Question Number 81968 by mr W last updated on 17/Feb/20 Commented by mr W last updated on 17/Feb/20 $$\left[{derived}\:{from}\:{Q}\mathrm{81804}\right] \\ $$$$ \\ $$$${given}: \\ $$$${string}\:{length}\:={s}\:\:\:\left(>\:\mathrm{0}.\mathrm{5}{L}\right)…
Question Number 81966 by naka3546 last updated on 17/Feb/20 $$\left(\frac{\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:\right)^{\mathrm{2020}} \:+\:\:\left(\frac{\mathrm{1}\:−\:{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:\right)^{\mathrm{2020}} \:\:=\:\:\:{A} \\ $$$${A}^{\mathrm{4}} \:\:=\:\:? \\ $$ Answered by TANMAY PANACEA last updated on 17/Feb/20…
Question Number 147500 by mathdanisur last updated on 21/Jul/21 $${x}^{\mathrm{2}} \:-\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:=\:\mathrm{22} \\ $$$${what}\:{is}\:{the}\:{number}\:{of}\:{complete} \\ $$$${solutions}\:{that}\:{satisf}\:{the}\:{equation} \\ $$$$\left({x};{y}\right).? \\ $$ Commented by Mrsof last updated…
Question Number 16430 by Tinkutara last updated on 22/Jun/17 $$\mathrm{In}\:\Delta{ABC}\:\mathrm{with}\:\mathrm{usual}\:\mathrm{notation} \\ $$$$\frac{{r}_{\mathrm{1}} }{{bc}}\:+\:\frac{{r}_{\mathrm{2}} }{{ca}}\:+\:\frac{{r}_{\mathrm{3}} }{{ab}}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{{R}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{{R}} \\ $$…
Question Number 81963 by jagoll last updated on 17/Feb/20 $${if}\:\mathrm{tan}\:\left({x}\right)+\mathrm{sec}\:\left({x}\right)\:=\:\frac{\mathrm{7}}{\mathrm{8}} \\ $$$${find}\:\mathrm{cot}\:\left({x}\right)+\mathrm{cosec}\:\left({x}\right)\:=\: \\ $$ Commented by john santu last updated on 17/Feb/20 $${let}\:\mathrm{cot}\:\left({x}\right)+\mathrm{cosec}\:\left({x}\right)\:=\:{t} \\ $$$$\left({i}\right)\:\left\{\mathrm{tan}\:\left({x}\right)+\mathrm{sec}\:\left({x}\right)\right\}×\left\{\mathrm{cot}\:\left({x}\right)+\mathrm{cosec}\:\left({x}\right)\right\}=\:\frac{\mathrm{7}{t}}{\mathrm{8}}…
Question Number 147493 by henderson last updated on 21/Jul/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{dears}}\:\boldsymbol{\mathrm{masters}}\:! \\ $$$$\boldsymbol{\mathrm{A}}\:=\:\left\{\boldsymbol{{au}}+\boldsymbol{{bv}},\:\left(\boldsymbol{{a}},\:\boldsymbol{{b}},\:\boldsymbol{{u}},\:\boldsymbol{{v}}\right)\:\in\:\mathbb{Z}^{\mathrm{4}} \right\}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{{a}}\:\neq\:\boldsymbol{{b}}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{ideal}}\:\boldsymbol{\mathrm{of}}\:\:\mathbb{Z}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{let}}\:\boldsymbol{\lambda}\mathbb{Z}\:=\:\left\{\boldsymbol{\lambda{n}}\:,\:\boldsymbol{{n}}\:\in\:\mathbb{Z}\right\}.\: \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{smaller}}\:\boldsymbol{\mathrm{element}}\:\boldsymbol{\lambda}\:\boldsymbol{\mathrm{strictly}}\: \\ $$$$\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{A}}\:=\:\boldsymbol{\lambda}\mathbb{Z}. \\ $$$$\mathrm{3}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\lambda}\:=\:\boldsymbol{\mathrm{gcd}}\left(\boldsymbol{{a}},\boldsymbol{{b}}\right). \\ $$…
Question Number 147492 by Kunal12588 last updated on 21/Jul/21 $$\mathrm{2}\sqrt{\mathrm{19}}\:\mathrm{cos}\:\left[\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{45}\sqrt{\mathrm{3}}}{\mathrm{28}}\right)\right] \\ $$$${it}\:{is}\:{equal}\:{to}\:\mathrm{8}.\:{How}? \\ $$ Commented by MJS_new last updated on 21/Jul/21 $$\mathrm{just}\:\mathrm{an}\:\mathrm{idea} \\ $$$${x}^{\mathrm{3}}…
Question Number 81954 by Cmr 237 last updated on 16/Feb/20 $$\left.\:{soit}\:\alpha\in\right]\mathrm{0};\pi\left[.\:{determiner}:\right. \\ $$$$\left.\mathrm{1}\right){le}\:{module}\:{et}\:{l}'{argument}\:{de}: \\ $$$$\left.\boldsymbol{{a}}\left.\right)\mathrm{1}−\boldsymbol{{e}}^{\boldsymbol{{i}}\alpha} ,\boldsymbol{{b}}\right)\mathrm{1}+\boldsymbol{{e}}^{\boldsymbol{{i}\alpha}} \\ $$$$\left.\mathrm{2}\right)\boldsymbol{{deduire}}\:\boldsymbol{{le}}\:\boldsymbol{{module}}\:\boldsymbol{{et}}\:\boldsymbol{{l}}'\boldsymbol{{argument}}\:\boldsymbol{{de}} \\ $$$$\left.\:\left.\boldsymbol{{a}}\right)\:\frac{\mathrm{1}−\boldsymbol{{e}}^{\boldsymbol{{i}}\alpha} }{\mathrm{1}+{e}^{{i}\alpha} },\:{b}\right)\left(\mathrm{1}−{e}^{{i}\alpha} \right)\left(\mathrm{1}+{e}^{{i}\alpha} \right) \\…
Question Number 147488 by mathdanisur last updated on 21/Jul/21 $${if}\:\:{x};{y};{z}\geqslant\mathrm{1}\:\:{then}: \\ $$$$\frac{\mathrm{1}}{\mathrm{3}{xy}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{yz}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{zx}−\mathrm{1}}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}{xyz}} \\ $$ Answered by mindispower last updated on 21/Jul/21 $${in}\:{firt} \\ $$$$\mathrm{1}\leqslant{xy},{xy}=\frac{{xyz}}{{z}} \\…
Question Number 147487 by mnjuly1970 last updated on 21/Jul/21 $$ \\ $$$$ \\ $$$$\left({a}\:,\:\mathrm{2}{a}\:+\mathrm{1}\:\right]\cap\left[\:{a}^{\:\mathrm{2}} \:−{a}\:,\:{a}^{\:\mathrm{2}} +\:\mathrm{4}{a}\:+\mathrm{1}\:\right)\neq\:\varnothing \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:\in\:? \\ $$$$ \\ $$ Answered…