Area of the triangle formed by points $(102, -4)$,$(105, -2)$,and $(103, -3)$ is:

If $\Delta_k=\left|\begin{array}{ccc}1 & 0 & 0 \\ 0 & k & k-1 \\ 0 & k-1 & k\end{array}\right|$,then $\Delta_1+\Delta_2+\ldots+\Delta_{20}$ is equal to

An equilateral triangle has each of its sides of length $6 \text{ cm}$. If $(x_1, y_1), (x_2, y_2), \text{ and } (x_3, y_3)$ are its vertices,then the value of the determinant $\left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right|^2$ is equal to:

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 the value of $B$ is

If $a x^{4}+b x^{3}+c x^{2}+d x+e = \left|\begin{array}{ccc}x^{3}+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x\end{array}\right|$,then $e$ is equal to

If $\Delta = \begin{vmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{vmatrix}$,then $\Delta$ lies in the interval