If $a \ne 6,b,c$ satisfy $\left| {\,\begin{array}{*{20}{c}}a&{2b}&{2c}\\3&b&c\\4&a&b\end{array}\,} \right| = 0,$then $abc = $

If the system of equation $2x + 3y =\, -1; 3x + y = 2; \lambda x + 2y = \mu $ is consistent, then

If $C = 2\cos \theta $, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}C&1&0\\1&C&1\\6&1&C\end{array}\,} \right|$ is

The values of $\theta, \lambda$ for which the following equations $\sin \theta x – cos\theta y + (\lambda +1)z = 0$; $\cos\theta x + \sin\theta\, y – \lambda z = 0$;$ \lambda x +(\lambda + 1)y + \cos\theta z = 0$ have non trivial solution, is

If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda  + \mu $ is