Đáp án đúng: C
Giải chi tiết:\(\begin{array}{l}\;\;\;\;\;\cos 7x + \sqrt 3 \sin 7x = \sqrt 3 \cos 5x + \sin 5x\,\,\,\,\,\\ \Leftrightarrow \frac{1}{2}\cos 7x + \frac{{\sqrt 3 }}{2}\sin 7x = \frac{{\sqrt 3 }}{2}\cos 5x + \frac{1}{2}\sin 5x\,\\ \Leftrightarrow \cos \frac{\pi }{3}\cos 7x + \sin \frac{\pi }{3}\sin 7x = \cos \frac{\pi }{6}\cos 5x + \sin \frac{\pi }{6}\sin 5x\,\,\,\,\,\,\,\,\,\end{array}\)
\(\begin{array}{l} \Leftrightarrow \cos \left( {7x - \frac{\pi }{3}} \right) = \cos \left( {5x - \frac{\pi }{6}} \right)\\ \Leftrightarrow \left[ \begin{array}{l}7x - \frac{\pi }{3} = 5x - \frac{\pi }{6} + k2\pi \\7x - \frac{\pi }{3} = \pi - \left( {5x - \frac{\pi }{6}} \right) + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}2x = \frac{\pi }{6} + k2\pi \\12x = \frac{{3\pi }}{2} + m2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{{12}} + k\pi \\x = \frac{\pi }{8} + \frac{{m\pi }}{6}\end{array} \right.\,\,,\,\,\,\,\,\,\,\,k,\;m \in Z\end{array}\)
Ta có: \(\left[ \begin{array}{l}0 < \frac{\pi }{{12}} + k\pi < 2\pi \Leftrightarrow - 0,083 < k < 1,91 \Rightarrow k \in \left\{ {0;\;1} \right\}\\0 < \frac{\pi }{8} + \frac{{m\pi }}{6} < 2\pi \Leftrightarrow - 0,75 < m < 11,25 \Rightarrow k \in \left\{ {0;\;1;\;2;..;\;11} \right\}\end{array} \right.\)
Vậy có tất cả 14 nghiệm thõa mãn.
Chọn C.
