Question Number 21293 by Tinkutara last updated on 19/Sep/17 $$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{{n}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{let}\:\alpha_{\mathrm{0}} ,\:\alpha_{\mathrm{1}} ,\:….,\:\alpha_{{n}−\mathrm{1}} \:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{complex}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({a}\:+\:{b}\alpha_{{k}} ^{\mathrm{2}} \right)\:=\:\left({a}^{\frac{{n}}{\mathrm{2}}} \:+\:{b}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}}…
Question Number 152358 by Jonathanwaweh last updated on 27/Aug/21 Commented by JDamian last updated on 27/Aug/21 $$\mathrm{5} \\ $$ Answered by Olaf_Thorendsen last updated on…
Question Number 86825 by M±th+et£s last updated on 31/Mar/20 $${a}^{\mathrm{3}} +\frac{\mathrm{1}}{{a}^{\mathrm{3}} }=\mathrm{18} \\ $$$${a}^{\mathrm{4}} +\frac{\mathrm{1}}{{a}^{\mathrm{4}} }=? \\ $$ Commented by Ar Brandon last updated on…
Question Number 152326 by mathdanisur last updated on 27/Aug/21 $$\int\sqrt{\frac{\mathrm{1}\:+\:\mathrm{sin}\boldsymbol{\mathrm{x}}}{\mathrm{cos}\boldsymbol{\mathrm{x}}}}\:\mathrm{dx}\:=\:? \\ $$ Commented by puissant last updated on 27/Aug/21 $$=\int\sqrt{{secx}+{tanx}}{dx} \\ $$$${Q}\mathrm{151568}\: \\ $$ Answered…
Question Number 21248 by Tinkutara last updated on 17/Sep/17 $$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{which} \\ $$$$\mathrm{touches}\:\mathrm{the}\:\mathrm{given}\:\mathrm{circles}\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:= \\ $$$$\mid\mathrm{3}\:+\:\mathrm{4}{i}\mid\:\mathrm{and}\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}\mid\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{hyperbola},\:\mathrm{then}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{transverse}\:\mathrm{axis}\:\mathrm{is} \\ $$ Terms of Service…
Question Number 152323 by mathdanisur last updated on 27/Aug/21 $$\mathrm{x}^{\mathrm{2}} \centerdot\mathrm{y}=\frac{\mathrm{1}}{\mathrm{18}}\:\:\mathrm{and}\:\:\mathrm{x}\centerdot\mathrm{y}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{12}} \\ $$$$\mathrm{find}\:\:\left(\mathrm{xy}\right)^{−\mathrm{2}} \:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 27/Aug/21 $${x}^{\mathrm{2}}…
Question Number 86779 by M±th+et£s last updated on 31/Mar/20 $${ssolve} \\ $$$$\left.\mathrm{1}\right){x}−\left[{x}\right]\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right){x}−\left[{x}\right]\leqslant\mathrm{0} \\ $$$$\left.\mathrm{3}\right){x}+\left[{x}\right]\geqslant\mathrm{0} \\ $$$$\left.\mathrm{4}\right){x}+\left[{x}\right]\leqslant\mathrm{0}\: \\ $$ Answered by mr W last…
Question Number 21247 by Tinkutara last updated on 17/Sep/17 $$\mathrm{If}\:\left[\:\right]\:\mathrm{represents}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer} \\ $$$$\mathrm{function}\:\mathrm{and}\:{f}\left({x}\right)\:=\:{x}\:−\:\left[{x}\right]\:\mathrm{then} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}\:\mathrm{are}\:\mathrm{infinite}. \\ $$$$\boldsymbol{\mathrm{True}}/\boldsymbol{\mathrm{False}} \\ $$ Answered by dioph last updated…
Question Number 152314 by mathdanisur last updated on 27/Aug/21 Answered by ghimisi last updated on 27/Aug/21 $$\frac{{a}^{\mathrm{2}} }{{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} }{{c}\left({a}+{b}+{c}\right)}\geqslant\frac{\left({a}+{c}\right)^{\mathrm{2}} }{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{ab}+{bc}+{ac}}…
Question Number 21235 by Tinkutara last updated on 17/Sep/17 $$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{k},\:\mathrm{let}\:\alpha_{{k}} \:=\:\mathrm{cos}\:\left(\frac{{k}\pi}{\mathrm{7}}\right)\:+ \\ $$$${i}\:\mathrm{sin}\:\left(\frac{{k}\pi}{\mathrm{7}}\right),\:\mathrm{where}\:{i}\:=\:\sqrt{−\mathrm{1}}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\sum}}\mid\alpha_{{k}+\mathrm{1}} \:−\:\alpha_{{k}} \mid}{\underset{{k}=\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\mid\alpha_{\mathrm{4}{k}−\mathrm{1}} \:−\:\alpha_{\mathrm{4}{k}−\mathrm{2}} \mid}\:\mathrm{is} \\ $$…