The cubic equation $\left| \begin{array}{ccc} 0 & a-x & b-x \\ -a-x & 0 & c-x \\ -b-x & -c-x & 0 \end{array} \right| = 0$ has a repeated root in $x$. Then:

If $a$ is a non-real complex number for which the system of equations $ax - a^2y + a^3z = 0$,$-a^2x + a^3y + az = 0$,and $a^3x + ay - a^2z = 0$ has non-trivial solutions,then $|a|$ is:

If $a, b, c$ are positive real numbers each distinct from unity, then the value of the determinant $\left|\begin{array}{ccc}1 & \log _a b & \log _a c \\ \log _b a & 1 & \log _b c \\ \log _c a & \log _c b & 1\end{array}\right|$ is

If $f(x) = \left| \begin{array}{ccc} 1 & 6+x & 36+x^2 \\ 0 & x-3 & 3x^2-27 \\ 0 & 2x-4 & 8x^2-32 \end{array} \right|$,then $\lim_{x \rightarrow 1} \frac{f(x)}{f(-x)} = $

Find the equation of the line joining $(3, 1)$ and $(9, 3)$ using determinants.

For real numbers $x, y$ and $z$,if $x \neq y \neq z$,$\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$ and $\left|\begin{array}{ccc}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{array}\right| \neq 0$,then $xyz = $ . . . . . . .