If the system of equations  $x +y + z = 6$ ; $x + 2y + 3z= 10$ ; $x + 2y + \lambda z = 0$ has a unique solution, then $\lambda $ is not equal to

if $\left| \begin{gathered}
   - 6\ \ \,\,1\ \ \,\,\lambda \ \  \hfill \\
  \,0\ \ \,\,\,\,3\ \ \,\,7\ \  \hfill \\
   - 1\ \ \,\,0\ \ \,\,5\ \  \hfill \\ 
\end{gathered}  \right| = 5948 $, then $\lambda $  is

If $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}}   x&0&1 \\    0&y&0 \\    0&0&z  \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$,  $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ 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

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

If the system of linear equations $x - 2y + kz = 1$ ; $2x + y + z = 2$ ;  $3x - y - kz = 3$ Has a solution $(x, y, z) \ne 0$, then $(x, y)$ lies on the straight line whose equation is