In a triangle $\mathrm{ABC}, l(\mathrm{AB})=\sqrt{23}$ units, $l(\mathrm{BC})=3$ units, $l(\mathrm{CA})=4$ units, then $\frac{\cot A+\cot C}{\cot B}$ is

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