Question Number 204041 by hardmath last updated on 04/Feb/24 $$\mathrm{Find}:\:\:\:\begin{vmatrix}{\mathrm{1}}&{\mathrm{7}}&{−\mathrm{1}}\\{\mathrm{9}}&{−\mathrm{3}}&{\mathrm{5}}\\{−\mathrm{1}}&{\mathrm{5}}&{\mathrm{3}}\end{vmatrix}=\:? \\ $$ Answered by AST last updated on 04/Feb/24 $$=\mathrm{1}\begin{vmatrix}{−\mathrm{3}}&{\mathrm{5}}\\{\mathrm{5}}&{\mathrm{3}}\end{vmatrix}−\mathrm{7}\begin{vmatrix}{\mathrm{9}}&{\mathrm{5}}\\{−\mathrm{1}}&{\mathrm{3}}\end{vmatrix}−\mathrm{1}\begin{vmatrix}{\mathrm{9}}&{−\mathrm{3}}\\{−\mathrm{1}}&{\mathrm{5}}\end{vmatrix} \\ $$$$=−\mathrm{9}−\mathrm{25}−\mathrm{7}\left(\mathrm{27}+\mathrm{5}\right)−\mathrm{1}\left(\mathrm{45}−\mathrm{3}\right)=−\mathrm{300} \\ $$ Answered…
Question Number 204038 by hardmath last updated on 04/Feb/24 $$\mathrm{y}\:=\:\left(\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}\right)\:\centerdot\:\mathrm{3}^{\boldsymbol{\mathrm{x}}} \\ $$$$\Rightarrow\:\mathrm{y}^{'} \:=\:? \\ $$ Answered by AST last updated on 04/Feb/24 $${y}^{'} =\mathrm{3}^{{x}+\mathrm{1}}…
Question Number 204039 by hardmath last updated on 04/Feb/24 $$\begin{vmatrix}{\mathrm{1}}&{\mathrm{7}}&{−\mathrm{1}}\\{\mathrm{9}}&{−\mathrm{3}}&{\boldsymbol{\mathrm{x}}}\\{−\mathrm{1}}&{\mathrm{5}}&{\mathrm{3}}\end{vmatrix}=\:\mathrm{0}\:\:\:\Rightarrow\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$ Answered by AST last updated on 04/Feb/24 $$\mathrm{1}\left(−\mathrm{9}−\mathrm{5}{x}\right)−\mathrm{7}\left(\mathrm{27}+{x}\right)−\mathrm{1}\left(\mathrm{45}−\mathrm{3}\right)=\mathrm{0} \\ $$$$\Rightarrow−\mathrm{9}−\mathrm{5}{x}−\mathrm{189}−\mathrm{7}{x}−\mathrm{42}=\mathrm{0}\Rightarrow\mathrm{12}{x}=−\mathrm{240} \\ $$$$\Rightarrow{x}=−\mathrm{20} \\…
Question Number 204018 by mnjuly1970 last updated on 04/Feb/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:{Aut}\:\left(\mathbb{Z}\:\right)=\:? \\ $$$$\:\:\:\:\:\:\:{where}\:,\:{Aut}\:\left({G}\:\right)=\:\left\{\:{f}\:\mid\:{f}\::{G}\:\underset{{G}\:{is}\:{a}\:{group}} {\overset{{f}\:{is}\:{a}\:{isomorphism}} {\rightarrow}}\:{G}\right\} \\ $$ Answered by witcher3 last updated on 04/Feb/24…
Question Number 204019 by mnjuly1970 last updated on 04/Feb/24 $$ \\ $$$$\:\:\:\:\:{G}\:{is}\:{a}\:{group}\:: \\ $$$$\:\:\:\:\:{prove}\:{that}\::\:\:\frac{{G}}{{Z}\:\left({G}\:\right)}\:\cong\:{Inn}\left({G}\:\right) \\ $$$$\:\:\:\:{Where}\:,\:{Inn}\left({G}\right)=\:\left\{{f}\:\mid\:{f}:\:{G}\:\overset{{f}\:{is}\:{an}\:{Automorphism}} {\rightarrow}\:{G}\right\} \\ $$$$ \\ $$ Commented by mokys last…
Question Number 203964 by mnjuly1970 last updated on 03/Feb/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Advanced}\:\:{calculus}\:… \\ $$$$\:\:{Q}:\:\:\:{If}\:,\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\mathrm{ln}\left({x}\right)\mathrm{ln}^{\mathrm{2}} \left({y}\:\right)}{\mathrm{1}−{xy}}\:{dxdy}\:=\:\lambda\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:\mathrm{3}} \left(\:{n}+\mathrm{1}\:\right)^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{Find}\:,\:\:\:\:\:\lambda\:=?…
Question Number 203994 by Davidtim last updated on 03/Feb/24 $${d}\left(\frac{{x}!}{{dx}}\right)=? \\ $$ Answered by Frix last updated on 03/Feb/24 $$\forall{x}\in\mathbb{N}:\:{x}!:=\underset{{n}=\mathrm{1}} {\overset{{x}} {\prod}}{n};\:\mathrm{0}!:=\mathrm{1}\:\Rightarrow \\ $$$$\Rightarrow\:{x}!\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous}\:\Rightarrow \\…
Question Number 203995 by 0977460907 last updated on 03/Feb/24 $$\mathrm{3}{x}+\mathrm{4}{x}=\mathrm{5} \\ $$ Answered by SonGoku last updated on 03/Feb/24 $$\: \\ $$$${x}\left(\mathrm{3}+\mathrm{4}\right)=\mathrm{5} \\ $$$${x}\left(\mathrm{7}\right)=\mathrm{5} \\…
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Question Number 203985 by ajfour last updated on 03/Feb/24 Commented by mr W last updated on 03/Feb/24 $${equilibrium}\:{is}\:{not}\:{possible}! \\ $$$$\Sigma{M}_{{P}} \neq\mathrm{0} \\ $$ Commented by…