A) \[\left| \begin{align} & \frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\sqrt{3}}{2} \\ & -\frac{\sqrt{3}}{2}\,\,\,\,\,\,\,\,\frac{1}{2} \\ \end{align} \right|\]
B) \[\left| \begin{align} & \frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,-\,\frac{\sqrt{3}}{2} \\ & \frac{\sqrt{3}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2} \\ \end{align} \right|\]
C) \[\left| \begin{align} & \frac{\sqrt{3}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2}\,\,\, \\ & -\frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\sqrt{3}}{2} \\ \end{align} \right|\]
D) \[\left| \begin{align} & \frac{\sqrt{3}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{2}\,\,\, \\ & \frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\sqrt{3}}{2} \\ \end{align} \right|\]
Correct Answer: C
Solution :
\[A\,\,=\,\,\left[ \begin{align} & \cos \,\theta \,\,\,\,-\,\sin \,\theta \\ & \sin \,\theta \,\,\,\,\,\,\,\,\cos \,\theta \\ \end{align} \right]\] \[{{A}^{50}}\,\,=\,\,\left[ \begin{align} & \cos \,50\,\theta \,\,\,\,\,\,\,-\,\sin \,50\,\theta \\ & -\sin \,50\,\theta \,\,\,\,\,\,\,\cos \,50\,\theta \\ \end{align} \right]\] \[{{({{A}^{-1}})}^{50}}\,\,=\,\,\left[ \begin{align} & \cos \,50\,\theta \,\,\,\,\,\,\,\,\,\,\sin \,50\,\theta \\ & -\sin \,50\,\theta \,\,\,\,\,\,\,\cos \,50\,\theta \\ \end{align} \right]\] At \[\theta \,\,=\,\,\frac{\pi }{12}\] \[{{({{A}^{-1}})}^{50}}\,\,=\,\,\left[ \begin{align} & \cos \,\frac{50\,\pi }{12}\,\,\,\,\,\,\,\,\,\,\sin \,\frac{50\,\pi }{12} \\ & -\sin \,\frac{50\,\pi }{12}\,\,\,\,\,\,\,\cos \,\frac{50\,\pi }{12}\, \\ \end{align} \right]\] \[=\,\,\left[ \begin{align} & \frac{\sqrt{3}}{2}\,\,\,\,\,\,\,\,\,\,\frac{1}{2} \\ & -\frac{1\,}{2}\,\,\,\,\,\,\,\,\frac{\sqrt{3}}{2}\, \\ \end{align} \right]\]
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