If in a triangle $ABC$,$a, b, c$ are sides and angle $A$ is given,and $c \sin A < a < c$,and $b_1$ and $b_2$ are two possible values of $b$,then:

In a triangle $ABC$,if $\frac{r}{r_1} = \frac{1}{2}$,then $4 \tan \frac{A}{2} \left( \tan \frac{B}{2} + \tan \frac{C}{2} \right) = $

In a $\Delta ABC$,$a = 5, b = 4$ and $\cos(A - B) = \frac{31}{32}$,then side $c$ is equal to

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 $\Delta ABC$,if $\sin^{2} A + \sin^{2} B + \sin^{2} C = 2$,then the triangle is:

If the angles $A, B$ and $C$ of a triangle are in $A.P.$ and if $a, b$ and $c$ denote the length of the sides opposite to $A, B$ and $C$ respectively,then the value of $\frac{a}{b} \sin 2B + \frac{b}{a} \sin 2A$ is