Question Number 8013 by Yozzia last updated on 28/Sep/16 $${Prove}\:{that},\:\forall{m}\in\mathbb{N}, \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{m}} {\sum}}\begin{pmatrix}{\mathrm{2}{m}}\\{\mathrm{2}{r}−\mathrm{1}}\end{pmatrix}\:\left(\mathrm{2}{r}−\mathrm{1}\right)^{\mathrm{2}{m}−\mathrm{1}} =\underset{{r}=\mathrm{1}} {\overset{{m}} {\sum}}\begin{pmatrix}{\mathrm{2}{m}}\\{\mathrm{2}{r}}\end{pmatrix}\:\left(\mathrm{2}{r}\right)^{\mathrm{2}{m}−\mathrm{1}} . \\ $$ Commented by FilupSmith last updated…
Question Number 139081 by BHOOPENDRA last updated on 22/Apr/21 Commented by BHOOPENDRA last updated on 22/Apr/21 $${help}\:{me}\:{out}\:{this}\: \\ $$ Commented by BHOOPENDRA last updated on…
Question Number 73544 by Rio Michael last updated on 13/Nov/19 $${expressf}\left(\theta\right)=\:\mathrm{8}{cos}\theta\:−\mathrm{15}{sin}\theta\:{in}\:{the}\:{form} \\ $$$$\:{rcos}\left(\theta\:+\:\alpha\right),\:{where}\:{r}>\mathrm{0}\:{and}\:\alpha\:{is}\:{a}\:{positive}\:{acute}\:{angle} \\ $$$${hence} \\ $$$${find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{equation} \\ $$$$\:\:\mathrm{80}{cos}\:\theta\:−\mathrm{150}{sin}\theta\:=\:\mathrm{13} \\ $$$${the}\:{maximum}\:{and}\:{minimum}\:{value}\:{of}\:\:\frac{\mathrm{5}}{{f}\left(\theta\right)\:+\:\mathrm{3}} \\ $$ Answered by…
Question Number 73545 by Rio Michael last updated on 13/Nov/19 $${evaluate}\:\:\int{lnx}\:{dx} \\ $$ Commented by Tony Lin last updated on 13/Nov/19 $${integration}\:{by}\:{part} \\ $$$$\int{f}\:'\left({x}\right){g}\left({x}\right)={f}\left({x}\right){g}\left({x}\right)−\int{f}\left({x}\right){g}\:'\left({x}\right) \\…
Question Number 73542 by Rio Michael last updated on 13/Nov/19 $${Given}\:{that}\:{the}\:{function}\:{f}\left({x}\right)=\:{x}^{\mathrm{3}} \:{is}\:{differentiable} \\ $$$${in}\:{the}\:{interval}\:\left(−\mathrm{2},\mathrm{2}\right),\:{Use}\:{the}\:{mean}\:{value}\:{theorem} \\ $$$${to}\:{find}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{the}\:{tangent}\:{to}\:{the}\:{curve} \\ $$$${is}\:{parallel}\:{to}\:{the}\:{chord}\:{through}\:{the}\:{point}\:\left(−\mathrm{2},\mathrm{8}\right)\:{and}\:\left(\mathrm{2},\mathrm{8}\right) \\ $$ Terms of Service Privacy Policy…
Question Number 139076 by bramlexs22 last updated on 22/Apr/21 $$\lceil\:\mathrm{3x}\:+\:\mathrm{5}\lfloor\:\mathrm{x}\:\rfloor\:\rceil\:=\:\mathrm{9} \\ $$ Answered by mathmax by abdo last updated on 22/Apr/21 $$\Rightarrow\left[\mathrm{3x}\right]+\mathrm{5}\left[\mathrm{x}\right]=\mathrm{9}\:\:\:\mathrm{let}\:\left[\mathrm{x}\right]=\mathrm{n}\:\Rightarrow\mathrm{n}\leqslant\mathrm{x}<\mathrm{n}+\mathrm{1}\:\Rightarrow\mathrm{3n}\leqslant\mathrm{3x}<\mathrm{3n}+\mathrm{3} \\ $$$$\mathrm{if}\:\mathrm{3n}\leqslant\mathrm{3x}<\mathrm{3n}+\mathrm{1}\:\Rightarrow\left[\mathrm{3x}\right]=\mathrm{3n}\:\:\mathrm{and}\:\mathrm{e}\Rightarrow\mathrm{3n}+\mathrm{5n}=\mathrm{9}\:\Rightarrow\mathrm{8n}=\mathrm{9}\:\:\mathrm{impossible} \\…
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Question Number 73538 by Rio Michael last updated on 13/Nov/19 $${find}\:{the}\:{general}\:{solution}\:{of}\: \\ $$$$\:{sin}\mathrm{4}{x}\:+\:{cos}\mathrm{2}{x}\:=\:\mathrm{0} \\ $$ Answered by ajfour last updated on 13/Nov/19 $$\mathrm{cos}\:\mathrm{2}{x}\left(\mathrm{2sin}\:\mathrm{2}{x}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow\:\:\mathrm{2}{x}=\left(\mathrm{2}{n}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}}\:\:\:{or}…
Question Number 8003 by Yozzia last updated on 27/Sep/16 $$\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\left(\underset{{r}=\mathrm{0}} {\overset{{m}−\mathrm{1}} {\sum}}\begin{pmatrix}{\mathrm{2}{m}}\\{\mathrm{2}{r}+\mathrm{1}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{m}−{r}} \right)^{\mathrm{2}} +\left(\underset{{r}=\mathrm{0}} {\overset{{m}} {\sum}}\begin{pmatrix}{\mathrm{2}{m}}\\{\mathrm{2}{r}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{m}−{r}} \right)^{\mathrm{2}} }\right)=? \\ $$ Commented by prakash…
Question Number 139075 by bramlexs22 last updated on 22/Apr/21 Answered by mr W last updated on 22/Apr/21 $$\mathrm{tan}\:\alpha=\frac{{x}}{\mathrm{144}} \\ $$$$\mathrm{tan}\:\beta=\frac{{x}}{\mathrm{144}+\mathrm{81}}=\frac{{x}}{\mathrm{225}} \\ $$$$\mathrm{tan}\:\gamma=\frac{{x}}{\mathrm{225}+\mathrm{99}}=\frac{{x}}{\mathrm{324}} \\ $$$$\mathrm{tan}\:\gamma=\mathrm{tan}\:\left(\mathrm{90}°−\alpha−\beta\right)=\frac{\mathrm{1}}{\mathrm{tan}\:\left(\alpha+\beta\right)} \\…