Question Number 107122 by aurpeyz last updated on 08/Aug/20 Answered by 1549442205PVT last updated on 09/Aug/20 $$\mathrm{Denote}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{be}\:\mathrm{number}\:\mathrm{of}\:\mathrm{persons}\:\mathrm{get} \\ $$$$\mathrm{medal}\:\mathrm{in}\:\mathrm{only}\:\mathrm{one}\:\mathrm{category} \\ $$$$\mathrm{dance},\mathrm{dramatics},\mathrm{music}\:\mathrm{respectively} \\ $$$$\mathrm{m}−\mathrm{be}\:\mathrm{number}\:\mathrm{of}\:\mathrm{persons}\:\mathrm{got} \\ $$$$\mathrm{exactly}\:\mathrm{medal}\:\mathrm{in}\:\mathrm{two}\:\mathrm{categories}\:\mathrm{dance}…
Question Number 172662 by SANOGO last updated on 29/Jun/22 Answered by Mathspace last updated on 29/Jun/22 $$\left.{a}\right)\:\Delta\left({x},{y}\right)=\frac{{xy}}{{M}\left({x},{y}\right)}=\frac{\mathrm{1512}}{\mathrm{252}}=\mathrm{6} \\ $$$${x}={pd}\:{and}\:{y}={qd}\:\:{with}\left({p},{d}\right)=\mathrm{1} \\ $$$${x}+{y}=\mathrm{1512}\:\Rightarrow{d}\left({p}+{q}\right)=\mathrm{1512} \\ $$$$\Rightarrow{p}+{q}=\frac{\mathrm{1512}}{\mathrm{6}}=\mathrm{252} \\ $$$$\Rightarrow\left({p},{q}\right)\in\left\{\left(\mathrm{1},\mathrm{251}\right),\left(\mathrm{2},\mathrm{250}\right),\left(\mathrm{3},\mathrm{249}\right)…\left(\mathrm{251},\mathrm{1}\right)\right\}…
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Question Number 107080 by Khalmohmmad last updated on 08/Aug/20 Answered by abdomsup last updated on 08/Aug/20 $${A}_{{n}} =\frac{{n}^{{n}} \left\{\frac{{n}}{{n}^{{n}} }+\frac{{n}^{\mathrm{2}} }{{n}^{{n}} }+….+\mathrm{1}\right\}}{{n}^{{n}} \left\{\frac{\mathrm{1}}{{n}^{{n}} }\:+\left(\frac{\mathrm{2}}{{n}}\right)^{{n}\:} +\left(\frac{\mathrm{3}}{{n}^{}…
Question Number 107088 by mathdave last updated on 08/Aug/20 Commented by mohammad17 last updated on 08/Aug/20 $$\int_{−\mathrm{1}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}=\left({sin}^{−\mathrm{1}} {x}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right)_{−\mathrm{1}} ^{\:\mathrm{1}} =\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{2}}=\pi \\…
Question Number 41536 by mondodotto@gmail.com last updated on 09/Aug/18 $$\boldsymbol{\mathrm{if}}\:\boldsymbol{{y}}=\sqrt{\frac{\mathrm{1}+\boldsymbol{\mathrm{sin}{x}}}{\mathrm{1}−\boldsymbol{\mathrm{sin}{x}}}\:}\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}=\frac{\mathrm{1}}{\mathrm{1}−\boldsymbol{\mathrm{sin}{x}}} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 09/Aug/18 $${sinx}=\frac{\mathrm{2}{tan}\frac{{x}}{\mathrm{2}}}{\mathrm{1}+{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}} \\ $$$${y}=\sqrt{\frac{\mathrm{1}+\frac{\mathrm{2}{tan}\frac{{x}}{\mathrm{2}}}{\mathrm{1}+{tan}^{\mathrm{2}}…
Question Number 41534 by mondodotto@gmail.com last updated on 09/Aug/18 $$\boldsymbol{\mathrm{Three}}\:\boldsymbol{\mathrm{chidren}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{playing}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{game}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{1}\boldsymbol{\mathrm{sec}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{second}}\: \\ $$$$\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{10}\boldsymbol{\mathrm{sec}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claps}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{5}\boldsymbol{\mathrm{sec}}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{long}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{all}} \\ $$$$\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{children}}\:\boldsymbol{\mathrm{will}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{together}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{time}}? \\ $$ Answered by alex041103…
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Question Number 107069 by mathdave last updated on 08/Aug/20 Answered by bemath last updated on 09/Aug/20 $$\:\:\:\:\:\:\:@{bemath}@ \\ $$$$\left(\sqrt{\mathrm{cos}\:{x}}\:+\sqrt{\mathrm{sin}\:{x}}\right)^{\mathrm{5}} =\left(\sqrt{\mathrm{cos}\:{x}}\left(\mathrm{1}+\sqrt{\mathrm{tan}\:{x}}\right)^{\mathrm{5}} \right. \\ $$$$=\:\mathrm{cos}\:^{\mathrm{2}} \:{x}\:\sqrt{\mathrm{cos}\:{x}}\:\left(\mathrm{1}+\sqrt{\mathrm{tan}\:{x}}\right)^{\mathrm{5}} \\…