CBSE Class 10 Mathematics Standard 2025: question 29

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Step-by-step solution

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1. Assume, to the contrary, that $\frac{1}{\sqrt{5}}$ is rational. Then $\frac{1}{\sqrt{5}}=\frac{p}{q}$, where $p,q$ are positive integers, $q\ne 0$, and $p,q$ are co-prime.

   ½ mark

2. Then $q=\sqrt{5}p$. Squaring gives $q^2=5p^2$. So 5 divides $q^2$. Since 5 is prime, 5 divides $q$ (if a prime divides $a^2$, it divides $a$). ...(i)

   1 mark

3. Let $q=5a$ for some integer $a$. Then $25a^2=5p^2$, so $p^2=5a^2$. So 5 divides $p^2$, and therefore 5 divides $p$. ...(ii)

   1 mark

4. From (i) and (ii), 5 is a common factor of $p$ and $q$. This contradicts the fact that $p$ and $q$ are co-prime. So our assumption is wrong, and $\frac{1}{\sqrt{5}}$ is an irrational number.

   ½ mark

Answer: $\frac{1}{\sqrt{5}}$ is irrational (proved by contradiction).

Revise the chapter

• Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
• Topic: Proving Numbers Irrational