The value of $\theta$ lying between $- \,\frac{\pi }{4}$ and $\frac{\pi }{2}$ and $0 \le A \le \frac{\pi }{2}$ and satisfying the equation $\left| {\begin{array}{*{20}{c}}{1\, + \,{{\sin }^2}A}&{{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{1\, + \,{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{{{\cos }^2}A}&{1\, + \,2\,\sin \,4\theta }\end{array}} \right|$ $= 0$ are :

For what values of $x:\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=\mathrm{O} ?$

Let $A = \left( {\begin{array}{*{20}{c}}
{\cos \,\alpha }&{ – \sin \,\alpha }\\
{\sin \,\alpha }&{\cos \,\alpha }
\end{array}} \right)$, $\left( {\alpha  \in R} \right)$ such that ${A^{32}} = \left( {\begin{array}{*{20}{c}}
0&{ – 1}\\
1&0
\end{array}} \right)$. Then a value of $\alpha $ is

Let $A = \left[ {\begin{array}{*{20}{c}}1&2\\3&4\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}a&0\\0&b\end{array}} \right]\;,a,b \in N$ then

Let $A =\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]$ and $B = A – I$. If $\omega=\frac{\sqrt{3} i -1}{2}$ then the number of elements in the set $\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }+(\omega B )^{ n }= A + B \right\}$ is equal to $……….$