Câu 1: Giải pt lượng giác:
[TEX]1 +4cosx.cos3x =\frac{1}{2sin.\frac{x}{2}}[/TEX]
Câu 2: Giải hệ pt
[TEX] \left\{ x+y-z=7 \\ x^2 +y^2 -z^2 =37 \\ x^3 +y^3 -z^3 =1 [/TEX]
Câu 3: Giải pt :
[TEX] \sqrt[3]{2x+3 }= x^3 +3x^2 +2x -1 [/TEX]
Bài 1
[TEX]\begin{array}{l}DK:x \ne k2\pi \\\sin x = 0:ko - la - no\\ = > pt \Leftrightarrow \sin x\left( {1 + 2\cos 2x + 2\cos 4x} \right) = \frac{{\sin x}}{{2\sin \frac{x}{2}}}\\\Leftrightarrow \sin 5x = c{\rm{os}}\frac{x}{2} = \sin \left( {\frac{\pi }{2} - \frac{x}{2}} \right) \Leftrightarrow \left[ \begin{array}{l}x = \frac{{\left( {4k + 1} \right)\pi }}{{11}}\\x = \frac{{\left( {4k + 1} \right)\pi }}{9}\end{array} \right.\end{array}\][/TEX]
Bài 2
[TEX]\begin{array}{l}hpt \Leftrightarrow \left\{ \begin{array}{l}x + y = 7 + z(1)\\{x^2} + {y^2} - {z^2} = 37(2)\\{x^3} + {y^3} - {z^3} = 1\end{array} \right.\\(1) + (2) = > xy = 6 + 7z\\hpt <= > \left\{ \begin{array}{l}xy = 6 + 7z\\x + y = z + 7\\{x^3} + {y^3} - {z^3} = 1\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}xy = 6 + 7z\\x + y = z + 7\\{\left( {z + 7} \right)^3} - 3\left( {6 + 7z} \right)\left( {z + 7} \right) - {z^3} = 1\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}z = 12\\x = 10;y = 9\\x = 9;y = 10\end{array} \right.\end{array}[/TEX]
Bài 3
[TEX]\begin{array}{l}\sqrt[3]{{2x + 3}} = {\left( {x + 1} \right)^3} - x - 2\\y + 1 = \sqrt[3]{{2x + 3}}\\ = > \left\{ \begin{array}{l}{\left( {x + 1} \right)^3} - x - 2 - y - 1 = 0\\{\left( {y + 1} \right)^3} - 2x - 3 = 0\end{array} \right. < = > \left\{ \begin{array}{l}{\left( {x + 1} \right)^3} - {\left( {y + 1} \right)^3} + \left( {x - y} \right) = 0\\{\left( {y + 1} \right)^3} - 2x - 3 = 0\end{array} \right.\\\Leftrightarrow \left\{ \begin{array}{l}\left( {x - y} \right)\left[ {{{\left( {x + 1} \right)}^2} + \left( {x + 1} \right)\left( {y + 1} \right) + {{\left( {y + 1} \right)}^2}} \right] = 0\\{\left( {y + 1} \right)^3} - 2x - 3 = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = y\\{x^3} + 3{x^2} + x - 2 = 0\end{array} \right. \Leftrightarrow x = - 2;x = \frac{{ - 1 + \sqrt 5 }}{2};x = \frac{{ - 1 - \sqrt 5 }}{2}\end{array}[/TEX]
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