Q75 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics
If each of the coefficients $a, b, c$ in the equation $a x^2+b x$ $+c=0$ is determined by throwing a die, then the probability that the equation will have equal roots, is
- A$\frac{1}{36}$
- B$\frac{1}{72}$
- C$\frac{7}{216}$
- Correct answerD$\frac{5}{216}$
Solution
$a x^2+b x+c=0$
For equal roots, $b^2=4 a c$ $\begin{array}{|c|c|c|} \hline \mathbf{b} & (\mathbf{a}, \mathbf{c}) & \text { Total } \\ \hline 1 & - & 0 \\ \hline 2 & (1,1) & 1 \\ \hline 3 & - & 0 \\ \hline 4 & (1,4)(4,1)(2,2) & 3 \\ \hline 5 & - & 0 \\ \hline 6 & (3,3) & 1 \\ \hline \end{array}$ Required probability $=\frac{5}{216}$
For equal roots, $b^2=4 a c$ $\begin{array}{|c|c|c|} \hline \mathbf{b} & (\mathbf{a}, \mathbf{c}) & \text { Total } \\ \hline 1 & - & 0 \\ \hline 2 & (1,1) & 1 \\ \hline 3 & - & 0 \\ \hline 4 & (1,4)(4,1)(2,2) & 3 \\ \hline 5 & - & 0 \\ \hline 6 & (3,3) & 1 \\ \hline \end{array}$ Required probability $=\frac{5}{216}$
Hence, the correct answer is option: 'D'.