CBSE Class 10 Mathematics Standard 2026: question 21

Hints

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Step-by-step solution

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1. Assume, to the contrary, that $2-5\sqrt{3}$ is rational. Then $2-5\sqrt{3}=\frac{a}{b}$, where $a,b$ are integers and $b\ne 0$.

   ½ mark

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

   ½ mark

3. Since $a,b$ are integers and $b\ne 0$, $\frac{2b-a}{5b}$ is rational. This would make $\sqrt{3}$ rational, which contradicts the given fact that $\sqrt{3}$ is irrational. So our assumption is wrong, and $2-5\sqrt{3}$ is irrational.

   1 mark

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