Let S be the set of all values of θ ∈ [ - π, π ] for which the...

Given,

$S$ be the set of all values of $\theta \in [ - \pi, \pi ]$ for which the system of linear equations
$x + y + \sqrt{3}z = 0$
$- x + ( tan\theta ) y + \sqrt{7}z = 0$
$x + y + ( tan\theta ) z = 0$
has non-trivial solution,

So, by condition of non-trivial solution we get,

\left|\begin{cases}1 & 1 & \sqrt{3} \\ - 1 & tan\theta & \sqrt{7} \\ 1 & 1 & tan\theta\end{cases}\right| = 0

$⇒ \left(\tan^{2}\theta - \sqrt{7}\right) - \left(- tan\theta - \sqrt{7}\right) + \sqrt{3}\left(- 1 - tan\theta\right) = 0$

$⇒ \tan^{2}\theta + \left(1 - \sqrt{3}\right)tan\theta - \sqrt{3} = 0$

$⇒ \left(tan\theta + 1\right)\left(tan\theta - \sqrt{3}\right) = 0$

$⇒ tan\theta = \sqrt{3}, - 1$

$⇒ \theta = \dfrac{\pi}{3}, \dfrac{- 2\pi}{3}, \dfrac{- \pi}{4}, \dfrac{3\pi}{4}$

Hence,

$\dfrac{120}{\pi} ∑ \theta = 120\left(\dfrac{1}{3} - \dfrac{2}{3} - \dfrac{1}{4} + \dfrac{3}{4}\right) = 20$