QJEEM23Mac5e6: JEE Main 2023 Mathematics - Class 12 JEEM23Mac5e6 Solution: Let S be the set of all values of for which the system of…

Question

Verified Solution • 2025–26 Session moderate

Let 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. Then $\frac{120}{π} \sum_{θ \in s} θ$ 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 &amp; 1 &amp; \sqrt{3} \\-1 &amp; \tan \theta &amp; \sqrt{7} \\1 &amp; 1 &amp; \tan \theta\end{array}\right|=0

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

Step-by-Step Verified Solution