Question Number 22154 by Joel577 last updated on 12/Oct/17 $$\mathrm{Given}\:{a},{b},{c}\:\mathrm{real}\:\mathrm{and}\:\mathrm{positive}\:\mathrm{numbers},\:\mathrm{and} \\ $$$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{1} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\:\frac{{a}\:+\:{b}}{{abc}} \\ $$ Answered by ajfour last updated on 12/Oct/17 $${let}\:{a}={cx}\:\:\:\:{and}\:\:{b}={cy} \\…
Question Number 153227 by Eric002 last updated on 05/Sep/21 $${let}\:{D}=\begin{bmatrix}{{v}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}}\\{\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:{m}}\end{bmatrix}\:{find}\:{number}\:\left({v}\right)\:{and} \\ $$$$\left({m}\right)\:{such}\:{that}\:{D}^{\mathrm{2}} =\mathrm{5}{I}\:\:\:\:\:\left({I}={identity}\:{matrix}\right) \\ $$ Answered by puissant last updated on 05/Sep/21 $${D}^{\mathrm{2}} =\begin{bmatrix}{{v}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\mathrm{5}{v}+\mathrm{5}{m}}\\{\frac{{v}}{\mathrm{3}}+\frac{{m}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{5}}{\mathrm{3}}+{m}^{\mathrm{2}}…
Question Number 153226 by ajfour last updated on 05/Sep/21 Commented by mr W last updated on 05/Sep/21 $${x}^{\mathrm{3}} +{x}−{c}=\mathrm{0} \\ $$$$\Rightarrow{x}=\sqrt[{\mathrm{3}}]{\sqrt{\frac{\mathrm{1}}{\mathrm{27}}+\frac{{c}^{\mathrm{2}} }{\mathrm{4}}}+\frac{{c}}{\mathrm{2}}}−\sqrt[{\mathrm{3}}]{\sqrt{\frac{\mathrm{1}}{\mathrm{27}}+\frac{{c}^{\mathrm{2}} }{\mathrm{4}}}−\frac{{c}}{\mathrm{2}}} \\ $$$${any}\:{other}\:{ways}\:{to}\:{solve}?…
Question Number 87686 by M±th+et£s last updated on 05/Apr/20 $$\int\sqrt{\frac{{ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$ Answered by TANMAY PANACEA. last updated on 05/Apr/20 $${t}^{\mathrm{2}} ={ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\right)…
Question Number 22151 by j.masanja06@gmail.com last updated on 12/Oct/17 $${integrate} \\ $$$$\int\frac{{cosx}−{cos}\mathrm{2}{x}}{\mathrm{1}+{cosx}}{dx} \\ $$ Answered by $@ty@m last updated on 12/Oct/17 $$=\int\frac{{cosx}−\left(\mathrm{2}{cos}^{\mathrm{2}} {x}−\mathrm{1}\right)}{\mathrm{1}+{cosx}}{dx} \\ $$$$=\int\frac{\mathrm{1}+\mathrm{cos}{x}\:−\mathrm{2cos}\:^{\mathrm{2}}…
Question Number 153220 by talminator2856791 last updated on 05/Sep/21 $$\: \\ $$$$\:\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{number}\:\: \\ $$$$\:{n}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{three}\:\mathrm{positive}\:\: \\ $$$$\:\mathrm{integers}\:\mathrm{if}\:\mathrm{representations}\:\mathrm{differing}\:\: \\ $$$$\:\mathrm{in}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{the}\:\mathrm{terms}\:\mathrm{are}\:\mathrm{considered}\:\: \\ $$$$\:\mathrm{to}\:\mathrm{be}\:\mathrm{different}?\:\: \\ $$$$\: \\ $$ Answered…
Question Number 87687 by ~blr237~ last updated on 05/Apr/20 $${Let}\:\:{w}=\left[\mathrm{1};\frac{\pi}{{n}}\right]\:,{n}\in\mathbb{N}^{\ast} \: \\ $$$$\:{a}_{{n}} =\underset{{p}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\:\frac{\mathrm{2}{p}+\mathrm{1}}{\mathrm{1}−{w}^{\mathrm{2}{p}+\mathrm{1}} }\:\:\:\:{and}\:\:\:{b}_{{n}} =\underset{{p}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\:\frac{{n}}{\mathrm{1}+{w}^{{p}} }\: \\ $$$${Find}\:\:{all}\:{integer}\:{n}\:{such}\:{as}\:\:{a}_{{n}} ={b}_{{n}} \:…
Question Number 87682 by john santu last updated on 05/Apr/20 $$\mathrm{what}\:\mathrm{is}\:\bigtriangledown^{\mathrm{2}} \left(\frac{\mathrm{1}}{\overset{\rightarrow} {\mathrm{r}}}\right)\:\mathrm{if}\: \\ $$$$\overset{\rightarrow} {\bigtriangledown}\:=\:\hat {\mathrm{i}}\:\frac{\partial}{\partial\mathrm{x}}+\hat {\mathrm{j}}\frac{\partial}{\partial\mathrm{y}}+\hat {\mathrm{k}}\:\frac{\partial}{\partial\mathrm{z}} \\ $$$$\mathrm{and}\:\overset{\rightarrow} {\mathrm{r}}\:=\:\hat {\mathrm{i}x}\:+\:\hat {\mathrm{j}y}\:+\:\hat {\mathrm{k}z}\:…
Question Number 153216 by talminator2856791 last updated on 05/Sep/21 $$ \\ $$ Commented by talminator2856791 last updated on 05/Sep/21 $$\:\mathrm{appears}\:\mathrm{to}\:\mathrm{be}\:\mathrm{a}\:\mathrm{bug}. \\ $$$$\:\mathrm{image}\:\mathrm{not}\:\mathrm{load}. \\ $$ Terms…
Question Number 22145 by Tinkutara last updated on 11/Oct/17 $$\mathrm{A}\:\mathrm{particle}\:{P}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{under} \\ $$$$\mathrm{the}\:\mathrm{action}\:\mathrm{of}\:\mathrm{only}\:\mathrm{one}\:\mathrm{force}\:\mathrm{acting} \\ $$$$\mathrm{always}\:\mathrm{towards}\:\mathrm{fixed}\:\mathrm{point}\:{O}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{circumference}.\:\mathrm{Find}\:\mathrm{ratio}\:\mathrm{of}\:\frac{{d}^{\mathrm{2}} \phi}{{dt}^{\mathrm{2}} }\:\mathrm{and} \\ $$$$\left(\frac{{d}\phi}{{dt}}\right)^{\mathrm{2}} . \\ $$ Commented by…