The general value of $\theta$ that satisfies both the equations $\cot^3\theta + 3\sqrt{3} = 0$ and $\csc^5\theta + 32 = 0$ is $(n \in I)$.

Assertion $(A)$: If $A=15^{\circ}, B=17^{\circ}$ and $C=13^{\circ}$,then $\cot 2A + \cot 2B + \cot 2C = \cot 2A \cot 2B \cot 2C$.
Reason $(R)$: In a $\triangle PQR$,$\tan \frac{P}{2} \tan \frac{Q}{2} + \tan \frac{Q}{2} \tan \frac{R}{2} + \tan \frac{P}{2} \tan \frac{R}{2} = 1$.
The correct option among the following is:

Let $\frac{\sin A}{\sin B} = \frac{\sin (A-C)}{\sin (C-B)}$,where $A, B, C$ are angles of a triangle $ABC$. If the lengths of the sides opposite these angles are $a, b, c$ respectively,then:

The product of the distances of the incentre $I$ from the angular points $A, B, C$ of a $\Delta ABC$ is:

Statement $-1$: The number of common solutions of the trigonometric equations $2\sin^2\theta - \cos 2\theta = 0$ and $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, 2\pi]$ is two.
Statement $-2$: The number of solutions of the equation $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, \pi]$ is two.

In a triangle with one of the angles $120^{\circ}$,the lengths of the sides form an $A$.$P$. If the length of the greatest side is $7 \ m$,then the area of the triangle is