Question Number 55788 by gunawan last updated on 04/Mar/19 $$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{last}\:\mathrm{eight}\:\mathrm{coefficients}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{16}} \:\mathrm{is}\:\mathrm{2}^{\mathrm{15}} \:. \\ $$ Commented by gunawan last updated on 04/Mar/19 $$\mathrm{True}\:\mathrm{or}\:\mathrm{false} \\…
Question Number 55787 by gunawan last updated on 04/Mar/19 $$\mathrm{The}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{5}^{\mathrm{99}} \mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{13}\:\mathrm{is} \\ $$ Answered by MJS last updated on 04/Mar/19 $$\mathrm{remainders}\:\mathrm{of}\:\frac{\mathrm{5}^{{n}} }{\mathrm{13}} \\ $$$${n}=\mathrm{1}+\mathrm{4}{k}:\:\mathrm{5} \\…
Question Number 55786 by gunawan last updated on 04/Mar/19 $$\mathrm{If}\:\mathrm{the}\:\mathrm{third}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{1}}{{x}}\:+\:{x}^{\mathrm{log}_{\mathrm{10}} {x}} \right)^{\mathrm{5}} \:\mathrm{is}\:\mathrm{1000},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}\:\mathrm{is} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 55785 by gunawan last updated on 04/Mar/19 $$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{4}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)^{\mathrm{11}} \:\mathrm{is} \\ $$ Commented by Tawa1 last updated on 04/Mar/19…
Question Number 55784 by gunawan last updated on 04/Mar/19 $$\mathrm{If}\:{n}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{natural}\:\mathrm{number},\:\mathrm{then} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{r}} }{\:^{{n}} {C}_{{r}} \:}\:\mathrm{equals} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 55780 by gunawan last updated on 04/Mar/19 $$\int\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}}\:{dx}=\:{A}\:\mathrm{log}\:\mid{x}+\mathrm{1}\mid+{C}\:\mathrm{for} \\ $$$${x}>−\mathrm{1},\:\mathrm{then}\:{A}=\_\_\_\_\_. \\ $$ Commented by maxmathsup by imad last updated on 04/Mar/19 $$\int\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}}…
Question Number 55779 by gunawan last updated on 04/Mar/19 $$\mathrm{If}\:\int\:{x}\:\mathrm{log}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right){dx}\: \\ $$$$\:\:\:\:\:\:=\:{f}\left({x}\right)\:\centerdot\:\mathrm{log}\:\left({x}+\mathrm{1}\right)+{g}\left({x}\right)\:\centerdot\:{x}^{\mathrm{2}} +{Ax}+{C}, \\ $$$$\mathrm{then} \\ $$ Commented by gunawan last updated on 04/Mar/19 $$\mathrm{Find}\:{A}…
Question Number 55778 by gunawan last updated on 04/Mar/19 $$\int\:\:\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:= \\ $$ Answered by MJS last updated on 04/Mar/19 $$\int\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx}= \\…
Question Number 55776 by gunawan last updated on 04/Mar/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\frac{{r}^{\mathrm{3}} }{{r}^{\mathrm{4}} +{n}^{\mathrm{4}} }\right)\:= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19…
Question Number 55777 by gunawan last updated on 04/Mar/19 $$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\mathrm{cos}^{\mathrm{7}} {x}\:\mathrm{sin}\:\mathrm{3}{x}\:{dx}\:=\:\mathrm{0} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19 $${I}=\int_{\mathrm{0}} ^{\pi} \left\{{cos}\left(\pi−{x}\right)\right\}^{\mathrm{7}}…