A) \[\theta =n\pi ,\,\lambda \in R-\{0\}\]
B) \[\theta =2n\pi ,\lambda \]is any rational number
C) \[\theta =(2n+1)\pi ,\lambda \in {{R}^{+}},n\in I\]
D) \[\theta =(2n+1)\frac{\pi }{2},\lambda \in R,n\in I\]
Correct Answer: D
Solution :
For non trivial solution \[\left| \begin{matrix} sin\theta & -\cos \theta & \lambda +1 \\ \cos \theta & \sin \theta & -\lambda \\ \lambda & \lambda +1 & \cos \theta \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[{{\sin }^{2}}\theta \cos \theta +{{\lambda }^{2}}\cos \theta +{{(\lambda +1)}^{2}}\cos \theta -\] \[\sin \theta \lambda (\lambda +1)+co{{s}^{3}}\theta +\sin \theta \lambda (\lambda +1)=0\] \[\Rightarrow \]\[2\cos \theta ({{\lambda }^{2}}+\lambda +1)=0\] \[\Rightarrow \]\[\cos \theta =0\] \[\Rightarrow \]\[\theta =(2n+1)\frac{\pi }{2},\lambda \in R,\,n\in I\]
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