Question

The values of $\theta$ and $\lambda$ for which the following equations $\left(\sin \theta \right)x-\left(\cos \theta \right)y+(\lambda +1)z=0;\left(\cos \theta \right)x-\left(\sin \theta \right)y-\lambda z=0;\lambda x+(\lambda +1)y-\cos \theta z=0$ have non-trivial solution, is

• A. $\theta =(2n+1)\pi ,\lambda \in R^{+},n\in I$
• B. $\theta =2n\pi ,\lambda$ is any rational number, $n\in I$
• C. $\theta =(2n+1)\frac{\pi }{2},\lambda \in R,n\in I$
• D. $\theta =n\pi ,\lambda \ne 0,n\in I$

Answer 1

The correct option is C $\theta =(2n+1)\frac{\pi }{2},\lambda \in R,n\in I$

For non-trivial solutions
$\left|\begin{array}{ccc}\sin \theta & -\cos \theta & \lambda +1\\ \cos \theta & -\sin \theta & -\lambda \\ \lambda & \lambda +1& -\cos \theta \end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}\sin \theta +\cos \theta & -(\sin \theta +\cos \theta )& 1\\ \cos \theta & -\sin \theta & -\lambda \\ \lambda & \lambda +1& -\cos \theta \end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}0& -(\sin \theta +\cos \theta )& 1\\ \cos \theta -\sin \theta & -\sin \theta & -\lambda \\ 2\lambda & \lambda +1& -\cos \theta \end{array}\right|=0$
$\Rightarrow \cos \theta =0\Rightarrow \theta =(2n+1)\frac{\pi }{2},n\in I$ and $\lambda \in R$