Xác định $a$ để hệ sau có nghiệm duy nhất :
$\left\{ \begin{array}{l}
{2^{\left| x \right|}} + \left| x \right| = y + {x^2} + a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
{x^2} + {y^2} = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
\end{array} \right.$
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Giải và biện luận theo $a$ hệ sau :
$\left\{ \begin{array}{l}
2cos\,x + a. \sin\,y = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
{\log _z}\sin y = {\log _z}a.{\log _a}\left( {2 - 3cos\,x} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\\
{\log _a}z + {\log _a}\left( {\frac{1}{{2a}} - 1} \right) = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)
\end{array} \right.$
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Cho hệ phương trình :
$\left\{ \begin{array}{l}
9{x^2} - 4{y^2} = 5\\
{\log _m}\left( {3x + 2y} \right) - {\log _3}\left( {3x - 2y} \right) = 1
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$
$1)$ Giải $(1)$ khi $m = 5$
$2)$ Tìm giá trị lớn nhất của tham số $m$ sao cho hệ $(1)$ có nghiệm $\left( {x,\,y} \right)$ thỏa mãn $3x + 2y \le 5$
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Giải hệ :
$1)\,\,\,\left\{ \begin{array}{l}
y\,\sin \,x = {\log _2}\left| {\frac{{y\,\sin \,x}}{{1 + 3y}}} \right|\\
\left( {6{y^2} + 2y} \right)\left( {{4^{{{\sin }^2}x}} + {4^{co{s^2}x}}} \right) = 25{y^2} + 6y + 1\\
\left| y \right| \le 1
\end{array} \right.$
$2)\,\,\,\left\{ \begin{array}{l}
{\left( {3 - y} \right)}co{s^2}x = {\log _3}\left| {\frac{{8 + y}}{{y\left( {1 - \sin {\,^3}x} \right)}}} \right|\\
\left( {{y^2} + 8y} \right)\left( {{3^{2 + 2{{\sin }^4}x}} + {3^{2co{s^4}x + {{\sin }^2}2x - 4}}} \right) = 2{y^2} + 16y + 64\\
1 \le y < 10
\end{array} \right.$
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Giải hệ :
$\left\{ \begin{array}{l}
{\log _6}\left( {\sqrt x + \sqrt[4] x} \right) = \frac{1}{4}{\log _2}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
\frac{{\sin \,\frac{{16\pi }}{x}}+1}{{cos\,\frac{{x\pi }}{{16}}}} < 1 - cos\,\frac{{\sqrt x \pi }}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
\end{array} \right.$
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Giải các hệ :
$\begin{array}{l}
1)\,\,\,\left\{ \begin{array}{l}
{\log _{2 - x}}\left( {2 - y} \right) > 0\\
{\log _{4 - y}}\left( {2x - 2} \right) > 0
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3)\,\,\left\{ \begin{array}{l}
{\log _{x - 2}}\left( {2y - 4} \right) > 0\\
{\log _{3 - y}}\left( {x - 4} \right) > 0
\end{array} \right.\\
2)\,\,\,\left\{ \begin{array}{l}
{\log _{x - 1}}\left( {5 - y} \right) < 0\\
{\log _{2 - y}}\left( {4 - x} \right) < 0
\end{array} \right.
\end{array}$
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Giải các hệ :
$\begin{array}{l}
1)\,\,\,\,\left\{ \begin{array}{l}
{2^{{{\log }_{\frac{1}{2}}}\left( {x + y} \right)}} = {5^{{{\log }_5}\left( {x - y} \right)}}\\
{\log _2}x + {\log _2}y = \frac{1}{2}
\end{array} \right.\\
2)\,\,\,\left\{ \begin{array}{l}
{\log _2}xy.{\log _2}\frac{x}{y} = - 3\\
\log _2^2x + \log _2^2y = 5
\end{array} \right.\\
3)\,\,\,\left\{ \begin{array}{l}
{x^2} = 1 + 6{\log _4}x\\
{y^2} = {2^x}.y + {2^{2x + 1}}
\end{array} \right.\\
4)\,\,\,\left\{ \begin{array}{l}
{\log _2}x + {\log _4}y + {\log _4}z = 2\\
{\log _3}y + {\log _9}z + {\log _9}x = 2\\
{\log _4}z + {\log _{16}}x + {\log _{16}}y = 2
\end{array} \right.
\end{array}$
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Giải các hệ :
$\begin{array}{l}
1)\,\,\,\left\{ \begin{array}{l}
{y^{1 - \frac{2}{5}{{\log }_x}y}} = {x^{\frac{2}{5}}}\\
1 + {\log _x}\left( {1 - \frac{{3y}}{x}} \right) = {\log _x}4
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
2)\,\,\,\left\{ \begin{array}{l}
y.{x^{{{\log }_y}x}} = {x^{\frac{5}{2}}}\\
{\log _4}y.{\log _y}\left( {y - 3x} \right) = 1
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
\end{array}$
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Giải các hệ :
$\begin{array}{l}
1)\,\,\,\left\{ \begin{array}{l}
\left( {1 + 2{{\log }_{\left| {xy} \right|}}2} \right).{\log _{x + y}}\left| {xy} \right| = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
x - y = 2\sqrt {3\,} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
\end{array} \right.\\
2)\,\,\,\left\{ \begin{array}{l}
{\log _{\left| {xy} \right|}}\left( {x - y} \right) = 1\\
2{\log _5}\left| {xy} \right|.{\log _{\left| {xy} \right|}}\left( {x + y} \right) = 1
\end{array} \right.
\end{array}$
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Giải các hệ :
$\begin{array}{l}
1)\,\,\,\left\{ \begin{array}{l}
x + {2^{y + 1}} = 3\\
4x + {4^y} = 32
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3)\,\,\,\left\{ \begin{array}{l}
x + {3^{y - 1}} = 2\\
3x + {9^y} = 18
\end{array} \right.\\
2)\,\,\left\{ \begin{array}{l}
\sqrt y + \log {x^2} = 2\\
y + 4\log x = 28
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\,\,\left\{ \begin{array}{l}
\sqrt y + 2\log x = 3\\
y - 3\log {x^2} = 1
\end{array} \right.
\end{array}$
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Giải các hệ phương trình:
$\begin{array}{l}
1)\,\,\,\left\{ \begin{array}{l}
{x^{x + y}} = {y^{12}}\\
{y^{x + y}} = {x^3}
\end{array} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x,y > 0)\\
2)\,\,\,\left\{ \begin{array}{l}
{\log _2}\left( {x + y} \right) - {\log _3}\left( {x - y} \right) = 1\\
{x^2} - {y^2} = 1
\end{array} \right.
\end{array}$
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