Question Number 215520 by CrispyXYZ last updated on 09/Jan/25 $$\mathrm{Prove}\:\mathrm{that}\:\begin{cases}{{x}\:=\:\frac{\mathrm{7}\:\mathrm{cos}\:{t}\:−\:\mathrm{2}}{\mathrm{2}\:−\:\mathrm{cos}\:{t}}}\\{{y}\:=\:\frac{\mathrm{4}\sqrt{\mathrm{3}}\:\mathrm{sin}\:{t}}{\mathrm{2}\:−\:\mathrm{cos}\:{t}}}\end{cases}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$ Answered by alephnull last updated on 09/Jan/25 $$=\:{x}\left(\mathrm{2}−\mathrm{cos}\left({t}\right)\right)=\mathrm{7cos}\left({t}\right)−\mathrm{2},\:{y}\left(\mathrm{2}−\mathrm{cos}\:\left({t}\right)\right)=\mathrm{4}\sqrt{\mathrm{3}}\mathrm{sin}\:\left({t}\right) \\ $$$$\mathrm{Rearrange} \\ $$$$\mathrm{2}{x}−{x}\mathrm{cos}\:\left({t}\right)=\mathrm{7cos}\:\left({t}\right)−\mathrm{2} \\…
Question Number 215523 by Ismoiljon_008 last updated on 09/Jan/25 $$ \\ $$$$\:\:\:\mathcal{T}{wo}\:{friends}\:{set}\:{off}\:\:{by}\:{train}\:{at}\:{dawn}\:{to}\:{visit} \\ $$$$\:\:\:{each}\:{other}.\:{The}\:{two}\:{friends}\:{caught}\:{sight}\:{of} \\ $$$$\:\:\:{each}\:{other}\:{through}\:{the}\:{window}\:{as}\:{the}\:{trains}\: \\ $$$$\:\:\:{passed}\:{in}\:{opposite}\:{direction}\:{on}\:{adjacent}\:{tracks}− \\ $$$$\:\:\:{it}\:{was}\:\mathrm{12}^{{oo}} \:{hours}.\:{The}\:{friends}\:{helplessly}\:{reached}\: \\ $$$$\:\:\:{their}\:{destinations}.\:{If}\:{the}\:{first}\:{of}\:{them}\:{reached}\: \\ $$$$\:\:\:{their}\:{destination}\:{at}\:\mathrm{16}^{{oo}}…
Question Number 215469 by Abdullahrussell last updated on 08/Jan/25 Answered by alephnull last updated on 08/Jan/25 $$\left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }\right)=\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{27}}\right) \\ $$$$=\left(\frac{\mathrm{27}}{\mathrm{108}}+\frac{\mathrm{4}}{\mathrm{108}}\right) \\ $$$${the}\:{root}\:{is}\:\frac{\mathrm{31}}{\mathrm{108}} \\ $$$$…
Question Number 215496 by MATHEMATICSAM last updated on 08/Jan/25 $$\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{tan}{x}}{\mathrm{sec}{x}\:+\:\mathrm{tan}{x}}\:{dx} \\ $$$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{integral}. \\ $$ Commented by mr W last updated on 09/Jan/25 $$=\frac{\pi^{\mathrm{2}}…
Question Number 215498 by alephnull last updated on 08/Jan/25 $$\int\left({e}^{−\omega{u}} +\mathrm{cos}\left({u}\right)−\frac{\mathrm{sin}\left(\omega{u}\right)}{{e}^{{u}} }\right){du} \\ $$ Answered by MathematicalUser2357 last updated on 10/Jan/25 $$−\frac{{e}^{−{u}\omega} }{\omega}+\frac{{e}^{−{u}} \mathrm{sin}\:{u}\omega}{\left(\omega−{i}\right)\left(\omega+{i}\right)}+\frac{{e}^{−{u}} \omega\mathrm{cos}\:{u}\omega}{\left(\omega−{i}\right)\left(\omega+{i}\right)}+\mathrm{sin}\:{u}+{C}…
Question Number 215492 by BaliramKumar last updated on 08/Jan/25 Commented by BaliramKumar last updated on 08/Jan/25 $${Decimal}\:\:{to}\:{binary} \\ $$ Answered by khorshidi17 last updated on…
Question Number 215476 by Ismoiljon_008 last updated on 08/Jan/25 $$\:\:\:{I}\:{need}\:{help}\:{for}\:{this}: \\ $$$$\:\:\:\mathcal{T}{wo}\:{friends}\:{set}\:{off}\:\:{by}\:{train}\:{at}\:{dawn}\:{to}\:{visit} \\ $$$$\:\:\:{each}\:{other}.\:{The}\:{two}\:{friends}\:{caught}\:{sight}\:{of} \\ $$$$\:\:\:{each}\:{other}\:{through}\:{the}\:{window}\:{as}\:{the}\:{trains}\: \\ $$$$\:\:\:{passed}\:{in}\:{opposite}\:{direction}\:{on}\:{adjacent}\:{tracks}− \\ $$$$\:\:\:{it}\:{was}\:\mathrm{12}^{{oo}} \:{hours}.\:{The}\:{friends}\:{helplessly}\:{reached}\: \\ $$$$\:\:\:{their}\:{destinations}.\:{If}\:{the}\:{first}\:{of}\:{them}\:{reached}\: \\ $$$$\:\:\:{their}\:{destination}\:{at}\:\mathrm{16}^{{oo}}…
Question Number 215504 by alephnull last updated on 08/Jan/25 $$\mathrm{simplify}\:{a}−\left\{{b}^{{c}−{b}} +\langle\left({b}^{{c}−{b}} \right)^{\mathrm{2}} +\left(\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} \rangle+\frac{{a}}{\mathrm{2}}\right\}+{c}^{{ab}−{c}} −{c}^{{a}−{c}} \\ $$ Answered by MrGaster last updated on 08/Jan/25 $$\mathrm{simplify}\:{a}−\left\{{b}^{{c}−{b}}…
Question Number 215473 by alephnull last updated on 08/Jan/25 $${Solve}\:{for}\:{x} \\ $$$$ \\ $$$$\mathrm{2}{sin}^{\mathrm{2}} {x}+\mathrm{3}{sin}\left({x}\right)+\mathrm{1}=\mathrm{0}\:{for}\:\mathrm{0}\:\leqslant\:{x} \\ $$ Answered by Rasheed.Sindhi last updated on 08/Jan/25 $$\mathrm{2}{sin}^{\mathrm{2}}…
Question Number 215468 by alephnull last updated on 07/Jan/25 $$\frac{\partial{x}\omega}{\partial{y}\omega}\centerdot\frac{\partial{e}}{\partial\omega}=? \\ $$$${x}={f}\left({y}\right) \\ $$$$\omega={g}\left({y}\right) \\ $$$${e}={h}\left(\omega\right) \\ $$ Answered by MrGaster last updated on 08/Jan/25…