Evaluate the determinant: $\left|\begin{array}{ccc}1 & a & b \\ 1 & a+b & b \\ 1 & a & a+b\end{array}\right| = $ . . . . . . .

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:

$A$ value of $\theta$ for which the following system of equations has a non-trivial solution is:
$(4 \sin \theta) x - 3y + z = 0$
$x - (6 \cos 2\theta) y + z = 0$
$3x - 12y + 4z = 0$

Evaluate the determinant: $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

Let $A = \begin{bmatrix} 3-t & 1 & 0 \\ -1 & 3-t & 1 \\ 0 & -1 & 0 \end{bmatrix}$ and $\det(A) = 5$, then find the value of $t$.

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