The value of the determinant $\left| {\,\begin{array}{*{20}{c}}2&8&4\\{ – 5}&6&{ – 10}\\1&7&2\end{array}\,} \right|$is

Let the system of linear equations

$x+y+\alpha z=2$

$3 x+y+z=4$

$x+2 z=1$

have a unique solution $\left(x^{*}, y^{*}, z^{*}\right)$. If $\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)$ and $\left(x^{*},-y^{*}\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

If $a > 0$and discriminant of $a{x^2} + 2bx + c$is negative, then $\left| {\,\begin{array}{*{20}{c}}a&b&{ax + b}\\b&c&{bx + c}\\{ax + b}&{bx + c}&0\end{array}\,} \right|$ is

Set of equations $a + b – 2c = 0,$ $2a – 3b + c = 0$ and $a – 5b + 4c = \alpha $ is consistent for $\alpha$ equal to

The value of $k \in R$, for which the following system of linear equations

$3 x-y+4 z=3$

$x+2 y-3 x=-2$

$6 x+5 y+k z=-3$

has infinitely many solutions, is: