In a triangle ABC , with usual notations, $2 \mathrm{ac} \sin \left(\frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2}\right)$ is equal to

In a triangle ABC , with usual notations, $\frac{\cos \mathrm{B}+\cos \mathrm{C}}{\mathrm{b}+\mathrm{c}}+\frac{\cos \mathrm{A}}{\mathrm{a}}$ has the value

The angles of a triangle are in the ratio $5: 1: 6$, then ratio of the smallest side to the greatest side is

In a $\triangle \mathrm{PQR}, \mathrm{m} \angle \mathrm{R}=\frac{\pi}{2}$. If $\tan \left(\frac{\mathrm{P}}{2}\right)$ and $\tan \left(\frac{\mathrm{Q}}{2}\right)$ are the roots of the equation $a x^2+b x+c=0(a \neq 0)$, then