The existence of the unique solution of the system $x + y + z = \lambda ,$ $5x - y + \mu z = 10$, $2x + 3y - z = 6$ depends on

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 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:

If $C = 2\cos \theta $, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}C&1&0\\1&C&1\\6&1&C\end{array}\,} \right|$ is

For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$