CBSE Class 10 Mathematics Basic sample paper 2025-26: question 26

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Step-by-step solution

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1. Suppose, to the contrary, that $\sqrt{2}-\sqrt{5}=x$, where $x$ is rational ($x\ne 0$). Then $\sqrt{2}=x+\sqrt{5}$; squaring, $2=x^2+5+2\sqrt{5}x$.

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2. So $2\sqrt{5}x=-x^2-3\Rightarrow \sqrt{5}=\frac{-x^2-3}{2x}$ ...(1)

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3. Since $x$ is rational, $\frac{-x^2-3}{2x}$ is rational, but $\sqrt{5}$ is irrational (square root of a prime). A rational number cannot equal an irrational number, so our assumption is wrong. Hence $\sqrt{2}-\sqrt{5}$ is irrational.

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Answer: $\sqrt{2}-\sqrt{5}$ 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