Question Number 152178 by mathdanisur last updated on 26/Aug/21 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system} \\ $$$$\begin{cases}{\mathrm{y}\sqrt{\mathrm{x}}\:+\:\mathrm{x}\sqrt{\mathrm{y}}\:=\:\mathrm{x}\:+\:\mathrm{y}}\\{\sqrt{\mathrm{x}}\:+\:\sqrt{\mathrm{y}}\:=\:\mathrm{xy}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{other}\:\mathrm{than} \\ $$$$\mathrm{x}\:=\:\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{y}\:=\:\mathrm{0} \\ $$ Commented by john_santu last updated on 26/Aug/21…
Question Number 152150 by timila1127 last updated on 26/Aug/21 Answered by puissant last updated on 26/Aug/21 $$\left({AB}\right){C}=\begin{bmatrix}{\mathrm{4}\:\:\:\:\:\:\mathrm{3}}\\{\mathrm{8}\:\:\:\:\:\:\mathrm{7}}\end{bmatrix}\begin{bmatrix}{\mathrm{3}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\mathrm{2}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\begin{bmatrix}{\mathrm{12}+\mathrm{8}\:\:\:\:\:\:\:\:\mathrm{9}+\mathrm{7}}\\{\mathrm{4}+\mathrm{16}\:\:\:\:\:\:\mathrm{3}+\mathrm{14}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\begin{bmatrix}{\mathrm{20}\:\:\:\:\:\mathrm{16}}\\{\mathrm{20}\:\:\:\:\:\mathrm{17}}\end{bmatrix} \\ $$ Terms of…
Question Number 86610 by TawaTawa1 last updated on 29/Mar/20 Commented by TawaTawa1 last updated on 29/Mar/20 $$\mathrm{X}\:\:\mathrm{and}\:\:\mathrm{Y}\:\:\mathrm{are}\:\mathrm{two}\:\mathrm{cylindrical}\:\mathrm{tanks}\:\mathrm{with}\:\mathrm{base}\:\:\mathrm{2r}\:\mathrm{cm}\:\:\mathrm{and}\:\:\mathrm{r}\:\mathrm{cm} \\ $$$$\mathrm{respectively}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{water}\:\mathrm{level}\:\mathrm{in}\:\mathrm{Y}\:\mathrm{is}\:\mathrm{10}\:\mathrm{cm}.\:\mathrm{What}\:\mathrm{level}\:\mathrm{will} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{quantity}\:\mathrm{of}\:\mathrm{water}\:\mathrm{be}\:\mathrm{in}\:\:\mathrm{X}.\:\:\left(\pi\:\:=\:\:\frac{\mathrm{22}}{\mathrm{7}}\right) \\ $$ Commented by…
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Question Number 86586 by Power last updated on 29/Mar/20 Commented by Power last updated on 29/Mar/20 $$\left(\mathrm{x}_{\mathrm{1}} ^{\mathrm{5}} −\mathrm{20}\right)\left(\mathrm{3x}_{\mathrm{2}} ^{\mathrm{4}} −\mathrm{2x}_{\mathrm{2}} −\mathrm{35}\right)=? \\ $$ Commented…
Question Number 152091 by rexford last updated on 25/Aug/21 $${Given}\:{tbat}\:{Arg}\left({z}+\mathrm{1}\right)=\frac{\Pi}{\mathrm{6}}\:{and}\: \\ $$$${Arg}\left({z}−\mathrm{1}\right)=\frac{\mathrm{2}\Pi}{\mathrm{3}}.{Find}\:{z}. \\ $$$${please}\:{help}\:{me} \\ $$ Answered by Olaf_Thorendsen last updated on 25/Aug/21 $$\mathrm{Let}\:{z}\:=\:{x}+{iy} \\…
Question Number 152082 by Tawa11 last updated on 25/Aug/21 Answered by Ar Brandon last updated on 25/Aug/21 $$=\underset{\mathrm{1}\leqslant\mathrm{r}\leqslant\mathrm{80}} {\sum}\mathrm{r}=\mathrm{40}\left(\mathrm{81}\right)=\mathrm{3240} \\ $$ Commented by Tawa11 last…
Question Number 21005 by Tinkutara last updated on 10/Sep/17 $$\mathrm{If}\:{z}_{\mathrm{1}} \:=\:{a}\:+\:{ib}\:\mathrm{and}\:{z}_{\mathrm{2}} \:=\:{c}\:+\:{id}\:\mathrm{are}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{Re}\left({z}_{\mathrm{1}} \bar {{z}}_{\mathrm{2}} \right)\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\omega_{\mathrm{1}} \:=\:{a}\:+\:{ic}\:\mathrm{and}\:\omega_{\mathrm{2}} \:=\:{b}\:+\:{id}…
Question Number 21006 by Tinkutara last updated on 10/Sep/17 $$\mathrm{Let}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{and}\:\mathrm{let}\:{z}\:=\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{1}} \:+\:{tz}_{\mathrm{2}} \:\mathrm{for} \\ $$$$\mathrm{some}\:\mathrm{real}\:\mathrm{number}\:{t}\:\mathrm{with}\:\mathrm{0}\:<\:{t}\:<\:\mathrm{1}.\:\mathrm{If} \\ $$$$\mathrm{arg}\left({w}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{principal}\:\mathrm{argument} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{non}-\mathrm{zero}\:\mathrm{complex}\:\mathrm{number}\:{w},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:+\:\mid{z}\:−\:{z}_{\mathrm{2}}…
Question Number 86541 by peter frank last updated on 29/Mar/20 Commented by jagoll last updated on 29/Mar/20 $$\mathrm{Q}^{\mathrm{2}} \:=\:\mathrm{PR} \\ $$$$\mathrm{4Q}\:=\:\mathrm{P}+\mathrm{R}\:\Rightarrow\:\mathrm{4ar}\:=\:\mathrm{a}+\mathrm{ar}^{\mathrm{2}} \\ $$$$\mathrm{a}\left(\mathrm{r}^{\mathrm{2}} −\mathrm{4r}+\mathrm{1}\right)\:=\:\mathrm{0} \\…