CBSE Class 10 Mathematics Standard sample paper 2024-25: question 31

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 $\sqrt{3}$ is rational. Then $\sqrt{3}=\frac{p}{q}$, where $p,q$ are integers, $q\ne 0$, and $p,q$ are co-prime.

   ½ mark

2. Squaring: $3q^2=p^2$. So 3 divides $p^2$, and since 3 is prime, 3 divides $p$ …(i) [if a prime divides $a^2$, it divides $a$].

   1 mark

3. Let $p=3a$ for some integer $a$. Then $3q^2=9a^2\Rightarrow q^2=3a^2$. So 3 divides $q^2$, and hence 3 divides $q$ …(ii)

   1 mark

4. From (i) and (ii), 3 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 $\sqrt{3}$ is irrational.

   ½ mark

Answer: $\sqrt{3}$ 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