A) \[\alpha \]
B) \[\beta \]
C) \[\alpha \]and\[\beta \]
D) Neither \[\alpha \]nor \[\beta \]
Correct Answer: A
Solution :
We have the determinant \[\left| \begin{matrix} \cos (\alpha +\beta ) & -\sin (\alpha +\beta ) & \cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \\ \end{matrix} \right|\] \[{{R}_{1}}\to {{R}_{1}}+(sin\beta ){{R}_{2}}+(cos\beta ){{R}_{3}},\] \[\left| \begin{matrix} 0 & 0 & 1+\cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \\ \end{matrix} \right|\] \[=(1+\cos 2\beta )\,({{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha )\] \[=1+\cos 2\beta \] which is independent of \[\alpha \].
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