Question
माना $A =\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right),(\alpha \in R )$ इस प्रकार है कि $A ^{32}=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)$ तो $\alpha$ का एक मान है
${A^2} = \left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{ - \sin 2\alpha }\\
{\sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$
${A^3} = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{ - \sin 2\alpha }\\
{\sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}}
{\cos 3\alpha }&{ - \sin 3\alpha }\\
{\sin 3\alpha }&{\cos 3\alpha }
\end{array}} \right]$
Simiarly ${A^{32}} = \left[ {\begin{array}{*{20}{c}}
{\cos 32\alpha }&{ - \sin 32\alpha }\\
{\sin 32\alpha }&{\cos 32\alpha }
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
0&{ - 1}\\
1&0
\end{array}} \right]$
$ \Rightarrow \cos 32\alpha = 0$ and $\sin 32\alpha = 0$
$ \Rightarrow 32\alpha = \left( {4n + 1} \right)\frac{\pi }{2},n \in 1$
$\alpha = \left( {4n + 1} \right)\frac{\pi }{{64}},n \in 1$
$\alpha = \frac{\pi }{{64}}$ for $n = 0$
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यदि $\lim _{\mathrm{x} \rightarrow \infty} \mathrm{I}(\mathrm{x})=0$ है, तो $\mathrm{I}(1)$ बराबर है।