Let S be the set of all values of [- , ] for which the system of linear equations x+y+√3z=…

Question

Let S be the set of all values of $	heta \in [-\pi ,\pi ]$ for which the system of linear equations

$x+y+\sqrt{3}z=0$

$-x+(	an 	heta )y+\sqrt{7}z=0$

$x+y+(	an 	heta )z=0$

has non-trivial solution. Then $\frac{120}{\pi }\sum _{	heta \in s}	heta $ is equal to :

Accepted Answer

Since, the given system has a non trivial solution,

$$
\text { So, } \Delta=0
$$

$$
\Rightarrow \Delta=\left|\[\begin{array}{ccc} 1 & 1 & \sqrt{3} \ -1 & \tan \theta & \sqrt{7} \ 1 & 1 & \tan \theta \end{array}$$\right|=0
\]

$$
\begin{aligned}
& \Rightarrow 1\left(\tan ^2 \theta-\sqrt{7}\right)-1(-\tan \theta-\sqrt{7})+\sqrt{3}(-1-\tan \theta)=0 \\
& \Rightarrow \tan ^2 \theta-\sqrt{7}+\tan \theta+\sqrt{7}-\sqrt{3}-\sqrt{3} \tan \theta=0 \\
& \Rightarrow \tan \theta(\tan \theta-\sqrt{3})+1(\tan \theta-\sqrt{3})=0 \\
& \Rightarrow \tan \theta=\sqrt{3} \text { or } \tan \theta=-1 \\
& \therefore \theta=\left\{\frac{\pi}{3}, \frac{-2 \pi}{3}, \frac{-\pi}{4}, \frac{3 \pi}{4}\right\} \\
& \text { So, } \frac{120}{\pi} \sum \theta=\frac{120}{\pi}\left\{\frac{4 \pi-8 \pi-3 \pi+9 \pi}{12}\right\} \\
& =\frac{120}{\pi}\left[\frac{2 \pi}{12}\right]=20
\end{aligned}
$$

Solutions · 1