NCERT Class 10 Maths Solutions Chapter - 1 Real Numbers, Ex-1.3

Ex - 1.3

Question 1. 

Solution

Let us assume, on the contrary that  is a rational number.

Therefore, we can find two integers a,b (b # 0) such that  = 

Where a and b are co-prime integers.

Therefore, a^₂ is divisible by 5 then a is also divisible by 5.

So a = 5k, for some  integer k.




This means that b² is divisible by 5 and hence, b is divisible by 5.
This implies that a and b have 5 as a common factor.
And this is a contradiction to the fact that a and b are co-prime.



Concept Insight: There are various ways of proving in mathematics proof by contradiction is one of them. In this approach we assume something which is contrary to what needs to be proved and arrive at a fact which contradicts something which is true in general. Key result used here is "If P is a prime number and it divides a^₂ then it divides a as well".

Question 2. 

Solution

Therefore, we can find two integers a, b (b  0) such that




Concept Insight: This problem is solved using proof by contradiction. The key concept used is if p is prime number then  is irrational. Do not prove this question by assuming sum of rational and irrational is irrational.

Question 3. Prove that the following are irrationals:

Solution

Concept Insight: This problem is solved using proof by contradiction. The key concept used is if p is prime number then  is irrational.Do not prove this question by assuming sum or product of rational and irrational is irrational.

Ex-1.4