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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 $$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{is}\:\mathrm{intrinsic}\:\mathrm{by}\:\mathrm{showing} \\ $$$${K}=\frac{\begin{vmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}{E}_{{vv}} +{F}_{{uv}} −{G}_{{uu}} }&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{u}} }&{{F}_{{u}} −\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }\\{\:\:\:\:\:\:\:\:\:\:\:\:{F}_{{v}} −\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }&{\:\:\:{E}}&{\:\:\:\:\:\:{F}}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}{G}_{{v}} }&{\:\:\:{F}}&{\:\:\:\:\:{G}}\end{vmatrix}−\begin{vmatrix}{\:\:\:\:\mathrm{0}}&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }&{\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }\\{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}}…
Question Number 225604 by mr W last updated on 04/Nov/25 Commented by mr W last updated on 04/Nov/25 $${An}\:{uniform}\:{cylinder}\:{with}\:{radius}\:{R} \\ $$$${is}\:{spinned}\:{about}\:{its}\:{axis}\:{to}\:{the} \\ $$$${angular}\:{velocity}\:\omega_{\mathrm{0}} \:{and}\:{then}\:{placed} \\…
Question Number 225601 by gregori last updated on 04/Nov/25 $$\:\:\: \mathrm{2}^{\mathrm{2025}} ×\mathrm{3}^{\mathrm{2025}} \: \\ $$ Commented by mr W last updated on 04/Nov/25 $${the}\:{unit}\:{digit}\:{of}\:\mathrm{6}^{{n}} {is}\:{always}\:\mathrm{6}.…
Question Number 225581 by Lara2440 last updated on 03/Nov/25 $$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{intrinsic} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{same}\:\mathrm{thing}\:\mathrm{as}\:\mathrm{saying}; \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{can}\:\mathrm{only}\:\mathrm{consist}\:\mathrm{of} \\ $$$$\mathrm{First}\:\mathrm{Fundamental}\:\mathrm{Form}\:\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{Derivatives}. \\ $$ Terms of Service Privacy Policy…
Question Number 225568 by necx122 last updated on 03/Nov/25 $${A}\:{farmer}\:{produces}\:{seeds}\:{in}\:{packets} \\ $$$${for}\:{sale}.\:{The}\:{probability}\:{that}\:{a}\:{seed} \\ $$$${selected}\:{at}\:{random}\:{will}\:{grow}\:{is}\:\mathrm{0}.\mathrm{8}. \\ $$$${If}\:{there}\:{are}\:\mathrm{20}\:{seeds},\:{what}\:{is}\:{the} \\ $$$${probability}\:{that}\:{less}\:{than}\:\mathrm{2}\:{will}\:{not} \\ $$$${grow}? \\ $$ Commented by necx122…