Question Number 151561 by peter frank last updated on 21/Aug/21 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{x}+\mathrm{iy}=\frac{\mathrm{a}}{\mathrm{b}+\mathrm{sin}\:\theta+\mathrm{icos}\:\theta} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{b}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)+\mathrm{a}^{\mathrm{2}} =\mathrm{2abx} \\ $$ Answered by peter frank…
Question Number 151560 by peter frank last updated on 21/Aug/21 $$\mathrm{show}\:\mathrm{that}\:\:\mathrm{i}^{\mathrm{i}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{real} \\ $$ Answered by puissant last updated on 21/Aug/21 $${i}^{\mathrm{2}} =−\mathrm{1}\:\Rightarrow\:{i}=\sqrt{−\mathrm{1}} \\ $$$${i}^{{i}}…
Question Number 86024 by MJS last updated on 26/Mar/20 $$\mathrm{last}\:\mathrm{three}\:\mathrm{digits}\:\mathrm{of} \\ $$$$\mathrm{951413}^{\mathrm{314159}} =? \\ $$ Commented by naka3546 last updated on 26/Mar/20 $$\mathrm{277} \\ $$…
Question Number 86025 by Maclaurin Stickker last updated on 26/Mar/20 $${find}\:{the}\:{three}\:{last}\:{digits}\:{of}\:\:\mathrm{7}^{\mathrm{2020}} . \\ $$ Commented by MJS last updated on 26/Mar/20 $$\mathrm{you}\:\mathrm{can}\:\mathrm{do}\:\mathrm{it}\:\mathrm{yourself}.\:\mathrm{find}\:\mathrm{out},\:\mathrm{after}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{steps}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{repeat} \\…
Question Number 20488 by xing last updated on 27/Aug/17 Commented by ajfour last updated on 27/Aug/17 $${Question}\:{is}: \\ $$$${Given}\:\:\:{a}=−\sqrt{\mathrm{99}}+\sqrt{\mathrm{999}}+\sqrt{\mathrm{9999}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}=\sqrt{\mathrm{99}}−\sqrt{\mathrm{999}}+\sqrt{\mathrm{9999}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}=\sqrt{\mathrm{99}}+\sqrt{\mathrm{999}}−\sqrt{\mathrm{9999}} \\ $$$${Find}\:{the}\:{value}\:{of}…
Question Number 151559 by peter frank last updated on 21/Aug/21 $$\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}+\mathrm{cos}\:\theta+\mathrm{isin}\:\theta\right)^{\mathrm{n}} \\ $$$$+\:\left(\mathrm{1}+\mathrm{cos}\:\theta−\mathrm{isin}\:\theta\right)^{\mathrm{n}} =\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{cos}\:\frac{\theta}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{n}\theta}{\mathrm{2}} \\ $$ Answered by Olaf_Thorendsen last updated on…
Question Number 86021 by mathocean1 last updated on 26/Mar/20 $${E}\:{is}\:{a}\:{vectorial}\:{plan}\:{in}\:\mathbb{R}\:{with}\:{a}\:{base} \\ $$$${B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{of}\:{E} \\ $$$${defined}\:\forall\:\overset{\rightarrow} {{u}}={x}\overset{\rightarrow} {{i}}+{y}\overset{\rightarrow} {{j}}\:{by}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\left(−\mathrm{7}{x}−\mathrm{12}{y}\right)\overset{\rightarrow} {{i}}+\left(\mathrm{4}{x}+\mathrm{7}{y}\right)\overset{\rightarrow} {{j}}. \\ $$$$\left.\mathrm{1}\right)\:{Determinate}\:{f}\left(\overset{\rightarrow} {{i}}\right)\:{and}\:{f}\left(\overset{\rightarrow}…
Question Number 86018 by Maclaurin Stickker last updated on 26/Mar/20 $${Find}\:{the}\:{three}\:{last}\:{digits}\:{of}\:\mathrm{5}^{\mathrm{9999}} . \\ $$ Answered by MJS last updated on 26/Mar/20 $$\mathrm{5}^{\mathrm{1}} \:\:\:\:\:\mathrm{005} \\ $$$$\mathrm{5}^{\mathrm{2}}…
Question Number 86016 by mathmax by abdo last updated on 26/Mar/20 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{7}} }{dx} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 151554 by mathdanisur last updated on 21/Aug/21 Answered by MJS_new last updated on 21/Aug/21 $${f}\:\left({x},\:{y}\right)\:=\frac{{x}\left({x}+{y}\right)}{\mathrm{4}{x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} } \\ $$$$\frac{{d}}{{dy}}\left[{f}\:\left({x},\:{y}\right)\right]=\mathrm{0} \\ $$$$\frac{{x}\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{xy}−{y}^{\mathrm{2}} \right)}{\left(\mathrm{4}{x}^{\mathrm{2}}…