Question Number 214683 by MATHEMATICSAM last updated on 16/Dec/24 Commented by MATHEMATICSAM last updated on 16/Dec/24 $$\mathrm{Circles}\:\mathrm{C1}\:\mathrm{and}\:\mathrm{C2}\:\mathrm{have}\:\mathrm{equal}\:\mathrm{radii}\:\mathrm{and} \\ $$$$\mathrm{are}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}\:\mathrm{XY}.\:\mathrm{Circle} \\ $$$$\mathrm{C3}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{C1}\:\mathrm{and}\:\mathrm{C2}.\:\mathrm{Find} \\ $$$$\mathrm{distance}\:{h},\:\mathrm{from}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{C3}\:\mathrm{to}\:\mathrm{line} \\ $$$$\mathrm{XY}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{x}\:\mathrm{and}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circles}.…
Question Number 214678 by issac last updated on 16/Dec/24 $$\mathrm{solve} \\ $$$$\mathrm{partial}\:\mathrm{differantial}\:\mathrm{equation} \\ $$$${x}\frac{\partial{f}\left({x},{y}\right)}{\partial{x}}+{y}\frac{\partial{f}\left({x},{y}\right)}{\partial{y}}={f}\left({x},{y}\right)\mathrm{ln}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$$\frac{\partial^{\mathrm{2}} {f}\left({x},{y}\right)}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {f}\left({x},{y}\right)}{\partial{y}^{\mathrm{2}} }=\mathrm{0} \\ $$ Answered…
Question Number 214675 by cherokeesay last updated on 16/Dec/24 Answered by mr W last updated on 16/Dec/24 Commented by mr W last updated on 16/Dec/24…
Question Number 214667 by issac last updated on 16/Dec/24 $$\mathrm{let}'\mathrm{s}\:\mathrm{define}\:\mathrm{linear}\:\mathrm{differantial}\:\mathrm{operator}\:\mathcal{D} \\ $$$$\mathrm{as}\:\mathcal{D}={z}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\left({z}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\right)+{z}\left(\mathrm{1}−\left(\frac{\alpha}{{z}}\right)^{\mathrm{2}} \right) \\ $$$$\mathrm{when} \\ $$$$\mathcal{D}{f}\left({z}\right)=\left\{{z}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\left({z}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\right)+{z}\left(\mathrm{1}−\left(\frac{\alpha}{{z}}\right)^{\mathrm{2}} \right)\right\}{f}\left({z}\right)=\mathrm{0} \\ $$$${f}\left({z}\right)=? \\ $$ Terms of Service…
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Question Number 214661 by efronzo1 last updated on 15/Dec/24 Answered by Rasheed.Sindhi last updated on 15/Dec/24 $${x}.{f}\left({x}\right)+{x}.{f}\left(\frac{\mathrm{1}}{{x}}\right)={x}+\mathrm{1} \\ $$$${f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)=\frac{{x}+\mathrm{1}}{{x}}……{A} \\ $$$${Replace}\:{x}\:{by}\:\:\frac{\mathrm{1}}{{x}}: \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)+\:{f}\left({x}\right)=\frac{\frac{\mathrm{1}}{{x}}+\mathrm{1}}{\frac{\mathrm{1}}{{x}}}={x}+\mathrm{1}…{B} \\ $$$${A}\:\&\:{B}:\:\:\:\frac{{x}+\mathrm{1}}{{x}}={x}+\mathrm{1}…
Question Number 214656 by mr W last updated on 15/Dec/24 Commented by mr W last updated on 17/Dec/24 $${same}\:{conditions}\:{as}\:{in}\:{Q}\mathrm{214449} \\ $$$${find}\:{the}\:{final}\:{speed}\:{of}\:{the}\:{lower} \\ $$$${cylinder}. \\ $$…
Question Number 214658 by ChantalYah last updated on 15/Dec/24 $$\left.\mathrm{1}\right)\:\mathrm{The}\:\mathrm{function}\:\mathrm{H}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{H}\left(\mathrm{x}\right)\:=\mathrm{3cosh}\frac{\mathrm{x}}{\mathrm{3}}+\mathrm{sinh}\frac{\mathrm{x}}{\mathrm{3}}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\lambda\:\mathrm{for}\:\mathrm{which}\:\mathrm{H}\left(\mathrm{ln}\lambda^{\mathrm{3}} \right)=\mathrm{4} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{2}} ^{\mathrm{4}} \:\frac{\mathrm{6x}+\mathrm{1}}{\left(\mathrm{2x}−\mathrm{3}\right)\left(\mathrm{3x}−\mathrm{2}\right)}\mathrm{dx}=\:\mathrm{ln}\:\mathrm{10}. \\ $$$$\left.\mathrm{3}\right)\mathrm{Show}\:\mathrm{that}\:\frac{\mathrm{sin}\theta\:+\:\mathrm{sin2}\theta}{\mathrm{1}+\:\mathrm{cos}\theta+\:\mathrm{cos2}\theta}\equiv\:\mathrm{tan}\theta \\ $$$$\left.\mathrm{4}\right)\:\mathrm{If}\:\mathrm{z}=\mathrm{cos}\theta+\:\mathrm{i}\:\mathrm{sin}\theta,\:\mathrm{Show}\:\mathrm{that}\:\mathrm{z}+\frac{\mathrm{1}}{\mathrm{z}}=\mathrm{2cos}\theta\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{z}^{\mathrm{n}} +\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{n}} }=\mathrm{2cos}\:\mathrm{n}\theta\:\mathrm{hence}\:\mathrm{or}\:\mathrm{otherwise}\:\mathrm{show}\:\mathrm{that}\:…
Question Number 214638 by mr W last updated on 14/Dec/24 $${if}\:{the}\:{sum}\:{of}\:{three}\:{prime}\:{numbers} \\ $$$${is}\:\mathrm{130},\:{what}\:{is}\:{the}\:{possible}\: \\ $$$${maximum}\:{of}\:{their}\:{product}? \\ $$ Answered by BaliramKumar last updated on 14/Dec/24 $$\mathrm{2}\:+\:{x}\:+\:{y}\:=\:\mathrm{130}…
Question Number 214644 by asifumer658 last updated on 14/Dec/24 $$\frac{−\mathrm{6}}{\mathrm{7}}/\frac{−\mathrm{7}}{\mathrm{6}} \\ $$ Answered by Frix last updated on 17/Dec/24 $$\frac{−\mathrm{6}}{\mathrm{7}}/\frac{−\mathrm{7}}{\mathrm{6}}=\left(−\frac{\mathrm{6}}{\mathrm{7}}\right)/\left(−\frac{\mathrm{7}}{\mathrm{7}}\right)=\left(−\frac{\mathrm{6}}{\mathrm{7}}\right)×\left(−\frac{\mathrm{6}}{\mathrm{7}}\right)= \\ $$$$=\frac{\mathrm{36}}{\mathrm{49}} \\ $$ Terms…