CBSE Class 10 Mathematics Basic 2023: question 27

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Step-by-step solution

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1. Assume, to the contrary, that $7-2\sqrt{3}$ is rational. Then $7-2\sqrt{3}=\frac{a}{b}$, where $a,b$ are integers and $b\ne 0$.

   1 mark

2. Rearranging: $2\sqrt{3}=7-\frac{a}{b}\Rightarrow \sqrt{3}=\frac{7b-a}{2b}$

   1 mark

3. Since $a,b$ are integers, $\frac{7b-a}{2b}$ is rational, so $\sqrt{3}$ would be rational. This contradicts the fact that $\sqrt{3}$ is irrational. So our assumption is wrong and $7-2\sqrt{3}$ is irrational.

   1 mark

Answer: $7-2\sqrt{3}$ 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