In a $\triangle A B C, A D$ and $B E$ are medians. If $A D=4, \angle D A B=\frac{\pi}{6}$ and $\angle A B E=\frac{\pi}{3}$, then the area of $\triangle A B C$ is

If $S$ is the circumentre of a $\triangle A B C, a=5, b=6, c=9$ and $S B=\frac{27}{4 \sqrt{2}}$, then $\sin 2 C=$

In a $\triangle A B C$, if $\frac{r}{r_1}=\frac{1}{2}$, then $4 \tan \frac{A}{2}\left(\tan \frac{B}{2}+\tan \frac{C}{2}\right)=$

If the sides of a $\triangle A B C$ whose perimeter is 42 are in arithmetic progression, its circumradius is $\frac{65}{8}$ and $B