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AllQuestion and Answers: Page 660
Question Number 150432 Answers: 1 Comments: 0
$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2n}+\mathrm{1}\right)!}{\mathrm{8}^{\mathrm{n}} \centerdot\left(\mathrm{n}!\right)^{\mathrm{2}} }=?\:\:\:\:\:\mathrm{Help}\:\mathrm{please} \\ $$
Question Number 150429 Answers: 1 Comments: 0
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{tangents}\:\mathrm{to}\:\mathrm{the}\:\mathrm{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{parabola}\:{x}^{\mathrm{2}} =\mathrm{2}{y}−\mathrm{3}. \\ $$
Question Number 150425 Answers: 0 Comments: 6
Question Number 150421 Answers: 0 Comments: 0
Question Number 150418 Answers: 2 Comments: 3
Question Number 150413 Answers: 0 Comments: 3
Question Number 150410 Answers: 1 Comments: 1
Question Number 150392 Answers: 0 Comments: 0
Question Number 150376 Answers: 1 Comments: 2
Question Number 150374 Answers: 1 Comments: 0
$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$
Question Number 150372 Answers: 0 Comments: 1
Question Number 150450 Answers: 0 Comments: 0
$$\mathrm{show}\:\mathrm{the}?\mathrm{connection}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{beta}\:\mathrm{distribution}\left(\mathrm{n},\mathrm{p}\right)\:\mathrm{and}\:\mathrm{hypergeometric} \\ $$$$\mathrm{distribution}\left(\mathrm{N},\mathrm{k},\mathrm{n}\right)\mathrm{in}\:\mathrm{a}\:\mathrm{limiting}\:\mathrm{case} \\ $$
Question Number 150366 Answers: 0 Comments: 1
Question Number 150405 Answers: 2 Comments: 0
Question Number 150404 Answers: 4 Comments: 0
$$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{log}\left(\mathrm{x}\right)}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$
Question Number 150402 Answers: 1 Comments: 1
$$\mathrm{For}\:\:\mathrm{m}\geqslant\mathrm{1} \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{x}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\left(\mathrm{x}\right)}{\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{1}}\:=\:\mathrm{2}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\boldsymbol{\zeta}\left(\mathrm{3}\right) \\ $$
Question Number 150400 Answers: 0 Comments: 0
Question Number 150363 Answers: 2 Comments: 5
Question Number 150469 Answers: 1 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{4}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}}\:\:\:\mathrm{find}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{17}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{16}}{\mathrm{17}}\right)\:\overset{?} {=} \\ $$
Question Number 150351 Answers: 1 Comments: 2
Question Number 150350 Answers: 0 Comments: 0
Question Number 150346 Answers: 0 Comments: 2
$$\mathrm{If}\:\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}}\:\mathrm{and}\:\boldsymbol{\mathrm{z}}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\boldsymbol{\mathrm{N}}=\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{xy}+\mathrm{yz}+\mathrm{zx},\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{bigger}\:\mathrm{than}\:\mathrm{2022}. \\ $$
Question Number 150332 Answers: 0 Comments: 0
Question Number 150331 Answers: 0 Comments: 0
$${ultimately}\:\:{Q}.\mathrm{149894}\:\:{boils}\: \\ $$$${down}\:{to}\:{finding}\:{s}_{{max}} ,\:{s}_{{min}} \:\forall \\ $$$$\:\:\:\:\left\{{h}−{s}\mathrm{cos}\:\left(\theta−\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} \\ $$$$+\left\{{k}−{s}\mathrm{sin}\:\left(\theta+\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$ \\ $$
Question Number 150328 Answers: 0 Comments: 0
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} \:=\:\boldsymbol{\mathrm{xy}}\:+\:\mathrm{61} \\ $$
Question Number 150327 Answers: 2 Comments: 0
$$\left.\mathrm{1}\right)\:\mathrm{33}\boldsymbol{\mathrm{x}}\:\:\equiv\:\:\mathrm{48}\:\left(\mathrm{mod}\:\mathrm{654}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{5}^{\mathrm{1000}\:\mathrm{000}} \:\:\equiv\:\:\boldsymbol{\mathrm{x}}\:\left(\mathrm{mod}\:\mathrm{41}\right) \\ $$$$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$
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