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 \((2 + \sqrt{3})\) and \((2 - \sqrt{3})\) in mind to quickly verify these properties during exams!

Answer 1

The Correct Option is C

Step 1: Prove the Assertion using a difference-of-squares trick.
Suppose $\sqrt3+\sqrt5=r$ were rational. Since $(\sqrt5-\sqrt3)(\sqrt5+\sqrt3)=5-3=2$, we'd get $\sqrt5-\sqrt3=\frac{2}{r}$, also rational.
Step 2: Combine the two equations.
Adding $\sqrt3+\sqrt5=r$ and $\sqrt5-\sqrt3=\frac2r$ gives $2\sqrt5=r+\frac2r$, a rational number, forcing $\sqrt5$ to be rational, a contradiction. So Assertion (A) is true.
Step 3: Test the Reason with a fresh counterexample.
Take $2+\sqrt3$ and $2-\sqrt3$, both irrational. Their sum is $4$, which is rational, so the Reason's claim is false.
Step 4: Conclude.
Assertion (A) is true but Reason (R) is false, matching option (C).
\[ \boxed{\text{Option (C)}} \]