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AlgebraQuestion and Answers: Page 67
Question Number 190611 Answers: 0 Comments: 2
$$\mathrm{Montrer}\:\mathrm{que}: \\ $$$$\mathrm{1}\bullet\boldsymbol{\mathrm{c}}^{\mathrm{2}} =\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\mathrm{2}\bullet\:\mathrm{rayon}\:\:\:\:\:\boldsymbol{\mathrm{r}}=\frac{\boldsymbol{\mathrm{c}}}{\mathrm{1}+\sqrt{\mathrm{2}}}−\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\mathrm{2}+\sqrt{\mathrm{2}}} \\ $$$$ \\ $$
Question Number 190607 Answers: 0 Comments: 0
Question Number 190536 Answers: 1 Comments: 0
$$\:\mathrm{If}\:\mathrm{p},\mathrm{q}\:\mathrm{and}\:\mathrm{r}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:\mathrm{x}^{\mathrm{3}} −\mathrm{3x}^{\mathrm{2}} +\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\:\mathrm{of}\:\sqrt[{\mathrm{3}}]{\mathrm{3p}−\mathrm{2}}\:+\sqrt[{\mathrm{3}}]{\mathrm{3q}−\mathrm{2}}+\sqrt[{\mathrm{3}}]{\mathrm{3r}−\mathrm{2}}\: \\ $$
Question Number 190523 Answers: 1 Comments: 0
Question Number 190522 Answers: 1 Comments: 0
Question Number 190520 Answers: 2 Comments: 0
$${if}\:{a},{b}\:{and}\:{c}\:{root}\:{of}\:{the} \\ $$$${x}^{\mathrm{3}} −\mathrm{16}{x}^{\mathrm{2}} −\mathrm{57}{x}+\mathrm{1}=\mathrm{0} \\ $$$${thi}\:{find}\:{thd}\:{volue}\:{of} \\ $$$${a}^{\frac{\mathrm{1}}{\mathrm{5}}} +{b}^{\frac{\mathrm{1}}{\mathrm{5}}} +{c}^{\frac{\mathrm{1}}{\mathrm{5}}} =? \\ $$
Question Number 190448 Answers: 1 Comments: 0
Question Number 190435 Answers: 2 Comments: 0
$$ \\ $$$$\mathrm{how}\:\mathrm{is}\:\mathrm{solution} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{10}} \centerdot\mathrm{sin}^{\mathrm{4}} \mathrm{x}\centerdot\mathrm{cos}^{\mathrm{8}} \mathrm{x}\centerdot\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{3x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} +\mathrm{x}}=? \\ $$$$ \\ $$
Question Number 190407 Answers: 2 Comments: 0
Question Number 190392 Answers: 1 Comments: 1
$$\mathrm{If}\:\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{3} \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{2a}\:−\:\mathrm{2b}}{\mathrm{a}^{\mathrm{2}} \:−\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{4a}\:+\:\mathrm{4}} \\ $$
Question Number 190357 Answers: 0 Comments: 1
Question Number 190325 Answers: 1 Comments: 0
Question Number 190324 Answers: 1 Comments: 1
Question Number 190321 Answers: 0 Comments: 1
$${What}\:{is}\:{the}\:{length}\:\:\boldsymbol{{x}}\:\:{of}\:{triangle} \\ $$$${equilateral}\:{A}^{'} {B}'{C}',{such}\:{that} \\ $$$${Area}\left({triangle}\:{ABC}\right)={Area}\left({triangle}\:{A}'{B}^{'} {C}'\right) \\ $$$$ \\ $$
Question Number 190303 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\mathrm{I}{f},\:\:{f}\left({x}\right)=\:\frac{\lfloor−{x}\:\rfloor}{{x}}\:+\mathrm{1}\:\:\Rightarrow\:\:{critical}\:{points}\:\:=\:? \\ $$$$ \\ $$
Question Number 190297 Answers: 1 Comments: 0
Question Number 190284 Answers: 2 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:\mathrm{4}^{\mathrm{2023}} \: \\ $$$$\mathrm{divides}\:\mathrm{by}\:\mathrm{7} \\ $$
Question Number 190275 Answers: 1 Comments: 0
$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{i}^{\mathrm{n}} \:\mathrm{for}\:\mathrm{every}\:\mathrm{positive} \\ $$$$\:\mathrm{integer}\:\mathrm{n},\:\mathrm{where}\:\mathrm{i}^{\mathrm{2}} \:=\:−\mathrm{1},\:\mathrm{i}^{\mathrm{3}} =\:\mathrm{i}^{\mathrm{2}} \mathrm{i},\:\mathrm{i}^{\mathrm{4}} \:=\:\mathrm{i}^{\mathrm{2}} \mathrm{i}^{\mathrm{2}} \:,\:{etc}. \\ $$
Question Number 190241 Answers: 1 Comments: 0
$${show}\:{that}\:{a}\circledast{b}={a}+{ab}+{b}\:{is}\:{a}\:{monoid}\:{when}\:{G}={Z} \\ $$
Question Number 190239 Answers: 0 Comments: 0
Question Number 190238 Answers: 0 Comments: 0
Question Number 190237 Answers: 0 Comments: 0
Question Number 190137 Answers: 2 Comments: 0
$$\begin{cases}{\mathrm{fog}^{−\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{3x}+\mathrm{2}}\\{\mathrm{gof}\left(\mathrm{x}\right)=\mathrm{2x}−\mathrm{1}}\end{cases} \\ $$$${find}\:\:\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)=?\:\:\:\mathrm{and}\:\:\:\:\mathrm{fof}\left(\mathrm{3}\right)=? \\ $$
Question Number 190131 Answers: 1 Comments: 0
$$\:\mathrm{if}:\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \:+\:\mathrm{14}\:=\:\mathrm{2}\left(\mathrm{x}\:+\:\mathrm{2y}\:+\:\mathrm{3z}\right) \\ $$$$\:\mathrm{find}:\:\:\mathrm{T}=\frac{\mathrm{xyz}}{\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} +\mathrm{z}^{\mathrm{3}} }\: \\ $$
Question Number 190115 Answers: 1 Comments: 0
$$\mathrm{if}:\:\:\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{a}+\mathrm{1}\right)\:=\:\mathrm{b} \\ $$$$\:\mathrm{find}:\:\:\mathrm{P}\:=\:\:\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} −\mathrm{3ab}} \\ $$
Question Number 190093 Answers: 2 Comments: 3
$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$${c}=\:\left(\:\sqrt{\mathrm{5}}\:+\mathrm{2}\right)^{\:\frac{\mathrm{1}}{\mathrm{3}}} \:−\:\left(\sqrt{\mathrm{5}}\:−\mathrm{2}\right)^{\:\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\mathrm{is}\:\:\:\mathrm{a}\:\:{rational}\:\:\mathrm{number}. \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$
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