If the system of equations $x + y + z = 5$,$x + 2y + 3z = 9$,and $x + 3y + \alpha z = \beta$ has infinitely many solutions,then $\beta - \alpha$ equals:

The ordered pair $(a, b)$,for which the system of linear equations $3x - 2y + z = b$,$5x - 8y + 9z = 3$,and $2x + y + az = -1$ has no solution,is

If the system of linear equations $(\sin \theta) x - y + z = 0$, $x - (\cos \theta) y + z = 0$, and $x + y + (\sin \theta) z = 0$ has a non-trivial solution, then the least positive value of $\theta$ is

The system of equations $x+y+z=6$,$x+2y+5z=9$,$x+5y+\lambda z=\mu$ has no solution if

Let $M$ be a $3 \times 3$ matrix such that $M \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, $M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}$ and $M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$. If $M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}$, then $x + y + z$ equals :

If $(x, y, z)=(\alpha, \beta, \gamma)$ is the unique solution of the system of simultaneous linear equations $3x - 4y + z + 7 = 0$, $2x + 3y - z = 10$, and $x - 2y - 3z = 3$, then $\alpha = $