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

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

The existence of the unique solution of the system $x + y + z = \lambda ,$ $5x – y + \mu z = 10$, $2x + 3y – z = 6$ depends on

The value of the determinant $\left| {\,\begin{array}{*{20}{c}}2&8&4\\{ – 5}&6&{ – 10}\\1&7&2\end{array}\,} \right|$is

If $\omega $ be a complex cube root of unity, then $\left| {\,\begin{array}{*{20}{c}}1&\omega &{ – {\omega ^2}/2}\\1&1&1\\1&{ – 1}&0\end{array}\,} \right| = $