Question
Statement-1 (Assertion): \( \sqrt{2} \) is an irrational number.
Statement-2 (Reason): The decimal expansion of \( \sqrt{2} \) is non-terminating and non-recurring.
Options:
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Solution
\( \sqrt{2} \) cannot be expressed as a ratio of two integers, so it is an irrational number.
Its decimal expansion is:
\[ \sqrt{2} = 1.4142135\ldots \]
This is non-terminating and non-repeating, which is the defining property of irrational numbers.
Thus:
- Statement-1 is true
- Statement-2 is true
- Statement-2 correctly explains Statement-1
Final Answer
✔ Correct option: (a)