Question

(i) If m is prime then `sqrt(m)` is an irrational number.

(ii) There are infinitely many irrational numbers between any two rational numbers.

विकल्प

• Only (i) 

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Both (i) and (ii)

Explanation:

(i) If m is a prime number, then `sqrt(m)` is irrational. 

This is proved by contradiction: assuming `sqrt(m)` is rational leads to the conclusion that (a) and (b) from the form `sqrt(m) = a/b` share a common factor (m), contradicting that they have no common factors except 1. 

Therefore, `sqrt(m)` must be irrational if m is prime.

(ii) There are infinitely many irrational numbers between any two rational numbers.

This is a well-known fact about the density of irrational numbers on the number line.

Thus, both statements are valid and true.