CBSE Class 10 Mathematics Standard 2026: question 24(a)

Hints

Open one hint at a time and try again before the next.

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, on the contrary, that $2+3\sqrt{5}$ is rational. Then $2+3\sqrt{5}=\frac{p}{q}$, where $p,q$ are integers and $q\ne 0$.

   ½ mark

2. Rearranging: $3\sqrt{5}=\frac{p}{q}-2$, so $\sqrt{5}=\frac{p-2q}{3q}$.

   ½ mark

3. Since $p,q$ are integers, $\frac{p-2q}{3q}$ is rational, so $\sqrt{5}$ would be rational. This contradicts the given fact that $\sqrt{5}$ is irrational. So the assumption is wrong, and $2+3\sqrt{5}$ is irrational.

   1 mark

Answer: $2+3\sqrt{5}$ 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