Question

Assertion (A) : \((a + \sqrt{b})(a - \sqrt{b})\) is a rational number, where a and b are positive integers.
Reason (R) : Product of two irrationals is always rational.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A)
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true

Answer 1

The Correct Option is C

Given:
- Assertion (A): \((a + \sqrt{b})(a - \sqrt{b})\) is a rational number, where \(a\) and \(b\) are positive integers.
- Reason (R): Product of two irrationals is always rational.

Step 1: Analyze Assertion (A)
\[ (a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b \] Since \(a\) and \(b\) are integers, \(a^2 - b\) is an integer, hence rational.
So, Assertion (A) is true.

Step 2: Analyze Reason (R)
- Product of two irrational numbers is not always rational.
- For example, \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\) (irrational).
- So Reason (R) is false.

Final Answer:
\[ \boxed{ \text{Assertion (A) is true, but Reason (R) is false.} } \]