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AlgebraQuestion and Answers: Page 230
Question Number 116529 Answers: 1 Comments: 1
Question Number 116509 Answers: 2 Comments: 0
$$\mathrm{x}^{\mathrm{4}} −\mathrm{48x}^{\mathrm{2}} +\mathrm{x}+\mathrm{565}=\mathrm{0}\: \\ $$$$\mathrm{x}=? \\ $$
Question Number 116507 Answers: 1 Comments: 0
Question Number 116452 Answers: 2 Comments: 0
Question Number 116358 Answers: 2 Comments: 0
$$\mathrm{Given}\:\mathrm{that}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}−\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}}\:\mathrm{and}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}+\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{ax}+\mathrm{b}\:=\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{ab}. \\ $$
Question Number 116323 Answers: 2 Comments: 0
$$\left(\mathrm{1}\right)\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:=\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{bc}}{\mathrm{b}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{and}\:\frac{\mathrm{ac}}{\mathrm{a}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{5}}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{24abc}}{\mathrm{ab}+\mathrm{ac}+\mathrm{bc}}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Let}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{be}\:\mathrm{two}\:\mathrm{real}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{satisfy}\:\mathrm{p}.\mathrm{q}=\mathrm{2013}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\left(\mathrm{p}+\mathrm{q}\right)^{\mathrm{2}} \:? \\ $$
Question Number 116301 Answers: 2 Comments: 0
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{for}\:\mathrm{a} \\ $$$$\mathrm{given}\:\mathrm{line}\:\mathrm{to}\:\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{not}\:\mathrm{to}\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve}\: \\ $$
Question Number 116259 Answers: 1 Comments: 0
Question Number 116253 Answers: 0 Comments: 0
$$\left(\mathrm{1}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right)\:\:\:=\:\:\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right).\:\mathrm{x}_{\mathrm{i}} ^{\mathrm{k}} \:\:\:=\:\:\:\mathrm{x}^{\mathrm{k}} ,\:\:\:\:\:\:\:\:\mathrm{k}\:\leqslant\:\mathrm{n} \\ $$
Question Number 116221 Answers: 3 Comments: 0
$$\mathrm{Given}\:\alpha,\beta\:\mathrm{and}\:\varphi\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{pq}\:=\:\mathrm{0}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\varphi}+\frac{\varphi}{\beta}+\frac{\alpha}{\varphi}+\frac{\varphi}{\alpha}=? \\ $$
Question Number 116192 Answers: 2 Comments: 1
Question Number 116173 Answers: 0 Comments: 2
$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\:\mathrm{DENKENMATHEMATICAL} \\ $$$$\mathrm{be}\:\mathrm{arranged}\:\mathrm{if}\:\mathrm{no}\:\mathrm{same}\:\mathrm{letters}\:\mathrm{must}\:\mathrm{be}\:\mathrm{together} \\ $$
Question Number 116163 Answers: 1 Comments: 0
$$\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{general}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{factorising} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} \\ $$$$\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}. \\ $$
Question Number 116138 Answers: 3 Comments: 1
$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{2}=\mathrm{0}\:\mathrm{and}\:\alpha\:>\beta,\:\mathrm{find}: \\ $$$$\left(\mathrm{i}\right)\:\frac{\mathrm{1}}{\beta}−\frac{\mathrm{1}}{\alpha} \\ $$$$\left(\mathrm{ii}\right)\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} \\ $$
Question Number 116226 Answers: 2 Comments: 1
$$\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}}\:+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{4}}}+...+\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{99}}+\mathrm{99}\sqrt{\mathrm{100}}} \\ $$
Question Number 116092 Answers: 1 Comments: 2
$${Solve}\:{for}\:{x}\:{and}\:{y}\:{if} \\ $$$${x}^{{y}} =\mathrm{36} \\ $$
Question Number 116091 Answers: 2 Comments: 0
$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{formular}\:\mathrm{to}\:\mathrm{tell}\:\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{a}\:\mathrm{digit}\:\mathrm{occur}\:\mathrm{in}\:\mathrm{an}\:\mathrm{interval}. \\ $$$$ \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{digits}\:\:\mathrm{2}\:\:\mathrm{occur}\:\mathrm{between}\:\:\mathrm{1}\:−\:\mathrm{100} \\ $$
Question Number 116059 Answers: 2 Comments: 0
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}!}{\mathrm{3}^{{n}+\mathrm{1}} } \\ $$
Question Number 116057 Answers: 2 Comments: 0
$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{3}}{\mathrm{3}{n}+\mathrm{1}}=? \\ $$
Question Number 116056 Answers: 2 Comments: 0
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{5}^{{n}} }{{n}!}=? \\ $$
Question Number 115986 Answers: 1 Comments: 0
Question Number 115916 Answers: 1 Comments: 0
Question Number 115909 Answers: 1 Comments: 0
$${solve}\:{the}\:{system}\:{of}\:{equations} \\ $$$${x}+\frac{\mathrm{3}{x}−{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{3}\:,\:{y}−\frac{{x}+\mathrm{3}{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{0} \\ $$
Question Number 115908 Answers: 1 Comments: 2
$${what}\:{is}\:{the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{4}{x}\right)......\left(\mathrm{1}+\mathrm{2}^{{n}} {x}\right) \\ $$
Question Number 115906 Answers: 1 Comments: 0
$${find}\:{all}\:{pairs}\:{of}\:{integers}\:\left({x},{y}\right)\:{such}\:{that} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\left({x}+{y}\right)^{\mathrm{2}} \\ $$
Question Number 115897 Answers: 2 Comments: 1
$$\mathrm{Given} \\ $$$${a}_{{n}} \:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{2015}} {\sum}}\left(\frac{\mathrm{4}}{{a}_{{n}} }\right)\:\mathrm{is}\:... \\ $$
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