JEE Main 2026 · 2 Apr · Shift 1, Mathematics question 23

Condition. With $a>0$, the quadratic is positive for all $x$ iff $8b^2-4ac<0$, i.e. $ac>2b^2$.

Count ($a,c\in\{1,2,3,4\}$, 16 pairs for each $b$):

• $b=1$: $ac\ge3$, i.e. all except $(1,1),(1,2),(2,1)$: 13.
• $b=2$: $ac>8$: $(3,3),(3,4),(4,3),(4,4)$: 4.
• $b=3,4$: $ac\le16<18$: none.

Probability $\frac{17}{64}$, so $m+n=81$.

Answer: 81