The set of equations $\mathrm{\lambda x}-\mathrm{y}+(\cos \mathrm{\theta })\mathrm{z}=0,3\mathrm{x}+\mathrm{y}+2\mathrm{z}=0,(\cos \mathrm{\theta })\mathrm{x}+\mathrm{y}+2\mathrm{z}=0,0\le \mathrm{\theta }<2\mathrm{\pi }$ has non -
trivial solution (s)

$$\begin{gathered} \text{ where }\mathrm{A}=\left[\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right],\mathrm{X}=\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right],\mathrm{O}=\left[\begin{array}{l}0\\ 0\\ 0\end{array}\right] \\ For non-trivial solution \\ \Rightarrow D=0 \\ \begin{array}{r}\Rightarrow \left|\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right|=0\end{array} \end{gathered}$$

$\mathrm{D}=\cos \mathrm{\theta }-{\cos }^2\mathrm{\theta }+6>0,\text{ Since }\mathrm{D}>0$ only trivial solution is possible

$\text{ Hence, there is no value of }\theta \text{ and }\lambda \text{ for non-trivial solution}$