Let S be the set of all values of $\mathrm{\theta }\in [-\mathrm{\pi },\mathrm{\pi }]$ for which the system of linear equations $\mathrm{x}+\mathrm{y}+\sqrt{3}\mathrm{z}=0;-\mathrm{x}+(\tan \mathrm{\theta })\mathrm{y}+\sqrt{7}\mathrm{z}=0;\mathrm{x}+\mathrm{y}+(\tan \mathrm{\theta })\mathrm{z}=0$has non –trivial solution. Then $\frac{120}{\mathrm{\pi }}\sum _{\mathrm{\theta }\in \mathrm{S}} \mathrm{\theta }$ is equal to

For non trivial solution |A| = 0
$\begin{array}{ll}\Rightarrow \left|\begin{array}{ccc}1& 1& \sqrt{3}\\ -1& \tan \mathrm{\theta }& \sqrt{7}\\ 1& 1& \tan \mathrm{\theta }\end{array}\right|=0& \end{array}$
$\begin{array}{l}\left({\tan }^2\theta -\sqrt{7}\right)-(-\tan \theta -\sqrt{7})+\sqrt{3}(-1-\tan \theta )=0\\ {\tan }^2\theta +(1-\sqrt{3})\tan \theta -\sqrt{3}=0\end{array}$
$\begin{array}{l}\begin{array}{l}\Rightarrow (\tan \mathrm{\theta }+1)(\tan \mathrm{\theta }-\sqrt{3})=0\\ \Rightarrow \tan \mathrm{\theta }=-1\text{ (or) }\sqrt{3},\mathrm{\theta }\in [-\mathrm{\pi },\mathrm{\pi }]\\ \mathrm{\theta }=\left\{\frac{-\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{3},\frac{-2\mathrm{\pi }}{3}\right\}\\ \frac{120}{\mathrm{\pi }}\sum _{\mathrm{\theta }\in \mathrm{S}} \mathrm{\theta }=\frac{120}{\mathrm{\pi }}\left(\frac{\mathrm{\pi }}{6}\right)=20\end{array}\end{array}$