Tính tổng S các nghiệm trên đoạn \[\left[ {{\rm{0; \pi }}} \right]\] của phương trình \[\left( {{\rm{1 + cosx}}} \right)\left( {{\rm{2sinx}} - {\rm{cosx}}} \right){\rm{ = si}}{{\rm{n}}^{\rm{2}}}{\rm{x}}\]

\[{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{x + co}}{{\rm{s}}^{\rm{2}}}{\rm{4x = 1}} \Leftrightarrow {\mathop{\rm c}\nolimits} {\rm{o}}{{\rm{s}}^{\rm{2}}}{\rm{4x = co}}{{\rm{s}}^{\rm{2}}}{\rm{x}} \Leftrightarrow {\mathop{\rm c}\nolimits} {\rm{os8x = cos2x}} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x\,{\rm{ = }}\,\frac{{k\pi }}{3}}\\{x\,{\rm{ = }}\,\frac{{l\pi }}{5}}\end{array};\left( {k,l \in \mathbb{Z}} \right)} \right.\]

Mà \(x \in \left( {0;\pi } \right) \Rightarrow \left[ {\begin{array}{*{20}{c}}{0 < k\frac{\pi }{3} < \pi }\\{0 < k\frac{\pi }{5} < \pi }\end{array}} \right. \Rightarrow \left[ {\begin{array}{*{20}{c}}{0 < k < 3 \Rightarrow k\,{\rm{ = }}1;\,k\,{\rm{ = }}\,2}\\{0 < l < 5 \Rightarrow l = 1;l{\rm{ = }}2;\,l\,{\rm{ = }}\,4}\end{array}} \right.\)

\[ \Rightarrow {\rm{x}} \in \left\{ {\frac{{\rm{\pi }}}{{\rm{3}}}{\rm{; }}\frac{{{\rm{2\pi }}}}{{\rm{3}}}{\rm{; }}\frac{{\rm{\pi }}}{{\rm{5}}}{\rm{; }}\frac{{{\rm{2\pi }}}}{{\rm{5}}}{\rm{; }}\frac{{{\rm{3\pi }}}}{{\rm{5}}}{\rm{; }}\frac{{{\rm{4\pi }}}}{{\rm{5}}}} \right\}\]

Suy ra nghiệm lớn nhất là \[{{\rm{x}}_{\rm{0}}}{\rm{ = }}\frac{{{\rm{4\pi }}}}{{\rm{5}}} \Leftrightarrow {\rm{P = 41}}\]Đáp án cần chọn là: B

\[{\rm{si}}{{\rm{n}}^{\rm{3}}}{\rm{x + sinxcosx = 1}} - {\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{x}} \Leftrightarrow {\rm{si}}{{\rm{n}}^{\rm{3}}}{\rm{x + co}}{{\rm{s}}^{\rm{3}}}{\rm{x = 1}} - {\rm{sinxcosx}}\]

\[ \Leftrightarrow \left( {{\rm{sinx + cosx}}} \right)\left( {{\rm{1}} - {\rm{sinxcosx}}} \right){\rm{ = 1}} - {\rm{sinxcosx}}\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{1 - sinxco{\rm{sx = 0}}}\\{{\rm{sinx + cosx = 1}}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{sin2x = 2(VN)}}}\\{{\rm{sin}}\left( {{\rm{x + }}\frac{{\rm{\pi }}}{{\rm{4}}}} \right){\rm{ = }}\frac{{\rm{1}}}{{\sqrt {\rm{2}} }}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{x = k2\pi }}}\\{{\rm{x = }}\frac{{\rm{\pi }}}{{\rm{2}}}{\rm{ + k2\pi }}}\end{array}} \right.,k \in \mathbb{Z}\)

Vì \[{\rm{x}} \in \left[ {{\rm{0; }}\frac{{\rm{\pi }}}{{\rm{2}}}} \right] \Rightarrow {\rm{x}} \in \left\{ {{\rm{0; }}\frac{{\rm{\pi }}}{{{\rm{ 2}}}}} \right\}\]Đáp án cần chọn là: C