Question Number 142246 by mathdanisur last updated on 28/May/21 Answered by mr W last updated on 28/May/21 $${V}=\frac{\mathrm{1}}{\mathrm{6}}\begin{pmatrix}{{a}}&{{a}}&{{a}}\\{{b}}&{\mathrm{1}}&{{b}}\\{{a}}&{{b}}&{{b}}\end{pmatrix} \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}\left[{a}\left({b}−{b}^{\mathrm{2}} \right)−{a}\left({b}^{\mathrm{2}} −{ab}\right)+{a}\left({b}^{\mathrm{2}} −{a}\right)\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}{a}\left({b}−{a}\right)\left(\mathrm{1}−{b}\right)…
Question Number 76711 by aliesam last updated on 29/Dec/19 Answered by MJS last updated on 30/Dec/19 $${x}>\mathrm{0}\wedge{y}>\mathrm{0}\:\Rightarrow\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} >\mathrm{0}\wedge{x}^{\mathrm{2}} +{y}^{\mathrm{2}} >\mathrm{0} \\ $$$$\mathrm{0}<{x}^{\mathrm{3}} +{y}^{\mathrm{3}} \leqslant{x}−{y}\:\Rightarrow\:\mathrm{0}<{x}−{y}\:\Leftrightarrow\:{y}<{x}…
Question Number 76700 by Mikael_786 last updated on 29/Dec/19 Commented by benjo 1/2 santuyy last updated on 29/Dec/19 $${what}\:{are}\:{you}\:{looking}\:{for}\:,{sir}? \\ $$ Commented by Mikael_786 last…
Question Number 142232 by liberty last updated on 28/May/21 Answered by MJS_new last updated on 28/May/21 $$\mathrm{1}.\:{x}=\mathrm{0} \\ $$$$\Rightarrow\:{y}^{\mathrm{3}} =\mathrm{1}\wedge{y}^{\mathrm{4}} −{y}=\mathrm{0}\:\Rightarrow\:{y}=\mathrm{1}\vee{y}=−\frac{\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i} \\ $$$$\mathrm{2}.\:{y}=\mathrm{0}\:…\:\Rightarrow\:{x}=\mathrm{1}\vee{x}=−\frac{\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i} \\ $$$$\mathrm{3}.\:{y}={x}\:…\:\Rightarrow\:{x}={y}=\mathrm{1}\vee{x}={y}=−\frac{\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}…
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Question Number 11131 by Joel576 last updated on 13/Mar/17 $$\mathrm{Proof}\:\mathrm{with}\:\mathrm{induction},\:\mathrm{for}\:\mathrm{every}\:{n}\:\in\:\mathbb{N} \\ $$$$\underset{{i}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{2}^{{i}\:+\:\mathrm{2}} \:=\:\mathrm{2}^{{n}\:+\:\mathrm{3}} \:−\:\mathrm{8} \\ $$ Commented by Joel576 last updated on 13/Mar/17…
Question Number 11114 by ABD last updated on 12/Mar/17 Answered by ajfour last updated on 12/Mar/17 Commented by ajfour last updated on 12/Mar/17 $$\frac{{AB}}{{BC}}=\frac{\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta}\:\Rightarrow{BC}=\frac{\mathrm{4cos}\:\theta}{\mathrm{sin}\:\theta} \\…
Question Number 11095 by FilupS last updated on 12/Mar/17 $$\mathrm{You}\:\mathrm{have}\:\mathrm{a}\:\mathrm{line}\:\mathrm{of}\:\mathrm{length}\:\mathrm{1}. \\ $$$$\mathrm{You}\:\mathrm{place}\:\mathrm{two}\:\mathrm{random}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}. \\ $$$$\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{average}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{points}?\:\mathrm{Please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working}. \\ $$ Commented by FilupS last updated…
Question Number 76622 by Sathvik last updated on 28/Dec/19 $$\mathrm{Find}\:\left(\mathrm{a}.\mathrm{b}.\mathrm{c}\right)\:\mathrm{for}\:\mathrm{equation}\:\mathrm{acos}\:\mathrm{2x}+\mathrm{bsin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by}\:\mathrm{every}\:\mathrm{x} \\ $$ Answered by MJS last updated on 28/Dec/19 $${a}\mathrm{cos}\:\mathrm{2}{x}\:+{b}\mathrm{sin}^{\mathrm{2}} \:{x}\:+{c}=\mathrm{0} \\ $$$${a}\mathrm{cos}\:\mathrm{2}{x}\:+{b}×\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}\right)+{c}=\mathrm{0} \\…
Question Number 142148 by islamo last updated on 27/May/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!}=? \\ $$ Answered by mathmax by abdo last updated on 27/May/21 $$\mathrm{e}^{\mathrm{x}} =\sum_{\mathrm{n}=\mathrm{0}}…