If the solution of the system of simultaneous linear equations $x+y-z=6$,$3x+2y-z=5$ and $2x-y-2z+3=0$ is $x=\alpha, y=\beta, z=\gamma$,then $\alpha+\beta=$

Consider the system of equations in $x, y$ and $z$:
$12x + by + cz = 0$
$ax + 24y + cz = 0$
$ax + by + 36z = 0$
(where $a, b, c$ are real numbers,$a \ne 12, b \ne 24, c \ne 36$).
If the system of equations has a non-trivial solution $(z \ne 0)$,then the value of $\frac{1}{a - 12} + \frac{2}{b - 24} + \frac{3}{c - 36}$ is:

$A$ trust fund has Rs. $30,000$ that must be invested in two different types of bonds. The first bond pays $5 \%$ interest per year,and the second bond pays $7 \%$ interest per year. Using matrix multiplication,determine how to divide Rs. $30,000$ among the two types of bonds if the trust fund must obtain an annual total interest of Rs. $2000$.

For a matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$, if $U_{1}, U_{2}$, and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $A U_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$, $A U_{3}=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$, and $U$ is a $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$, and $U_{3}$, then the sum of the elements of $U^{-1}$ is:

If the system of equations $\begin{bmatrix} 1 & -2 & 5 \\ 2 & -1 & 1 \\ 11 & -7 & p \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ q \end{bmatrix}$ has infinitely many solutions,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?