If the system of equation $3x - 2y + z = 0$, $\lambda x - 14y + 15z = 0$, $x + 2y + 3z = 0$ have a non-trivial solution, then $\lambda = $

If the system of equations

$ 11 x+y+\lambda z=-5 $

$ 2 x+3 y+5 z=3 $

$ 8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

If the lines $ax + y + 1 = 0$, $x + by + 1 = 0$ and $x + y + c = 0$ (where $a, b$ and $c$ are distinct and different from $1$ ) are concurrent, then the value of $\frac{1}{{1 - a}} + \frac{1}{{1 - b}} + \frac{1}{{1 - c}} =$ 

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set

The greatest value of $c \in R$ for which the system of linear equations

$x - cy - cz = 0 \,\,;\,\, cx - y + cz = 0 \,\,;\,\, cx + cy - z = 0 $ has a non -trivial solution, is

The system of equations ${x_1} - {x_2} + {x_3} = 2,$ $\,3{x_1} - {x_2} + 2{x_3} = - 6$ and $3{x_1} + {x_2} + {x_3} = - 18$ has