CBSE Class 10 Mathematics Standard 2026: question 28

Hints

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Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

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

   ½ mark

2. Squaring: $5q^2=p^2$. So $p^2$ is divisible by 5.
   Since 5 is prime, $p$ is divisible by 5 ...(1) [if a prime divides $p^2$, it divides $p$].

   1 mark

3. Let $p=5a$ for some integer $a$. Then $5q^2=25a^2\Rightarrow q^2=5a^2$.
   So $q^2$ is divisible by 5, and hence $q$ is divisible by 5 ...(2).

   1 mark

4. From (1) and (2), 5 is a common factor of $p$ and $q$. This contradicts the fact that $p$ and $q$ are coprime.
   So our assumption is wrong, and $\sqrt{5}$ is irrational.

   ½ mark

Answer: $\sqrt{5}$ 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