The system of equations ${x_1} – {x_2} + {x_3} = 2,$ $\,3{x_1} – {x_2} + 2{x_3} = – 6$ and $3{x_1} + {x_2} + {x_3} = – 18$ has

For the system of linear equations $a x+y+z=1$, $x+a y+z=1, x+y+a z=\beta$, which one of the following statements is NOT correct ?

If $n$  be the number of values of $x$ for which
matrix $\Delta (x) =\left[ {\begin{array}{*{20}{c}}
{ – x}&x&2\\
2&x&{ – x}\\
x&{ – 2}&{ – x}
\end{array}} \right]$ will be singular, then $det(\Delta\,(n))$ is

$($ where $det(B)$ denotes determinant of Matrix $B) -$

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{{\omega ^2}}&\omega \\1&\omega &{{\omega ^2}}\end{array}\,} \right| = $

If $\alpha ,\beta \ne 0$ and $f\left( n \right) = {\alpha ^n} + {\beta ^n}$ and $\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 – \alpha } \right)^2}$ ${\left( {1 – \beta } \right)^2}{\left( {\alpha – \beta } \right)^2}$ ,then $K=$ . . . . . .