Câu hỏi 11

Điều kiện:  \(\left\{ \begin{array}{l}\sin \left( {3x - \frac{\pi }{2}} \right) \ne 0\\\sin x \ne 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}3x - \frac{\pi }{2} \ne m\pi \\x \ne n\pi \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \ne \frac{\pi }{6} + \frac{{m\pi }}{3}\\x \ne n\pi \end{array} \right.\,\,\,\,\,(m,\;n \in \mathbb{Z}).\)

 \(\begin{array}{l}\,\,\,\,\,\cot \left( {3x - \frac{\pi }{2}} \right) = \cot x\\ \Leftrightarrow 3x - \frac{\pi }{2} = x + k\pi \\ \Leftrightarrow x = \frac{\pi }{4} + \frac{{k\pi }}{2}\,\;\;\left( {tm} \right)\,\,\,\,\,(k \in \mathbb{Z}).\end{array}\)

Phương trình có nghiệm thuộc \(\left[ {0;\;20} \right]\)  

\( \Leftrightarrow 0 \le \frac{\pi }{4} + \frac{{k\pi }}{2} \le 20 \Leftrightarrow \frac{{ - 1}}{2} \le k \le \frac{{20 - \frac{\pi }{4}}}{{\frac{\pi }{2}}} \approx 12.23 \Rightarrow k \in \{ 0;\;1;\;2;...;\;11;\;12\} \).

Tổng các nghiệm là:\(\sum\limits_{k = 0}^{12} {\left( {\frac{\pi }{4} + \frac{{k\pi }}{2}} \right)}  = 13 \cdot \frac{\pi }{4} + \frac{\pi }{2}(0 + 1 + 2 + ... + 12) = \frac{{169\pi }}{4}\)

Chọn A.