If for some $\alpha$ and $\beta$ in $R,$ the intersection of the following three planes  $x+4 y-2 z=1$ ; $x+7 y-5 z=\beta$ ; $x+5 y+\alpha z=5$ is a line in $\mathrm{R}^{3},$ then $\alpha+\beta$ is equal to

If ${\Delta _r} = \left| {\begin{array}{*{20}{c}}
  r&{2r - 1}&{3r - 2} \\ 
  {\frac{n}{2}}&{n - 1}&a \\ 
  {\frac{1}{2}n\left( {n - 1} \right)}&{{{\left( {n - 1} \right)}^2}}&{\frac{1}{2}\left( {n - 1} \right)\left( {3n - 4} \right)} 
\end{array}} \right|$ then the value of $\sum\limits_{r = 1}^{n - 1} {{\Delta _r}} $

${x_1} + 2{x_2} + 3{x_3} = a2{x_1} + 3{x_2} + {x_3} = $ $b3{x_1} + {x_2} + 2{x_3} = c$ this system of equations has

The system of equations $kx + 2y\,-z = 1$  ;  $(k\,-\,1)y\,-2z = 2$  ;  $(k + 2)z = 3$ has unique solution, if $k$ is equal to

If the system of equations, $x + 2y -3z = 1, (k + 3) z = 3, (2k + 1)x + z = 0$ is inconsistent, then the value of $k$ is :-

Let the system of equations $x+2 y+3 z=5$, $2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :