Question

\(\sqrt p+\sqrt q\) is an irrational number, where \(p, q\) are

• A. Even numbers
• B. Prime numbers
• C. Rational numbers
• D. None

Answer 1

The Correct Option is B

To solve the problem, we need to understand the condition under which an expression of the form $ \sqrt{p} + \sqrt{q} $ is irrational.

1. Understanding Irrational Numbers:
An irrational number is a number that cannot be expressed as a ratio of two integers. For example, $\sqrt{2}$ and $\sqrt{3}$ are irrational.

2. When is $ \sqrt{p} + \sqrt{q} $ Irrational?
If $p$ and $q$ are both positive rational numbers but not perfect squares, then $\sqrt{p}$ and $\sqrt{q}$ are irrational.
If $\sqrt{p}$ and $\sqrt{q}$ are irrational and not conjugates, then their sum is also irrational.

3. Evaluating the Options:
- If $p$ and $q$ are rational numbers (but not perfect squares), then $\sqrt{p} + \sqrt{q}$ is irrational. Hence, option (3) seems valid.
- However, the most specific and commonly considered case in problems like these is when $p$ and $q$ are **prime numbers**, because square roots of prime numbers are always irrational, and their sum is also irrational.

Final Answer:
Since $\sqrt{p} + \sqrt{q}$ is irrational when $p$ and $q$ are prime numbers, the correct answer is $ {\text{Prime numbers}} $.