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

Hints

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Step-by-step solution

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1. Suppose, on the contrary, that $2+3\sqrt{3}$ is rational.

   Then $2+3\sqrt{3}=\frac{a}{b}$, where $a,b$ are integers and $b\ne 0$.

   So $3\sqrt{3}=\frac{a}{b}-2$, which gives $\sqrt{3}=\frac{a-2b}{3b}$.

   1 mark

2. Since $a$ and $b$ are integers, $a-2b$ and $3b$ are integers, with $3b\ne 0$.

   So $\frac{a-2b}{3b}$ is rational, and therefore $\sqrt{3}$ would be rational.

   1 mark

3. This contradicts the given fact that $\sqrt{3}$ is irrational. So our assumption is false.

   Hence $2+3\sqrt{3}$ is irrational.

   1 mark

Answer: Proved: $2+3\sqrt{3}$ is irrational.

Revise the chapter

• Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
• Topic: Proving Numbers Irrational