$\cos \mathrm{\alpha }$ is a root of the equation $25{\mathrm{x}}^2+5\mathrm{x}-12=0, -1<\mathrm{x}<0,$ the value of $\sin 2\mathrm{\alpha }$ is 

$\because \mathrm{x}=\frac{-5\pm \sqrt{25+1200}}{50}=\frac{-5\pm 35}{50}=\frac{30}{50},\frac{-40}{50}$
$$\begin{gathered} \mathrm{or} \cos \mathrm{\alpha }=\frac{3}{5},\frac{-4}{5} \\ \mathrm{but} -1<\mathrm{x}<0 \end{gathered}$$
$\text{∴}\cos \mathrm{\alpha }=-\frac{4}{5} [\mathrm{lies} \mathrm{in} \mathrm{II} \mathrm{and} \mathrm{III} \mathrm{quadrants}]$
$$\begin{gathered} \therefore \sin \mathrm{\alpha }=\frac{3}{5} [\mathrm{lies} \mathrm{in} \mathrm{II} \mathrm{quadrant}] \\ \therefore \sin \mathrm{\alpha }=-\frac{3}{5} [\mathrm{lies} \mathrm{in} \mathrm{III} \mathrm{quadrant}] \\ \therefore \sin 2\mathrm{\alpha }=2\cdot \sin \mathrm{\alpha }\cdot \cos \mathrm{\alpha }=-\frac{24}{25} [\mathrm{lies} \mathrm{in} \mathrm{II} \mathrm{quadrant}] \\ \therefore \sin 2\mathrm{\alpha }=2\cdot \sin \mathrm{\alpha }\cdot \cos \mathrm{\alpha }=\frac{24}{25} [\mathrm{lies} \mathrm{in} \mathrm{III} \mathrm{quadrant}] \end{gathered}$$