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AlgebraQuestion and Answers: Page 226
Question Number 121127 Answers: 3 Comments: 0
$$\sqrt{\mathrm{3}+\mathrm{2}{i}}=? \\ $$
Question Number 121099 Answers: 1 Comments: 0
$${Let}\:\alpha\:{be}\:{a}\:{root}\:{of}\:\:{x}^{\mathrm{5}} −{x}^{\mathrm{3}} +{x}−\mathrm{2}=\mathrm{0} \\ $$$${Then}\:{prove}\:{that}\:\:\:\left[\alpha^{\mathrm{6}} \right]=\mathrm{3}\:\:\:\:\:\:\:{where}\left[\lambda\right]\:\:{denotes}\:{greatest}\:{integer} \\ $$$${less}\:{than}\:{or}\:\:{equal}\:\lambda \\ $$
Question Number 121041 Answers: 1 Comments: 0
$$\:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} .\mathrm{k}\:=?\: \\ $$
Question Number 120993 Answers: 0 Comments: 0
Question Number 120984 Answers: 2 Comments: 0
$$\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{4}\centerdot\mathrm{7}}\:+\:\frac{\mathrm{1}}{\mathrm{4}\centerdot\mathrm{7}\centerdot\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{7}\centerdot\mathrm{10}\centerdot\mathrm{13}}+...+\frac{\mathrm{1}}{\mathrm{25}\centerdot\mathrm{28}\centerdot\mathrm{31}}=\:? \\ $$
Question Number 120960 Answers: 0 Comments: 0
$${If}\:{x},{y},{z}>\mathrm{0}\:\:{then}\:{prove}\:{following} \\ $$$${inequality} \\ $$$$\left({x}^{\mathrm{2}} +\mathrm{2}\right)\left({y}^{\mathrm{2}} +\mathrm{2}\right)\left({z}^{\mathrm{2}} +\mathrm{2}\right)\geqslant\mathrm{9}\left({xy}+{yz}+{xz}\right) \\ $$
Question Number 120953 Answers: 4 Comments: 0
$$\mathrm{solve}\:\mathrm{in}\:\mathrm{x}\in\mathbb{R} \\ $$$$\mid\:\mathrm{3x}−\mathrm{4}\:\mid\:=\:\mathrm{x}−\mathrm{5}\: \\ $$
Question Number 120943 Answers: 3 Comments: 1
$$\int\left({x}^{\mathrm{3}} +\mathrm{3}\right)^{\mathrm{4}} {dx}=? \\ $$
Question Number 120929 Answers: 5 Comments: 2
Question Number 120904 Answers: 1 Comments: 1
Question Number 120882 Answers: 1 Comments: 0
Question Number 120855 Answers: 2 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:...\:{elementary}\:\:{calculus}... \\ $$$$\:\:::\:\alpha,\beta\:{are}\:{roots}\:{of}\:\:{equation} \\ $$$$\:\:\:\:\:{of}\::\:{x}^{\mathrm{2}} −\mathrm{6}{x}−\mathrm{2}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:{define}\:::\:{t}_{{n}} =\alpha^{{n}} −\beta^{{n}} \:\left({n}\geqslant\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{then}\:\:{evaluate}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{A}=\frac{{t}_{\mathrm{10}} −\mathrm{2}{t}_{\mathrm{8}} }{\mathrm{2}{t}_{\mathrm{9}} }\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$
Question Number 120839 Answers: 3 Comments: 0
Question Number 120809 Answers: 0 Comments: 0
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{found}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way} \\ $$$$\mathrm{that}\:\mathrm{any}\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:{a}\:\mathrm{and}\:{b}\:\left(\:{a}\neq{b}\right)\: \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{next}\:\mathrm{inequality}\:\mid{a}−{b}\mid\geqslant\frac{{ab}}{\mathrm{100}} \\ $$
Question Number 120780 Answers: 2 Comments: 2
$${solve}\:{x}^{\mathrm{2}^{{x}} } =−\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Question Number 120711 Answers: 3 Comments: 0
$${show}\:{that}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}...=\frac{−\mathrm{1}}{\mathrm{8}} \\ $$
Question Number 120636 Answers: 0 Comments: 23
$${selective}\:{Binomial} \\ $$
Question Number 120601 Answers: 1 Comments: 0
$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\: \\ $$$$\:\mathrm{which}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\:\left(\mathrm{2}\:−\:\mathrm{3}{x}\right)^{−\mathrm{4}} \:\mathrm{is}\:\mathrm{valid}.\: \\ $$$$\:{I}\:{need}\:{help}\:{with}\:{explanation}\:{please} \\ $$
Question Number 120583 Answers: 0 Comments: 0
Question Number 120558 Answers: 3 Comments: 0
Question Number 120556 Answers: 0 Comments: 2
Question Number 120545 Answers: 1 Comments: 2
$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$
Question Number 120529 Answers: 1 Comments: 1
Question Number 120480 Answers: 4 Comments: 0
$${show}\:{that}\:\forall\:{n}\:\in\mathbb{N}^{\ast} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left({n}−{k}\right)=\frac{\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$
Question Number 120475 Answers: 1 Comments: 0
Question Number 120472 Answers: 0 Comments: 0
$${A}=\overline {\mathrm{5}{x}\mathrm{23}}\:^{\mathrm{6}} \:{show}\:{that}\:{A}\equiv{x}−\mathrm{4}\left[\mathrm{7}\right] \\ $$$${deduct}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{A}\:{is}\: \\ $$$${divisible}\:{by}\:\mathrm{7} \\ $$
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