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
  $- 6 + \frac{5\pi}{3} - \frac{\pi^2}{3}$
• B
  $\frac{5\pi}{3} - \frac{\pi^2}{3} - 5$
• C
  $\frac{5\pi}{3} + \frac{\pi^2}{3} + 6$
• D
  $- 5 + \frac{\pi^3}{3} - \frac{5\pi^2}{3}$