If $r_1, r_2$ and $r_3$ of a $\triangle ABC$ are in Harmonic progression,then $a, b$ and $c$ will be in

In a triangle $ABC$,$\angle B < \angle C$ and the values of $B$ and $C$ satisfy the equation $2 \tan x - k (1 + \tan^2 x) = 0$,where $0 < k < 1$. Then the measure of angle $A$ is:

The equation $\sin ^4 x-(k+3) \sin ^2 x-k-4=0$ has a solution if

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 $OE = \frac{1}{6}$

In a $\Delta ABC$,the sides $a$,$b$,$c$ are the roots of the equation $x^3 - 11x^2 + 38x - 40 = 0$; then $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = $

In $\triangle ABC$,if $a, b, c$ are $5, 12$ and $13$ respectively,then $b^2 \sin 2C + c^2 \sin 2B =$