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Question Number 103736 Answers: 0 Comments: 0
$$\mathrm{Solve}:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} \:\:+\:\:\mathrm{c}^{\mathrm{2}} \:\:=\:\:\:\mathrm{196}\:\:\:\:\:\:...\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{b}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{c}\:\:−\:\:\mathrm{a}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{169}\:\:\:\:\:...\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{c}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{b}\:\:−\:\:\mathrm{c}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{225}\:\:\:\:\:...\:\left(\mathrm{iii}\right) \\ $$
Question Number 103723 Answers: 0 Comments: 0
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{n},\:\mathrm{such}\:\mathrm{that}; \\ $$$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\centerdot\centerdot\centerdot+\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{1}+\mathrm{x}}\mathrm{dx}−\mathrm{ln2}−\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} \\ $$
Question Number 103716 Answers: 1 Comments: 3
Question Number 103703 Answers: 1 Comments: 0
$${if}\:{an}\:{object}\:{moves}\:{along}\:{a}\:{straight}\:{line}\: \\ $$$${according}\:{to}\:{the}\:{relationship}\left(\:{x}\left({t}\right)=\left(\frac{\mathrm{1}}{\mathrm{2}}{t}^{\mathrm{2}} −{t}+\mathrm{2}\right)\right) \\ $$$${find}\: \\ $$$$\left(\mathrm{1}\right)\:{the}\:{average}\:{speed}\:{between}\:\left({x}=\frac{\mathrm{3}}{\mathrm{2}}\:,\:{x}=\frac{\mathrm{7}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{2}\right)\:{the}\:{pelvic}\:{velocity}\:{between}\:\left({x}=\frac{\mathrm{7}}{\mathrm{2}}\right) \\ $$
Question Number 103700 Answers: 1 Comments: 1
Question Number 103691 Answers: 1 Comments: 0
$${solve}\:{for}\:{x} \\ $$$${x}^{{x}} ={s} \\ $$
Question Number 103686 Answers: 1 Comments: 0
$${x}^{{x}^{\mathrm{6}} } =\sqrt{\mathrm{2}}\:^{\sqrt{\mathrm{2}}} \:{x}=? \\ $$
Question Number 103685 Answers: 1 Comments: 0
Question Number 103683 Answers: 1 Comments: 1
$$\int_{\mathrm{0}} ^{\mathrm{1}} {tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{1}+{x}−{x}^{\mathrm{2}} }\right){dx} \\ $$
Question Number 103673 Answers: 1 Comments: 0
$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{4095}} {\sum}}\frac{\mathrm{1}}{\left(\sqrt{{k}}+\sqrt{{k}+\mathrm{1}}\right)\left(\sqrt[{\mathrm{4}}]{{k}}+\sqrt[{\mathrm{4}}]{{k}+\mathrm{1}}\right)}\:? \\ $$
Question Number 103672 Answers: 1 Comments: 3
Question Number 103670 Answers: 4 Comments: 0
$${Given}\:{b}_{{n}} \:=\:\mathrm{3}.\mathrm{2}^{{n}} \:{is}\:{a}\:{GP}\:.\:{find}\:{the}\:{value} \\ $$$${of}\:\frac{\mathrm{1}}{{b}_{\mathrm{1}} }+\frac{\mathrm{1}}{{b}_{\mathrm{2}} }+\frac{\mathrm{1}}{{b}_{\mathrm{3}} }+...+\frac{\mathrm{1}}{{b}_{\mathrm{10}} }\:?\: \\ $$
Question Number 103669 Answers: 2 Comments: 0
$${prove}\:{that}\:: \\ $$$$\left.{a}\right)\:\int_{−\mathrm{3}} ^{−\mathrm{1}} {x}^{\mathrm{2}} {dx}\:\geqslant\int_{\mathrm{1}} ^{\mathrm{3}} \left(\mathrm{2}{x}−\mathrm{1}\right){dx} \\ $$$$\left.{b}\right)\int_{−\mathrm{2}} ^{\mathrm{0}} {xdx}\:\leqslant\int_{\mathrm{0}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} \:+\:{x}\:\right){dx} \\ $$$$\left.{c}\right)\int_{\mathrm{1}} ^{\mathrm{4}} \left({x}^{\mathrm{2}} \:+\:\mathrm{2}\right){dx}\:\:\geqslant\int_{\mathrm{2}} ^{\mathrm{5}} \left(\mathrm{2}{x}\:−\mathrm{5}\right){dx} \\ $$$$\left.{d}\right)\int_{−\pi} ^{−\frac{\mathrm{3}\pi}{\mathrm{4}}} \mathrm{cos}\:\mathrm{2}{x}\:{dx}\:\geqslant\int_{\frac{\mathrm{3}\pi}{\mathrm{4}}} ^{\pi} \mathrm{sin}\:\mathrm{2}{x}\:{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Question Number 103665 Answers: 0 Comments: 0
Question Number 103664 Answers: 1 Comments: 0
Question Number 103659 Answers: 2 Comments: 1
$$\mathrm{When}\:\mathrm{y}=\mathrm{ax}+\mathrm{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{curve}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} \:\mathrm{passing}\:\mathrm{through}\:\left(\mathrm{0};\:−\mathrm{2}\right), \\ $$$$\mathrm{find}\:\mathrm{a}+\mathrm{b}? \\ $$
Question Number 103652 Answers: 0 Comments: 1
Question Number 103649 Answers: 0 Comments: 0
Question Number 103648 Answers: 2 Comments: 0
$$\boldsymbol{\mathrm{Evaluate}}\:\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{3}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{5}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}+...+\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.} \\ $$
Question Number 103647 Answers: 0 Comments: 0
Question Number 103644 Answers: 1 Comments: 1
Question Number 103643 Answers: 4 Comments: 0
$${if}\:\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\:=\:\frac{\mathrm{5}}{\mathrm{6}} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\:?\: \\ $$
Question Number 103633 Answers: 2 Comments: 1
$$ \\ $$$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{question}}\:\boldsymbol{\mathrm{is}} \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.}\right)=... \\ $$
Question Number 103624 Answers: 0 Comments: 0
Question Number 103623 Answers: 2 Comments: 0
$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{series}}\:\boldsymbol{\mathrm{whose}}\:\boldsymbol{\mathrm{nth}} \\ $$$$\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{is}}\:\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.}. \\ $$$$\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{have}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{problem}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{need}} \\ $$$$\boldsymbol{\mathrm{help}}\:\boldsymbol{\mathrm{please}} \\ $$
Question Number 103622 Answers: 2 Comments: 0
$${Given}\:{a}\:=\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{24}} {\sum}}\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}+\sqrt{{n}}}\:{then}\:{the}\:{value}\:{of} \\ $$$${a}\:+\:\frac{\mathrm{1}}{\mathrm{log}\:_{{a}} \left({bc}\right)+\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{log}\:_{{b}} \left({ac}\right)+\mathrm{1}}\:+ \\ $$$$\frac{\mathrm{1}}{\mathrm{log}\:_{{c}} \left({ab}\right)+\mathrm{1}}\:=\:? \\ $$
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