Question Number 209304 by RoseAli last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $${a}^{{x}} +{b}^{{x}} =\mathrm{62} \\ $$$${a}+{b}=\mathrm{8};{ab}=\mathrm{1} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\left({a}+{b}\right)^{\mathrm{2}}…
Question Number 209307 by efronzo1 last updated on 06/Jul/24 Commented by mr W last updated on 06/Jul/24 $$\mid{QR}\mid<\frac{\mathrm{8}}{\mathrm{3}}−\mathrm{1}=\frac{\mathrm{5}}{\mathrm{3}} \\ $$$$\frac{{QR}}{{PR}}=\frac{\mathrm{2}}{\mathrm{1}}\:\Rightarrow\frac{{PQ}}{{QR}}=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\mid{PQ}\mid=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}×{QR}<\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}×\frac{\mathrm{5}}{\mathrm{3}}=\frac{\mathrm{5}\sqrt{\mathrm{5}}}{\mathrm{6}}<\sqrt{\mathrm{5}} \\ $$$$\Rightarrow{impossible}\:{that}\:\mid{PQ}\mid=\sqrt{\mathrm{5}}\:! \\…
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Question Number 209288 by Jubr last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $$\frac{{x}}{{r}}=\frac{\mathrm{8}}{{r}}\Rightarrow{x}=\mathrm{8} \\ $$$${r}^{\mathrm{2}} =\left(\mathrm{2}{x}\right)^{\mathrm{2}} +\left({r}−\mathrm{8}\right)^{\mathrm{2}} \Rightarrow{r}=\mathrm{20} \\ $$$$\left[{Yellow}\right]=\frac{\mathrm{90}}{\mathrm{360}}×\mathrm{400}\pi−\frac{\mathrm{20}×\mathrm{20}}{\mathrm{2}}=\mathrm{100}\pi−\mathrm{200} \\…
Question Number 209289 by Jubr last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $$\frac{{s}}{{x}}=\frac{\mathrm{7}}{\mathrm{21}}\Rightarrow{x}=\mathrm{3}{s} \\ $$$$\frac{{s}}{{y}}=\frac{\mathrm{14}}{\mathrm{21}}\Rightarrow{y}=\frac{\mathrm{3}{s}}{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{21}^{\mathrm{2}} \Rightarrow\mathrm{9}{s}^{\mathrm{2}} +\frac{\mathrm{9}{s}^{\mathrm{2}}…
Question Number 209290 by klipto last updated on 06/Jul/24 $$\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}−^{\mathrm{1000}} \sqrt{\left(\mathrm{1}+\mathrm{8000}\boldsymbol{\mathrm{a}}\right)}=\mathrm{1000} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{a}} \\ $$ Commented by mr W last updated on 06/Jul/24 $${you}\:{can}\:{only}\:{approximate}!…
Question Number 209316 by klipto last updated on 06/Jul/24 $$\boldsymbol{\mathrm{if}}\:\:\mathrm{2}\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{n}}^{\mathrm{3}} =\boldsymbol{\mathrm{n}}! \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{n}} \\ $$ Answered by A5T last updated on 07/Jul/24 $${n}\left(\mathrm{2}+\mathrm{3}{n}\right)=\left({n}−\mathrm{1}\right)! \\…
Question Number 209318 by essaad last updated on 06/Jul/24 Answered by Frix last updated on 06/Jul/24 $$\mathrm{We}\:\mathrm{know}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{k}}\:=\infty \\ $$$${k}>\mathrm{1}:\:\frac{\mathrm{1}}{\:\sqrt{{k}}}>\frac{\mathrm{1}}{\:{k}}\:\Rightarrow\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{{k}}}\:>\underset{{k}=\mathrm{1}} {\overset{\infty}…
Question Number 209281 by Tawa11 last updated on 06/Jul/24 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{r},\:\mathrm{if}\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{r}} \:\:=\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{2r}\:\:+\:\:\mathrm{1}} \\ $$ Commented by klipto last updated on 06/Jul/24 $$\:\:\:\:\:\:\:^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{C}}_{\mathrm{r}} =^{\mathrm{n}}…