Question Number 77132 by peter frank last updated on 03/Jan/20 Commented by kaivan.ahmadi last updated on 03/Jan/20 $${z}=\mathrm{4}\sqrt{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{2}}+{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)−\mathrm{4}\left(−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}+{i}\frac{\mathrm{1}}{\mathrm{2}}\right)= \\ $$$$\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{6}{i}+\mathrm{2}\sqrt{\mathrm{3}}−\mathrm{2}{i}=\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{4}{i}=\mathrm{4}\left(\sqrt{\mathrm{3}}+{i}\right)=\mathrm{8}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}+{i}\frac{\mathrm{1}}{\mathrm{2}}\right)= \\ $$$$\mathrm{8}{e}^{{i}\frac{\pi}{\mathrm{6}}} \\ $$$$\Rightarrow\frac{{z}}{\mathrm{8}}+{i}\left(\frac{{z}}{\mathrm{8}}\right)^{\mathrm{2}} +\left(\frac{{z}}{\mathrm{8}}\right)^{\mathrm{3}}…
Question Number 77131 by peter frank last updated on 03/Jan/20 Answered by mr W last updated on 03/Jan/20 $${perpendicular}\:{tangents}\:{from}\:{P}\left({u},{v}\right): \\ $$$${y}={v}+{m}\left({x}−{u}\right)\:\Rightarrow{mx}−{y}+\left({v}−{mu}\right) \\ $$$${y}={v}−\frac{\mathrm{1}}{{m}}\left({x}−{u}\right)\:\Rightarrow{x}+{my}−\left({mv}+{u}\right) \\ $$$${from}\:{Q}\mathrm{77127}\:{we}\:{have}:…
Question Number 77128 by peter frank last updated on 03/Jan/20 $${Find}\:{the}\:{value}\:{of}\:{constant} \\ $$$$“{a}''\:{such}\:{that}\:{axe}^{−{x}\:} {is} \\ $$$${a}\:{solution}\:{of}\:{Differential} \\ $$$${equation} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\mathrm{3}\frac{{dy}}{{dx}}+\mathrm{2}{y}=\mathrm{2}{e}^{−{x}} \\ $$$${solve}\:{D}.{E}\:{for}\:\:{which} \\…
Question Number 77126 by peter frank last updated on 03/Jan/20 $$\left.\mathrm{1}\right){Express}\:\frac{{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{4}} }\:\:\:{in} \\ $$$${partial}\:{fraction} \\ $$$$\left.\mathrm{2}\right)\:{Solve} \\ $$$${xdy}+{ydy}−\left(\frac{{xdx}−{ydy}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)=\mathrm{0} \\ $$$$ \\ $$ Answered…
Question Number 77127 by peter frank last updated on 03/Jan/20 $${Prove}\:{that}\:{line}\:{lx}+{my}+{n}=\mathrm{0} \\ $$$${is}\:{tangent}\:{to}\:{the}\:{ellipse} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}\:} }=\mathrm{1}\:{if}\:{a}^{\mathrm{2}} {l}^{\mathrm{2}} +{b}^{\mathrm{2}} {m}^{\mathrm{2}} ={n}^{\mathrm{2}} \\ $$…
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Question Number 76963 by peter frank last updated on 01/Jan/20 $$\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\left(\frac{\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}\right) \\ $$ Answered by MJS last updated on 02/Jan/20 $$\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\frac{\mathrm{cos}\:\theta\:+\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta\:−\mathrm{sin}\:\theta}\:{d}\theta= \\ $$$$=\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\frac{\mathrm{cos}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{sin}\:\mathrm{2}\theta}\:{d}\theta= \\ $$$$\:\:\:\:\:\mathrm{by}\:\mathrm{parts}…
Question Number 76965 by peter frank last updated on 02/Jan/20 $${Evaluate} \\ $$$${I}_{{ab}} =\int\mathrm{sin}\:{ax}\mathrm{cos}\:{bxdx} \\ $$$${if}\:{a}\neq{b}\:{and}\:{use}\:{it}\:{to} \\ $$$$\int_{\mathrm{0}} ^{{n}} \mathrm{sin}\:\mathrm{3}{x}\mathrm{cos}\:\mathrm{2}{xdx}=\frac{\mathrm{3}−\sqrt{\mathrm{3}}}{\mathrm{5}} \\ $$ Answered by mr…
Question Number 11399 by tawa last updated on 23/Mar/17 $$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{and}\:\mathrm{last}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}\:\mathrm{is}\:\mathrm{51}.\:\mathrm{And}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{progression}\:\mathrm{is}\:\mathrm{255}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{A}.\mathrm{P}. \\ $$ Answered by ajfour last updated on 23/Mar/17 $$\mathrm{last}\:\mathrm{term}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{51}+\mathrm{9d}\right) \\ $$$$\mathrm{where}\:\mathrm{d}\:\mathrm{is}\:\mathrm{whatever}\:\mathrm{common} \\…
Question Number 76929 by peter frank last updated on 01/Jan/20 $$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}\left(\mathrm{sin}\left({a}+{x}\right)\right)\:}} \\ $$ Answered by MJS last updated on 01/Jan/20 $$\int\frac{{dx}}{\:\sqrt{\mathrm{sin}^{\mathrm{3}} \:{x}\:\mathrm{sin}\:\left({a}+{x}\right)}}= \\ $$$$=\int\frac{{dx}}{\:\sqrt{\mathrm{sin}^{\mathrm{3}}…