CBSE Class 10 Mathematics Basic sample paper 2026-27: question 26

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

   ½ mark

2. $3\sqrt{5}=\frac{p}{q}-2\Rightarrow \sqrt{5}=\frac{p-2q}{3q}$

   1 mark

3. Since $p$ and $q$ are integers, $\frac{p-2q}{3q}$ is rational, so $\sqrt{5}$ would be rational.

   ½ mark

4. But $\sqrt{5}$ is irrational (given), which is a contradiction.

   ½ mark

5. Hence our assumption is wrong, and $2+3\sqrt{5}$ is irrational.

   ½ 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