$\left|\begin{array}{ccc}\cos 3\pi & \sin 5\pi & \tan 7\pi \\ \sqrt{3} & 1 & 0 \\ \sqrt{5} & 0 & 1\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:

If $x^{3}-2x^{2}-9x+18=0$ and $A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|$,then the maximum value of $A$ is

The number of positive integral solutions of the equation $\left| \begin{array}{ccc} 1 - \lambda & 2 & 1 \\ -3 & \lambda & -2 \\ 2 & -2 & 1 + \lambda \end{array} \right| = 0$ is

If $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & 1 \\ 3 & 2 & 6 \end{bmatrix}$ and $A_{ij}$ are cofactors of the elements $a_{ij}$ of $A$,then $a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$ is equal to

$\left| \begin{array}{ccc} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \end{array} \right| = $