JEE Main 2024 — Probability Question with Solution

Given quadratic equation is

$$a{x}^2+bx+c=0\text{ where }a,b,c\in \{1,2,3,\dots ,8\}$$

For repeated roots,

$$\begin{array}{rl}& b^2-4ac=0\\ & \Rightarrow b^2=4ac\end{array}$$

$$\Rightarrow ac$$ must be a perfect square

$$(a,c)\in \{(1,1),(1,4),(2,2),(2,8),(3,3),(4,1),(4,4),(5,5),(6,6),(7,7),(8,2),(8,8)\}$$

Corresponding $$b$$ must lie in set $$\{1,2,3,\dots 8\}$$

$$\begin{array}{rl}& (a,b,c)\in \{(1,2,1),(1,2,4),(2,4,2),(2,8,8)\\ & (3,6,3),(4,4,1),(4,8,4),(8,8,2)\}\\ & \therefore \text{ probability }=\frac{8}{8^3}\\ & =\frac{1}{64}\\ & \end{array}$$