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: Understanding the Question:
The topic of this question is Real Numbers.
This is an Assertion-Reason question where we must independently evaluate the mathematical truth of two statements: the Assertion (A) and the Reason (R).
If both are true, we check if the Reason provides the correct explanation for the Assertion.

Step 2: Key Formula or Approach:
- An irrational number cannot be expressed as a simple fraction of two integers.
- We can prove the irrationality of \(\sqrt{3} + \sqrt{5}\) using a proof by contradiction.
- We can check the truth of the Reason by searching for a counterexample where the sum of two irrationals is rational.

Step 3: Detailed Explanation:

• Analyze Assertion (A):
Assume for contradiction that \(\sqrt{3} + \sqrt{5} = x\) is a rational number.
Rearrange:
\[ x - \sqrt{3} = \sqrt{5} \] Square both sides:
\[ (x - \sqrt{3})^2 = 5 \] \[ x^2 + 3 - 2x\sqrt{3} = 5 \] Isolate the radical term:
\[ x^2 - 2 = 2x\sqrt{3} \] \[ \sqrt{3} = \frac{x^2 - 2}{2x} \] Since \(x\) is rational, \(\frac{x^2 - 2}{2x}\) is rational. However, \(\sqrt{3}\) is irrational.
This contradiction proves that \(\sqrt{3} + \sqrt{5}\) is indeed irrational. Thus, Assertion (A) is true.

• Analyze Reason (R):
The statement claims that the "Sum of any two irrational numbers is always irrational."
Let us test this with a counterexample:
Let \(a = \sqrt{3}\) (irrational) and \(b = -\sqrt{3}\) (irrational).
Their sum is:
\[ a + b = \sqrt{3} + (-\sqrt{3}) = 0 \] Since 0 is rational, the sum of these two irrationals is rational.
Thus, Reason (R) is false.


Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.
Therefore, the correct option is (C).