For how many different values of $a$ does the following system of linear equations have at least two distinct solutions?
$ax + y = 0$
$x + (a + 10)y = 0$

For what value of $k$ does the following system of equations possess a non-trivial solution?
$x + ky + 3z = 0$
$3x + ky - 2z = 0$
$2x + 3y - 4z = 0$

For which of the following ordered pairs $(\mu, \delta)$ is the system of linear equations $x+2y+3z=1$,$3x+4y+5z=\mu$,and $4x+4y+4z=\delta$ inconsistent?

Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix}$,$B = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix}$ and $X = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$. If $AX = B$,then the value of $2a + b + 2c$ is:

The number of real values of $\lambda$ for which the system of linear equations $2x + 4y - \lambda z = 0$,$4x + \lambda y + 2z = 0$,and $\lambda x + 2y + 2z = 0$ has infinitely many solutions is:

If the system of equations $2x + 3y - 3z = 3$,$x + 2y + \alpha z = 1$,and $2x - y + z = \beta$ has infinitely many solutions,then $\frac{\alpha}{\beta} - \frac{\beta}{\alpha} =$