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

The number of roots of the equation $(81)^{\sin ^{2} x} + (81)^{\cos ^{2} x} = 30$ in the interval $[0, \pi]$ is equal to

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

If the sides of a triangle $ABC$ whose perimeter is $42$ are in arithmetic progression,its circum-radius is $\frac{65}{8}$ and $B < A < C$,then $\sin A=$

If the incircle of the $\Delta ABC$ touches its sides at $L, M$ and $N$ respectively,and if $x, y, z$ are the circumradii of the triangles $\Delta MIN, \Delta NIL$ and $\Delta LIM$ respectively,where $I$ is the incentre,then the product $xyz$ is equal to:

In a $\triangle ABC$,the value of $\Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right)$ is equal to