a*is divisible by^{2}*5*and so, it can be said that*a*is divisible by*5*.
Now suppose,

*a*= 5

*k*, where

*k*is an integer.

From the above expression,

*b*

^{2}is divisible by 5 and hence,

*b*is divisible by

*5*.

This implies
that

*a*and*b*have at least*5*as a common factor.
And this
contradicts the fact that

*a*and*b*are co-prime.
Hence, √5 is an
irrational number.

**Q2: Prove that 3 + 2√5 is irrational.**

__Solution__:

Let us assume on the contrary
that 3 + 2√5 is rational.

Therefore, there exist co-prime positive integers say

*a*,*b*(*b*≠ 0) such thatThis contradicts the fact that √5 is irrational. So, our assumption that 3 + 2√5 is rational is false.

Therefore, 3 + 2√5 is
an irrational number.

Solution:

(i)

(ii) Let us assume on the contrary that 7√5 is rational.

Therefore, √5 must be rational.

This contradicts to the fact that √5 is irrational.

Or, our assumption that 7√5
is rational is false. Hence, 7√5 is
irrational.

(iii)

Let 6 + √2 be rational.

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