Question Number 215516 by MrGaster last updated on 09/Jan/25 $$\:\:\:\:\:\:\mathrm{prove}: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\pi^{\mathrm{2}} {n}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{1}}{{e}^{\mathrm{2}} −\mathrm{1}} \\ $$ Answered by mr W last updated…
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 $$=\:{x}\left(\mathrm{2}−\mathrm{cos}\left({t}\right)\right)=\mathrm{7cos}\left({t}\right)−\mathrm{2},\:{y}\left(\mathrm{2}−\mathrm{cos}\:\left({t}\right)\right)=\mathrm{4}\sqrt{\mathrm{3}}\mathrm{sin}\:\left({t}\right) \\ $$$$\mathrm{Rearrange} \\ $$$$\mathrm{2}{x}−{x}\mathrm{cos}\:\left({t}\right)=\mathrm{7cos}\:\left({t}\right)−\mathrm{2} \\…
Question Number 215523 by Ismoiljon_008 last updated on 09/Jan/25 $$ \\ $$$$\:\:\:\mathcal{T}{wo}\:{friends}\:{set}\:{off}\:\:{by}\:{train}\:{at}\:{dawn}\:{to}\:{visit} \\ $$$$\:\:\:{each}\:{other}.\:{The}\:{two}\:{friends}\:{caught}\:{sight}\:{of} \\ $$$$\:\:\:{each}\:{other}\:{through}\:{the}\:{window}\:{as}\:{the}\:{trains}\: \\ $$$$\:\:\:{passed}\:{in}\:{opposite}\:{direction}\:{on}\:{adjacent}\:{tracks}− \\ $$$$\:\:\:{it}\:{was}\:\mathrm{12}^{{oo}} \:{hours}.\:{The}\:{friends}\:{helplessly}\:{reached}\: \\ $$$$\:\:\:{their}\:{destinations}.\:{If}\:{the}\:{first}\:{of}\:{them}\:{reached}\: \\ $$$$\:\:\:{their}\:{destination}\:{at}\:\mathrm{16}^{{oo}}…
Question Number 215476 by Ismoiljon_008 last updated on 08/Jan/25 $$\:\:\:{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 215417 by MATHEMATICSAM last updated on 06/Jan/25 $${a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{b}\mid\:=\:\mathrm{1}.\:\mathrm{Find}\:\mid\frac{{b}\:−\:{a}}{\mathrm{1}\:−\:\overline {{a}b}}\mid \\ $$ Answered by MrGaster last updated on 06/Jan/25 $$\mathrm{Let}\:{a}={x}+{yi},{b}=\mathrm{cos}\theta+{i}\mathrm{sin}\theta \\ $$$${b}−{a}=\left(\mathrm{cos}\theta−{x}\right)+{i}\left(\mathrm{sin}\theta−{y}\right)…
Question Number 215227 by Ghisom last updated on 01/Jan/25 $$\left(\mathrm{20}+\mathrm{25}\right)^{\mathrm{2}} =\mathrm{2025} \\ $$$$\mathrm{only}\:\mathrm{3}\:\mathrm{other}\:\mathrm{4}\:\mathrm{digit}\:\mathrm{numbers}: \\ $$$$\left(\mathrm{00}+\mathrm{01}\right)^{\mathrm{2}} =\mathrm{0001} \\ $$$$\left(\mathrm{30}+\mathrm{25}\right)^{\mathrm{2}} =\mathrm{3025} \\ $$$$\left(\mathrm{98}+\mathrm{01}\right)^{\mathrm{2}} =\mathrm{9801} \\ $$ Answered…
Question Number 215193 by MATHEMATICSAM last updated on 31/Dec/24 Commented by MATHEMATICSAM last updated on 31/Dec/24 $$\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{smaller}\:\mathrm{circle}\:\mathrm{is}\:{a}\:\mathrm{sq}\:\mathrm{unit} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{larger}\:\mathrm{circle}\:\mathrm{is}\:{A}\:\mathrm{sq}\:\mathrm{unit} \\ $$$$\mathrm{find}\:\frac{{a}}{{A}}\:. \\ $$ Answered by…
Question Number 215200 by mr W last updated on 31/Dec/24 $$\boldsymbol{\mathcal{HAPPY}}\:\:\boldsymbol{\mathcal{NEW}}\:\:\boldsymbol{\mathcal{YEAR}}\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{9}} {\boldsymbol{\sum}}{n}}^{\mathrm{3}} \:! \\ $$ Commented by ajfour last updated on 01/Jan/25 $${Happy}\:{new}\:{Year}\:\bullet\bullet\mathrm{25}.\:{lets}\:{all}\:{be}\:\mathrm{25}{y} \\…
Question Number 215166 by MathematicalUser2357 last updated on 30/Dec/24 $$\mathrm{If}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{8}{x}+\mathrm{2}{k}=\mathrm{0}\:\mathrm{has}\:\mathrm{two}\:\mathrm{different}\:\mathrm{negative}\:\mathrm{real}\:\mathrm{roots}, \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{k}\:\left(\mathrm{For}\:\mathrm{example},\:\mathrm{0}<{k}<\mathrm{3}\right)\:\mathrm{or}\:\mathrm{I}\:\mathrm{will}\:\mathrm{force}\:\mathrm{you}\:\mathrm{to}\:\mathrm{determine} \\ $$ Answered by A5T last updated on 30/Dec/24 $$\mathrm{x}_{\mathrm{1},\mathrm{2}} =\frac{−\mathrm{8}\underset{−} {+}\sqrt{\mathrm{64}−\mathrm{24k}}}{\mathrm{6}}=\frac{−\mathrm{8}\underset{−}…