Question Number 165545 by HongKing last updated on 03/Feb/22 $$\mathrm{If} \\ $$$$\mathrm{13x}^{\mathrm{2}} +\mathrm{5y}^{\mathrm{2}} +\mathrm{9z}^{\mathrm{2}} +\mathrm{1}=\mathrm{4x}-\mathrm{6xy}-\mathrm{12yz} \\ $$$$\mathrm{Find} \\ $$$$\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)\left(\mathrm{xy}+\mathrm{xz}+\mathrm{yz}\right)=? \\ $$ Commented by MJS_new last…
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Question Number 165532 by mathlove last updated on 03/Feb/22 Answered by amin96 last updated on 03/Feb/22 $$\boldsymbol{\mathrm{S}}=\underset{\boldsymbol{\mathrm{n}}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} −\mathrm{1}}=\underset{\boldsymbol{\mathrm{n}}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\underset{\boldsymbol{\mathrm{n}}=\mathrm{2}} {\overset{\infty}…
Question Number 165505 by mathlove last updated on 02/Feb/22 Answered by TheSupreme last updated on 02/Feb/22 $$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}\left({n}−{i}\right){e}^{{i}} ={n}\frac{\mathrm{1}−{e}^{{n}+\mathrm{1}} }{\mathrm{1}−{e}}−\Sigma{ie}^{{i}} = \\ $$$$={n}\frac{\mathrm{1}−{e}^{{n}+\mathrm{1}} }{\mathrm{1}−{e}}−\frac{\mathrm{1}}{{e}}\Sigma{D}\left({e}^{{i}}…
Question Number 34429 by $@ty@m last updated on 06/May/18 $${Prove}\:{that} \\ $$$$\mathrm{3}^{{m}} +\mathrm{3}^{{n}} +\mathrm{1}\:{is}\:{not}\:{a}\:{perfect}\:{square}. \\ $$$${where}\:{m}\:{and}\:{n}\:{are}\:{positive}\:{integers}. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 06/May/18…
Question Number 34422 by math1967 last updated on 06/May/18 $${Show}\:{that} \\ $$$$\frac{\mathrm{1}+{x}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}}\:}\:+\frac{\mathrm{1}−{x}}{\mathrm{1}−\sqrt{\mathrm{1}−{x}\:}}\:=\mathrm{1}\:{when}\:{x}=\frac{\sqrt{\mathrm{3}\:}}{\mathrm{2}} \\ $$ Answered by Rio Mike last updated on 06/May/18 $$\frac{\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{\mathrm{1}+\sqrt{\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}}\:\:+\:\:\:\frac{\mathrm{1}−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{\mathrm{1}−\sqrt{\mathrm{1}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}} \\ $$$$\frac{\frac{\mathrm{2}+\sqrt{\mathrm{3}}}{\mathrm{2}}}{\mathrm{1}+\:\sqrt{\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}}\:\:\:+\:\:\:\frac{\frac{\mathrm{2}−\sqrt{\mathrm{3}}}{\mathrm{2}}}{\mathrm{1}−\sqrt{\mathrm{1}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}}…
Question Number 165483 by mathlove last updated on 02/Feb/22 Commented by MJS_new last updated on 04/Feb/22 $$\mathrm{for}\:{x}={y}={z}\:\mathrm{there}\:\mathrm{are}\:\mathrm{3}\:\mathrm{solutions}\:\mathrm{in}\:\mathbb{C} \\ $$$$\mathrm{for}\:{x}={y}\neq{z}\:\mathrm{there}\:\mathrm{are}\:\mathrm{9}\:\mathrm{solutions}\:\mathrm{in}\:\mathbb{C} \\ $$$$\:\:\:\:\:\left(\mathrm{and}\:\mathrm{of}\:\mathrm{course}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{interchangeable}\right) \\ $$$$\mathrm{for}\:{x}\neq{y}\neq{z}\:\mathrm{I}\:\mathrm{had}\:\mathrm{no}\:\mathrm{time}\:\mathrm{yet} \\ $$…
Question Number 99916 by Lordose last updated on 24/Jun/20 $$\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{anyone}}\:\boldsymbol{\mathrm{recommend}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{good}}\:\boldsymbol{\mathrm{textbook}} \\ $$$$\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{learn}}\:\boldsymbol{\mathrm{calculus}}.\hat {.} \\ $$ Answered by bobhans last updated on 24/Jun/20 Terms of Service…
Question Number 165442 by HongKing last updated on 01/Feb/22 Answered by Rasheed.Sindhi last updated on 02/Feb/22 $$\begin{cases}{\mathrm{25725}−{a}^{\mathrm{2}} ={b}^{\mathrm{3}} }\\{{c}^{\mathrm{2}} −\mathrm{25725}={d}^{\mathrm{2}} }\end{cases}\:\:;\:\:{a},{b},{c},{d}\in\mathbb{Z}^{+} \\ $$$$\underset{−} {\mathcal{T}{he}\:{smallest}\:{a}+{b}+{c}+{d}\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \\…
Question Number 99900 by Rio Michael last updated on 23/Jun/20 $$\mathrm{An}\:\mathrm{insulated}\:\mathrm{wire}\:\mathrm{of}\:\mathrm{diameter}\:\mathrm{1}.\mathrm{22}\:\mathrm{mm}\:\mathrm{carries}\:\mathrm{a}\:\mathrm{steady}\:\mathrm{current} \\ $$$$\mathrm{of}\:\mathrm{5}.\mathrm{4}\:\mathrm{A}.\:\mathrm{The}\:\mathrm{insulation}\:\mathrm{material}\:\mathrm{is}\:\mathrm{1}.\mathrm{22}\:\mathrm{mm}\:\mathrm{thick}\:\mathrm{and}\:\mathrm{has}\:\mathrm{a}? \\ $$$$\mathrm{coeffiecient}\:\mathrm{of}\:\mathrm{thermal}\:\mathrm{conductivity}\:\mathrm{of}\:\mathrm{0}.\mathrm{23}\:\mathrm{W}/\mathrm{Km}.\:\mathrm{the}\:\mathrm{electrical} \\ $$$$\mathrm{resistivity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{material}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{is}\:\mathrm{5}.\mathrm{2}\:×\mathrm{10}^{−\mathrm{7}} \Omega\mathrm{m}.\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{difference}\:\mathrm{between}\:\mathrm{the}\:\mathrm{inner}\:\mathrm{and}\:\mathrm{outer}\:\mathrm{surface}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{insulated}\:\mathrm{material}\:\mathrm{when}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{is}\:\mathrm{reached}. \\ $$ Terms…