Let $\alpha, \beta(\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1 ; x+2 y+4 z=m$ and $x+4 y+10 z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)$ is equal to :

$A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{array}\right],$ then show that $|3 A|=27|A|$.

Let the system of equations $x+2 y+3 z=5$, $2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :

If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
  {{a^2}}&{{d^2}}&x \\ 
  {{b^2}}&{{e^2}}&y \\ 
  {{c^2}}&{{f^2}}&z 
\end{array}} \right|$ depends on

The system of equations $4x + y - 2z = 0\ ,\ x - 2y + z = 0$ ; $x + y - z =0 $ has