The cubic $\left| {\begin{array}{*{20}{c}}
  0&{a - x}&{b - x} \\ 
  { - a - x}&0&{c - x} \\ 
  { - b - x}&{ - c - x}&0 
\end{array}} \right| = 0$ has a reperated root in $x$ then,

For positive numbers $x,y$ and $z$  the numerical value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{{{\log }_x}y}&{{{\log }_x}z}\\{{{\log }_y}x}&1&{{{\log }_y}z}\\{{{\log }_z}x}&{{{\log }_z}y}&1\end{array}\,} \right|$is

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations

$x+2 y+3 z=\alpha$

$4 x+5 y+6 z=\beta$

$7 x+8 y+9 z=\gamma-$

is consistent. Let $| M |$ represent the determinant of the matrix

$M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$

Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.

($1$) The value of $| M |$ is

($2$) The value of $D$ is

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 $\Delta = \left| {\,\begin{array}{*{20}{c}}x&y&z\\p&q&r\\a&b&c\end{array}\,} \right|,$ then $\left| {\,\begin{array}{*{20}{c}}x&{2y}&z\\{2p}&{4q}&{2r}\\a&{2b}&c\end{array}\,} \right|$equals

The determinant $\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&2&3\\1&3&6\end{array}\,} \right|$ is not equal to