In $\triangle ABC$,if $\cos 3A + \cos 3B + \cos 3C + \cos 3\pi = 0$,then the least value of the sum of two of its angles is

In $\Delta ABC$,if $\cos A + \cos C = 4 \sin^2 \frac{B}{2}$,then $a, b, c$ are in

In $\Delta ABC$,the expression $a(\cos^2 B + \cos^2 C) + \cos A(c \cos C + b \cos B)$ is equal to:

With usual notations,in a triangle $ABC$,$a \cos(B - C) + b \cos(C - A) + c \cos(A - B)$ is equal to

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 $\Delta ABC$,if $a, b, c$ are in $A.P.$ (with usual notations),identify the incorrect statement -