For what value of $\lambda $, the system of equations $x + y + z = 6,x + 2y + 3z = 10,$ $x + 2y + \lambda z = 12$ is inconsistent $\lambda =$ ........
$1$
$2$
$-2$
$3$
If $x + y - z = 0,\,3x - \alpha y - 3z = 0,\,\,x - 3y + z = 0$ has non zero solution, then $\alpha = $
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 equations : $2x\, \cos^2\theta + y\, \sin2\theta - 2\sin\theta = 0$ $x\, \sin2\theta + 2y\, \sin^2\theta = - 2\, \cos\theta$ $x\, \sin\theta - y \cos\theta = 0$ , for all values of $\theta$ , can
If $\left| {\,\begin{array}{*{20}{c}}{ - {a^2}}&{ab}&{ac}\\{ab}&{ - {b^2}}&{bc}\\{ac}&{bc}&{ - {c^2}}\end{array}\,} \right| = K{a^2}{b^2}{c^2},$ then $K = $
If $\left| {\,\begin{array}{*{20}{c}}{x - 1}&3&0\\2&{x - 3}&4\\3&5&6\end{array}\,} \right| = 0$, then $x =$