The set of values of $k$ for which the system of simultaneous equations $x+y+kz=1$,$2x+2y=3$,and $x+2y+2kz=k$ has no real solution is

An ordered pair $(\alpha, \beta)$ for which the system of linear equations $(1 + \alpha)x + \beta y + z = 2$; $\alpha x + (1 + \beta)y + z = 3$; $\alpha x + \beta y + 2z = 2$ has a unique solution,is

If the system of equations $2x + \lambda y + 3z = 5$,$3x + 2y - z = 7$,and $4x + 5y + \mu z = 9$ has infinitely many solutions,then $(\lambda^2 + \mu^2)$ is equal to:

The values of $p$ and $q$ so that the system of equations $2x + py + 6z = 8$,$x + 2y + qz = 5$ and $x + y + 3z = 4$ may have no solution are

The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,$ and $x - 3y + z = 0$ is

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 = $