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Question Number 55914 Answers: 1 Comments: 0
Question Number 55913 Answers: 1 Comments: 0
Question Number 55909 Answers: 0 Comments: 0
$$\mathrm{If}\:{E}=\left\{{f}\:\mid{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{continoues}\:\mathrm{function}\right. \\ $$$$\left.{f}\left({x}\right)\:\in\mathrm{Q},\:\forall{x}\:\in\mathbb{R}\right\}\:\mathrm{then}\:{E}=... \\ $$
Question Number 55908 Answers: 1 Comments: 0
$$\mathrm{known}\:{a}\:<\:\frac{\pi}{\mathrm{2}}\:. \\ $$$$\mathrm{If}\:\:\mathrm{M}<\mathrm{1}\:\mathrm{with}\:\mid\mathrm{cos}\:{x}−\mathrm{cos}\:{y}\mid\leqslant\mathrm{M}\:\mid{x}−{y}\mid \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:{y}\:\in\:\left[\mathrm{0},{a}\right],\:\mathrm{then}\:\mathrm{M}=.. \\ $$
Question Number 55907 Answers: 1 Comments: 1
$$\mathrm{for}\:\mathrm{every}\:{n}\:\in\:\mathbb{N}\:,\:{f}_{{n}} \left({x}\right)={nx}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} , \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\mathrm{and}\:{a}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right)\:{dx}. \\ $$$$\mathrm{If}\:\mathrm{S}_{\mathrm{n}} =\mathrm{sin}\:\left(\pi{a}_{{n}} \right),\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{n}\in\:\mathbb{N},\:\mathrm{then}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{s}_{\mathrm{n}} =... \\ $$
Question Number 55906 Answers: 1 Comments: 0
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\overset{{n}} {\underset{{k}=\mathrm{1}} {\sum}}\frac{\mathrm{8}{n}^{\mathrm{2}} }{{n}^{\mathrm{4}} +\mathrm{1}}=.. \\ $$
Question Number 55905 Answers: 1 Comments: 0
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}+\mathrm{1}} \right)=\mathrm{315} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}−\mathrm{1}} \right)=\mathrm{2016} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}_{\mathrm{2}{n}} }{{x}_{\mathrm{2}{n}+\mathrm{1}} }=... \\ $$
Question Number 55904 Answers: 2 Comments: 0
$$\mathrm{Let}\:{a}_{{i}} >\mathrm{0},\:\forall_{{i}} =\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\ldots\mathrm{2016} \\ $$$$\mathrm{If}\:\left({a}_{\mathrm{1}} {a}_{\mathrm{2}} \ldots{a}_{\mathrm{2016}} \right)^{\frac{\mathrm{1}}{\mathrm{2016}}} =\mathrm{2} \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)\ldots\left(\mathrm{1}+{a}_{\mathrm{2016}} \right)\geqslant... \\ $$
Question Number 55902 Answers: 1 Comments: 0
Question Number 55893 Answers: 1 Comments: 0
$$\mathrm{Three}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{in}\:\mathrm{G}.\mathrm{P}\:\mathrm{such}\:\mathrm{that}\:\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\:\boldsymbol{\mathrm{p}}\:\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{square}\:\mathrm{is}\:\:\boldsymbol{\mathrm{q}}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{G}.\mathrm{P} \\ $$
Question Number 55889 Answers: 1 Comments: 0
$$\:\:\:\:\:\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{bc}}+\boldsymbol{\mathrm{c}}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{ac}}+\boldsymbol{\mathrm{c}}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{c}}=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{ab}}+\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\::\:\:\boldsymbol{\mathrm{a}},\:\:\boldsymbol{\mathrm{b}},\:\:\boldsymbol{\mathrm{c}}. \\ $$
Question Number 55877 Answers: 0 Comments: 0
Question Number 55873 Answers: 2 Comments: 1
$${Integrate}..\int\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx} \\ $$
Question Number 55869 Answers: 2 Comments: 2
$${if}\:\boldsymbol{{a}}_{\boldsymbol{{n}}+\mathrm{2}} =\frac{\boldsymbol{{a}}_{\boldsymbol{{n}}+\mathrm{1}} ^{\mathrm{3}} }{\boldsymbol{{a}}_{\boldsymbol{{n}}} ^{\mathrm{2}} }\:{and}\:{a}_{\mathrm{1}} =\mathrm{2},\:{a}_{\mathrm{2}} =\mathrm{4} \\ $$$${find}\:\boldsymbol{{a}}_{\boldsymbol{{n}}} =? \\ $$
Question Number 55888 Answers: 1 Comments: 0
$$\:\:\:\boldsymbol{\mathrm{a}}=\mathrm{1}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\mathrm{b}}=\mathrm{1}+\boldsymbol{\mathrm{c}}+\boldsymbol{\mathrm{c}}^{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\mathrm{c}}=\boldsymbol{\mathrm{ab}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\::\:\:\boldsymbol{\mathrm{a}},\:\:\boldsymbol{\mathrm{b}},\:\:\boldsymbol{\mathrm{c}}. \\ $$
Question Number 55887 Answers: 2 Comments: 0
$$\:\:\:{a}^{\mathrm{2}} +\mathrm{1}={b}^{\mathrm{2}} \\ $$$$\:\:\:{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={b}^{\mathrm{4}} \\ $$$$\:\:\:{ab}={c} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\::\:\:\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}},\:\boldsymbol{\mathrm{c}}. \\ $$
Question Number 55860 Answers: 2 Comments: 0
Question Number 55859 Answers: 1 Comments: 0
Question Number 55858 Answers: 0 Comments: 0
Question Number 55857 Answers: 0 Comments: 0
$$\mathrm{Let}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{matrices}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2017}×\mathrm{2017}} \:\mathrm{such}\:\mathrm{that}\: \\ $$$${A}^{−\mathrm{1}} \:=\:\left({A}\:+\:{B}\right)^{−\mathrm{1}} \:−\:{B}^{−\mathrm{1}} \\ $$$$\mathrm{and}\: \\ $$$$\mathrm{det}\left({A}^{−\mathrm{1}} \right)\:=\:\mathrm{2017} \\ $$$$\mathrm{Find}\:\:\mathrm{det}\left({B}\right) \\ $$
Question Number 55856 Answers: 1 Comments: 1
$$\mathrm{For}\:\mathrm{what}\:\mathrm{values}\:\mathrm{of}\:{p}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{p}} \:\mathrm{ln}\:{x}\:{dx} \\ $$$$\mathrm{converge}? \\ $$
Question Number 55855 Answers: 0 Comments: 1
$$\mathrm{How}\:\mathrm{to}\:\mathrm{integrate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{sec}^{\mathrm{2}} \:{x}}{{x}\sqrt{{x}}}\:{dx}\:\:? \\ $$
Question Number 55853 Answers: 0 Comments: 0
Question Number 55846 Answers: 1 Comments: 0
$$\mathrm{If}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{determinant} \\ $$$$\begin{vmatrix}{{a}+\mathrm{2}}&{{a}+\mathrm{3}}&{{a}+\mathrm{2}{x}}\\{{a}+\mathrm{3}}&{{a}+\mathrm{4}}&{{a}+\mathrm{2}{y}}\\{{a}+\mathrm{4}}&{{a}+\mathrm{5}}&{{a}+\mathrm{2}{z}}\end{vmatrix}\:\mathrm{is} \\ $$
Question Number 55845 Answers: 0 Comments: 0
$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{3}×\mathrm{4}\:\mathrm{matrix}\:\mathrm{and}\:{B}\:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{such} \\ $$$$\mathrm{that}\:{A}'{B}\:\mathrm{and}\:{BA}'\:\mathrm{are}\:\mathrm{both}\:\mathrm{defined}.\:\mathrm{Then} \\ $$$${B}\:\:\:\mathrm{is}\:\mathrm{of}\:\mathrm{the}\:\mathrm{type} \\ $$
Question Number 55844 Answers: 1 Comments: 0
$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{an}\:\mathrm{involutory}\:\mathrm{matrix},\:\mathrm{then}\: \\ $$$$\left({I}+{A}\right)\left({I}−{A}\right)=\mathrm{0}. \\ $$
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