Question Number 67996 by TawaTawa last updated on 03/Sep/19 Commented by mathmax by abdo last updated on 03/Sep/19 $${we}\:{have}\:\:\mathrm{1}\leqslant{k}\leqslant{n}\:\Rightarrow{n}^{\mathrm{2}} +\mathrm{1}\leqslant{n}^{\mathrm{2}} \:+{k}\leqslant{n}^{\mathrm{2}} \:+{n}\:\Rightarrow \\ $$$$\sqrt{{n}^{\mathrm{2}} \:+\mathrm{1}}\leqslant\sqrt{{n}^{\mathrm{2}}…
Question Number 67997 by ajfour last updated on 03/Sep/19 $$\mathrm{5}{y}^{\mathrm{2}} +\mathrm{2}{axy}+{b}=\mathrm{0} \\ $$$${ay}^{\mathrm{2}} +\mathrm{2}{bx}+\mathrm{5}{c}=\mathrm{0} \\ $$$$\left(\mathrm{5}{x}+\mathrm{3}{a}\right){y}^{\mathrm{2}} +\left(\mathrm{4}{ax}^{\mathrm{2}} \right){y}−{bx}−\mathrm{5}{c}=\mathrm{0} \\ $$$$\mathrm{5}{y}^{\mathrm{2}} −{x}\left(\mathrm{5}{x}+\mathrm{2}{a}\right){y}−{ax}^{\mathrm{3}} −\mathrm{3}{b}=\mathrm{0} \\ $$$${Please}\:{solve}\:{simultaneously} \\…
Question Number 2458 by alib last updated on 20/Nov/15 $${The}\:{medians}\:{of}\:{a}\:{triangle} \\ $$$${are}\:{m}_{\mathrm{1}} ,\:{m}_{\mathrm{2}} ,\:{m}_{\mathrm{3}} . \\ $$$${Find}\:{the}\:{length}\:{of}\:{each}\:{sides}\: \\ $$$${the}\:{triangle}. \\ $$ Answered by prakash jain…
Question Number 67992 by MJS last updated on 03/Sep/19 $$\left(\mathrm{1}\right)\:{z}={a}+{b}\mathrm{i} \\ $$$$\left(\mathrm{2}\right)\:{z}={r}\mathrm{e}^{\mathrm{i}\theta} \\ $$$$\mathrm{express}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{real}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{real}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{b}\right)\:\mathrm{imag}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{imaginary}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{c}\right)\:\mathrm{abs}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{absolute}\:\mathrm{value}\right] \\ $$$$\left(\mathrm{d}\right)\:\mathrm{arg}\:\left({z}^{{z}}…
Question Number 133530 by snipers237 last updated on 22/Feb/21 $$\:{I}=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(−{lnx}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\: \\ $$ Commented by Dwaipayan Shikari last updated on 22/Feb/21 $$\int_{\mathrm{0}} ^{\infty}…
Question Number 67991 by ramirez105 last updated on 03/Sep/19 Commented by mathmax by abdo last updated on 03/Sep/19 $${xy}\:{dx}+\mathrm{2}\left({x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \right){dy}\:=\mathrm{0}\:\Rightarrow\mathrm{2}\left({x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \right){dy}\:=−{xydx}\:\Rightarrow \\ $$$$\mathrm{2}\frac{{dy}}{{y}}\:=\frac{−{xdx}}{{x}^{\mathrm{2}}…
Question Number 2454 by Yozzi last updated on 20/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}+{sinx}}{{a}+{siny}}\leqslant{e}^{{y}−{x}} \\ $$$${for}\:\forall\:{x}\leqslant{y}. \\ $$ Commented by Rasheed Soomro last updated on 21/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which}…
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Question Number 67983 by peter frank last updated on 03/Sep/19 Commented by Abdo msup. last updated on 03/Sep/19 $$\left.{b}\right)\:{let}\:{f}\left({x}\right)=\left(\frac{\mathrm{1}}{{x}}\right)^{{x}\:} \:\Rightarrow{f}\left({x}\right)={x}^{−{x}} \:={e}^{−{xln}\left({x}\right)} \\ $$$$\left.{f}\:{is}?{defined}\:{on}\:\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{x}\rightarrow\mathrm{0}\:{and}\:{x}>\mathrm{0}}…
Question Number 133518 by Abdoulaye last updated on 22/Feb/21 Answered by mnjuly1970 last updated on 22/Feb/21 $${ans}:\:\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$ Answered by Dwaipayan Shikari last…