Let S be the set of all values of θ ∈ [-π ,π ] for which the system of linear equations x+y+√3z=0 -x+( tanθ )y+√7z=0 x+y+( tanθ )z=0 has non-trivial…

Given,

$S$S be the set of all values of $\theta \in [-\pi ,\pi ]$θ∈[−π,π] for which the system of linear equations

$$x+y+\sqrt{3}z=0$$x+y+3​z=0

$$-x+(\mathrm{tan}\theta )y+\sqrt{7}z=0$$−x+(tanθ)y+7​z=0

$$x+y+(\mathrm{tan}\theta )z=0$$x+y+(tanθ)z=0

has non-trivial solution,

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

$$|\begin{matrix}1 & 1 & \sqrt{3} \\ -1 & \mathrm{tan}\theta & \sqrt{7} \\ 1 & 1 & \mathrm{tan}\theta \end{matrix}|=0$$∣1−11​1tanθ1​3​7​tanθ​∣=0

$$\Rightarrow ({\mathrm{tan}}^{2}\theta -\sqrt{7})-(-\mathrm{tan}\theta -\sqrt{7})+\sqrt{3}(-1-\mathrm{tan}\theta )=0$$⇒(tan2θ−7​)−(−tanθ−7​)+3​(−1−tanθ)=0

$$\Rightarrow {\mathrm{tan}}^{2}\theta +(1-\sqrt{3})\mathrm{tan}\theta -\sqrt{3}=0$$⇒tan2θ+(1−3​)tanθ−3​=0

$$\Rightarrow (\mathrm{tan}\theta +1)(\mathrm{tan}\theta -\sqrt{3})=0$$⇒(tanθ+1)(tanθ−3​)=0

$$\Rightarrow \mathrm{tan}\theta =\sqrt{3},-1$$⇒tanθ=3​,−1

$$\Rightarrow \theta =\frac{\pi }{3},\frac{-2\pi }{3},\frac{-\pi }{4},\frac{3\pi }{4}$$⇒θ=3π​,3−2π​,4−π​,43π​

Hence,

$$\frac{120}{\pi }\sum \theta =120(\frac{1}{3}-\frac{2}{3}-\frac{1}{4}+\frac{3}{4})=20$$π120​∑θ=120(31​−32​−41​+43​)=20