Question Number 193464 by Mingma last updated on 14/Jun/23 Commented by Frix last updated on 14/Jun/23 $$\mathrm{For}\:\mathrm{any}\:\mathrm{triangle} \\ $$$${r}_{{n}} =\frac{{c}\delta}{\mathrm{2}\left(\left({a}+{b}+{c}\right){c}+\left({n}−\mathrm{1}\right)\delta\right)} \\ $$$$\:\:\:\:\:\left[\delta=\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}\right. \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{rectangular}\:\mathrm{triangle}\:\mathrm{with}\:{c}=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}}…
Question Number 193467 by aba last updated on 14/Jun/23 $$\mathrm{Proof}\:: \\ $$$$\mathrm{cot}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+\mathrm{sint}}+\sqrt{\mathrm{1}−\mathrm{sint}}}{\:\sqrt{\mathrm{1}+\mathrm{sint}}−\sqrt{\mathrm{1}−\mathrm{sint}}}\right)=\frac{\mathrm{t}}{\mathrm{2}}\: \\ $$ Commented by MM42 last updated on 14/Jun/23 $${attention} \\ $$$${in}\:{all}\:\:{three}\:{arguments}\:\:{we}\:{used}\:{equalities}…
Question Number 193461 by SAMIRA last updated on 14/Jun/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}−\mathrm{1}}{\mathrm{sin}\:\mathrm{2x}}\right)\:=\:?? \\ $$ Answered by aba last updated on 14/Jun/23 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}+\mathrm{sinx}−\mathrm{1}}{\mathrm{sin}\left(\mathrm{2x}\right)\left(\sqrt{\mathrm{1}+\mathrm{sinx}}+\mathrm{1}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{2cosx}\left(\sqrt{\mathrm{1}+\mathrm{sinx}}+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{4}}\:\checkmark \\ $$…
Question Number 193458 by SAMIRA last updated on 14/Jun/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}\:\mathrm{sin}\left(\mathrm{x}\right)}\right)\:=\:\:\:??? \\ $$ Commented by aba last updated on 14/Jun/23 $$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }×\frac{\mathrm{x}}{\mathrm{sin}\left(\mathrm{x}\right)}=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{1}=\mathrm{0}.\mathrm{5} \\…
Question Number 193449 by Mingma last updated on 14/Jun/23 Commented by Frix last updated on 14/Jun/23 $$\mathrm{The}\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{10}^{{n}} −\mathrm{1}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{625} \\ $$$$\mathrm{is}\:\mathrm{624}\:\mathrm{for}\:{n}\geqslant\mathrm{4} \\ $$ Commented by Mingma…
Question Number 193448 by Mingma last updated on 14/Jun/23 Answered by cherokeesay last updated on 14/Jun/23 Commented by Mingma last updated on 14/Jun/23 Perfect �� Terms…
Question Number 193485 by Socracious last updated on 15/Jun/23 $$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} −\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}} \\ $$$$\:\:\:\:\:\:\:\mathrm{3}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{n}}. \\ $$ Answered by MM42 last updated on 15/Jun/23…
Question Number 193484 by Frix last updated on 15/Jun/23 $$\mathrm{Nice}\:\mathrm{problem}: \\ $$$$\mathrm{Find}\:\mathrm{8}\:\mathrm{distinctive}\:\mathrm{numbers}\:\in\mathbb{N}\backslash\left\{\mathrm{0}\right\}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{these}\:\mathrm{are}\:\mathrm{simultaniously}\:\mathrm{true}: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:{a}+{b}+{c}+{d}\:=\:{e}+{f}+{g}+{h} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{e}^{\mathrm{2}} +{f}^{\mathrm{2}} +{g}^{\mathrm{2}} +{h}^{\mathrm{2}}…
Question Number 193486 by Mingma last updated on 15/Jun/23 Answered by Subhi last updated on 15/Jun/23 $${x}^{{x}} .{ln}\left({x}\right)={ln}\left({y}\right) \\ $$$${ln}\left({x}^{{x}} .{ln}\left({x}\right)\right)={ln}\left({ln}\left({y}\right)\right)\Rrightarrow\:{xln}\left({x}\right)+{ln}\left({ln}\left({x}\right)\right)={ln}\left({ln}\left({y}\right)\right) \\ $$$${ln}\left({x}\right)+{x}.\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{x}.{ln}\left({x}\right)}=\frac{\mathrm{1}}{{y}.{ln}\left({y}\right)}.\frac{{dy}}{{dx}} \\ $$$$\frac{{dy}}{{dx}}={x}^{{x}}…
Question Number 65102 by da last updated on 25/Jul/19 $$\mathrm{The}\:\mathrm{polynomial}\:\:\:\mathrm{5}{x}^{\mathrm{5}} −\mathrm{3}{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} −{k}\: \\ $$$$\mathrm{gives}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{1},\:\mathrm{when}\:\mathrm{divided} \\ $$$$\mathrm{by}\:{x}+\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}. \\ $$ Commented by ~ À ® @…