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AlgebraQuestion and Answers: Page 91
Question Number 174735 Answers: 0 Comments: 0
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Question Number 174684 Answers: 1 Comments: 0
$$ \\ $$$$\boldsymbol{\mathrm{Q}}:\:\:\boldsymbol{\mathrm{I}}\:,\:\boldsymbol{\mathrm{J}}\:\:\boldsymbol{{are}}\:\boldsymbol{{two}}\:\boldsymbol{{ideals}}\:\boldsymbol{{of}}\:\:\boldsymbol{{commutative}}\: \\ $$$$\:\:\:\:\boldsymbol{{ring}}\:,\:\left(\:\boldsymbol{\mathrm{R}}\:,\oplus,\: \:\right)\:.\boldsymbol{{prove}}\:\boldsymbol{{that}}\:: \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\sqrt{\:\boldsymbol{\mathrm{I}}\:\cap\:\boldsymbol{\mathrm{J}}\:}\:\:\overset{?} {=}\:\sqrt{\:\boldsymbol{\mathrm{I}}\:}\:\:\cap\:\:\sqrt{\:\boldsymbol{\mathrm{J}}\:}\:\:\:\:\:\boldsymbol{{m}}.\boldsymbol{{n}} \\ $$$$\:\:\:\:\boldsymbol{{note}}\::\:\sqrt{\boldsymbol{\mathrm{I}}\:}\:=\:\left\{\:\boldsymbol{{x}}\:\in\:\boldsymbol{\mathrm{R}}\:\mid\:\exists\:\boldsymbol{{n}}\in\:\mathbb{N}\:,\:\boldsymbol{{x}}^{\:\boldsymbol{{n}}} \:\in\:\boldsymbol{\mathrm{I}}\:\right\}\: \\ $$$$\: \\ $$
Question Number 174674 Answers: 0 Comments: 0
$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship}\:\mathrm{holds}: \\ $$$$\mathrm{a}^{\mathrm{4}} \:<\:\left(\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \right)^{\mathrm{2}} \:+\:\mathrm{9R}^{\mathrm{4}} \\ $$
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$$\mathrm{Factorize}\:{a}^{\mathrm{3}} \:−\:\mathrm{16}{a}\:−\:\mathrm{3} \\ $$
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$$\mathrm{Factorize}\:{x}^{\mathrm{9}} \:+\:{x}^{\mathrm{7}} \:+\:\mathrm{1} \\ $$
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$$\:\:\:\:{find}\:{minimum}\:{value}\:{of}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{{a}}^{\mathrm{2}} \left(\boldsymbol{{b}}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{2}\boldsymbol{{ab}}+\left(\boldsymbol{{a}}−\mathrm{1}\right)^{\mathrm{2}} }{\boldsymbol{{a}}\left(\boldsymbol{{ab}}+\mathrm{1}\right)}\:\:\:\:\:\:\:\:\:\left[\boldsymbol{{all}}\in\boldsymbol{{R}}\right] \\ $$
Question Number 174569 Answers: 0 Comments: 0
$$\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}+\boldsymbol{{z}}=\mathrm{0}}\\{\boldsymbol{{y}}+\boldsymbol{{t}}+\boldsymbol{{xz}}=−\mathrm{2}\boldsymbol{{b}}}\\{\boldsymbol{{xt}}+\boldsymbol{{yz}}=\mathrm{1}}\\{\boldsymbol{{yt}}=\boldsymbol{{b}}^{\mathrm{2}} −\boldsymbol{{a}}}\end{cases}\:\:\:\:\:\:\left[\:\:\boldsymbol{{all}}\in\boldsymbol{{R}}\right] \\ $$$$\boldsymbol{{solve}}\:\boldsymbol{{for}}\:\:\::\:\:\:\boldsymbol{{x}},\:\boldsymbol{{y}}\:,\boldsymbol{{z}},\:\boldsymbol{{t}}\:\:\:\:\:\:\:\:\:\:. \\ $$
Question Number 174568 Answers: 1 Comments: 0
$$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{{a}\:+\:{b}\:+\:{c}}\:\mathrm{then}\: \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{3}} \:}\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{3}} }\:=\:? \\ $$
Question Number 174552 Answers: 2 Comments: 3
Question Number 174538 Answers: 1 Comments: 3
$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{and} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:``\mathrm{x}''\:\mathrm{if} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:=\:\mathrm{6},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{18}, \\ $$$$\mathrm{for}\:\mathrm{real}\:\mathrm{values}\:\mathrm{of}\:``\mathrm{x},\mathrm{y}\:\mathrm{and}\:\mathrm{z}'' \\ $$
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Question Number 174512 Answers: 1 Comments: 0
$${how}\:{many}\:{integer}\:{a},{b}\in{z}^{+} \\ $$$${a}^{\mathrm{5}} −{b}^{\mathrm{5}} =\mathrm{10}\left({b}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{9} \\ $$$$ \\ $$
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$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\mathrm{R}\in\left(\mathrm{AB}\right)\:,\:\mathrm{P}\in\left(\mathrm{BC}\right)\:,\:\mathrm{Q}\in\left(\mathrm{CA}\right) \\ $$$$\mathrm{AR}=\mathrm{3}\:,\:\mathrm{RB}=\mathrm{1}\:,\:\mathrm{BP}=\mathrm{6}\:,\:\mathrm{PC}=\mathrm{2}\:,\:\mathrm{CQ}=\mathrm{5}\:,\:\mathrm{QA}=\mathrm{4} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{PQ}\:+\:\mathrm{QR}\:+\:\mathrm{RP}\:>\:\frac{\mathrm{21}}{\mathrm{2}} \\ $$
Question Number 174451 Answers: 0 Comments: 0
$${please}\:{help} \\ $$$${If}\:{x}\:{is}\:{directly}\:{porportional}\:{to}\:{z}\:\:{and}\: \\ $$$${y}\:{is}\:{also}\:{directly}\:{porportional}\:{to}\:{z} \\ $$$$\:{then}\:{what}\:{is} \\ $$$${the}\:{value}\:{of}\:\:{xy}\:{propotional}\:{to}\:\:? \\ $$$$ \\ $$
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