All the real values of $p, q$ so that the system of equations $\begin{cases} 2x + py + 6z = 8 \\ x + 2y + qz = 5 \\ x + y + 3z = 4 \end{cases}$ may have no solution are

The equations $x+y+z=3$,$x+2y+2z=6$ and $x+ay+3z=b$ have

Statement $1$: If the system of equations $x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0$ has a nontrivial solution,then the value of $k$ is $\frac{31}{2}$.
Statement $2$: $A$ system of three homogeneous equations in three variables has a nontrivial solution if the determinant of the coefficient matrix is zero.

If a point $P(\alpha, \beta, \gamma)$ satisfying $(\alpha \ \beta \ \gamma)\begin{bmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{bmatrix} = (0 \ 0 \ 0)$ lies on the plane $2x + 4y + 3z = 5$,then $6\alpha + 9\beta + 7\gamma$ is equal to:

The system of equations $x+y+z=5, x+2y+az=9, x+2y+z=b$ is inconsistent if

If $p$ and $q$ are two distinct real values of $\lambda$ for which the system of equations $\begin{aligned} (\lambda-1) x+(3 \lambda+1) y+2 \lambda z &=0 \\ (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z &=0 \\ 2 x+(3 \lambda+1) y+3(\lambda-1) z &=0 \end{aligned}$ has a non-zero solution,then $p^2+q^2-p q=$