Question Number 135300 by bobhans last updated on 12/Mar/21 $$\left(\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{3}}\:+\mathrm{1}\right)\left(\sqrt{\mathrm{2}}\:−\sqrt{\mathrm{3}}\:+\mathrm{1}\right)\left(\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{3}}−\mathrm{1}\right)\left(\sqrt{\mathrm{2}}−\sqrt{\mathrm{3}}−\mathrm{1}\right)=? \\ $$ Answered by Olaf last updated on 12/Mar/21 $$\left(\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}+\mathrm{1}\right)\left(\sqrt{\mathrm{2}}−\sqrt{\mathrm{3}}−\mathrm{1}\right)\:=\:\mathrm{2}−\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$=\:−\mathrm{2}−\mathrm{2}\sqrt{\mathrm{3}}\:=\:−\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\left(\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}−\mathrm{1}\right)\left(\sqrt{\mathrm{2}}−\sqrt{\mathrm{3}}+\mathrm{1}\right)\:=\:\mathrm{2}−\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)^{\mathrm{2}} \\…
Question Number 4230 by prakash jain last updated on 03/Jan/16 $$\mathrm{Solve}\:\mathrm{for}\:+\mathrm{ve}\:\mathrm{integers}\:>\mathrm{0}. \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{4}} ={z}^{\mathrm{6}} \\ $$ Commented by Rasheed Soomro last updated on 04/Jan/16…
Question Number 135303 by bobhans last updated on 12/Mar/21 $$ \\ $$in how many ways can 10 people be seated in a round table if…
Question Number 69764 by Rio Michael last updated on 27/Sep/19 $${find}\:\:\:\frac{{dy}}{{dx}}\:\:{if}\:\:{x}\:=\:{sin}^{\mathrm{2}} {t}\:\:{and}\:\:{y}=\:{tan}\:{t}\:{at}\:\:{t}\:=\:\frac{\pi}{\mathrm{4}} \\ $$ Commented by kaivan.ahmadi last updated on 27/Sep/19 $$\frac{{dy}}{{dx}}=\frac{\frac{{dy}}{{dt}}}{\frac{{dx}}{{dt}}}=\frac{\mathrm{1}+{tan}^{\mathrm{2}} {t}}{{sin}\mathrm{2}{t}} \\ $$…
Question Number 69765 by Rio Michael last updated on 27/Sep/19 $${Given}\:{that}\:\:{y}\:=\:\sqrt{\mathrm{5}{x}^{\mathrm{2}} \:+\:\mathrm{3}}\:,\:{show}\:{that}\:\:{when}\:{x}^{\mathrm{2}} \:=\:\frac{\mathrm{6}}{\mathrm{5}}\:,\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}\:} }\:=\:\frac{\mathrm{125}}{\mathrm{8}} \\ $$ Answered by MJS last updated on 27/Sep/19 $${y}=\sqrt{\mathrm{5}{x}^{\mathrm{2}}…
Question Number 69762 by Rio Michael last updated on 27/Sep/19 $${prove}\:{by}\:{mathematical}\:{induction},\:{that}\:{for}\:{all}\:{positive}\:{integers}\:{n}, \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left(\:{n}\:+\:\mathrm{2}\right) \\ $$ Commented by mathmax by abdo last updated on…
Question Number 69763 by Rio Michael last updated on 27/Sep/19 $${find}\:\frac{{dy}}{{dx}}\:\:{if}\:\:{y}\:=\:\mathrm{3}^{{x}} {e}^{\mathrm{2}{x}\:+\:\mathrm{1}} ,\:{at}\:{x}\:=\mathrm{1} \\ $$ Commented by mathmax by abdo last updated on 16/Oct/19 $${y}\left({x}\right)={e}^{{xln}\mathrm{3}+\mathrm{2}{x}+\mathrm{1}}…
Question Number 4225 by Rasheed Soomro last updated on 03/Jan/16 $$\:^{\bullet} \mathrm{A}\:\mathrm{kite}\:\mathrm{is}\:\mathrm{a}\:\mathrm{quadrilateral}\:\mathrm{having}\:\mathrm{two} \\ $$$$\mathrm{pairs}\:\mathrm{of}\:\mathrm{adjacent}\:\mathrm{sides}\:\mathrm{equal}. \\ $$$$\mathrm{Draw}\:\mathrm{a}\:\mathrm{semi}-\mathrm{circle}\:\mathrm{inside}\:\mathrm{it}\:\mathrm{touching} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{using}\:\mathrm{Eucledian}\:\mathrm{tools}. \\ $$$$\:^{\bullet} \:\mathrm{Can}\:\mathrm{we}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\:\mathrm{above}\:\mathrm{semi}-\mathrm{circle} \\ $$$$\mathrm{is}\:\mathrm{of}\:\mathrm{the}\:\mathrm{laregest}\:\mathrm{possible}\:\mathrm{area}\:\mathrm{inside}\:\mathrm{the} \\ $$$$\mathrm{kite}?…
Question Number 135298 by Ahmed1hamouda last updated on 12/Mar/21 Answered by mr W last updated on 12/Mar/21 $$\mathrm{1} \\ $$$${z}={x}+{yi} \\ $$$$\frac{\mathrm{1}}{{z}}=\frac{\mathrm{1}}{{x}+{yi}}=\frac{{x}−{yi}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } \\…
Question Number 4222 by Rasheed Soomro last updated on 02/Jan/16 $${Six}\:{free}-{hand}\:{lines}\:{are}\:{drawn}\:{inside} \\ $$$${a}\:{circle}\:{in}\:{order}\:{to}\:{divide}\:{it}\:{into}\:{maximum} \\ $$$${number}\:{of}\:{parts}.{Determine}\:{this}\:{number}. \\ $$$${Note}\:{that}\:{a}\:{free}-{hand}\:{line}\:{has}\:{following} \\ $$$$\:{three}\:\:{properties}: \\ $$$$\left.{i}\right)\:{It}\:{doesn}'{t}\:{cut}\:\:{itself}. \\ $$$$\left.{ii}\right){It}\:{joins}\:{two}\:{points}\:{of}\:{circle}. \\ $$$$\left.{iii}\right)\:{It}\:{can}\:{cut}\:{another}\:{line}\:{at}\:{most}\:{at}\:{three}…