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Question Number 57407 Answers: 0 Comments: 1
$${let}\:{U}_{\mathrm{0}} ={cos}\left(\frac{\pi}{\mathrm{3}}\right)\:{and}\:{U}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}+{U}_{{n}} }{\mathrm{2}}} \\ $$$${find}\:{U}_{{n}} \:{interms}\:{of}\:{n}\:. \\ $$
Question Number 57406 Answers: 0 Comments: 1
$${let}\:{f}\left({x}\right)=\frac{{x}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:+{cos}\left(\pi{x}\right) \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\left({x}\right)=\mathrm{0}\:{have}\:{a}\:{solurion}\:\alpha\:{inside}\:\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right)\:{use}\:{newton}\:{method}\:{to}\:{find}\:{a}\:{approximate}\: \\ $$$${value}\:{of}\:\alpha\:. \\ $$
Question Number 57405 Answers: 1 Comments: 0
$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−{cosx}.{cos}\left(\mathrm{2}{x}\right)....{cos}\left({nx}\right)}{{x}^{\mathrm{2}} } \\ $$$${with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0}. \\ $$
Question Number 57404 Answers: 1 Comments: 5
$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{sin}\left(\pi\sqrt{{cosx}}\right)}{{x}^{\mathrm{2}} } \\ $$
Question Number 57400 Answers: 0 Comments: 1
Question Number 57390 Answers: 0 Comments: 0
$${a}^{\mathrm{2}} \:+\:{d}^{\mathrm{2}} \:−\:{ad}\:\:=\:\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:+\:{bc} \\ $$$${a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:\:=\:\:{c}^{\mathrm{2}} \:+\:{d}^{\mathrm{2}} \\ $$$$\frac{{ab}\:+\:{cd}}{{ad}\:+\:{bc}}\:\:=\:\:? \\ $$
Question Number 57389 Answers: 1 Comments: 0
$${If}\:{a}\epsilon{R}\:{and}\:{the}\:{equation}\:: \\ $$$$−\mathrm{3}\left\{{x}\right\}^{\mathrm{2}} +\mathrm{2}\left\{{x}\right\}+{a}^{\mathrm{2}} =\mathrm{0}\:{has}\:{no}\:{integral} \\ $$$${solution},\:{then}\:{all}\:{possible}\:{value}\:{of}\:{a} \\ $$$${lie}\:{in}\:{the}\:{interval}\:: \\ $$$$\left({a}\right)\left(−\mathrm{1},\mathrm{0}\right)\mathrm{U}\left(\mathrm{0},\mathrm{1}\right)\:\:\:\left({b}\right)\left(\mathrm{1},\mathrm{2}\right) \\ $$$$\left({c}\right)\:\left(−\mathrm{2},−\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\left({d}\right)\left(−\infty,−\mathrm{2}\right)\mathrm{U}\left(\mathrm{2},\infty\right) \\ $$
Question Number 57388 Answers: 0 Comments: 0
$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{2016}\right),\:\:\forall{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:{f}\left({x}\right)\:=\:\mathrm{30},\:\mathrm{then}\:\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:{f}\left({x}\:+\:\mathrm{2016}\right)\:=\:... \\ $$
Question Number 57387 Answers: 0 Comments: 0
Question Number 57385 Answers: 1 Comments: 1
Question Number 57383 Answers: 1 Comments: 1
Question Number 57381 Answers: 0 Comments: 0
Question Number 57377 Answers: 0 Comments: 0
$$\mathrm{Can}\:\mathrm{i}\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{a}\:\mathrm{product}\:\mathrm{to}\:\mathrm{infinity}\:? \\ $$$$\:\:\:\mathrm{e}.\mathrm{g}\:\:\:\:\:\:\:\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}.\mathrm{5}\:....\:\:\:\:\mathrm{infinity} \\ $$
Question Number 57373 Answers: 1 Comments: 1
$$\mathrm{tan}\:\mathrm{1}°+\mathrm{tan}\:\mathrm{5}°+\mathrm{tan}\:\mathrm{9}°+\ldots+\mathrm{tan}\:\mathrm{177}°=... \\ $$
Question Number 57368 Answers: 1 Comments: 0
$$\mathrm{5}^{\mathrm{3}{x}−\mathrm{3}} −\mathrm{5}^{\mathrm{3}{x}} −\mathrm{5}=\mathrm{615} \\ $$$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$
Question Number 57363 Answers: 0 Comments: 3
Question Number 57357 Answers: 3 Comments: 2
$$\mathrm{1}−\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\mathrm{4}\boldsymbol{\mathrm{x}}\right)<\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\mathrm{4}\boldsymbol{\mathrm{x}}\right) \\ $$$$\boldsymbol{\mathrm{solve}}. \\ $$
Question Number 57356 Answers: 0 Comments: 0
$$\boldsymbol{\mathrm{S}}=\frac{\boldsymbol{\pi\mathrm{R}}^{\mathrm{2}} }{\mathrm{360}°}×\boldsymbol{\alpha}° \\ $$$$\boldsymbol{\mathrm{prove}}. \\ $$
Question Number 57348 Answers: 1 Comments: 2
Question Number 57345 Answers: 1 Comments: 0
$$\left.\mathrm{1}\right)\boldsymbol{\mathrm{if}}:\:\:\:\:\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{tgx}}=\mathrm{1},\boldsymbol{\mathrm{then}}:\:\:\boldsymbol{\mathrm{sin}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{tg}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}=? \\ $$$$\left.\mathrm{2}\right)\boldsymbol{\mathrm{if}}:\:\:\:\:\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{tgx}}=\mathrm{2},\boldsymbol{\mathrm{then}}:\:\:\boldsymbol{\mathrm{sin}}\mathrm{4}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{tg}}\mathrm{4}\boldsymbol{\mathrm{x}}=? \\ $$$$\mathrm{3}.\boldsymbol{\mathrm{if}}:\:\:\:\:\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{tgx}}=\mathrm{3},\boldsymbol{\mathrm{then}}:\:\:\frac{\boldsymbol{\mathrm{sin}}\mathrm{4}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{sin}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}}+\frac{\boldsymbol{\mathrm{tg}}\mathrm{4}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{tg}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}}=? \\ $$
Question Number 57336 Answers: 2 Comments: 0
$$\mathrm{If}\:\:\mathrm{n}\:\mathrm{be}\:\mathrm{even},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{expression}\:\:\:\:\frac{\mathrm{n}\left(\mathrm{n}\:+\:\mathrm{2}\right)\left(\mathrm{n}\:+\:\mathrm{4}\right)\:...\:\left(\mathrm{2n}\:−\:\mathrm{2}\right)}{\mathrm{1}.\mathrm{3}.\mathrm{5}\:...\:\left(\mathrm{n}\:−\:\mathrm{1}\right)} \\ $$$$\mathrm{simplify}\:\mathrm{to}\:\:\mathrm{2}^{\mathrm{n}\:−\:\mathrm{1}} \\ $$
Question Number 57332 Answers: 0 Comments: 1
$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{such}\:\mathrm{that} \\ $$$${k}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\left({y}−\mathrm{2}{x}+\mathrm{1}\right)\left({y}+\mathrm{2}{x}+\mathrm{3}\right)=\mathrm{0} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{hence}\:\mathrm{obtain}\: \\ $$$$\mathrm{the}\:\mathrm{centre}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{resulting}\:\mathrm{circle}. \\ $$
Question Number 57330 Answers: 0 Comments: 0
Question Number 57329 Answers: 1 Comments: 1
$$\underset{−\mathrm{1}} {\int}\overset{\mathrm{2}} {\:}\:\mid{x}\mid\:\lfloor{x}\rfloor\:{dx}\:\:=\:\:\:? \\ $$
Question Number 57328 Answers: 1 Comments: 0
Question Number 57325 Answers: 0 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{ln}\left(\mathrm{1}+{sinx}\right)}{{sinx}}{dx} \\ $$
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