If $ 5$  is one root of the equation $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&{ - 2}\\7&8&x\end{array}\,} \right| = 0$, then other two roots of the equation are

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + x}&1\\1&1&{1 + x}\end{array}\,} \right| = 0$   are

The value of $k$ for which the set of equations $x + ky + 3z = 0,$ $3x + ky - 2z = 0,$ $2x + 3y - 4z = 0$ has a non trivial solution over the set of rationals is

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$,then $\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}$ is equal to :

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