If in a $\Delta ABC$,$b = 2$,$c = \sqrt{3}$,$\angle A = \frac{\pi}{6}$,then $R = $?

In a triangle $PQR$,let $\angle PQR = 30^{\circ}$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and $10$,respectively. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ $\angle QPR = 45^{\circ}$
$(B)$ The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QRP = 120^{\circ}$
$(C)$ The radius of the incircle of the triangle $PQR$ is $10\sqrt{3} - 15$
$(D)$ The area of the circumcircle of the triangle $PQR$ is $100\pi$

In any $\triangle ABC$, the expression $\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4b^2c^2}$ is equal to:

The sides of a $\triangle ABC$ are given by $a=3, b=5$ and $c=3$. Then,$\cos A=$

If $s$ is the semi-perimeter of $\triangle ABC$ and if $\frac{s-a}{4}=\frac{s-b}{5}=\frac{s-c}{6}$,then $\sum \sin^2\left(\frac{A}{2}\right) = $

In a $\triangle ABC$,if $r_1 = 2r_2 = 3r_3$,then the ratio $a : b$ is: