Question:
The coefficients $a, b$ and $c$ of the quadratic equation, $a x^{2}+b x+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :
Correct Option: , 4
Solution:
$a x^{2}+b x+c=0$
$\mathrm{a}, \mathrm{b}, \mathrm{c} \in\{1,2,3,4,5,6\}$
$\mathrm{n}(\mathrm{s})=6 \times 6 \times 6=216$
$\mathrm{D}=0 \Rightarrow \mathrm{b}^{2}=4 \mathrm{ac}$
$\mathrm{ac}=\frac{\mathrm{b}^{2}}{4} \quad$ If $\mathrm{b}=2, \mathrm{ac}=1 \quad \Rightarrow \quad \mathrm{a}=1, \mathrm{c}=1$
If $b=4, a c=4 \quad \Rightarrow \quad a=1, c=4$
$\mathrm{a}=4, \mathrm{c}=1$
Ifb $=6, \mathrm{ac}=9 \Rightarrow \quad 2, \mathrm{c}=2$
$\therefore$ probability $=\frac{5}{216}$