CBSE Class 10 Mathematics Basic 2023: question 27

Hints

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Step-by-step solution

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1. Assume, to the contrary, that $4+2\sqrt{3}$ is rational. Then $4+2\sqrt{3}=\frac{p}{q}$, where p, q are integers and $q\ne 0$.

   1 mark

2. Rearranging: $2\sqrt{3}=\frac{p}{q}-4=\frac{p-4q}{q}\Rightarrow \sqrt{3}=\frac{p-4q}{2q}$

   1 mark

3. Since p, q are integers, $\frac{p-4q}{2q}$ is rational, so $\sqrt{3}$ would be rational. This contradicts the given fact that $\sqrt{3}$ is irrational. So our assumption is wrong, and $4+2\sqrt{3}$ is irrational.

   1 mark

Answer: $4+2\sqrt{3}$ 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