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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.
Options
Only (i)
Only (ii)
Both (i) and (ii)
Neither (i) nor (ii)
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Solution
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.
