1.

√2 is(a) an integer (b) an irrational number (c) a rational number (d) none of these

Answer»

Let √2 is a rational number.

∴ √2 = \(\frac{p}q\), where p and q are some integers and HCF(p,q) = 1 ….(1)

⇒ √2q = p 

⇒ (√2q)2 = p

⇒2q2 = p

⇒ p2 is divisible by 2 

⇒ p is divisible by 2 …..(2)

Let p = 2m, where m is some integer. 

∴ √2q = p 

⇒ √2q = 2m 

⇒ (√2q)2 = (2m)2 

⇒2q2 = 4m2 

⇒ q2 = 2m2 

⇒ q2 is divisible by 2 

⇒ q is divisible by 2 …..(3)

From (2) and (3), 2 is a common factor of both p and q, which contradicts (1). 

Hence, our assumption is wrong. 

Thus, √2 is an irrational number. 

Hence, the correct answer is option (b).



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