Question Number 204583 by hardmath last updated on 22/Feb/24 $$\mathrm{For}\:\:\:\mathrm{z}\:=\:\mathrm{a}\:−\:\mathrm{bi} \\ $$$$\mathrm{If}\:\:\:\left(\mid\mathrm{z}\mid\:−\:\mathrm{z}\right)\centerdot\left(\mid\mathrm{z}\mid\:+\:\overline {\mathrm{z}}\right)\:=\:\mathrm{4bi} \\ $$$$\mathrm{Find}\:\:\:\mid\mathrm{z}\mid\:=\:? \\ $$ Answered by A5T last updated on 22/Feb/24 $$\left(\mid{z}\mid−{z}\right)\left(\mid{z}\mid+\overset{−}…
Question Number 204545 by hardmath last updated on 21/Feb/24 $$\mathrm{If} \\ $$$$\mathrm{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:\:+\:\:…\:\:+\:\:\frac{\mathrm{1}}{\mathrm{100}^{\mathrm{2}} } \\ $$$$\mathrm{b}\:=\:\mathrm{0},\mathrm{99} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:<\:\mathrm{b} \\ $$ Answered by…
Question Number 204544 by hardmath last updated on 21/Feb/24 $$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} }\:\:+\:\:…\:\:+\:\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} }\:\:<\:\:\frac{\mathrm{1}}{\mathrm{12}} \\ $$ Answered by mr W last updated on 21/Feb/24…
Question Number 204511 by Ghisom last updated on 20/Feb/24 $$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$$${x}^{\mathrm{2}} −\mathrm{10}\lfloor{x}\rfloor+\frac{\mathrm{57}}{\mathrm{4}}=\mathrm{0} \\ $$ Answered by mr W last updated on 20/Feb/24 $$\lfloor{x}\rfloor={n}\:\in\:\mathbb{Z}^{+} \\…
Question Number 204510 by Ghisom last updated on 20/Feb/24 $$\mathrm{solve}\:\mathrm{for}\:{x}\neq{y}\wedge{y}\neq{z}\wedge{z}\neq{x} \\ $$$$\left(\mathrm{exact}\:\mathrm{solutions}\:\mathrm{required}\right) \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){x}}={y} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){y}}={z} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){z}}={x} \\ $$ Answered by Frix last updated…
Question Number 204512 by Ghisom last updated on 20/Feb/24 $$\mathrm{solve}\:\mathrm{for}\:{x}\in\mathbb{C} \\ $$$$\mathrm{3}^{\mathrm{2i}{x}} −\mathrm{3}^{\mathrm{i}{x}} \mathrm{2}+\mathrm{5}=\mathrm{0} \\ $$ Answered by Rasheed.Sindhi last updated on 20/Feb/24 $$\left(\mathrm{3}^{{ix}} \right)^{\mathrm{2}}…
Question Number 204500 by DeArtist last updated on 19/Feb/24 $$\mathrm{Given}\:\mathrm{that}\:{I}\:=\:\int\int_{{R}} \left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\frac{\mathrm{5}}{\mathrm{2}}} {dxdy}\:\mathrm{where}\:{R} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{region}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:\leqslant\:{a}^{\mathrm{2}} \\ $$$$\mathrm{use}\:\mathrm{a}\:\mathrm{suitable}\:\mathrm{transformation}\:\mathrm{to}\:\mathrm{evaluate}\:{I} \\ $$ Answered by witcher3…
Question Number 204499 by DeArtist last updated on 19/Feb/24 $$\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{derivative}\:\mathrm{of}\: \\ $$$${F}\left({x},{y},{z}\right)\:=\:\mathrm{2}{xy}−{z}^{\mathrm{2}} \:\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{2},−\mathrm{1},\mathrm{1}\right)\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{direction}\:\mathrm{towards}\:\left(\mathrm{3},\mathrm{1},−\mathrm{1}\right)\: \\ $$$$\mathrm{in}\:\mathrm{what}\:\mathrm{direction}\:\mathrm{is}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{derivative} \\ $$$$\mathrm{maximum}?\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{this}\:\mathrm{maximum}? \\ $$ Answered by TonyCWX08 last…
Question Number 204461 by MATHEMATICSAM last updated on 18/Feb/24 $$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$$\left({x}\:−\:\mathrm{12}\right)\left({x}\:−\:\mathrm{13}\right)\:=\:\frac{\mathrm{34}}{\mathrm{33}^{\mathrm{2}} } \\ $$ Answered by TonyCWX08 last updated on 18/Feb/24 $${x}^{\mathrm{2}} −\mathrm{25}{x}+\mathrm{156}\:=\:\frac{\mathrm{34}}{\mathrm{1089}} \\…
Question Number 204480 by EJJDJX last updated on 18/Feb/24 $${x}\:+\:\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{2}.\mathrm{05} \\ $$$${x}\:=\:? \\ $$ Commented by TonyCWX08 last updated on 19/Feb/24 $$ \\ $$$${x}^{\mathrm{2}} +\mathrm{1}=\:\mathrm{2}.\mathrm{05}{x}…