Question Number 21200 by Tinkutara last updated on 15/Sep/17 $$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$ Commented by mrW1 last updated…
Question Number 152265 by mathdanisur last updated on 26/Aug/21 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{k}}\:-\:\sqrt{\mathrm{k}\:-\:\mathrm{1}}}{\:\sqrt{\mathrm{k}\left(\mathrm{k}\:+\:\mathrm{1}\right.}}\right)\:=\:? \\ $$ Answered by mindispower last updated on 27/Aug/21 $${sin}^{−} \left({a}\right)−{sin}^{−} \left({b}\right)={sin}^{−}…
Question Number 152266 by mathdanisur last updated on 26/Aug/21 Answered by qaz last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx} \\ $$$$=\left[\left(\mathrm{1}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)−\left(\mathrm{1}+\mathrm{x}\right)\right]\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mid_{\mathrm{0}} ^{\mathrm{1}} +\int_{\mathrm{0}}…
Question Number 86723 by A8;15: last updated on 30/Mar/20 Commented by jagoll last updated on 30/Mar/20 $$\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} \\…