The roots of the determinant equation (in $x$) $\left| {\,\begin{array}{*{20}{c}}a&a&x\\m&m&m\\b&x&b\end{array}\,} \right| = 0$

If the system of equations $2x + 3y – z = 0$, $x + ky – 2z = 0$ and  $2x – y + z = 0$ has a non -trivial solution $(x, y, z)$, then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

If the system of equations  $2 x+3 y-z=5$  ;  $x+\alpha y+3 z=-4$  ;  $3 x-y+\beta z=7$ has infinitely many solutions, then $13 \alpha \beta$ is equal to

The value of the determinant$\left| {\,\begin{array}{*{20}{c}}{ – 1}&1&1\\1&{ – 1}&1\\1&1&{ – 1}\end{array}\,} \right|$is equal to

Let $A =$ $\left[ {\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ – \sin \theta }&1&{\sin \theta }\\{ – 1}&{ – \sin \theta }&1\end{array}} \right]$, where $0 \le \theta < 2\pi$ , then