If $D = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{array} \right|$ for $x \neq 0, y \neq 0$,then $D$ is

$\begin{aligned} & \text { If }\left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ (n+2)^2 & (n+3)^2 & (n+4)^2\end{array}\right|=\Delta \text { and } \\ & \left|\begin{array}{ccc}1 & -4 & 7 \\ -2 & 3 & -5 \\ 3 & x & -3\end{array}\right|=2 \Delta+1, \text { then } x=\end{aligned}$

Consider three points $P(\cos \alpha, \sin \beta)$, $Q(\sin \alpha, \cos \beta)$, and $R(0,0)$, where $0 < \alpha, \beta < \frac{\pi}{4}$. Then:

If matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$,then the value of $a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33}$ is

If $x = cy + bz$,$y = az + cx$,$z = bx + ay$ (where $x, y, z$ are not all zero) have a solution other than $x = 0, y = 0, z = 0$,then $a, b$,and $c$ are connected by the relation:

Let $A = \begin{bmatrix} 2 & 0 & 3 \\ 4 & 7 & 11 \\ 5 & 4 & 8 \end{bmatrix}$. Then