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: Understanding the Question:
The topic of this question is Real Numbers.
We need to evaluate the mathematical truth of two statements: the Assertion (A) about the irrationality of the sum of two square roots of prime numbers, and the Reason (R) about a general property of operations on irrational numbers.

Step 2: Key Formula or Approach:
- An irrational number is a real number that 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.
- To evaluate the truth of the Reason, we will search for a counterexample that shows the sum of two irrational numbers can be rational.

Step 3: Detailed Explanation:

• Let us analyze Assertion (A):
Assume for contradiction that \(\sqrt{3} + \sqrt{5} = x\) is a rational number.
Rearrange the equation:
\[ x - \sqrt{3} = \sqrt{5} \] Square both sides of the equation:
\[ (x - \sqrt{3})^2 = (\sqrt{5})^2 \] \[ x^2 + 3 - 2x\sqrt{3} = 5 \] Rearrange the terms to isolate the radical term:
\[ x^2 + 3 - 5 = 2x\sqrt{3} \] \[ x^2 - 2 = 2x\sqrt{3} \] \[ \sqrt{3} = \frac{x^2 - 2}{2x} \] Since we assumed \(x\) is a non-zero rational number, the term \(\frac{x^2 - 2}{2x}\) must also be a rational number.
However, we know that \(\sqrt{3}\) is an irrational number.
An irrational number cannot be equal to a rational number.
This contradiction proves that our initial assumption was false, meaning \(\sqrt{3} + \sqrt{5}\) must be an irrational number. Thus, Assertion (A) is true.

• Let us analyze Reason (R):
The statement claims that the "Sum of any two irrational numbers is always irrational."
Let us test this statement using a counterexample:
Let the first irrational number be \(a = \sqrt{3}\).
Let the second irrational number be \(b = -\sqrt{3}\).
Now, calculate their sum:
\[ a + b = \sqrt{3} + (-\sqrt{3}) = 0 \] Since 0 is a rational number, the sum of these two irrational numbers is rational.
This counterexample proves that the sum of two irrational numbers is not always irrational. Thus, Reason (R) is false.


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