JEE Main 2024 — 9 April, Shift 1 — Question 21

a, b, c $\in\{1,2,3,4\}$

Tetrahedral dice

$a x^{2}+b x+c=0$ has all real roots $\Rightarrow \mathrm{D} \geq 0$

$\Rightarrow \mathrm{b}^{2}-4 \mathrm{ac} \geq 0$

Let $\mathrm{b}=1 \Rightarrow 1-4 \mathrm{ac} \geq 0$ (Not feasible)

$\mathrm{b}=2 \Rightarrow 4-4 \mathrm{ac} \geq 0$

$1 \geq \mathrm{ac} \Rightarrow \mathrm{a}=1, \mathrm{c}=1$,

$\mathrm{b}=3 \Rightarrow 9-4 \mathrm{ac} \geq 0$

$\frac{9}{4} \geq$ ac $\Rightarrow \mathrm{a}=1, \mathrm{c}=1$

$\Rightarrow \mathrm{a}=1, \mathrm{c}=2$

$\Rightarrow \mathrm{a}=2, \mathrm{c}=1$

$\mathrm{b}=4 \Rightarrow 16-4 \mathrm{ac} \geq 0$

$4 \geq$ ac $\Rightarrow \mathrm{a}=1, \mathrm{c}=1$

$\Rightarrow \mathrm{a}=1, \mathrm{c}=2 \quad \Rightarrow \mathrm{a}=2, \mathrm{c}=1$

$\Rightarrow \mathrm{a}=1, \mathrm{c}=3 \quad \Rightarrow \mathrm{a}=3, \mathrm{c}=1$

$\Rightarrow \mathrm{a}=1, \mathrm{c}=4 \quad \Rightarrow \mathrm{a}=4, \mathrm{c}=1$

$\Rightarrow \mathrm{a}=2, \mathrm{c}=2$

Probability $=\frac{12}{(4)(4)(4)}=\frac{3}{16}=\frac{m}{m}$

$\mathrm{m}+\mathrm{n}=19$