Question

Assertion (A) : \((\sqrt{3} + \sqrt{5})\) is an irrational number.
Reason (R) : Sum of the any two irrational numbers is always irrational.

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

Hint:

In Assertion-Reason questions, always check if the Reason is a universally true statement first. If you find one counter-example for the Reason, it is false, and you likely only have one option left.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Irrational numbers are numbers that cannot be expressed as a simple fraction.
The sum of two irrational numbers can sometimes be rational.
Step 3: Detailed Explanation:
Assertion (A): \(\sqrt{3} + \sqrt{5}\) is indeed irrational.
If it were rational, say \(r\), then \(\sqrt{5} = r - \sqrt{3}\). Squaring both sides gives \(5 = r^2 + 3 - 2r\sqrt{3}\), which implies \(\sqrt{3} = (r^2 - 2)/(2r)\). Since the RHS is rational and \(\sqrt{3}\) is irrational, this is a contradiction. Thus, (A) is true.
Reason (R): This statement is false.
Counter-example: Consider two irrational numbers \(\sqrt{2}\) and \(-\sqrt{2}\).
Sum: \(\sqrt{2} + (-\sqrt{2}) = 0\), which is a rational number.
Since the Reason is false, the option is (C).
Step 4: Final Answer: Assertion (A) is true, but Reason (R) is false.