CBSE Class 10 Mathematics Standard sample paper 2023-24: question 21

Hints

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

   ½ mark

2. Then $b\sqrt{2}=a$. Squaring gives $2b^2=a^2$. So 2 divides $a^2$, and hence 2 divides $a$ (2 is prime).

   ½ mark

3. Write $a=2c$. Then $2b^2=4c^2$, so $b^2=2c^2$. So 2 divides $b^2$, and hence 2 divides $b$.

   ½ mark

4. Now 2 is a common factor of $a$ and $b$. This contradicts the fact that $a,b$ are coprime. The contradiction comes from our assumption, so $\sqrt{2}$ is irrational.

   ½ mark

Answer: $\sqrt{2}$ 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