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Question Number 206473 Answers: 4 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{4}^{\boldsymbol{\mathrm{p}}} \:=\:\mathrm{5} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{2}^{\mathrm{3}\boldsymbol{\mathrm{p}}} \:=\:? \\ $$
Question Number 206471 Answers: 1 Comments: 0
$$\mathrm{If}\:{a}\mathrm{sin}\theta\:=\:{b}\mathrm{cos}\theta\:=\:\frac{\mathrm{2}{c}\mathrm{tan}\theta}{\mathrm{1}\:−\:\mathrm{tan}^{\mathrm{2}} \theta}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\left({a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} \right)^{\mathrm{2}} \:=\:\mathrm{4}{c}^{\mathrm{2}} \left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right). \\ $$
Question Number 206466 Answers: 1 Comments: 1
Question Number 206458 Answers: 3 Comments: 0
$${if}\:{f}\left({x}\right)=\sqrt{{x}−{x}^{\mathrm{2}} }\:\:\:\:\:{then}\:\:{f}^{−\mathrm{1}} \left({x}\right)=? \\ $$
Question Number 206452 Answers: 1 Comments: 0
Question Number 206451 Answers: 0 Comments: 0
Question Number 206449 Answers: 1 Comments: 0
$${solve}\:{the}\:{first}\:{order}\:{differential} \\ $$$${equation}: \\ $$$$ \\ $$$${xdy}\:−\:{ydx}\:=\:\left({xy}\right)^{\mathrm{1}/\mathrm{2}} {dx} \\ $$
Question Number 206443 Answers: 0 Comments: 4
Question Number 206442 Answers: 1 Comments: 0
Question Number 206434 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{tan}^{\mathrm{2}} \theta\:=\:\mathrm{1}\:−\:{x}^{\mathrm{2}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{sec}\theta\:+\:\mathrm{tan}^{\mathrm{3}} \theta\mathrm{cosec}\theta\:=\:\sqrt{\left(\mathrm{2}\:−\:{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:. \\ $$
Question Number 206433 Answers: 2 Comments: 0
$$\:\:\:\:\:\mathrm{let}\:\mathrm{f}:\left[\mathrm{0},\infty\right)\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:\mathrm{if} \\ $$$$\:\:\:\:\underset{\mathrm{n}\rightarrow\infty\:} {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}+\mathrm{n}\right)\mathrm{dx}\:=\:\mathrm{2} \\ $$$$\:\mathrm{then}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{nx}\right)\:=\:? \\ $$$$\: \\ $$
Question Number 206430 Answers: 2 Comments: 0
Question Number 206425 Answers: 1 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{cos}\boldsymbol{\alpha}\:=\:\frac{\mathrm{3}}{\mathrm{5}}\:\:\:\left(\mathrm{0}<\boldsymbol{\alpha}<\frac{\boldsymbol{\pi}}{\mathrm{2}}\right) \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{tan}^{\mathrm{2}} \:\left(\mathrm{45}°\:+\:\frac{\boldsymbol{\alpha}}{\mathrm{2}}\right)}{\mathrm{3}}\:=\:? \\ $$
Question Number 206421 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{tan}{p}\theta\:=\:{p}\mathrm{tan}\theta\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{sin}^{\mathrm{2}} {p}\theta}{\mathrm{sin}^{\mathrm{2}} \theta}\:=\:\frac{{p}^{\mathrm{2}} }{\mathrm{1}\:+\:\left({p}^{\mathrm{2}} \:−\:\mathrm{1}\right)\mathrm{sin}^{\mathrm{2}} \theta}\:.\: \\ $$
Question Number 206399 Answers: 2 Comments: 1
Question Number 206396 Answers: 3 Comments: 0
Question Number 206394 Answers: 0 Comments: 1
Question Number 206393 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{S}=\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(−\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}−\frac{\mathrm{1}}{\ell+\mathrm{1}}\right)\:,\:\ell\in\left[\mathrm{1},\infty\right) \\ $$$$\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(−\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}−\frac{\mathrm{1}}{\ell+\mathrm{1}}\right) \\ $$$$\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\right)−\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}\right)+\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{5}}\right)−...... \\ $$
Question Number 206391 Answers: 2 Comments: 0
$$\mathrm{Find}: \\ $$$$\int_{−\mathrm{3}} ^{\:−\mathrm{2}} \:\left(\mid\mathrm{x}\mid\:+\:\mid\mathrm{x}\:−\:\mathrm{4}\mid\right)\:\mathrm{dx}\:=\:? \\ $$
Question Number 206365 Answers: 2 Comments: 4
$$\mathrm{Number}\:\mathrm{series}: \\ $$$$\mathrm{a}_{\mathrm{3}} \:=\:\mathrm{2a}\:+\:\mathrm{b}\:−\:\mathrm{6} \\ $$$$\mathrm{a}_{\mathrm{9}} \:=\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{5} \\ $$$$\mathrm{a}_{\mathrm{15}} \:=\:\mathrm{3a}\:+\:\mathrm{b}\:−\:\mathrm{7} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}\:=\:? \\ $$
Question Number 206364 Answers: 2 Comments: 0
Question Number 206363 Answers: 0 Comments: 2
$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}<\mathrm{1} \\ $$$$\mathrm{Compare}: \\ $$$$\frac{\mathrm{1}}{\mathrm{a}−\mathrm{1}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{a}−\mathrm{1}}\:\:,\:\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{1}−\mathrm{a}}\:\:,\:\:\frac{\mathrm{a}}{\mathrm{2a}} \\ $$
Question Number 206357 Answers: 2 Comments: 0
$$\mathrm{Find}: \\ $$$$\mathrm{1}\:+\:\mathrm{cos444}°\:−\:\mathrm{cos84}°\:+\:\mathrm{cot45}°\:=\:? \\ $$
Question Number 206355 Answers: 2 Comments: 0
$${if}\:{the}\:{sum}\:{of}\:{three}\:{positive}\:{real}\: \\ $$$${numbers}\:{is}\:{equal}\:{to}\:{their}\:{product}, \\ $$$${prove}\:{that}\:{at}\:{least}\:{one}\:{of}\:{the}\: \\ $$$${numbers}\:{is}\:{larger}\:{than}\:\mathrm{1}.\mathrm{7}. \\ $$
Question Number 206353 Answers: 0 Comments: 6
Question Number 206351 Answers: 0 Comments: 1
$${expression}\:{of}\:{the}\:{sequence}\:\left({a}_{{n}} \right)\:{defined} \\ $$$${by}\: \\ $$$$\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{2}}−\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}} \left(\mathrm{2}{n}+\mathrm{3}\right)}{{n}+\mathrm{2}}{a}_{{n}+\mathrm{1}} +\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}{a}_{{n}} }\end{cases} \\ $$
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