In a triangle $ABC$,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 in a right-angled triangle the hypotenuse is four times as long as the perpendicular drawn to it from the opposite vertex,then one of its acute angles is......$^o$

In a $\Delta ABC$,if two sides $b$ and $c$ and an angle $B$ are given.
Statement $-1$: Let $b = 5 \ cm, c = 3 \ cm$ and $\angle B = 60^\circ$. The total number of possible triangles is $2$.
Statement $-2$: If $c \sin B < b < c$ and $B$ is an acute angle,then there are two possible values of $\angle C$.

If $a = 2, b = 3, c = 5$ in $\Delta ABC$,then $C = $

If $R$ is the circumradius of $\triangle ABC$,then the area of $\triangle ABC$ is equal to $\ldots$.

If two angles of $\triangle ABC$ are $45^{\circ}$ and $60^{\circ}$,then the ratio of the smallest side to the greatest side is