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AlgebraQuestion and Answers: Page 268
Question Number 83590 Answers: 0 Comments: 3
$${transform}\:{the}\:{ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}\:{to} \\ $$$${the}\:{polar}\:{equation}\:{r}=\:\frac{{a}\left(\mathrm{1}−{e}^{\mathrm{2}} \right)}{\mathrm{1}+{ecos}\theta} \\ $$$${a}:\:{semimajor}\:{axis} \\ $$$${e}:\:{eccentricity} \\ $$
Question Number 83570 Answers: 2 Comments: 3
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{a}\:\mathrm{point}\:\mathrm{which}\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mathrm{its} \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{line}\:\:\:\mathrm{y}\:\:=\:\:\mathrm{4}\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\:\:\mathrm{k}. \\ $$
Question Number 83554 Answers: 1 Comments: 0
Question Number 83543 Answers: 1 Comments: 2
Question Number 83542 Answers: 2 Comments: 0
Question Number 83621 Answers: 2 Comments: 1
$$\mid\:{x}+\frac{\mathrm{1}}{{x}}\mid\:<\:\mathrm{4}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution} \\ $$
Question Number 83512 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\: \\ $$$$\frac{\mathrm{4x}^{\mathrm{2}} }{\left(\mathrm{1}−\sqrt{\mathrm{2x}+\mathrm{1}}\right)^{\mathrm{2}} }\:<\:\mathrm{2x}+\mathrm{9} \\ $$
Question Number 83477 Answers: 0 Comments: 1
$${a}^{{b}} +{b}^{{a}} =\mathrm{1}\:\:\:{a}=?\:,\:{b}=? \\ $$$${a}\neq{b}\neq\mathrm{0} \\ $$
Question Number 83473 Answers: 1 Comments: 0
$${solve}\:{in}\:{R} \\ $$$${sin}\left(\pi{ln}\left({x}\right)\right)+{cos}\left(\pi{ln}\left({x}\right)\right)=\mathrm{1} \\ $$
Question Number 83378 Answers: 0 Comments: 2
$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =\mathrm{3} \\ $$$${find} \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =? \\ $$
Question Number 83373 Answers: 0 Comments: 4
Question Number 83372 Answers: 0 Comments: 1
Question Number 83370 Answers: 0 Comments: 3
Question Number 83341 Answers: 1 Comments: 1
$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${of}\:{the}\:{expression}\:\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{a}}_{\boldsymbol{{i}}} \boldsymbol{{x}}_{\boldsymbol{{i}}} \:{with} \\ $$$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}}\left(\boldsymbol{{x}}_{\boldsymbol{{i}}} −\boldsymbol{{b}}_{\boldsymbol{{i}}} \right)^{\mathrm{2}} =\boldsymbol{{c}}^{\mathrm{2}} ,\:{where}\:\boldsymbol{{a}}_{\boldsymbol{{i}}} ,\:\boldsymbol{{b}}_{\boldsymbol{{i}}} \:{and}\:\boldsymbol{{c}}\:{are} \\ $$$${constants}. \\ $$$$ \\ $$$$\left({extracted}\:{and}\:{modified}\:{from}\:{Q}\mathrm{83331}\right) \\ $$
Question Number 83264 Answers: 1 Comments: 2
Question Number 83262 Answers: 1 Comments: 0
Question Number 83229 Answers: 0 Comments: 4
Question Number 83050 Answers: 0 Comments: 1
Question Number 83049 Answers: 0 Comments: 0
Question Number 82991 Answers: 1 Comments: 2
Question Number 82952 Answers: 1 Comments: 0
$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\:\:\mathrm{y}\:\:+\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\:\geqslant\:\:\mathrm{1}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{0}\:\:<\:\:\mathrm{y}\:\:−\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\leqslant\:\:\mathrm{1} \\ $$$$\mathrm{if}\:\:\mathrm{y}\:\:\geqslant\:\mathrm{1} \\ $$
Question Number 82897 Answers: 1 Comments: 2
Question Number 82881 Answers: 1 Comments: 2
$$ \\ $$$$\sqrt{\sqrt{...\sqrt{\mathrm{6561}}}}\:=\:\mathrm{3}^{\mathrm{8}^{\mathrm{x}} } \:\left(\mathrm{60}\:\mathrm{times}\right) \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$
Question Number 82873 Answers: 1 Comments: 2
Question Number 82843 Answers: 1 Comments: 0
$${show}\:{that} \\ $$$$\frac{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{i}\right)^{\mathrm{4}} \left(\mathrm{1}+{i}\right)^{\mathrm{8}} }{\left({cos}\mathrm{100}°−{i}\:{sin}\mathrm{100}\right)^{\mathrm{3}} }=−\mathrm{256} \\ $$
Question Number 82877 Answers: 1 Comments: 1
$$\left.\mathrm{1}\right){find}\:{xy}\in{R} \\ $$$$\left.\mathrm{2}\right){find}\:{x},{y}\in{Z} \\ $$$$\left({x}+\mathrm{2}{yi}\right)^{\mathrm{6}} =\mathrm{8}{i} \\ $$
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