CBSE Class 10 Mathematics Standard 2023: question 21

Hints

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Step-by-step solution

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

   ½ mark

2. Rearranging gives $\sqrt{3}=\frac{p}{q}-2=\frac{p-2q}{q}$.

   ½ mark

3. Since $p$ and $q$ are integers, $p-2q$ is an integer and $q\ne 0$. So $\frac{p-2q}{q}$ is a rational number.

   ½ mark

4. This makes $\sqrt{3}$ rational, which contradicts the given fact that $\sqrt{3}$ is irrational. Hence our assumption is wrong, and $2+\sqrt{3}$ is irrational.

   ½ mark

Answer: $2+\sqrt{3}$ 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