Question Number 20298 by mondodotto@gmail.com last updated on 25/Aug/17 Answered by Tinkutara last updated on 25/Aug/17 $$\mathrm{45}\:=\:\mathrm{36}\:+\:\mathrm{12}\:+\:\mathrm{18}\:−\:{n}\left(\mathrm{medals}\:\mathrm{in}\:\mathrm{two}\right. \\ $$$$\left.\mathrm{categories}\right)\:+\:\mathrm{4} \\ $$$${n}\left(\mathrm{medals}\:\mathrm{in}\:\mathrm{two}\:\mathrm{categories}\right)\:=\:\mathrm{25} \\ $$$${n}\left(\mathrm{exactly}\:\mathrm{two}\:\mathrm{categories}\right)\:=\:\mathrm{25}−\mathrm{3}×\mathrm{4} \\ $$$$=\:\mathrm{13}…
Question Number 151371 by mathdanisur last updated on 20/Aug/21 $$\mathrm{a};\mathrm{b}\in\mathbb{N} \\ $$$$\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} \centerdot\left(\mathrm{a}-\mathrm{b}\right)=\mathrm{128} \\ $$$$\mathrm{ab}=? \\ $$ Answered by nimnim last updated on 20/Aug/21 $$\mathrm{a};\mathrm{b}\in\mathrm{N}…
Question Number 85832 by jagoll last updated on 25/Mar/20 $$\left(\mathrm{x}+\mathrm{x}^{−\mathrm{1}} \right)^{\mathrm{2}} +\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{−\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{−\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$+\:…\:+\:\left(\mathrm{x}^{\mathrm{10}} +\mathrm{x}^{−\mathrm{10}} \right)^{\mathrm{2}} \:=\: \\ $$ Commented…
Question Number 20297 by Tinkutara last updated on 25/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{expression}\:{ax}^{\mathrm{2}} \:+\:\mathrm{2}{hxy} \\ $$$$+\:{by}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{resolved}\:\mathrm{into}\:\mathrm{two}\:\mathrm{linear}\:\mathrm{rational}\:\mathrm{factors} \\ $$$$\mathrm{if}\:\Delta\:=\:{abc}\:+\:\mathrm{2}{fgh}\:−\:{af}^{\mathrm{2}} \:−\:{bg}^{\mathrm{2}} \:−\:{ch}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$ Answered by…
Question Number 20296 by Tinkutara last updated on 25/Aug/17 $$\mathrm{If}\:\left({m}_{{r}} \:,\:\frac{\mathrm{1}}{{m}_{{r}} }\right)\:;\:{r}\:=\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\mathrm{be}\:\mathrm{four}\:\mathrm{pairs} \\ $$$$\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{and}\:{y}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{m}_{\mathrm{1}} .{m}_{\mathrm{2}} .{m}_{\mathrm{3}} .{m}_{\mathrm{4}} \:=\:\mathrm{1}. \\…
Question Number 85828 by M±th+et£s last updated on 25/Mar/20 $$\int\frac{\mathrm{1}}{{x}+{cot}\left({x}\right)}\:{dx} \\ $$ Answered by mind is power last updated on 26/Mar/20 $${not}\:{found}\:\:{only}\:{withe}\:{Series}\: \\ $$$${have}\:{you}\:{a}\:{solutions}\:{sir}\:? \\…
Question Number 20293 by tammi last updated on 25/Aug/17 $$\int\sqrt{\frac{{a}+{x}}{{x}}{dx}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{{a}+{x}}{\:\sqrt{{x}\left({a}+{x}\right)}}{dx} \\ $$$$=\int\frac{{a}+{x}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+{a}+{a}}{\:\sqrt{{ax}+{x}^{\mathrm{2}}…
Question Number 20292 by tammi last updated on 25/Aug/17 $$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} +\sqrt{{x}−\mathrm{1}}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{{dx}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{{x}−\mathrm{1}}} \\ $$$$=\int\frac{\sqrt{{x}+\mathrm{1}}−\sqrt{{x}−\mathrm{1}}}{\left({x}+\mathrm{1}\right)−\left({x}−\mathrm{1}\right)}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}+\mathrm{1}}{dx}−\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}−\mathrm{1}}{dx}…
Question Number 151361 by mathdanisur last updated on 20/Aug/21 $$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\mathrm{45}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{x}\right)}{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{y}\right)}\:=\:\mathrm{tan}\left(\mathrm{2y}\right) \\ $$ Answered by Ar Brandon last updated on 20/Aug/21 $${x}+{y}=\frac{\pi}{\mathrm{4}} \\ $$$$\frac{\mathrm{1}−\mathrm{tan}{x}}{\mathrm{1}−\mathrm{tan}{y}}=\frac{\mathrm{1}−\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}−{y}\right)}{\mathrm{1}−\mathrm{tan}{y}}…
Question Number 85826 by jagoll last updated on 25/Mar/20 $$\:^{\mathrm{x}} \mathrm{log}\:\left(\mathrm{xy}\right).\:^{\mathrm{y}} \mathrm{log}\:\left(\mathrm{xy}\right)\:+\:^{\mathrm{x}} \mathrm{log}\:\left(\mathrm{x}−\mathrm{y}\right).^{\mathrm{y}} \mathrm{log}\:\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{x}+\mathrm{y}\: \\ $$ Commented by john santu last updated on…