The following system of linear equations  $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$  ;$x-y+4 z=8$

If $a\, -\, 2b + c = 1$ , then value of $\left| {\begin{array}{*{20}{c}}
  {x + 1}&{x + 2}&{x + a} \\ 
  {x + 2}&{x + 3}&{x + b} \\ 
  {x + 3}&{x + 4}&{x + c} 
\end{array}} \right|$ is

The value of $\left| {\,\begin{array}{*{20}{c}}{{1^2}}&{{2^2}}&{{3^2}}\\{{2^2}}&{{3^2}}&{{4^2}}\\{{3^2}}&{{4^2}}&{{5^2}}\end{array}\,} \right|$ is

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 

Evaluate the determinants

$\left|\begin{array}{ccc}
3 & -4 & 5 \\
1 & 1 & -2 \\
2 & 3 & 1
\end{array}\right|$

If the system of linear equations $x + ky + 3z = 0;3x + ky - 2z = 0$ ; $2x + 4y - 3z = 0$  has a non-zero solution $\left( {x,y,z} \right)$ then $\frac{{xz}}{{{y^2}}} = $. . . . .