Question Number 214050 by CrispyXYZ last updated on 25/Nov/24 $$\mathrm{If}\:{A}:\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:…,\:{a}_{\mathrm{62}} \:\mathrm{and}\:{B}:\:{b}_{\mathrm{1}} ,\:{b}_{\mathrm{2}} ,\:…,\:{b}_{\mathrm{62}} \:\mathrm{are}\:\mathrm{two} \\ $$$$\mathrm{strictly}\:\mathrm{increasing}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{sequences} \\ $$$$\mathrm{such}\:\mathrm{that}\:{a}_{\mathrm{62}} \leqslant\mathrm{755}\:\mathrm{and}\:{b}_{\mathrm{62}} \leqslant\mathrm{755}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{of}\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{62}}…
Question Number 214051 by issac last updated on 25/Nov/24 $$\mathrm{evaluate}\:\int_{\mathrm{0}} ^{\:\pi} \:{e}^{\mathrm{sin}^{\mathrm{2}} \left({u}\right)} \mathrm{d}{u}…\: \\ $$$$\mathrm{i}\:\mathrm{use}\:\mathrm{Feynman}'\mathrm{s}\:\mathrm{trick}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{integral} \\ $$$$\int_{\mathrm{0}} ^{\:\pi} \:{e}^{\mathrm{sin}^{\mathrm{2}} \left({u}\right)} \mathrm{d}{u}={I} \\ $$$${I}\left({t}\right)=\int_{\mathrm{0}} ^{\:\pi}…
Question Number 214012 by ajfour last updated on 24/Nov/24 Commented by ajfour last updated on 24/Nov/24 $${Outer}\:{circle}\:{radius}\:{is}\:{R}.\:{Circle}\:{with} \\ $$$${center}\:{A}\:{has}\:{radius}\:{r}={R}/\mathrm{2}. \\ $$$${If}\:\bigtriangleup{ABC}\:{is}\:{equilateral},\:{find}\:{its} \\ $$$${edge}\:{length}\:\left({say}\:{a}\right). \\ $$…
Question Number 214030 by RoseAli last updated on 24/Nov/24 Answered by A5T last updated on 24/Nov/24 $${x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow\left({x}+\mathrm{1}\right)^{\mathrm{2}} \equiv\mathrm{6}\left({mod}\:\mathrm{10}\right) \\ $$$$\Rightarrow{x}=\mathrm{3},\mathrm{5} \\ $$…
Question Number 213999 by issac last updated on 24/Nov/24 $$\int\int…\int_{\:\mathcal{D}} \:\:{e}^{−\left({z}_{\mathrm{1}} ^{\mathrm{2}} +{z}_{\mathrm{2}} ^{\mathrm{2}} …+{z}_{{n}} ^{\mathrm{2}} \right)} \mathrm{da} \\ $$$$\mathcal{D}=\underset{\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{times}}} {\left[\mathrm{0},\infty\right)×\left[\mathrm{0},\infty\right)……\left[\mathrm{0},\infty\right)} \\ $$$$\int_{\mathrm{0}} ^{\:\pi} \:{e}^{−\mathrm{sin}^{\mathrm{2}}…
Question Number 214020 by ajfour last updated on 24/Nov/24 Commented by ajfour last updated on 24/Nov/24 $${Red}\:{area}\:=\:{Blue}\:{area}.\:{Find}\:{r}\:{in} \\ $$$${terms}\:{of}\:{k},\:\phi. \\ $$ Answered by dionigi last…
Question Number 214005 by RoseAli last updated on 24/Nov/24 $${find}\:{all}\:{zero}\:{divisors}\:{of}\:{Z}_{\mathrm{24}} \\ $$ Answered by TonyCWX08 last updated on 24/Nov/24 $$\mathrm{1}×\mathrm{24} \\ $$$$\mathrm{2}×\mathrm{12} \\ $$$$\mathrm{3}×\mathrm{8} \\…
Question Number 214000 by golsendro last updated on 24/Nov/24 $$\:\:\mathrm{Let}\:\mathrm{y}\left(\mathrm{x}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{diff}\:\mathrm{eq}. \\ $$$$\:\:\mathrm{y}\:'=\:\frac{\mathrm{cos}\:\mathrm{x}+\mathrm{y}}{\mathrm{cos}\:\mathrm{x}}\:,\:\mathrm{y}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\:\:\mathrm{Find}\:\mathrm{y}\left(\frac{\pi}{\mathrm{6}}\right). \\ $$ Commented by mr W last updated on 24/Nov/24 $${do}\:{you}\:{mean}\:\mathrm{cos}\:\left({x}+{y}\right)\:{instead}\:{of}…
Question Number 214001 by RoseAli last updated on 24/Nov/24 $$\mathrm{find}\:\mathrm{all}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} ={x}\:\mathrm{in}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rings}\: \\ $$$${Z}_{\mathrm{2}} \: \\ $$$${Z}_{\mathrm{3}} \\ $$$$\mathrm{and}\:\mathrm{Z}_{\mathrm{6}} \\ $$ Answered by TonyCWX08 last updated…
Question Number 214002 by RoseAli last updated on 24/Nov/24 $${find}\:{the}\:{integers}\:{x}\:{that}\:{satisfies}\:{a}\:{congruence}\:\mathrm{3}{x}=\mathrm{4}\:\left({mod}\:\mathrm{11}\right)\:. \\ $$ Answered by Rasheed.Sindhi last updated on 24/Nov/24 $$\mathrm{3}{x}\equiv\mathrm{4}\left({mod}\:\mathrm{11}\right) \\ $$$$\mathrm{3}{x}\equiv\mathrm{4}+\mathrm{11}\left({mod}\:\mathrm{11}\right) \\ $$$$\mathrm{3}{x}\equiv\mathrm{15}\left({mod}\:\mathrm{11}\right) \\…