CBSE Class 10 Mathematics Standard 2025: question 27

Hints

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Step-by-step solution

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

   1 mark

2. Rearrange: $5\sqrt{3}=\frac{a}{b}-\frac{2}{3}=\frac{3a-2b}{3b}$, so $\sqrt{3}=\frac{3a-2b}{15b}$

   1 mark

3. $3a-2b$ and $15b$ are integers and $15b\ne 0$. So $\frac{3a-2b}{15b}$ is rational, which makes $\sqrt{3}$ rational.

   ½ mark

4. This contradicts the given fact that $\sqrt{3}$ is irrational. So our assumption is wrong, and $5\sqrt{3}+\frac{2}{3}$ is irrational.

   ½ mark

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