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 $ab+bc+ca=$

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 $\alpha\beta+\beta\gamma+\gamma\alpha=$

Let $f(x) = 2x^2 + 5x + 1$. If we write $f(x)$ as $f(x) = a(x+1)(x-2) + b(x-2)(x-1) + c(x-1)(x+1)$ for real numbers $a, b, c$,then:

If the system of linear equations $x + ky + 3z = 0$,$3x + ky - 2z = 0$,and $2x + 4y - 3z = 0$ has a non-zero solution $(x, y, z)$,then $\frac{xz}{y^2} = \dots$

If the matrix equation is given,then $x=$ . . . . . . and $y=$ . . . . . . .

For a system of simultaneous linear equations,if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$,$\operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$,then $X=$