Question 26 of 26intermediate🔍 AnalyzeShort Answer2 marks

Given that p\sqrt{p} is irrational for all primes pp and also suppose that 3721 is a prime. Can you conclude that 3721\sqrt{3721} is an irrational number? Is your conclusion correct? Why or why not?

Correct Answer

Yes, we can conclude that 3721\sqrt{3721} is an irrational number using deductive reasoning. Given the hypothesis that p\sqrt{p} is irrational for all primes pp and assuming that 3721 is a prime, the conclusion follows logically.

However, this conclusion may not be correct. As stated in the context, we haven't checked whether 3721 is actually a prime or not. We are assuming it to be a prime for the sake of our argument. The context clearly states that if we start with an incorrect premise (or hypothesis), we may arrive at a wrong conclusion. The process of reasoning is correct, but the conclusion depends on the trueness of the hypothesis.

Exercise: EXERCISE A1.2 | Q: 7 | (Chapter: 6)
For More Understanding

Explanation

This question tests understanding of deductive reasoning from the NCERT chapter on Mathematical Reasoning. The textbook context (Example 9) shows a similar problem with 19423. The key insight is that deductive reasoning gives valid conclusions from given premises, but the conclusion's correctness depends on whether the premises are true. The textbook explicitly states that if we start with an incorrect premise, we may arrive at a wrong conclusion. Students should recognize that the reasoning process is valid regardless, but the conclusion's truth value depends on the hypothesis being true.

Solution Steps

  1. Step 1: Apply deductive reasoning: If p\sqrt{p} is irrational for all primes pp, and if 3721 is prime, then 3721\sqrt{3721} must be irrational.

  2. Step 2: Recognize that we haven't verified whether 3721 is actually prime - we're assuming it.

  3. Step 3: Conclude that the reasoning is correct, but the conclusion may be wrong if the premise (3721 is prime) is false.

Previous