Properties of Triangles question bank (Maths)

In a triangle $\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}$ and $\cos \mathrm{A}=\frac{2}{3}$. If $49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to________

For a triangle $ABC$, the value of $\cos 2A+\cos 2B+\cos 2C$ is least. If its inradius is $3$ and incentre is $M$, then which of the following is NOT correct?