• # question_answer The set of equations: $\lambda x-y+\left( \cos \theta \right)z=0;3x+y+2z=0;$ $\left( cos\theta \right)x+y+2z=0; 0\le \theta <2\pi ,$ has non -trivial solutions A) For no value of $\lambda \operatorname{and}\theta$ B) For all values of $\lambda$and$\theta$ C) For all values of$\lambda$, and only two values of$\theta$ D) For only one value of $\lambda$ and all values of $\theta$

determinant of coefficients $=\left| \begin{matrix} \lambda & -1 & \cos \theta \\ 3 & 1 & 2 \\ \cos \theta & 1 & 2 \\ \end{matrix} \right|=\cos \theta -{{\cos }^{2}}\theta +6$ and this is positive for all $\theta$ since $\left| \cos \theta \right|\le 1.$the only solution is therefore the trivial solution.