The system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = \mu$ has no solution for:

If the system of equations $2x - y + z = 4$,$5x + \lambda y + 3z = 12$,and $100x - 47y + \mu z = 212$ has infinitely many solutions,then $\mu - 2\lambda$ is equal to

If $\begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ 0 & -6 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 2 \\ 1 \end{bmatrix}$,then $(x, y, z) = $

If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2$, and $x+2y+2z=1$, then $ab+bc+ca=$

If $[x]$ denotes the greatest integer $\leq x$,then the system of linear equations
$[\sin \theta ] x + [-\cos \theta ] y = 0$
$[\cot \theta ] x + y = 0$

If the system of simultaneous linear equations $x+y-z=6$,$4x+y+z=2$,and $x+ky+z=-8$ has a unique solution $x=2$,$y=\beta$,$z=\gamma$,then the value of $k$ satisfies which of the following quadratic equations?