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

The number of real values $\lambda$, such that the system of linear equations $2 x-3 y+5 z=9$  ;  $x+3 y-z=-18$    ; $3 x-y+\left(\lambda^{2}-1 \lambda \mid\right) z=16$ has no solution, is :-

If $n \ne 3k$ and 1, $\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^{2n}}}&1&{{\omega ^n}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\end{array}\,} \right|$ has the value

If the system of equations $x + ay = 0,$ $az + y = 0$ and $ax + z = 0$ has infinite solutions, then the value of $a$ is

Let $A = \left[ {\begin{array}{*{20}{c}}5&{5\alpha }&\alpha \\0&\alpha &{5\alpha }\\0&0&5\end{array}} \right]$, If ${\left| A \right|^2} = 25$, then $\left| \alpha \right|$ equals

The greatest value of $c \in R$ for which the system of linear equations

$x - cy - cz = 0 \,\,;\,\, cx - y + cz = 0 \,\,;\,\, cx + cy - z = 0 $ has a non -trivial solution, is