Question Number 86723 by A8;15: last updated on 30/Mar/20 Commented by jagoll last updated on 30/Mar/20 $$\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:=\:\mathrm{tan}\:\mathrm{x}\:×\:\mathrm{cot}\:\mathrm{x} \\ $$$$\mathrm{cos}\:\mathrm{2x}\:=\:\mathrm{1}\:\Rightarrow\:\begin{cases}{\mathrm{x}\:=\:\mathrm{0}}\\{\mathrm{x}\:=\:\pi}\end{cases} \\ $$ Terms of…
Question Number 21168 by Tinkutara last updated on 15/Sep/17 $$\mathrm{Let}\:{f}\left({x}\right)\:=\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:−\:\mathrm{2}\mid\:+\:\mid{x}\:−\:\mathrm{3}\mid, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{for}\:\mathrm{which}\:{f}\left({x}\right) \\ $$$$=\:{k}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{2}.\:\mathrm{only}\:\mathrm{one}\:\mathrm{solution} \\ $$$$\mathrm{3}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{same}\:\mathrm{sign} \\ $$$$\mathrm{4}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sign} \\ $$ Answered…
Question Number 152226 by mathdanisur last updated on 26/Aug/21 $$\mathrm{16}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}} \:+\:\mathrm{16}^{\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}} \:=\:\mathrm{1}\:\:\Rightarrow\:\:\mathrm{x};\mathrm{y}=? \\ $$ Commented by john_santu last updated on 26/Aug/21 $$\mathrm{x}=\mathrm{y}=−\frac{\mathrm{1}}{\mathrm{2}} \\…
Question Number 21137 by Tinkutara last updated on 14/Sep/17 $$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$ Answered by Tinkutara last updated on 15/Sep/17 $$\mid{z}_{\mathrm{1}} \mid+\mid{z}_{\mathrm{2}}…
Question Number 152187 by Tawa11 last updated on 26/Aug/21 $$\mathrm{Please}\:\mathrm{formular}\:\mathrm{for}\:\:\:\:\:\Gamma\left(\frac{\mathrm{8}}{\mathrm{3}}\right) \\ $$ Commented by Tawa11 last updated on 26/Aug/21 $$\mathrm{Or}\:\mathrm{generally}\:\:\:\:\:\:\:\:\Gamma\left(\frac{\mathrm{x}}{\mathrm{y}}\right) \\ $$ Answered by Olaf_Thorendsen…
Question Number 21111 by Tinkutara last updated on 13/Sep/17 $$\mathrm{STATEMENT}-\mathrm{1}\::\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{4}} ^{\mathrm{2}} \:= \\ $$$$\mathrm{0}\:\mathrm{where}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{and}\:{z}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{fourth} \\…
Question Number 21108 by Tinkutara last updated on 13/Sep/17 $$\mathrm{Let}\:{A},\:{B},\:{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{as}\:\mathrm{defined}\:\mathrm{below} \\ $$$${A}\:=\:\left\{{z}\::\:\mathrm{Im}\:{z}\:\geqslant\:\mathrm{1}\right\} \\ $$$${B}\:=\:\left\{{z}\::\:\mid{z}\:−\:\mathrm{2}\:−\:{i}\mid\:=\:\mathrm{3}\right\} \\ $$$${C}\:=\:\left\{{z}\::\:\mathrm{Re}\left(\left(\mathrm{1}\:−\:{i}\right){z}\right)\:=\:\sqrt{\mathrm{2}}\right\}. \\ $$$$\mathrm{Let}\:{z}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:{A}\:\cap\:{B}\:\cap\:{C}\:\mathrm{and}\:\mathrm{let} \\ $$$${w}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{satisfying}\:\mid{w}\:−\:\mathrm{2}\:−\:{i}\mid\:< \\ $$$$\mathrm{3}.\:\mathrm{Then},\:\mid{z}\mid\:−\:\mid{w}\mid\:+\:\mathrm{3}\:\mathrm{lies}\:\mathrm{between} \\…
Question Number 152178 by mathdanisur last updated on 26/Aug/21 $$\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…