# Roots of the equation $12x^2+mx+5=0$ are in the ratio $3 : 2$,then $m$ equals

$\begin{array}{1 1}(A)\;\large\frac{5}{12}&(B)\;\large\frac{1}{12}\\(C)\;\large\frac{5\sqrt{10}}{12}&(D)\;5\sqrt{10}\end{array}$

Let the roots be $\alpha,\beta$
$\large\frac{\alpha}{\beta}=\frac{3}{2}$
$\alpha+\beta=-\large\frac{m}{12}$
$\alpha\beta=\large\frac{5}{12}$
$\alpha=\large\frac{3}{2}$$\beta\Rightarrow\large\frac{3}{2}$$\beta^2=\large\frac{5}{12}$
$\Rightarrow \beta^2=\large\frac{10}{36}\Rightarrow \beta=-\large\frac{\sqrt{10}}{6}$
$\alpha=\large\frac{3}{2}$$\beta=-\large\frac{\sqrt{10}}{4}$
$m=-12(\alpha+\beta)$
$\;\;\;=12(\large\frac{\sqrt{10}}{6}+\frac{\sqrt{10}}{4})$
$\;\;\;=5\sqrt{10}$
Hence (D) is the correct answer.