The area of a triangle is $5$ and two of its vertices are $A(2, 1), B(3, -2)$. The third  vertex which lies on line $y = x + 3$ is-

$\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{m{a_1}}&{{b_1}}\\{{a_2}}&{m{a_2}}&{{b_2}}\\{{a_3}}&{m{a_3}}&{{b_3}}\end{array}\,} \right| = $

Let $A =$ $\left[ {\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ – \sin \theta }&1&{\sin \theta }\\{ – 1}&{ – \sin \theta }&1\end{array}} \right]$, where $0 \le \theta < 2\pi$ , then

Evaluate the determinants : $\left|\begin{array}{ll}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$

$\left| {\,\begin{array}{*{20}{c}}{1 + i}&{1 – i}&i\\{1 – i}&i&{1 + i}\\i&{1 + i}&{1 – i}\end{array}\,} \right| = $