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Question Number 192345 by Mingma last updated on 15/May/23 Answered by aleks041103 last updated on 15/May/23 $$\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({f}\left(\mathrm{1}+{x}\right)+{f}\left(\mathrm{1}−{x}\right)\right){dx}= \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left(\mathrm{1}+{x}\right){dx}\:−\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {f}\left(\mathrm{1}−{x}\right){d}\left(−{x}\right)=…
Question Number 192344 by Mingma last updated on 15/May/23 Commented by malwan last updated on 17/May/23 $${I}\:{think}\:{its}\:{greet}\:{question} \\ $$$${please}\:{any}\:{one}\:{try}\:{to}\:{solve}\:{it} \\ $$ Commented by Mingma last…
Question Number 61274 by alphaprime last updated on 31/May/19 $${If}\:{p}\:,\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,…{x}_{{i}} \:{and}\:{q},{y}_{\mathrm{1}} ,{y}_{\mathrm{2}} ,…{y}_{{i}} \:{form}\:{two}\: \\ $$$${infinite}\:{arithmetic}\:{sequences}\:{with}\:{common}\: \\ $$$${difference}\:\:{a}\:{and}\:{b}\:{respectively}\:, \\ $$$${then}\:{find}\:{the}\:{locus}\:{of}\:{the}\:{point}\:\left(\:\alpha\:,\:\beta\:\right)\: \\ $$$${where}\:\alpha\:=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}}…
Question Number 126808 by sdfg last updated on 24/Dec/20 Answered by JMZ last updated on 24/Dec/20 $${Since}\:\phi\:{is}\:{a}\:{homomorphism} \\ $$$$\phi\left({a}+{b}\right)=\phi\left({a}\right)+\phi\left({b}\right). \\ $$$${The}\:{kernel}\:{contais}\:{every}\:{x}\:{which} \\ $$ Answered by…
Question Number 61273 by alphaprime last updated on 31/May/19 $${Suppose}\:\alpha\:,\beta,\gamma,\delta\:{are}\:{real}\:{numbers} \\ $$$${such}\:{that}\:\alpha+\beta+\gamma+\delta\:=\:\alpha^{\mathrm{7}} +\beta^{\mathrm{7}} +\gamma^{\mathrm{7}} +\delta^{\mathrm{7}} =\mathrm{0} \\ $$$${Prove}\:{that}\:\alpha\left(\alpha+\beta\right)\left(\alpha+\gamma\right)\left(\alpha+\delta\right)=\mathrm{0} \\ $$ Commented by Rasheed.Sindhi last updated…
Question Number 192346 by Rupesh123 last updated on 15/May/23 Answered by som(math1967) last updated on 15/May/23 $$\left(\frac{\mathrm{2}{cos}\mathrm{30}{cos}\mathrm{20}+{cos}\mathrm{70}+{cos}\left(\mathrm{180}−\mathrm{70}\right)}{{cos}\mathrm{20}}\right)^{\mathrm{8}} \\ $$$$=\left(\frac{\sqrt{\mathrm{3}}{cos}\mathrm{20}+\mathrm{cos}\:\mathrm{70}−\mathrm{cos}\:\mathrm{70}}{{cos}\mathrm{20}}\right)^{\mathrm{8}} \\ $$$$=\left(\frac{\sqrt{\mathrm{3}}{cos}\mathrm{20}}{{cos}\mathrm{20}}\right)^{\mathrm{8}} \\ $$$$=\left(\sqrt{\mathrm{3}}\right)^{\mathrm{8}} =\mathrm{81} \\…
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Question Number 192341 by Mastermind last updated on 15/May/23 $$\left.\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sign}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{or}\:\mathrm{even}\:\left(\mathrm{or}\:\mathrm{pality}\right) \\ $$$$\mathrm{of}\:\mathrm{permutation}\:\theta=\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\:\mathrm{6}\:\mathrm{7}\:\mathrm{8}\right) \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{permutation} \\ $$$$\theta:\mathrm{S}\rightarrow\mathrm{S}\:\mathrm{where}\:\mathrm{S}\:\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{set}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{written}\:\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\mathrm{of}\:\mathrm{disjoint} \\ $$$$\mathrm{cycle} \\ $$$$ \\…
Question Number 126806 by sdfg last updated on 24/Dec/20 Commented by sdfg last updated on 24/Dec/20 $${pleaes}\:{help} \\ $$ Commented by mahdipoor last updated on…