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

Option: 1

20


Option: 2

40


Option: 3

30


Option: 4

10


Answers (1)

For non trivial solutions $$ \begin{aligned} & \mathrm{D}=0 \\ & \left|\begin{array}{ccc} 1 & 1 & \sqrt{3} \\ -1 & \tan \theta & \sqrt{7} \\ 1 & 1 & \tan \theta \end{array}\right|=0 \\ & \tan ^2 \theta-(\sqrt{3}-1)-\sqrt{3}=0 \\ & \tan \theta=\sqrt{3},-1 \\ & \theta=\left\{\frac{\pi}{3}, \frac{-2 \pi}{3}, \frac{-\pi}{4}, \frac{3 \pi}{4}\right\} \\ & \left.\frac{120}{\pi}(\Sigma \theta)=\frac{120}{\pi} \times \frac{\pi}{6}=20 \text { (Option } 1\right) \end{aligned}

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Sumit Saini

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