Question Number 226641 by fantastic2 last updated on 08/Dec/25 $${if}\:\mathrm{log}\:_{\mathrm{8}} {a}+\mathrm{log}\:_{\mathrm{4}} {b}^{\mathrm{2}} =\mathrm{5} \\ $$$$\& \\ $$$$\mathrm{log}\:_{\mathrm{8}} ^{{b}} +\mathrm{log}\:_{\mathrm{4}} {a}^{\mathrm{2}} =\mathrm{7} \\ $$$${ab}=? \\ $$…
Question Number 226622 by fantastic2 last updated on 07/Dec/25 Commented by mr W last updated on 07/Dec/25 $$\lambda=\frac{{m}}{{n}}=\frac{{x}−{x}_{\mathrm{1}} }{{x}_{\mathrm{2}} −{x}}=\frac{{y}−{y}_{\mathrm{1}} }{{y}_{\mathrm{2}} −{y}} \\ $$$${if}\:\lambda<\mathrm{0},\:{there}\:{are}\:{only}\:{two}\:{cases}: \\…
Question Number 226619 by fantastic2 last updated on 07/Dec/25 $${factorise} \\ $$$${x}^{\mathrm{2}} −{qx}−{p}^{\mathrm{2}} +\mathrm{5}{pq}−\mathrm{6}{q}^{\mathrm{2}} \\ $$ Commented by Frix last updated on 07/Dec/25 $${x}^{\mathrm{2}} −{qx}−{p}^{\mathrm{2}}…
Question Number 226598 by fantastic2 last updated on 07/Dec/25 $$\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{5}} ={a}_{\mathrm{5}} {x}^{\mathrm{5}} +{a}_{\mathrm{4}} {x}^{\mathrm{4}} +{a}_{\mathrm{3}} {x}^{\mathrm{3}} ..+{a}_{\mathrm{1}} {x}+{a}_{\mathrm{0}} \\ $$$${find}\:{a}_{\mathrm{5}} +{a}_{\mathrm{3}} +{a}_{\mathrm{1}} {and}\:{a}_{\mathrm{4}} +{a}_{\mathrm{2}} +{a}_{\mathrm{0}}…
Question Number 226624 by Lara2440 last updated on 07/Dec/25 Commented by Lara2440 last updated on 08/Dec/25 $$\: \\ $$$$\mathrm{Pseudo}-\mathrm{sphere}\:\mathrm{is}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{revolving}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tractrix} \\ $$$$\mathrm{paremetric}\:\mathrm{by}\:\:\hat {\boldsymbol{{v}}}\left({t}\right)\rightarrow\left(\alpha\left({t}−\mathrm{tanh}\left({t}\right)\right),\alpha\centerdot\mathrm{sech}\left({t}\right)\right) \\ $$$$\alpha>\mathrm{0}\:,\:\mathrm{0}\leq{t}<\infty \\…
Question Number 226581 by klipto last updated on 06/Dec/25 $$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{polar}}\:\boldsymbol{\mathrm{of}} \\ $$$$\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\right)\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\sqrt{\mathrm{3}}\right) \\ $$ Answered by Frix last updated on 06/Dec/25 $$=\sqrt{\mathrm{2}}\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} ×\mathrm{2e}^{\mathrm{i}\frac{\pi}{\mathrm{3}}} =\mathrm{2}\sqrt{\mathrm{2}}\mathrm{e}^{\mathrm{i}\frac{\mathrm{7}\pi}{\mathrm{12}}} \\…
Question Number 226573 by Lara2440 last updated on 06/Dec/25 $$\mathrm{Parametric}\:\mathrm{Surface}\:\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right);\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right)=\begin{cases}{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{cos}\left({v}\right)}\\{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{sin}\left({v}\right)}\\{{a}\centerdot\mathrm{cos}\left({u}\right)}\end{cases}\:\:\:{a}>\mathrm{0}\:,\:\mathrm{0}\leq{u}\leq\pi\:,\:\mathrm{0}\leq{v}\leq\mathrm{2}\pi \\ $$$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Direction} \\ $$$$\mathrm{2}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Curvature} \\ $$$$\mathrm{3}.\:\mathrm{Find}\:\mathrm{Gauss}\:\mathrm{Curvature} \\ $$$$\mathrm{4}.\:\mathrm{Find}\:\mathrm{Euler}\:\mathrm{Characteristic}\: \\ $$$$\mathrm{Hint}\:…
Question Number 226562 by thetpainghtun_111 last updated on 04/Dec/25 $$\mathrm{If}\:\mathrm{x}\:+\:\mathrm{y}\:{i}\:=\:\frac{\mathrm{a}\:+\:{i}}{\mathrm{a}\:−\:{i}}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{ay}\:−\:\mathrm{1}\:=\:\mathrm{x}. \\ $$$$\:\left(\mathrm{x}+\mathrm{y}{i}\right)\left(\mathrm{a}−{i}\right)=\mathrm{a}+{i} \\ $$$$\:\:\:\mathrm{ax}\:−\:\mathrm{x}{i}\:+\:\mathrm{ay}{i}\:−\:\mathrm{y}{i}^{\mathrm{2}} \:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\left(\mathrm{ax}\:+\:\mathrm{y}\right)\:+\:\left(\mathrm{ay}\:−\:\mathrm{x}\right){i}\:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\mathrm{ay}\:−\:\mathrm{x}\:=\:\mathrm{1} \\ $$$$\:\:\mathrm{x}\:=\:\mathrm{ay}\:−\mathrm{1} \\ $$ Answered by…
Question Number 226542 by Lara2440 last updated on 03/Dec/25 Commented by Lara2440 last updated on 04/Dec/25 $$\: \\ $$$$\mathrm{Smooth}\:\mathrm{Manifold}\:{M},{N}\:\mathrm{and}\:\mathrm{differentiable}\:\mathrm{Smooth}\:\mathrm{function}\: \\ $$$$\overset{\rightarrow} {\phi}\left({u},{v}\right);{M}\rightarrow{N} \\ $$$$\: \\…
Question Number 226507 by Lara2440 last updated on 01/Dec/25 Answered by Lara2440 last updated on 01/Dec/25 $$\mathrm{Smooth}\:\mathrm{manifold}\:{M},{N}\:\mathrm{and} \\ $$$$\mathrm{Differentiable}\:\mathrm{smooth}\:\mathrm{function}\:\:\phi;{M}\rightarrow{N} \\ $$$$\: \\ $$$$\phi\left({u},{v}\right)=\begin{cases}{−\mathrm{sin}\left({u}\right)−\mathrm{3sin}\left({v}\right)}\\{\:\:\:\:\mathrm{cos}\left({u}\right)+\mathrm{3cos}\left({v}\right)\:\:\:\:\:\:,\:{u}\in\left[−\pi,\pi\right]\:,\:{v}\in\left[−\mathrm{2}\pi,\mathrm{2}\pi\right]}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}{v}}\end{cases} \\ $$$$\:…