If $p + q + r = 0$ and $a + b + c = 0$,then the value of the determinant $\left| \begin{array}{ccc} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{array} \right|$ is

If $\left|\begin{array}{ccc}\cos (A+B) & -\sin (A+B) & \cos 2 B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{array}\right|=0$, then $B$ is equal to

The area of the triangle whose vertices are $(3,5), (2,2)$ and $(k, 2)$ is $3$ sq. unit. Then,the value of $k$ is . . . . . . .

Given that, $a \alpha^2+2 b \alpha+c \neq 0$ and that the system of equations
$\begin{aligned} & (a \alpha+b) x+a y+b z=0 \\ & (b \alpha+c) x+b y+c z=0 \\ & (a \alpha+b) y+(b \alpha+c) z=0\end{aligned}$
has a non-trivial solution, then $a, b$ and $c$ lie in

Let $[.]$,$\{.\}$ and $\operatorname{sgn}(.)$ denote the greatest integer function,fractional part function,and signum function respectively. Then,the value of the determinant $\left| {\begin{array}{*{20}{c}} {[ \pi ]} & {\operatorname{amp}(1 + i\sqrt 3 )} & 1 \\ 1 & 0 & 2 \\ {\operatorname{sgn} (\cot^{ - 1}x)} & 1 & {\{ \pi \} } \end{array}} \right|$ is:

If $A = \begin{bmatrix} \lambda & i \\ i & -\lambda \end{bmatrix}$ and $A^{-1}$ does not exist,then $\lambda = $ (where $i = \sqrt{-1}$)