The system of equations $4x + y - 2z = 0\ ,\ x - 2y + z = 0$ ; $x + y - z =0 $ has

The system of equations $kx + y + z =1$ $x + ky + z = k$ and $x + y + zk = k ^{2}$ has no solution if $k$ is equal to

If $a \ne b \ne c,$ the value of $x$ which satisfies the equation $\left| {\,\begin{array}{*{20}{c}}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}\,} \right| = 0$, is

The set of all values of $\lambda $ for which the system of linear equations $x - 2y - 2z = \lambda x$ ; $x + 2y + z = \lambda y$ ; $-x - y = \lambda z$ has non zero solutions.

If the system of equations

$x+y+z=2$

$2 x+4 y-z=6$

$3 x+2 y+\lambda z=\mu$ has infinitely many solutions, then 

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $