If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ $\operatorname{matrix} A$ and $|A|=4$, then $\alpha$ is equal to

If $A=\left|\begin{array}{cc}x & 1 \\ 1 & x\end{array}\right|$ and $B=\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|$, then $\frac{d B}{d x}$ is

$$\text { The value of }\left|\begin{array}{ccc} \sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ \sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ \tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ} \end{array}\right|$$ is

If $$x\left[\begin{array}{l}3 \\ 2\end{array}\right]+y\left[\begin{array}{r}1 \\ -1\end{array}\right]=\left[\begin{array}{l}15 \\ 5\end{array}\right]$$, then the value of $$x$$ and $$y$$ are