CBSE Class 10 Mathematics Standard 2025: question 26(A)

Hints

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Step-by-step solution

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1. Assume, to the contrary, that $\sqrt{2}$ is rational. Then $\sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are co-prime integers and $q\ne 0$.

   ½ mark

2. Then $\sqrt{2}q=p$, so $2q^2=p^2$. Hence 2 divides $p^2$, so 2 divides $p$ [if a prime divides $a^2$, it divides $a$]. Let $p=2m$ for some integer $m$.

   1 mark

3. Then $2q^2=4m^2$, so $q^2=2m^2$. Hence 2 divides $q^2$, so 2 divides $q$. Thus 2 is a common factor of $p$ and $q$, which contradicts that $p$ and $q$ are co-prime.

   1 mark

4. So our assumption is wrong, and $\sqrt{2}$ is irrational.

   ½ mark

Answer: √2 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