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:

The sum, difference, product, or quotient of two irrational numbers is not always irrational.
Keep simple counterexamples like \(\sqrt{a}\) and \(-\sqrt{a}\) in mind to quickly evaluate such statements!

Answer 1

The Correct Option is C

Step 1: Test Assertion (A) by squaring the sum.
\[ (\sqrt{3} + \sqrt{5})^2 = 3 + 5 + 2\sqrt{15} = 8 + 2\sqrt{15} \]
Step 2: Decide whether this square is rational or irrational.
Since $15$ is not a perfect square, $\sqrt{15}$ is irrational, so $8 + 2\sqrt{15}$ is irrational. If $\sqrt{3} + \sqrt{5}$ were rational, squaring it would give a rational result, which contradicts what we just found. So Assertion (A) is true.
Step 3: Test Reason (R) with a counterexample.
Take $\sqrt{3}$ and $-\sqrt{3}$, both irrational; their sum is $0$, which is rational. So Reason (R), which claims the sum of any two irrationals is always irrational, is false.
Assertion (A) is true but Reason (R) is false.
\[ \boxed{\text{Option (C)}} \]