If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2$, and $x+2y+2z=1$, then $ab+bc+ca=$

The existence of the unique solution of the system of equations $2x + y + z = \beta$,$10x - y + \alpha z = 10$ and $4x + 3y - z = 6$ depends on

If the system of equations $kx + y + 2z = 1$,$3x - y - 2z = 2$,and $-2x - 2y - 4z = 3$ has infinitely many solutions,then $k$ is equal to ..........

In matrix notation, if the system of equations $\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ -5 \\ 10 \end{bmatrix}$ has an infinite number of solutions, then all these solutions lie on

If the solution for the system of equations $x+2y-z=3$,$3x-y+2z=1$ and $2x-2y+3z=2$ is $(\alpha, \beta, \gamma)$,then $\alpha^2+\beta^2+\gamma^2=$

If the system of equations $x + 5y + 6z = 4$, $2x + 3y + 4z = 7$, and $x + 6y + az = b$ has infinitely many solutions, then the point $(a, b)$ lies on the line: