If the system of equations

$ 11 x+y+\lambda z=-5 $

$ 2 x+3 y+5 z=3 $

$ 8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

$\Delta = \left| {\,\begin{array}{*{20}{c}}a&{a + b}&{a + b + c}\\{3a}&{4a + 3b}&{5a + 4b + 3c}\\{6a}&{9a + 6b}&{11a + 9b + 6c}\end{array}\,} \right|$where $a = i,b = \omega ,c = {\omega ^2}$, then $\Delta $is equal to

Evaluate $\left|\begin{array}{cc}x & x+1 \\ x-1 & x\end{array}\right|$

The least value of the product $xyz$ for which the determinant $\left| {\begin{array}{*{20}{c}}
  x&1&1 \\ 
  1&y&1 \\ 
  1&1&z 
\end{array}} \right|$ is non-negative, 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