In a $\triangle ABC$,if $\tan A : \tan B : \tan C = 1 : 2 : 3$ and $\sin A : \sin B : \sin C = \sqrt{5} : 2\sqrt{2} : k$,then $k =$

In $\triangle ABC$, $A, B$ and $C$ are in arithmetic progression and $a: c = 1: 2$. If $b = 4 \sqrt{3} \text{ cm}$, then the area of $\triangle ABC$ (in $\text{sq. cm}$) is (in $\sqrt{3}$)

The number of solutions of $\sin ^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0$,where $-\pi \leq x \leq \pi$,is....................

In a $\triangle ABC$,$a, b, c$ are the sides of the triangle opposite to the angles $A, B, C$ respectively. Then,the value of $a^{3} \sin (B-C) + b^{3} \sin (C-A) + c^{3} \sin (A-B)$ is equal to

In $\triangle ABC$,with usual notations,if $a, b, c$ are in $A.P.$,then $a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right) = $

In a non-right-angled triangle $\triangle PQR$, let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $PQ$ at $S$, the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $O$. If $p=\sqrt{3}, q=1$, and the radius of the circumcircle of the $\triangle PQR$ equals $1$, then which of the following options is/are correct?
$(1)$ Area of $\triangle SOE = \frac{\sqrt{3}}{48}$
$(2)$ Radius of incircle of $\triangle PQR = \frac{\sqrt{3}}{2}(2-\sqrt{3})$
$(3)$ Length of $RS = \frac{\sqrt{7}}{2}$
$(4)$ Length of $OE = \frac{1}{6}$