Question Number 3228 by Rasheed Soomro last updated on 08/Dec/15 $$\mathcal{W}{hy}\:{is}\:{circumference}\:{of}\:{a}\:{circle}\:{derivative}\:{of} \\ $$$${its}\:{area}? \\ $$$$\frac{{d}}{{dr}}\left(\pi{r}^{\mathrm{2}} \right)=\mathrm{2}\pi{r}={circumference}. \\ $$ Commented by 123456 last updated on 08/Dec/15…
Question Number 134297 by mathlove last updated on 02/Mar/21 Answered by Ñï= last updated on 02/Mar/21 $$\frac{{d}}{{dx}}{ln}\left({x}^{{x}} +\mathrm{2}^{{x}^{{x}} } \right)=\frac{\mathrm{1}}{{x}^{{x}} +\mathrm{2}^{{x}^{{x}} } }\left[{x}^{{x}} \left({lnx}+\mathrm{1}\right)+\left({ln}\mathrm{2}\right)\mathrm{2}^{{x}^{{x}} }…
Question Number 134296 by EDWIN88 last updated on 02/Mar/21 Answered by bramlexs22 last updated on 02/Mar/21 $$\mathrm{total}\:\mathrm{three}\:\mathrm{digit}\:\mathrm{numbers}\:=\:\mathrm{4}×\mathrm{5}^{\mathrm{2}} =\mathrm{100} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{has}\:\mathrm{identical}\:\mathrm{digit}? \\ $$$$\left[\:\mathrm{111},\:\mathrm{333},\:\mathrm{555},\:\mathrm{999}\:\right]\:=\:\mathrm{4}\:\mathrm{numbers} \\ $$$$\mathrm{the}\:\mathrm{p}\left(\mathrm{A}\right)=\:\frac{\mathrm{4}}{\mathrm{100}}\:=\:\frac{\mathrm{1}}{\mathrm{25}} \\…
Question Number 68761 by Rasheed.Sindhi last updated on 15/Sep/19 $$\mathrm{Two}\:\boldsymbol{\mathrm{arcs}}\:\mathrm{having}\:\mathrm{their}\:\mathrm{centers}\:\mathrm{on}\:\mathrm{a} \\ $$$$\boldsymbol{\mathrm{circle}}\:\mathrm{are}\:\mathrm{cutting}\:\mathrm{each}\:\mathrm{other}\:\mathrm{at}\:\mathrm{a}\: \\ $$$$\mathrm{single}\:\mathrm{point}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{and}\:\mathrm{thus} \\ $$$$\:\mathrm{dividing}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{four}\:\mathrm{regions}. \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{arcs}\:\mathrm{cut}\:\mathrm{each}\:\mathrm{other}\:\mathrm{in}\:\boldsymbol{\mathrm{a}}:\boldsymbol{\mathrm{b}}\:\&\:\boldsymbol{\mathrm{c}}:\boldsymbol{\mathrm{d}}\: \\ $$$$\mathrm{ratios}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{between}\:\mathrm{four} \\ $$$$\mathrm{regions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{when}\:\mathrm{the}\:\mathrm{circle} \\…
Question Number 134292 by bramlexs22 last updated on 02/Mar/21 $$ \\ $$The digits 1, 2, 3, 4, 5, 6, and 7 are written on separate…
Question Number 134289 by bramlexs22 last updated on 02/Mar/21 $$\mathrm{I}=\int\:\frac{\mathrm{x}^{\mathrm{n}} }{\:\sqrt{\mathrm{ax}+\mathrm{b}}}\:\mathrm{dx} \\ $$$$\mathrm{H}=\int\:\frac{\mathrm{x}^{\mathrm{4}} }{\:\sqrt{\mathrm{2x}+\mathrm{1}}}\:\mathrm{dx} \\ $$ Answered by EDWIN88 last updated on 02/Mar/21 $$\mathrm{I}_{\mathrm{n}} =\:\frac{\mathrm{2x}^{\mathrm{n}}…
Question Number 134291 by Lordose last updated on 02/Mar/21 $$ \\ $$$$\:\boldsymbol{\mathrm{Prove}}\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{x}^{\mathrm{a}} }{\mathrm{1}+\mathrm{e}^{\mathrm{x}} }\mathrm{dx}\:=\:\left(\mathrm{1}−\mathrm{2}^{−\mathrm{a}} \right)\boldsymbol{\zeta}\left(\mathrm{a}+\mathrm{1}\right)\boldsymbol{\Gamma}\left(\mathrm{a}+\mathrm{1}\right) \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…
Question Number 3216 by prakash jain last updated on 07/Dec/15 $$\mathrm{You}\:\mathrm{have}\:\mathrm{unlimited}\:\mathrm{number}\:\mathrm{of}\:\mathrm{1kg},\:\mathrm{5kg} \\ $$$$\mathrm{10kg}\:\mathrm{and}\:\mathrm{25}\:\mathrm{kg}\:\mathrm{weights}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways} \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{create}\:\mathrm{a}\:\mathrm{total}\:\mathrm{of}\:\mathrm{43kg}. \\ $$$$\mathrm{For}\:\mathrm{example} \\ $$$$\mathrm{43}×\mathrm{1} \\ $$$$\mathrm{5}×\mathrm{8}+\mathrm{3}×\mathrm{2} \\ $$$$\mathrm{etc}. \\ $$…
Question Number 134285 by bramlexs22 last updated on 02/Mar/21 $$\mathrm{Find}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\left(\mathrm{16}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{9}\right)}\: \\ $$ Answered by EDWIN88 last updated on 02/Mar/21 $$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)=\left(\mathrm{16}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}}…
Question Number 3215 by prakash jain last updated on 07/Dec/15 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{circles}\:\mathrm{perimeter}\:\mathrm{to} \\ $$$$\mathrm{its}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{constant}. \\ $$$$\mathrm{Proof}\:\mathrm{should}\:\mathrm{not}\:\mathrm{utilize}\:\mathrm{a}\:\mathrm{result}\:\mathrm{based} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{above}\:\mathrm{fact}. \\ $$ Answered by Yozzi last updated on…