The number of solutions of the following equations $x_2 - x_3 = 1$,$-x_1 + 2x_3 = -2$,$x_1 - 2x_2 = 3$ is

If the system $\begin{bmatrix} 2 & 8 \\ 3 & 7 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = k \begin{bmatrix} a \\ b \end{bmatrix}$ has a non-trivial solution,then the positive value of $k$ and a solution of the system for that value of $k$ are:

If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions,then $\alpha + \beta - \alpha \beta$ is equal to .... .

If $x=\alpha, y=\beta, z=\gamma$ is the solution for the system of equations:
$\begin{aligned} 2x-y+8z &= 13 \\ 3x+4y+5z &= 18 \\ 5x-2y+7z &= 20 \end{aligned}$
then $\alpha\beta+\beta\gamma+\gamma\alpha=$

If the system of equations $x+2y-3z=2$,$2x+\lambda y+5z=5$,$14x+3y+\mu z=33$ has infinitely many solutions,then $\lambda+\mu$ is equal to:

Let $a, b, c, d, e$ be five numbers satisfying the system of equations:
$2a + b + c + d + e = 6$
$a + 2b + c + d + e = 12$
$a + b + 2c + d + e = 24$
$a + b + c + 2d + e = 48$
$a + b + c + d + 2e = 96$
Then $|c|$ is equal to: