Let $\alpha_1, \alpha_2$ be two values of $\alpha$ for which the system $2\alpha x + y = 5$,$x - 6y = \alpha$,and $x + y = 2$ is consistent. Then $|2(\alpha_1 + \alpha_2)|$ is -

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:

The system of equations $-k x+3 y-14 z=25$,$-15 x+4 y-k z=3$,and $-4 x+y+3 z=4$ is consistent for all $k$ in the set

The equations $x-y+2z=4$,$3x+y+4z=6$,and $x+y+z=1$ have

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|, \Delta_2=\left|\begin{array}{ll}a & m \\ c & n\end{array}\right|, \Delta_3=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$,then the values of $x$ and $y$ are respectively ($e$ is the base of natural logarithm).

$A$ 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. $2000$.