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…
Question Number 165433 by mathlove last updated on 01/Feb/22 Answered by aleks041103 last updated on 01/Feb/22 $$ \\ $$$$\frac{\mathrm{3}}{\mathrm{2}^{{x}} }+\frac{\mathrm{3}}{\mathrm{3}^{{x}} }=\mathrm{2}\frac{\left(\mathrm{3}/\mathrm{2}\right)^{{x}} }{\mathrm{2}^{{x}} }+\mathrm{2}\frac{\left(\mathrm{3}/\mathrm{2}\right)^{−{x}} }{\mathrm{3}^{{x}} }…
Question Number 165435 by mathlove last updated on 01/Feb/22 Answered by Kamel last updated on 01/Feb/22 Commented by mindispower last updated on 02/Feb/22 $${Barak}\:{Alah}\:{fik}\:{Nice}\:{Solution}\:{Sir} \\…
Question Number 34357 by math1967 last updated on 05/May/18 $${If}\:{a}+{b}+{c}=\mathrm{0}\:\:{prove}\:{that} \\ $$$$\left.{i}\right)\left(\frac{{a}}{{b}+{c}}+\:\frac{{b}}{{c}+{a}}\:+\frac{{c}}{{a}+{b}}\right)\left(\frac{{b}+{c}}{{a}}\:+\frac{{c}+{a}}{{b}}\:+\frac{{a}+{b}}{{c}}\right)=\mathrm{9} \\ $$$$\left.{ii}\right)\left(\frac{{a}}{{b}−{c}}\:+\frac{{b}}{{c}−{a}}\:+\frac{{c}}{{a}−{b}}\right)\left(\frac{{b}−{c}}{{a}}\:+\frac{{c}−{a}}{{b}}\:+\frac{{a}−{b}}{{c}}\right)=\mathrm{9} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 05/May/18 $$\left.{i}\right)\left\{\left(\frac{{a}}{{b}+{c}}+\mathrm{1}−\mathrm{1}\right)+\left(\frac{{b}}{{c}+{a}}+\mathrm{1}−\mathrm{1}\right)+\left(\frac{{c}}{{a}+{b}}+\mathrm{1}−\mathrm{1}\right)\right\}× \\…
Question Number 165419 by mathlove last updated on 01/Feb/22 Answered by TheSupreme last updated on 01/Feb/22 $$\left({a}+\frac{\mathrm{1}}{{a}}\right)={x} \\ $$$${a}^{\mathrm{2}} +\frac{\mathrm{1}}{{a}^{\mathrm{2}} }=\left({a}+\frac{\mathrm{1}}{{a}}\right)^{\mathrm{2}} −\mathrm{2}={x}^{\mathrm{2}} −\mathrm{2} \\ $$$${a}^{\mathrm{3}}…
Question Number 99877 by I want to learn more last updated on 23/Jun/20 Commented by bobhans last updated on 24/Jun/20 $$\left(\mathrm{3}.\mathrm{1}\right)\mathrm{gradient}\:\mathrm{of}\:\mathrm{PQ}\:=\:\mathrm{tan}\:\mathrm{45}^{\mathrm{o}} \:=\:\mathrm{1} \\ $$$$\left(\mathrm{3}.\mathrm{2}\right)\mathrm{MN}\:\mathrm{line}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{PQ}\:\Rightarrow\mathrm{then}\:\mathrm{equation} \\…