A) \[f(\theta )=0\]
B) \[f(\theta )=1\]
C) \[f(\theta )=-1\]
D) None of these
Correct Answer: B
Solution :
Given, \[f(\theta )=\left| \begin{matrix} {{\cos }^{2}}\theta & \cos \theta \sin \theta & -\sin \theta \\ \cos \theta \sin \theta & {{\sin }^{2}}\theta & \cos \theta \\ \sin \theta & -\cos \theta & 0 \\ \end{matrix} \right|\] \[={{\cos }^{2}}\theta (0+{{\cos }^{2}}\theta )-\cos \theta \] \[.\sin \theta (0-\sin \theta .\cos \theta )\] \[-\sin \theta (-{{\cos }^{2}}\theta .\sin \theta -{{\sin }^{3}}\theta )\] \[={{\cos }^{4}}\theta +2{{\sin }^{2}}\theta .{{\cos }^{2}}\theta +{{\sin }^{4}}\theta \] \[={{({{\cos }^{2}}\theta +{{\sin }^{2}}\theta )}^{2}}\] \[=1\] \[\therefore \]For all\[\theta ,f(\theta )=1\]
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