Question Number 4077 by Rasheed Soomro last updated on 28/Dec/15 $$\mathrm{For}\:\mathrm{two}\:\boldsymbol{\mathrm{co}}-\boldsymbol{\mathrm{planer}}\:\mathrm{circles}\:\mathrm{to}\:\mathrm{be}\:\mathrm{tangent} \\ $$$$\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{condition}\:\mathrm{is}, \\ $$$$\mathrm{I}\:\mathrm{think} \\ $$$$\:\:\:\:\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{centers}\:\mathrm{of}\: \\ $$$$\:\:\:\:\mathrm{circles}\:\:\mathrm{must}\:\mathrm{be}\:\mathrm{equal}\:\mathrm{to}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:\mathrm{or}\:\mid\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{r}}_{\mathrm{2}} \mid\:, \\ $$$$\:\:\:\:\:\mathrm{where}\:\:\boldsymbol{\mathrm{r}}_{\mathrm{1}}…
Question Number 135145 by ZiYangLee last updated on 10/Mar/21 $$\int_{\mathrm{0}} ^{\mathrm{3}} {x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} +\mathrm{9}}\:{dx}=? \\ $$ Answered by bramlexs22 last updated on 10/Mar/21 $$\int_{\mathrm{0}} ^{\:\mathrm{3}}…
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Question Number 4075 by Yozzii last updated on 27/Dec/15 Answered by prakash jain last updated on 28/Dec/15 $$\:^{\mathrm{2}{r}} {C}_{{r}} =\frac{\mathrm{2}{r}!}{{r}!{r}!} \\ $$$$\mathrm{2}^{{r}} \centerdot{r}!=\mathrm{2}\centerdot\mathrm{4}\centerdot\mathrm{6}\centerdot…\centerdot\mathrm{2}{r} \\ $$$$\mathrm{2}{r}!=\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}\centerdot…\centerdot\left(\mathrm{2}{r}−\mathrm{1}\right)\centerdot\mathrm{2}{r}…
Question Number 69608 by TawaTawa last updated on 25/Sep/19 Commented by Prithwish sen last updated on 27/Sep/19 $$\left.\boldsymbol{\mathrm{a}}\right)\:\mid\boldsymbol{\mathrm{AB}}\mid=\mathrm{2}\sqrt{\mathrm{2}}=\:\mathrm{2}.\mathrm{8m} \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\:\boldsymbol{\mathrm{perimeter}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{ALB}}\:=\frac{\mathrm{2}×\pi×\mathrm{2}×\mathrm{270}}{\mathrm{360}}\mathrm{m}=\:\mathrm{9}.\mathrm{4m} \\ $$ Commented by TawaTawa…
Question Number 69609 by TawaTawa last updated on 25/Sep/19 Commented by Prithwish sen last updated on 25/Sep/19 $$\mathrm{I}\:\mathrm{think} \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{inner}\:\mathrm{section}\:=\:\pi\left(\mathrm{75}\right)^{\mathrm{2}} \mathrm{sq}.\:\mathrm{cm} \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{section}\:=\:\pi\left(\mathrm{100}−\mathrm{75}\right)\left(\mathrm{100}+\mathrm{75}\right) \\ $$$$=\:\pi×\mathrm{25}×\mathrm{175}\:\mathrm{sq}.\:\mathrm{cm}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…
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Question Number 4071 by Yozzii last updated on 27/Dec/15 Commented by Yozzii last updated on 27/Dec/15 $${Just}\:{sharing}\:{some}\:{problems}\:{I}\:{did} \\ $$$${today}. \\ $$ Terms of Service Privacy…
Question Number 69607 by malwaan last updated on 25/Sep/19 $$\boldsymbol{{without}}\:\boldsymbol{{using}}\:\boldsymbol{{lhospital}}\:\boldsymbol{{please}} \\ $$$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{x}}−\boldsymbol{{sin}}\:\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{want}}\:\boldsymbol{{every}}\:\boldsymbol{{method}} \\ $$$$\boldsymbol{{possible}}\:\boldsymbol{{because}}\:\boldsymbol{{someone}} \\ $$$$\boldsymbol{{challenge}}\:\boldsymbol{{me}}\: \\ $$ Commented…
Question Number 135143 by mnjuly1970 last updated on 10/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:\:{please}\:\:{calculate}:\downarrow\downarrow\downarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\::::\:\:\boldsymbol{\phi}\overset{??} {=}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{H}_{{n}} ^{\:^{\mathrm{2}} } }{{n}^{\mathrm{2}} }\:\: \\ $$$$\:\:\:\:\:\: \\ $$…