Question Number 86206 by behi83417@gmail.com last updated on 27/Mar/20 $$\mathrm{1}.\mathrm{line}:\boldsymbol{\mathrm{y}}=−\boldsymbol{\mathrm{x}}+\mathrm{4}\:\:,\mathrm{meets}\::\:\boldsymbol{\mathrm{xy}}=\mathrm{1}\:\mathrm{at}:\boldsymbol{\mathrm{A}},\boldsymbol{\mathrm{B}}. \\ $$$$\:\:\:\:\:\:\Rightarrow\:\:\mathrm{S}_{\mathrm{O}\overset{\bigtriangleup} {\mathrm{A}B}} =?\:\left(\mathrm{O}=\mathrm{origin}\:\mathrm{of}\:\mathrm{cordinates}\right) \\ $$$$\mathrm{2}.\mathrm{find}\::\mathrm{center}\:\mathrm{area}\:\mathrm{of}\:\mathrm{region}\:\mathrm{bonded}\:\mathrm{by} \\ $$$$\mathrm{corve}:\:\:\sqrt{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{b}}}}=\mathrm{1},\mathrm{and}\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\:\mathrm{axes}. \\ $$$$\left(\boldsymbol{\mathrm{a}}\neq\boldsymbol{\mathrm{b}}\right)\in\boldsymbol{\mathrm{R}}^{+} \\ $$ Commented by jagoll…
Question Number 151747 by mathdanisur last updated on 22/Aug/21 $$\mathrm{let}\:\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\lambda-\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{and}\:\:\lambda\geqslant\frac{-\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{solve}\:\mathrm{in}\:\mathbb{R}\:\:\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)\:\leqslant\:\mathrm{0} \\ $$ Answered by mr W last updated on 22/Aug/21 $${say}\:{g}\left({x}\right)={f}\left({f}\left({x}\right)\right) \\…
Question Number 20670 by Tinkutara last updated on 31/Aug/17 $${For}\:{the}\:{equation}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:\mathrm{3}\:=\:\mathrm{0}, \\ $$$${find}\:{the}\:{value}\left({s}\right)\:{of}\:{p}\:{if}\:{one}\:{root}\:{is} \\ $$$$\left({i}\right)\:{square}\:{of}\:{the}\:{other} \\ $$$$\left({ii}\right)\:{fourth}\:{power}\:{of}\:{the}\:{other}. \\ $$ Answered by dioph last updated on…
Question Number 20652 by Tinkutara last updated on 30/Aug/17 $$\mathrm{Suppose}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left\{{x}\right\},\:\left[{x}\right]\:\mathrm{and}\:{x}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{such}\:\mathrm{that}\:{x}^{{n}} \:>\:\mathrm{100}. \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:{x}\right. \\ $$$$\left.\mathrm{and}\:\left\{{x}\right\}\:=\:{x}\:−\:\left[{x}\right].\right) \\ $$ Commented by…
Question Number 20631 by Tinkutara last updated on 30/Aug/17 $${Solve}\:{the}\:{inequality} \\ $$$$\left({x}\:+\:\mathrm{3}\right)^{\mathrm{5}} \:−\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{5}} \:\geqslant\:\mathrm{244}. \\ $$ Answered by mrW1 last updated on 30/Aug/17 $$\mathrm{let}\:\mathrm{u}=\mathrm{x}+\mathrm{1} \\…
Question Number 151699 by iloveisrael last updated on 22/Aug/21 $$\:\:\:\:\underset{{x}^{\mathrm{3}} +\mathrm{2022}{x}−\mathrm{2021}=\mathrm{0}} {\sum}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)\:=? \\ $$ Answered by mr W last updated on 22/Aug/21 $${x}^{\mathrm{3}} +\mathrm{2022}{x}−\mathrm{2021}=\mathrm{0} \\…
Question Number 20619 by Tinkutara last updated on 29/Aug/17 $${Solve}\:{the}\:{equation}\:{z}^{{n}−\mathrm{1}} \:=\:\bar {{z}}\:\left({n}\:\in\:{N}\right) \\ $$ Answered by ajfour last updated on 29/Aug/17 $$\mid{z}\mid=\mathrm{1}\: \\ $$$$\:{z}^{{n}} =\mid{z}\mid^{\mathrm{2}}…
Question Number 151673 by mathdanisur last updated on 22/Aug/21 Answered by Kamel last updated on 22/Aug/21 $${L}=\underset{{n}\rightarrow+\infty} {{lim}}\underset{{k}={n}} {\overset{\mathrm{2}{n}} {\prod}}\frac{\pi}{\pi−{Arctan}\left(\frac{\mathrm{1}}{{k}}\right)}=\underset{{n}\rightarrow+\infty} {{lim}}\underset{{k}={n}} {\overset{\mathrm{2}{n}} {\prod}}\frac{\pi{k}}{\pi{k}−\mathrm{1}} \\ $$$$\:\:\:=\underset{{n}\rightarrow+\infty}…
Question Number 86141 by TawaTawa1 last updated on 27/Mar/20 $$\mathrm{A}\:\mathrm{number}\:\mathrm{n}\:\mathrm{leaves}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{of}\:\:\mathrm{22}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{24}\:\mathrm{and} \\ $$$$\mathrm{remainder}\:\:\mathrm{30}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{33}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{n} \\ $$ Commented by mr W last updated on 27/Mar/20 $${there}\:{is}\:{no}\:{such}\:{number}!…
Question Number 151660 by mathdanisur last updated on 22/Aug/21 Answered by Olaf_Thorendsen last updated on 22/Aug/21 $$\mathrm{ln}\left({e}+\mathrm{sin}{kx}\right)\:=\:\mathrm{1}+\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{sin}{kx}}{{e}}\right)\:\underset{\mathrm{0}} {\sim}\:\mathrm{1}+\frac{{kx}}{{e}} \\ $$$$\frac{\mathrm{1}−\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\mathrm{ln}\left({e}+\mathrm{sin}{kx}\right)}{{x}}\:\underset{\mathrm{0}} {\sim}\:\frac{\mathrm{1}−\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}+\frac{{kx}}{{e}}\right)}{{x}}\:\:\left(\mathrm{1}\right)…