QJEEM24Maa129: JEE Main 2024 Mathematics - Class 11 JEEM24Maa129 Solution: In a triangle and . If , where , then is equal to _________.

$\begin{aligned}& \cos A=\frac{2}{3}=\frac{\alpha^2+8^2-7^2}{2 \cdot \alpha \cdot 8} \\& \Rightarrow\left(\alpha^2+15\right) 3=32 \alpha \\& 3 \alpha^2-32 \alpha+45=0 \\& \Rightarrow \alpha=\frac{5}{3}, 9 \\& \because \alpha \in N \Rightarrow \alpha=9 \\& \cos C=\frac{7^2+8^2-9^2}{2 \cdot 7 \cdot 8}=\frac{2}{7}\end{aligned}$

$\begin{aligned}\cos 3 C & =4 \cos ^3 C-3 \cos C \\= & \frac{4 \times 8}{7^3}-\frac{6}{7} \\49 \cos 3 C & =\frac{32}{7}-42 \\\Rightarrow \quad & 49 \cos 3 C+42=\frac{32}{7} \\\Rightarrow \quad & m+n =39\end{aligned}$