Question Number 134002 by Dwaipayan Shikari last updated on 26/Feb/21 $${What}\:{will}\:{be}\:{the}\:{minimum}\:{area}\:{of}\:{a}\:{heptagon}\:{inscribed}\:{in} \\ $$$${an}\:{unit}\:{square}? \\ $$ Commented by Dwaipayan Shikari last updated on 26/Feb/21 Answered by…
Question Number 133997 by rs4089 last updated on 26/Feb/21 $${lim}_{{n}\rightarrow\infty} \left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…+\frac{\mathrm{1}}{{n}}}{{n}^{\mathrm{2}} }\right)^{{n}} \\ $$ Answered by mathmax by abdo last updated on 27/Feb/21 $$\mathrm{U}_{\mathrm{n}} =\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+….+\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{n}^{\mathrm{2}}…
Question Number 133996 by mr W last updated on 26/Feb/21 $${show}\:{that}\:\sqrt{\mathrm{2}}<\mathrm{log}_{\mathrm{2}} \:\mathrm{3}<\sqrt{\mathrm{3}} \\ $$ Answered by som(math1967) last updated on 26/Feb/21 $$\:\:\mathrm{2}^{\sqrt{\mathrm{2}}} <\mathrm{3}<\mathrm{2}^{\sqrt{\mathrm{3}}} \\ $$$${log}_{\mathrm{2}}…
Question Number 133995 by mr W last updated on 26/Feb/21 $${in}\:{how}\:{many}\:{ways}\:{can}\:{n}\:{men}\:{and} \\ $$$${n}\:{women}\:{be}\:{arranged}\:{in}\:{a}\:{row}\:{such} \\ $$$${that}\:{men}\:{and}\:{women}\:{alternate}? \\ $$ Commented by benjo_mathlover last updated on 26/Feb/21 $$=\:\mathrm{2}×\mathrm{n}!×\mathrm{n}!\:=\:\mathrm{2}×\left(\mathrm{n}!\right)^{\mathrm{2}}…
Question Number 133994 by benjo_mathlover last updated on 26/Feb/21 $$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\:\mathrm{satisfies} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=\begin{cases}{{x}+{a}\sqrt{\mathrm{2}}\:\mathrm{sin}\:{x}\:;\:\mathrm{0}\leqslant{x}<\frac{\pi}{\mathrm{4}}}\\{\mathrm{2}{x}\:\mathrm{cot}\:{x}\:+{b}\:;\:\frac{\pi}{\mathrm{4}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}}\\{{a}\:\mathrm{cos}\:\mathrm{2}{x}−{b}\mathrm{sin}\:{x}\:;\:\frac{\pi}{\mathrm{2}}<{x}\leqslant\pi}\end{cases} \\ $$$$\:\mathrm{continuous}\:\mathrm{in}\:\left[\:\mathrm{0},\pi\:\right],\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{and}\:{b}.\: \\ $$ Answered by EDWIN88 last updated on 26/Feb/21…
Question Number 133989 by Mamifere last updated on 26/Feb/21 $$\frac{{x}}{{x}\mathrm{2}+\mathrm{1}}<{arctan}\left({x}\right)<{x} \\ $$$$ \\ $$ Answered by Ñï= last updated on 26/Feb/21 $${Let}\:{f}\left({x}\right)={tan}^{−\mathrm{1}} {x} \\ $$$${we}\:{have}\:{f}\left({x}\right)−{f}\left(\mathrm{0}\right)={f}\left(\xi\right)'\left({x}−\mathrm{0}\right)…
Question Number 133988 by Dwaipayan Shikari last updated on 26/Feb/21 $$\left(\mathrm{2}+\frac{\pi}{{e}}\right)\left(\frac{\mathrm{17}}{\mathrm{16}}+\frac{\pi}{\mathrm{4}{e}}\right)\left(\frac{\mathrm{82}}{\mathrm{81}}+\frac{\pi}{\mathrm{9}{e}}\right)\left(\frac{\mathrm{257}}{\mathrm{256}}+\frac{\pi}{\mathrm{16}{e}}\right)… \\ $$ Commented by Olaf last updated on 26/Feb/21 $$\mathrm{P}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{?+\mathrm{1}}{?}+\frac{\pi}{{n}^{\mathrm{2}} {e}}\right) \\…
Question Number 133991 by EDWIN88 last updated on 26/Feb/21 $$\:\mid\:\mathrm{3x}−\mid\:\mathrm{4x}+\mathrm{2}\:\mid\mid\:\geqslant\:\mathrm{4}\:>\:\mid\:\mathrm{5x}+\mathrm{8}\:\mid\: \\ $$ Answered by benjo_mathlover last updated on 26/Feb/21 $$\:\mid\mathrm{5x}+\mathrm{8}\mid\:<\:\mathrm{4}\:\leqslant\:\mid\:\mathrm{3x}−\mid\mathrm{4x}+\mathrm{2}\mid\mid \\ $$$$\left(\bullet\right)\:\mid\mathrm{5x}+\mathrm{8}\mid\:<\:\mathrm{4}\:\Rightarrow−\mathrm{4}<\mathrm{5x}+\mathrm{8}\:<\:\mathrm{4} \\ $$$$\Rightarrow−\mathrm{12}\:<\:\mathrm{5x}\:<\:−\mathrm{4}\:;\:−\frac{\mathrm{12}}{\mathrm{5}}\:<\mathrm{x}\:<−\frac{\mathrm{4}}{\mathrm{5}} \\…
Question Number 2917 by Rasheed Soomro last updated on 30/Nov/15 $$\mathcal{W}{hat}\:{is}\:{DMAS}\:\:{rule}?\:{Where}\:{is}\:{it}\:{followed}? \\ $$$${Do}\:{we}\:{follow}\:{this}\:{rule}?\: \\ $$ Answered by 123456 last updated on 30/Nov/15 $$\mathrm{if}\:\mathrm{i}\:\mathrm{understanded}\:\mathrm{it}\:\mathrm{about}\:\mathrm{order}\:\mathrm{of}\:\mathrm{operation} \\ $$$$\mathrm{like}…
Question Number 68451 by mr W last updated on 10/Sep/19 Commented by mr W last updated on 11/Sep/19 $${if}\:{M}=\mathrm{0},\:{the}\:{block}\:{m}\:{just}\:{has}\:{free} \\ $$$${fall},\:{therefore}\:{T}=\sqrt{\mathrm{2}{gh}}=\sqrt{\mathrm{2}{gR}}. \\ $$ Commented by…