Category: Arithmetic

Question Number 77132 by peter frank last updated on 03/Jan/20 Commented by kaivan.ahmadi last updated on 03/Jan/20 $$z=4\sqrt{3}(\frac{1}{2}+i\frac{\sqrt{3}}{2})-4(-\frac{\sqrt{3}}{2}+i\frac{1}{2})=$$$$2\sqrt{3}+6i+2\sqrt{3}-2i=4\sqrt{3}+4i=4(\sqrt{3}+i)=8(\frac{\sqrt{3}}{2}+i\frac{1}{2})=$$$$8e^{i\frac{\pi }{6}}$$$$\Rightarrow\frac{{z}}{\mathrm{8}}+{i}\left(\frac{{z}}{\mathrm{8}}\right)^{\mathrm{2}} +\left(\frac{{z}}{\mathrm{8}}\right)^{\mathrm{3}}…

Question Number 77128 by peter frank last updated on 03/Jan/20 $$Findthevalueofconstant$$$$“a^{″}suchthat{axe}^{-x}is$$$$asolutionofDifferential$$$$equation$$$$\frac{d^{2}y}{{dx}^2}+3\frac{dy}{dx}+2y=2e^{-x}$$$${solve}\:{D}.{E}\:{for}\:\:{which} \…

Question Number 77127 by peter frank last updated on 03/Jan/20 $$Provethatlinelx+my+n=0$$$$istangenttotheellipse$$$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1if{a}^2{l}^2+b^2{m}^2=n^2$$…

Question Number 76963 by peter frank last updated on 01/Jan/20 $$\int \cos 2\theta \ln \left(\frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }\right)$$ Answered by MJS last updated on 02/Jan/20 $$\int \cos 2\theta \ln \frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }d\theta =$$$$=\int \cos 2\theta \ln \frac{\cos 2\theta }{1-\sin 2\theta }d\theta =$$$$\:\:\:\:\:\mathrm{by}\:\mathrm{parts}…

Question Number 11399 by tawa last updated on 23/Mar/17 $$\mathrm{The}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{and}\mathrm{last}\mathrm{term}\mathrm{of}\mathrm{an}\mathrm{A}.\mathrm{P}\mathrm{is}51.\mathrm{And}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}$$$$\mathrm{progression}\mathrm{is}255.\mathrm{Find}\mathrm{the}\mathrm{last}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{A}.\mathrm{P}.$$ Answered by ajfour last updated on 23/Mar/17 $$\mathrm{last}\mathrm{term}=\frac{1}{2}(51+9\mathrm{d})$$$$\mathrm{where}\:\mathrm{d}\:\mathrm{is}\:\mathrm{whatever}\:\mathrm{common} \…