If the system of equations, $x + 2y -3z = 1, (k + 3) z = 3, (2k + 1)x + z = 0$ is inconsistent, then the value of $k$ is :-

If $p{\lambda ^4} + q{\lambda ^3} + r{\lambda ^2} + s\lambda + t = $ $\left| {\,\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda }&{\lambda - 1}&{\lambda + 3}\\{\lambda + 1}&{2 - \lambda }&{\lambda - 4}\\{\lambda - 3}&{\lambda + 4}&{3\lambda }\end{array}\,} \right|,$ the value of $t$ is

$\left| {\,\begin{array}{*{20}{c}}{1/a}&{{a^2}}&{bc}\\{1/b}&{{b^2}}&{ca}\\{1/c}&{{c^2}}&{ab}\end{array}\,} \right| = $

For what value of $\lambda $, the system of equations $x + y + z = 6,x + 2y + 3z = 10,$ $x + 2y + \lambda z = 12$ is inconsistent  $\lambda =$ ........

Let for any three distinct consecutive terms $a, b, c$ of an $A.P,$ the lines $a x+b y+c=0$ be concurrent at the point $\mathrm{P}$ and $\mathrm{Q}(\alpha, \beta)$ be a point such that the system of equations $ x+y+z=6, $ $ 2 x+5 y+\alpha z=\beta$ and $x+2 y+3 z=4$, has infinitely many solutions. Then $(P Q)^2$ is equal to________.

If $\omega $ is a cube root of unity and $\Delta = \left| {\begin{array}{*{20}{c}}1&{2\omega }\\\omega &{{\omega ^2}}\end{array}} \right|$, then ${\Delta ^2}$ is equal to