The value of $ heta$ for which the system $(\sin 3 heta) x-y+z=0,(\cos 2 heta) x+4 y+3 z=0$, $2

Question

The value of $	heta$ for which the system $(\sin 3 	heta) x-y+z=0,(\cos 2 	heta) x+4 y+3 z=0$, $2 x+7 y+7 z=0$ has non-trivial solution is $\frac{5 \pi}{\lambda}$, where $\lambda$ is a positive integer greater than 1 then $\lambda$ must be equal to

Professor PC Thomas Classes · Answer

For non-trivial solution $ \ \begin{aligned} & \mathrm{D}=\left|\begin{array}{ccc} \sin 3 	heta & -1 & 1 \ \cos 2 	heta & 4 & 3 \ 2 & 7 & 7 \end{array}ight|=0 \ & \Rightarrow \sin 3 	heta+2 \cos 2 	heta=2 \ & \\Rightarrow 4 \sin ^3 	heta+4 \sin ^2 	heta=3 \sin 	heta \end{aligned} $ $\begin{aligned} & \Rightarrow \sin 	heta=0 	ext { or } \sin 	heta=\frac{1}{2}\left(\because \sin 	heta 
eq \frac{-3}{2}ight) \ & \Rightarrow \sin 	heta=\sin \frac{\pi}{6} \ & \Rightarrow 	heta=\frac{5 \pi}{6} \ & \Rightarrow \lambda=6\end{aligned}$