CBSE Class 10 Mathematics Standard 2024: question 30

Hints

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Step-by-step solution

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

   ½ mark

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

   1 mark

3. Since $p,q$ are integers, $\frac{2q-5p}{q}$ is rational, so the RHS is rational while the LHS $\sqrt{3}$ is irrational (given).

   ½ mark

4. This is a contradiction, so our assumption is wrong.

   ½ mark

5. Hence $\frac{2-\sqrt{3}}{5}$ is an irrational number.

   ½ mark

Answer: (2 − √3)/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