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.
Whenever you need to evaluate general statements about irrational numbers, test them using simple counterexamples such as \(\sqrt{3}\) and \(-\sqrt{3}\), or \((2 + \sqrt{3})\) and \((2 - \sqrt{3})\).
This helps you identify false statements in seconds!

Answer 1

The Correct Option is C

Step 1: Prove the Assertion by isolating the other root.
Assume \(\sqrt{3} + \sqrt{5} = r\) is rational, so \(r - \sqrt{5} = \sqrt{3}\).
Step 2: Square both sides and isolate the radical.
\[ r^2 - 2r\sqrt{5} + 5 = 3 \implies \sqrt{5} = \frac{r^2+2}{2r} \] The right side is rational if r is rational and nonzero, but \(\sqrt{5}\) is irrational, a contradiction, so \(\sqrt{3}+\sqrt{5}\) is irrational and Assertion (A) is true.
Step 3: Test the Reason with a fresh counterexample.
Take the two irrational numbers \(2 - \sqrt{7}\) and \(5 + \sqrt{7}\). Their sum is \((2-\sqrt{7}) + (5+\sqrt{7}) = 7\), which is rational.
Step 4: Conclude about the Reason.
This counterexample shows the sum of two irrational numbers is not always irrational, so Reason (R) is false. Hence Assertion (A) is true but Reason (R) is false, matching option (C).
\[ \boxed{\text{Option (C)}} \]