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√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 = p2 ⇒2q2 = p2 ⇒ 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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