CBSE Class 10 Maths (Standard) 2024 · Paper 30(B) · Question 26

Assuming $\sqrt{3}=\frac{a}{b}$ with $a$, $b$ coprime integers and $b\ne 0$ gives $a^2=3b^2$, from which $3$ divides $a$ and then $3$ divides $b$. This contradicts that $a$ and $b$ are coprime, so the assumption is wrong and $\sqrt{3}$ is irrational. Hence proved.