If the system of equations  $2 x+3 y-z=5$  ;  $x+\alpha y+3 z=-4$  ;  $3 x-y+\beta z=7$ has infinitely many solutions, then $13 \alpha \beta$ is equal to

If the system of equation $2x + 3y =\, -1; 3x + y = 2; \lambda x + 2y = \mu $ is consistent, then

If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
  {x + 2}&\omega &{{\omega ^2}} \\ 
  \omega &{x + 1 + {\omega ^2}}&1 \\ 
  {{\omega ^2}}&1&{x + 1 + \omega } 
\end{array}} \right| = 0$ is 

$\left| {\,\begin{array}{*{20}{c}}{{{({a^x} + {a^{ – x}})}^2}}&{{{({a^x} – {a^{ – x}})}^2}}&1\\{{{({b^x} + {b^{ – x}})}^2}}&{{{({b^x} – {b^{ – x}})}^2}}&1\\{{{({c^x} + {c^{ – x}})}^2}}&{{{({c^x} – {c^{ – x}})}^2}}&1\end{array}\,} \right| = $

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________.