The existence of unique solution of the system of equations,  $x+y+z=\beta $ , $5x-y+\alpha z=10$ , $2x+3y-z=6$ depends on 

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

Let $A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)$ . If $A ^{2}+\gamma A +18 I = O$, then $\operatorname{det}( A )$ is equal to

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 $\left| {{\kern 1pt} \begin{array}{*{20}{c}}1&2&3\\2&x&3\\3&4&5\end{array}\,} \right| = 0,$ then  $x =$