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

Let's analyze the assertion and reason provided:

1. Assertion (A): \((\sqrt{3} + \sqrt{5})\) is an irrational number.
   • The sum of two irrational numbers, \(\sqrt{3}\) and \(\sqrt{5}\), needs to be verified for its nature.
   • Both \(\sqrt{3}\) and \(\sqrt{5}\) are non-repeating, non-terminating decimals, which means they are irrational numbers.
   • Adding two irrational numbers often but not always results in an irrational number. In this specific case, the sum \((\sqrt{3} + \sqrt{5})\) remains irrational as it cannot be represented as a ratio of integers.
   • Hence, Assertion (A) is true.
2. Reason (R): Sum of any two irrational numbers is always irrational.
   • This statement is not always correct. For example, consider the addition of \(\sqrt{2}\) and \(-\sqrt{2}\), both of which are irrational numbers. However, their sum is \(0\) (a rational number).
   • Therefore, Reason (R) is false.

Based on the analysis:

• The Assertion (A) is true because \((\sqrt{3} + \sqrt{5})\) is indeed an irrational number.
• The Reason (R) is false because the sum of two irrational numbers is not necessarily irrational.

Therefore, the correct answer is: Assertion (A) is true, but Reason (R) is false.