$ \Rightarrow\left(\mathrm{BCB}^{-1}\right)\left(\mathrm{BCB}^{-1}\right)=\mathrm{A} \cdot \mathrm{A} $
$ \Rightarrow \mathrm{BCI} \mathrm{CB^{-1 } = \mathrm { A } ^ { 2 }} $
$ \Rightarrow \mathrm{BC}^2 \mathrm{~B}^{-1}=\mathrm{A}^2 $
$ \Rightarrow \mathrm{B}^{-1}\left(\mathrm{BC}^2 \mathrm{~B}^{-1}\right) \mathrm{B}=\mathrm{B}^{-1}(\mathrm{~A} \cdot \mathrm{A}) \mathrm{B}$
From equation $(1)$
$ \mathrm{C}^2=\mathrm{A}^{-1} \cdot \mathrm{A} \cdot \mathrm{B} $
$ \mathrm{C}^2=\mathrm{B} $
$ \text { Also } \mathrm{AB}^{-1}=\mathrm{A}^{-1} $
$ \Rightarrow \mathrm{AB}^{-1} \cdot \mathrm{A}=\mathrm{A}^{-1} \mathrm{~A}=\mathrm{I} $
$ \Rightarrow \mathrm{A}^{-1}\left(A \mathrm{~B}^{-1} \mathrm{~A}\right)=\mathrm{A}^{-1} \mathrm{I} $
$ \mathrm{B}^{-1} \mathrm{~A}=\mathrm{A}^{-1}$
Now characteristics equation of $\mathrm{C}^2$ is
$ \left|C_2-\lambda I\right|=0 $
$ |B-\lambda I|=0$
$\Rightarrow\left|\begin{array}{cc}1-\lambda & 3 \\ 1 & 5-\lambda\end{array}\right|=0$
$ \Rightarrow(1-\lambda)(5-1)-3=0 \Rightarrow\left(\lambda^2-6 \lambda+5\right)-3=0 $
$ \Rightarrow \lambda^2-6 \lambda+2=0 $
$ \Rightarrow \beta^2-6 B+2 I=0 $
$ \Rightarrow C^4-6 C^2+2 I=0 $
$ \alpha=-6 $
$ \beta=2 $
$ \therefore 2 \beta-\alpha=4+6=10$
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Let $E_2=E_1-S_1$ and $F_2=F_1 \cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements.
Let $G _2= G _1 \cup S _2$. Finally, two elements are chosen at random, without replacement, from the set $G _2$ and let $S _3$ denote the set of these chosen elements.
Let $E_3=E_2 \cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is
$f(x) = \left\{ {\begin{array}{*{20}{l}}
{\frac{{k\cos x}}{{\pi - 2x}},}&{{\rm{ if }}\,x\, \ne \,\frac{\pi }{2}}\\
{3,}&{{\rm{ if }}\,x\, = \,\frac{\pi }{2}}
\end{array}} \right.$ at $x = \frac{\pi }{2}$