solve
$Q1)2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6$
$x^{2}+5y^{2}=6xy$
$Q2)xy-x-y=5$
$2^{x-y}-(\frac{1}{2})^{x-y}=\frac{3}{2}$
$Q3)3^{x}5^{y}=75$
$ 3^{y}5^{x}=45$
$Q4)\log_{y}x+\log _{x}y=2$
$ x^{2}-y=20$
$Q5)5(\log_{y}x+\log _{x}y)=26$
$ xy=64$
عرض للطباعة
solve
$Q1)2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6$
$x^{2}+5y^{2}=6xy$
$Q2)xy-x-y=5$
$2^{x-y}-(\frac{1}{2})^{x-y}=\frac{3}{2}$
$Q3)3^{x}5^{y}=75$
$ 3^{y}5^{x}=45$
$Q4)\log_{y}x+\log _{x}y=2$
$ x^{2}-y=20$
$Q5)5(\log_{y}x+\log _{x}y)=26$
$ xy=64$
$Q5)5(\log_{y}x+\log _{x}y)=26$
$ xy=64$
$sol:$
$log_{y}x +\frac{1}{log_{y}x}=\frac{26}{5}$
$(log_{y}x)^{2}-\frac{26}{5}log_{y}x +1=0$
$(log_{y}x-5)(log_{y}x-\frac{1}{5})=0$
$either y^{5}=x\rightarrow y^{6}=64\rightarrow y=2,x=32$
$or y^{\frac{1}{5}}=x\rightarrow x^{5}=y\rightarrow x^{6}=64\rightarrow x=2,y=32$
$Q6)log_{2}(x+y)+log_{3}(x-y)=1$
$ x^{2}-y^{2}=2$
$Q7)y=1+log_{4}x$
$x^{y}=4^{6}$
$Q8)y^{\frac{1}{x}}=2$
$y^{x}=16$
$Q9)xy=16$
$ x^{log_{2}y}=8$
$Q10)x^{\frac{-1}{2}} + y^{\frac{-1}{2}}=6$
$log_{4}x+log_{4}y=-3$
$Q11)\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}=\frac{3}{2}$
$xy+x+y=9$
$Q12)log_{y}x=2$
$x^{logy}=100$
["]$\LARGE {\color{Red} Q10) \log_{4}x+\log_{4}y=-3\Rightarrow xy=\frac{1}{b4}\Rightarrow y=\frac{1}{b4x}}$
$\LARGE {\color{Red} x^{\frac{-1}{2}}+y^{\frac{-1}{2}}=b\Rightarrow x^{\frac{-1}{2}}+8x^{\frac{1}{2}}=b\rightarrow }$
$\LARGE {\color{Red} 8x-bx^{\frac{1}{2}}+1=0\Rightarrow (4x^{\frac{1}{2}}-1)(2x^{\frac{1}{2}}-1)=0\Rightarrow x=\frac{1}{1b} , x=\frac{1}{4}}$
$\LARGE {\color{Red} \Rightarrow y=\frac{1}{4} , y=\frac{1}{1b}}$ [/COLOR]
للتذكير سؤال 11و12 ؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟
$Q13)\sqrt{6}cosx-\sqrt{2}\left | sinx \right |=2$
\[Q13)\sqrt{6}cosx-\sqrt{2}|sinx|=2\left \{ 2\sqrt{2} \right \}\\\frac{\sqrt{3}}{2}cosx-\frac{1}{2}|sinx|=\frac{1}{\sqrt{2}}\Rightarrow \\(1)cos\frac{\pi }{6}cosx-sin\frac{\pi }{6}sinx=cos\frac{\pi }{4}\\cos\left ( \frac{\pi }{6} +x\right )=cos\frac{\pi }{4}=cos\frac{7\pi }{4}\Rightarrow x=\left \{ \frac{\pi }{12},\frac{19\pi }{12} \right \}\\(2)cos\frac{\pi }{6}cosx+sin\frac{\pi }{6}sinx=cos\frac{\pi }{4}=cos\frac{7\pi }{4}\\cos(x-\frac{\pi }{6})=cos\frac{\pi }{4}=cos\frac{7\pi }{4}\Rightarrow x=\left \{ \frac{5\pi }{12},\frac{23\pi }{12} \right \}\]اقتباس:
المشاركة الأصلية كتبت بواسطة فلاح الناصري
5pi/12 لاتحقق
$Q14)solve .in [0,\pi ]$
$cos2x+cos4x=\left \lfloor cos4x \right \rfloor$
$Q15)log_{2}x=log_{3}(5-x^{log_{2}3})$
$\large {\color{Blue} \log_{2}x=\log_{3}(a-x^{\log_{2}3})\rightarrow \log_{2}x=\log_{3}(a-3^{\log_{2}x})}$
$\large {\color{Blue} let ,y=\log_{2}x\rightarrow x=2^{y}}$
$\large {\color{Blue} \Rightarrow y=\log_{3}(a-3^{y})\rightarrow a-3^{y}=3^{y}\rightarrow 3^{y}=\frac{a}{2}}$
$\large {\color{Blue} \Rightarrow y\log3=\log(\frac{a}{2})\rightarrow y=\frac{\log(\frac{a}{2})}{\log3}}$
$\large {\color{Blue}\Rightarrow x=2^{(\frac{\log\frac{a}{2}}{\log3})},, {a=five}}$
$Q16)arccos(\frac{7x+5}{13})=arcsin(\frac{4x+1}{13 })$
$Q17)arcsin(\frac{\sqrt{3x+2}}{2})=arccot( \sqrt{\frac{2}{x+1}})$
$Q18)(arcsinx) (arcsiny)=\frac{\pi ^{2}}{12}$
$(arccosx)(arccosy)=\frac{\pi ^{2}}{24}$
\[Q16)arccos\left ( \frac{7x+5}{13} \right )=arcsin\left ( \frac{4x+1}{13} \right )\\sin\left ( arccos\left ( \frac{7x+5}{13} \right ) \right )=\frac{4x+1}{13}\\\sqrt{1-\left (\frac{7x+5}{13} \right )^{2}}=\frac{4x+1}{13}\\\therefore 65x^{2}+78x-143=0\Rightarrow{\color{Blue} x=1},{\color{Red} x=-\frac{11}{5}\times }\]
$Q19)solve .in.R$
$ tanx+tany=1$
$x+y=\frac{\pi }{4}$
$Q20)solve.in.[0,2\pi ]$
$sinx+siny=1$
$cos2x-cos2y=1$
$\large {\color{Red} x+y=\frac{\pi}{4}\Rightarrow \tan(x+y)=1\Rightarrow 1-tanxtany=1}$
$\large {\color{Red} \tan(x)\tan(y)=0\Rightarrow x=n\pi ,,OR,, y=n\pi}$
\$\large {\color{Red} \rightarrow y=(\pi/4)-n\pi,,OR,,x=(\pi/4)-n\pi}$
$\large {\color{Red} cos2x-cos2y=1\Rightarrow 1-2sin^{2}x-1+2sin^{2}y=1}$
${\color{Red} \Rightarrow 2(sin^{2}y-sin^{2}x)=1\rightarrow (siny-sinx)(siny+sinx)=1/2} $
${\color{Red} \Rightarrow sinx+siny=1,,siny-sinx=1/2\Rightarrow siny=3/4,,sinx=1/4}$
$ {\color{Red} \Rightarrow x=0.24,,x=\pi-0.24,,y=0,88,,y=\pi-0.88}$
$Q21)solve.in.[0,2\pi ]$
$sinx+cosy=1$
$cos2x-cos2y=1$
للتذكير ؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟؟
$Q22)1+\left | log_4{(9x^{2}-39x+43)} \right |=\left | cos((x-2)cosx) \right |$
$Q23)solve in [0,\pi ]$
$sin^{8}x+cos^{8}x-cos^{2}2x=0$
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$Q24)solve in [0,\pi ]$
$sin^{10}x+cos^{6}x=1$