• # question_answer The set of equations$\lambda x-y+(\cos \theta )z=0$$3x+y+2z=0$$(\cos \theta )x+y+2z=0$ where$0\le \theta <2\pi$, has non-trivial solutions. A)  for no value of $\lambda$ 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 $\theta$

Let A be the coefficient matrix of the given set of equations, then$A=\left[ \begin{matrix} \lambda & -1 & \cos \theta \\ 3 & 1 & 2 \\ \cos \theta & 1 & 2 \\ \end{matrix} \right]$ Then, $|A|=\left[ \begin{matrix} \lambda & -1 & \cos \theta \\ 3 & 1 & 2 \\ \cos \theta & 1 & 2 \\ \end{matrix} \right]$ $=\cos \theta -{{\cos }^{2}}\theta +6$ |A| is positive for all $\theta$ since, | cos $\theta$| $\le$ 1. The only solution is therefore the trivial solution.