3.Prove that the following are irrationals:
(i) 1/√2
(ii) 7√5
(iii) 6 + √2
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subject : maths
class 10th
Answers
Answer:
Step-by-step explanation:
Solution :
Let us assume 1/√2 is a rational number
Let us assume 1/√2 = r where r is a rational number
On further calculation we get
1/r = √2
Since r is a rational number, 1/r = √2 is also a rational number
But we know that √2 is an irrational number
So our supposition is wrong.
Hence, 1/√2 is an irrational number.
(ii) 7√5
Solution :
Let’s assume on the contrary that 7√5 is a rational number. Then, there exist positive integers a and b such that
7√5 = a/b where, a and b, are co-primes
⇒ √5 = a/7b
⇒ √5 is rational [∵ 7, a and b are integers ∴ a/7b is a rational number]
This contradicts the fact that √5 is irrational. So, our assumption is incorrect.
Hence, 7√5 is an irrational number.
(iii) 6 + √2
Solution :
Let’s assume on the contrary that 6+√2 is a rational number. Then, there exist co prime positive integers a and b such that
6 + √2 = a/b
⇒ √2 = a/b – 6
⇒ √2 = (a – 6b)/b
⇒ √2 is rational [∵ a and b are integers ∴ (a-6b)/b is a rational number]
This contradicts the fact that √2 is irrational. So, our assumption is incorrect.
Hence, 6 + √2 is an irrational number.