Question Number 27882 by das47955@mail.com last updated on 16/Jan/18 $$\left(\mathrm{2}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{indepen}}− \\ $$$$\boldsymbol{\mathrm{dent}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{6}} \\ $$ Answered by Rasheed.Sindhi last updated on 16/Jan/18 $$\mathrm{T}_{\mathrm{r}+\mathrm{1}}…
Question Number 158945 by LEKOUMA last updated on 10/Nov/21 $$\left.\mathrm{1}\right)\:{Prove}\:{by}\:{absurd}\:{that}\: \\ $$$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:\mathrm{3}}\:\:{is}\:{irrational} \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{by}\:{absurd}\:{that} \\ $$$$\sqrt{\mathrm{2}}+\sqrt{\mathrm{6}\:}\leqslant\sqrt{\mathrm{15}} \\ $$ Answered by mr W last updated on…
Question Number 93397 by Rio Michael last updated on 12/May/20 $$\mathrm{prove}\:\mathrm{that}\:\frac{\mathrm{1}−\mathrm{tan}^{\mathrm{3}} \theta}{\mathrm{1}\:+\:\mathrm{tan}^{\mathrm{3}} \theta}\:=\:\mathrm{1}−\mathrm{2sin}^{\mathrm{3}} \theta\: \\ $$ Commented by john santu last updated on 13/May/20 $$\theta\:=\:\mathrm{45}^{\mathrm{o}\:}…
Question Number 93368 by Rio Michael last updated on 12/May/20 $$\mathrm{given}\:\mathrm{that}\:{f}\left({r}\right)=\:\:\mathrm{sin}\:\left(\mathrm{1}\:+\:\mathrm{2}{r}\right)\theta \\ $$$$\mathrm{show}\:\mathrm{that}\:{f}\left({r}\right)−{f}\left({r}−\mathrm{1}\right)\:=\:\mathrm{2}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta \\ $$$$\mathrm{hence}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta\:\:=\:\mathrm{cos}\:\left({n}\:+\mathrm{1}\right)\theta\:\mathrm{sin}\:{n}\theta \\ $$ Answered by mr W…
Question Number 93367 by Rio Michael last updated on 12/May/20 $$\mathrm{show}\:\mathrm{using}\:\mathrm{demoives}\:\mathrm{theorem}\:\mathrm{that}\: \\ $$$$\mathrm{sin}^{\mathrm{2}} \theta\:\mathrm{cos}^{\mathrm{2}} \mathrm{4}\theta\:=\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{1}−\mathrm{cos4}\theta\right) \\ $$ Commented by prakash jain last updated on 14/May/20…
Question Number 93366 by Rio Michael last updated on 12/May/20 $$\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\left({z}−\mathrm{2}\right)^{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}−{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$ Answered by prakash jain last updated on 12/May/20 $$\frac{\mathrm{1}}{\mathrm{2}}−{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}…
Question Number 93365 by Rio Michael last updated on 12/May/20 $$\mathrm{given}\:\mathrm{that}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number},\:\mathrm{use}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{or} \\ $$$$\mathrm{otherwise}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{2}} }\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{3}} }\right)\:…\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{{n}} }\right)\:=\:\frac{\mathrm{sin}\:\alpha}{\mathrm{2}^{{n}} \:\mathrm{sin}\left(\frac{\alpha}{\mathrm{2}^{{n}} }\right)} \\ $$$$\mathrm{hence}\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\alpha}{\mathrm{2}^{\mathrm{2}}…
Question Number 27809 by das47955@mail.com last updated on 15/Jan/18 $$\left(\mathrm{1}\right)\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{independent}} \\ $$$$\boldsymbol{\mathrm{of}}\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}−\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{10}} \\ $$ Answered by 8/mln(naing)060691 last updated on 15/Jan/18 $$\left(\mathrm{r}+\mathrm{1}\right)^{\mathrm{th}} \mathrm{term}=^{\mathrm{10}}…
Question Number 158858 by LEKOUMA last updated on 09/Nov/21 $${I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}{x}\right){dx} \\ $$$${to}\:{integrating}\:{by}\:{piece}\:{for}\:{n}\geqslant\mathrm{2}\: \\ $$$${proven}\: \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{4}{b}^{\mathrm{2}} }{I}_{{n}\:} =\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} −\mathrm{4}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}}…
Question Number 158855 by LEKOUMA last updated on 09/Nov/21 $${Resolve}\:{the}\:{system}\:{d}'\:{unknow}\:\:\left({x},\:{y},{z}\right)\:\in\:\subset^{\mathrm{3}} \\ $$$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =−\mathrm{5} \\ $$ Answered…