The system of equations $x + y + z = 2$,$3x - y + 2z = 6$ and $3x + y + z = -18$ has

Let $A = \begin{bmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{bmatrix}$ and $B = \begin{bmatrix} a \\ -b \\ -c \end{bmatrix}$. If $A$ and $[A: B]$ have the same rank, then:

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?

Let $A = \{X = (x, y, z)^{T} : PX = 0 \text{ and } x^{2} + y^{2} + z^{2} = 1\}$ where $P = \begin{bmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{bmatrix}$,then the set $A$:

If $\frac{5}{m}+\frac{2}{n}=9$ and $\frac{3}{m}+\frac{4}{n}=11$ and $mn \neq 0$,then the values of $m$ and $n$ are . . . . . . respectively.

Let the system of linear equations $x + 2y + z = 2$,$\alpha x + 3y - z = \alpha$,and $-\alpha x + y + 2z = -\alpha$ be inconsistent. Then $\alpha$ is equal to: