If the system of equation $3x – 2y + z = 0$, $\lambda x – 14y + 15z = 0$, $x + 2y + 3z = 0$ have a non-trivial solution, then $\lambda = $

For $\alpha, \beta \in \mathrm{R}$ and a natural number $\mathrm{n}$, let

$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$. Then $2 A_{10}-A_8$

For the system of linear equations

$2 x-y+3 z=5$

$3 x+2 y-z=7$

$4 x+5 y+\alpha z=\beta$

Which of the following is NOT correct ?

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

$\left| {\,\begin{array}{*{20}{c}}0&{p – q}&{p – r}\\{q – p}&0&{q – r}\\{r – p}&{r – q}&0\end{array}\,} \right| = $