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AllQuestion and Answers: Page 551
Question Number 158899 Answers: 1 Comments: 0
$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{4}}= \\ $$
Question Number 158894 Answers: 0 Comments: 1
Question Number 158893 Answers: 1 Comments: 0
Question Number 158884 Answers: 1 Comments: 0
Question Number 158883 Answers: 1 Comments: 4
Question Number 158886 Answers: 0 Comments: 1
Question Number 158878 Answers: 0 Comments: 0
Question Number 158874 Answers: 2 Comments: 1
Question Number 158873 Answers: 0 Comments: 0
Question Number 158870 Answers: 0 Comments: 1
Question Number 158869 Answers: 0 Comments: 2
$$\:{The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{x}^{\mathrm{2}} +{px}+{p}=\mathrm{0}\:{are}\:\mathrm{2}\alpha+\beta\:{and} \\ $$$$\:\alpha+\mathrm{2}\beta.\:{Calculate}\:{the}\:{value}\:{of}\:{p} \\ $$
Question Number 158862 Answers: 1 Comments: 0
Question Number 158863 Answers: 0 Comments: 0
Question Number 158858 Answers: 0 Comments: 0
$${I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}{x}\right){dx} \\ $$$${to}\:{integrating}\:{by}\:{piece}\:{for}\:{n}\geqslant\mathrm{2}\: \\ $$$${proven}\: \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{4}{b}^{\mathrm{2}} }{I}_{{n}\:} =\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} −\mathrm{4}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} \\ $$$${proven}\:{by}\:{rearring}\:{that}\: \\ $$$$\left(\frac{{a}}{\mathrm{2}{b}}\right)^{\mathrm{2}{n}+\mathrm{1}} {I}_{{n}} ={n}!\left[{p}\left(\frac{{q}}{\mathrm{2}{b}}\right)\mathrm{sin}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)+{Q}\left(\frac{{a}}{\mathrm{2}{b}}\right)\mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)\right] \\ $$
Question Number 158855 Answers: 1 Comments: 0
$${Resolve}\:{the}\:{system}\:{d}'\:{unknow}\:\:\left({x},\:{y},{z}\right)\:\in\:\subset^{\mathrm{3}} \\ $$$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =−\mathrm{5} \\ $$
Question Number 158847 Answers: 1 Comments: 0
Question Number 158843 Answers: 2 Comments: 2
Question Number 158839 Answers: 2 Comments: 0
$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{following}\:\mathrm{integrals}\:\mathrm{using} \\ $$$$\mathrm{integration}\:\boldsymbol{\mathrm{By}}\:\boldsymbol{\mathrm{Parts}} \\ $$$$\mathrm{1}.\:\int_{\frac{\pi\:}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} {xcsc}^{\mathrm{2}} {xdx} \\ $$$$ \\ $$$$\mathrm{2}.\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} {arctan}\left(\frac{\mathrm{1}}{{x}}\right){dx} \\ $$
Question Number 158823 Answers: 0 Comments: 0
Question Number 158822 Answers: 1 Comments: 0
$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{arctan}\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}+\mathrm{1}}=? \\ $$
Question Number 158833 Answers: 1 Comments: 0
$${what}\:{is}\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{2}}\% \\ $$
Question Number 158829 Answers: 1 Comments: 2
$$\:{The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{x}^{\mathrm{2}} +{px}+{q}=\mathrm{0}\:{are}\:\mathrm{2}\alpha+\beta\:{and} \\ $$$$\:\alpha+\mathrm{2}\beta.\:{Calculate}\:{the}\:{values}\:{of} \\ $$$$\:{p}\:{and}\:{q} \\ $$
Question Number 158827 Answers: 1 Comments: 0
$${resolve}\:\int\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right){dx} \\ $$
Question Number 158816 Answers: 0 Comments: 0
$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{2017}^{\mathrm{2017}} \:\:\mathrm{and}\:\:\:\mathrm{2017}^{\mathrm{2018}} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{perfect}\:\mathrm{squares}. \\ $$$$ \\ $$
Question Number 158814 Answers: 2 Comments: 0
$$\mathrm{Compare}\:\mathrm{it}: \\ $$$$\left(\mathrm{log}_{\mathrm{4}} \mathrm{20}\right)^{\mathrm{2}} \:\:\:\mathrm{and}\:\:\:\mathrm{log}_{\mathrm{4}} \mathrm{320} \\ $$$$ \\ $$
Question Number 158813 Answers: 0 Comments: 0
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{value}\:\:\boldsymbol{\beta}>\mathrm{0}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:+\infty} {\int}}\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2021}\boldsymbol{\beta}} \:+\:\mathrm{ln}\centerdot\left(\mathrm{1}\:+\:\beta\mathrm{x}\right)}\:\mathrm{dx}\:<\:+\infty \\ $$
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