CBSE Class 10 Mathematics Standard 2024: question 26(b)

Hints

Open one hint at a time and try again before the next.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. $(\sqrt{2}+\sqrt{3}{)}^2=2+3+2\sqrt{6}=5+2\sqrt{6}$

   1 mark

2. Assume, to the contrary, that $5+2\sqrt{6}$ is rational. Then $5+2\sqrt{6}=\frac{a}{b}$, where $a,b$ are integers, $b\ne 0$.

   ½ mark

3. $\sqrt{6}=\frac{a-5b}{2b}$

   ½ mark

4. Since $a,b$ are integers, $\frac{a-5b}{2b}$ is rational, so $\sqrt{6}$ would be rational. But $\sqrt{6}$ is irrational (given) — a contradiction.

   ½ mark

5. Hence our assumption is wrong, and $5+2\sqrt{6}=(\sqrt{2}+\sqrt{3}{)}^2$ is irrational.

   ½ mark

Answer: $(\sqrt{2}+\sqrt{3}{)}^2=5+2\sqrt{6}$ is irrational.

Revise the chapter

• Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
• Topic: Proving Numbers Irrational