Question Number 225658 by Jubr last updated on 05/Nov/25 Answered by ajfour last updated on 06/Nov/25 $${p}={R}\mathrm{sin}\:\alpha \\ $$$$\pi−\alpha=\frac{\mathrm{5}}{\mathrm{2}}\:\:\Rightarrow\:\:\mathrm{2}\alpha=\mathrm{2}\pi−\mathrm{5} \\ $$$$\angle{AOC}=\mathrm{2}\left(\frac{\pi}{\mathrm{2}}−\alpha\right)=\pi−\mathrm{2}\alpha \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\pi−\left(\mathrm{2}\pi−\mathrm{5}\right)=\mathrm{5}−\pi \\ $$$${A}_{{shade}}…
Question Number 225652 by fantastic last updated on 05/Nov/25 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{i}\pi{x}} \:{dx} \\ $$ Answered by mahdipoor last updated on 05/Nov/25 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ae}^{\mathrm{i}\pi\mathrm{x}} \right)=\left(\mathrm{i}\pi\mathrm{a}\right)\mathrm{e}^{\mathrm{i}\pi\mathrm{x}} =\mathrm{e}^{\mathrm{i}\pi\mathrm{x}}…
Question Number 225664 by jklasd last updated on 05/Nov/25 $${If}\:{a}\left({x}\right)=\mathrm{1}\:{and}\:\underset{{n}=−\mathrm{1}} {\overset{{Qw}_{{fr}.} \left(\frac{\mathrm{1}}{{x}−\mathrm{1}}×{x}\right)} {{W}}a}''\left({ust}^{{n}} {x}\right)=\mathrm{0}; \\ $$$${What}\:{value}\:{of}\:{Tk}\left({x}^{\mathrm{2}} \right)? \\ $$$$\left({This}\:{is}\:{nonstandartmath}\:{exercise}\right) \\ $$ Terms of Service Privacy…
Question Number 225666 by mr W last updated on 05/Nov/25 Commented by mr W last updated on 05/Nov/25 $${an}\:{alternative}\:{solution} \\ $$ Commented by ajfour last…
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Question Number 225629 by sonukgindia last updated on 05/Nov/25 Answered by Spillover last updated on 05/Nov/25 $$\frac{\pi}{\mathrm{2}}\mathrm{log}\left(\frac{\mathrm{2}+\sqrt{\mathrm{3}}}{\mathrm{4}}\right)? \\ $$ Answered by Spillover last updated on…
Question Number 225613 by Jubr last updated on 04/Nov/25 Answered by mahdipoor last updated on 04/Nov/25 $$\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}=\mathrm{0}\:\Rightarrow\frac{−\mathrm{1}\pm\sqrt{\mathrm{1}^{\mathrm{2}} −\mathrm{4}.\mathrm{1}.\mathrm{1}}}{\mathrm{2}.\mathrm{1}}=\frac{−\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i} \\ $$$$=\mathrm{cos}\left(\pm\mathrm{120}\right)+\mathrm{i}.\mathrm{sin}\left(\pm\mathrm{120}\right)=\mathrm{e}^{\mathrm{i}\left(\pm\frac{\mathrm{2}\pi}{\mathrm{3}}\right)} \\ $$$$\mathrm{v}_{\mathrm{n}} =\mathrm{e}^{\mathrm{i}\theta\mathrm{n}} +\mathrm{e}^{\mathrm{i}\left(−\theta\right)\mathrm{n}}…
Question Number 225599 by Jubr last updated on 04/Nov/25 Commented by Frix last updated on 04/Nov/25 $$\mathrm{If}\:{a},\:{b},\:{c},\:{d}\:\in\mathbb{R}\:\mathrm{no}\:\mathrm{maximum}\:\mathrm{exists}. \\ $$$$\mathrm{Let}\:{a}={b}=−{r};\:{c}=\mathrm{1};\:{d}=\mathrm{2}{r} \\ $$$$\left(\mathrm{1}−{r}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{r}\right)=\mathrm{1}−\mathrm{3}{r}^{\mathrm{2}} +\mathrm{2}{r}^{\mathrm{3}} \\ $$$$\underset{{r}\rightarrow+\infty}…
Question Number 225625 by fantastic last updated on 04/Nov/25 $${quick}\:{short}\:{Q} \\ $$$$\mathrm{2}\:{things}\:{are}\:{mixed}\: \\ $$$$\left.\mathrm{1}\right){both}\:{same}\:{volume} \\ $$$$\left.\mathrm{2}\right){both}\:{same}\:{mass} \\ $$$${density}\Rightarrow{d}_{{v}} \:{and}\:{d}_{{m}} \\ $$$${d}_{{v}} \:?\:{d}_{{m}} \left[=,<\:{or}>\right] \\ $$…
Question Number 225610 by Lara2440 last updated on 06/Nov/25 $$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{is}\:\mathrm{intrinsic}\:\mathrm{by}\:\mathrm{showing} \\ $$$${K}=\frac{\begin{vmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}{E}_{{vv}} +{F}_{{uv}} −{G}_{{uu}} }&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{u}} }&{{F}_{{u}} −\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }\\{\:\:\:\:\:\:\:\:\:\:\:\:{F}_{{v}} −\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }&{\:\:\:{E}}&{\:\:\:\:\:\:{F}}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}{G}_{{v}} }&{\:\:\:{F}}&{\:\:\:\:\:{G}}\end{vmatrix}−\begin{vmatrix}{\:\:\:\:\mathrm{0}}&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }&{\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }\\{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}}…