$\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{m{a_1}}&{{b_1}}\\{{a_2}}&{m{a_2}}&{{b_2}}\\{{a_3}}&{m{a_3}}&{{b_3}}\end{array}\,} \right| = $

If the system of linear equation $x - 4y + 7z = g,\,3y - 5z = h, \,-\,2x + 5y - 9z = k$ is
consistent, then

The values of $a$ and $b$, for which the system of equations    $2 x+3 y+6 z=8$   ;  $x+2 y+a z=5$     ;  $3 x+5 y+9 z=b$  has no solution, are:

The values of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\alpha - \beta )}&{\cos \alpha }\\{\cos (\alpha - \beta )}&1&{\cos \beta }\\{\cos \alpha }&{\cos \beta }&1\end{array}\,} \right|$ is

If the lines $x + 2ay + a = 0, x + 3by + b = 0$ and $x + 4cy + c = 0$ are concurrent, then $a, b$  and $c$ are in :-

The system of linear equations $x + \lambda y - z = 0,\lambda x - y - z = 0\;,\;x + y - \lambda z = 0$ has a non-trivial solution for: