JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Probability

Throwing the die three times gives $6^3=216$ equally likely triples $(a,b,c)$, and equal roots need $b^2=4ac$. Since $b^2$ is then a multiple of $4$, $b$ is even. For $b=2$, $ac=1$ gives $(a,c)=(1,1)$. For $b=4$, $ac=4$ gives $(1,4)$, $(2,2)$ and $(4,1)$. For $b=6$, $ac=9$ gives $(3,3)$ alone, as $9$ cannot appear on a die. That is $1+3+1=5$ favourable triples, so the probability of equal roots is $\frac{5}{216}$.