The system of equations $x + ky - z = 0$,$3x - ky - z = 0$,and $x - 3y + z = 0$ has a non-zero solution for $k =$

If $x = \alpha, y = \beta, z = \gamma$ is the unique solution of the system of equations $5x - 2y + 3z = 0$,$7x + 10y - 8z = 3$ and $2x + 3y - 4z = -4$,then $\beta =$

If the point $P(\alpha, \beta, \gamma)$ lies on the plane $2x + y + z = 1$ and $\begin{bmatrix} \alpha & \beta & \gamma \end{bmatrix} \begin{bmatrix} 1 & 9 & 1 \\ 8 & 2 & 1 \\ 7 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$,then $\alpha^2 + \beta^2 + \gamma^2 = $

Let $\lambda, \mu \in R$. If the system of equations
$3x + 5y + \lambda z = 3$
$7x + 11y - 9z = 2$
$97x + 155y - 189z = \mu$
has infinitely many solutions,then $\mu + 2\lambda$ is equal to :

The system of equations $x - 2y + 3z = 5$,$2x - 2y + z = 0$,and $-x + 2y - 3z = 6$ has

If the system of linear equations $2x + y - z = 7$,$x - 3y + 2z = 1$,and $x + 4y + \delta z = k$,where $\delta, k \in R$,has infinitely many solutions,then $\delta + k$ is equal to