Question Number 127460 by ajfour last updated on 30/Dec/20 Commented by ajfour last updated on 30/Dec/20 $${Q}.\mathrm{127127}\:\:\:\left({Revisit}\right) \\ $$$${Given}\:\:{p},{q}\:\:\:;\:\:{find}\:{R}. \\ $$ Answered by ajfour last…
Question Number 61923 by naka3546 last updated on 12/Jun/19 $${Find}\:\:{all}\:\:{solutions}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{3}} \:−\:\mathrm{12}{x}\:+\:\mathrm{8}\:=\:\:\mathrm{0} \\ $$ Answered by MJS last updated on 12/Jun/19 $${x}^{\mathrm{3}} +{px}+{q}=\mathrm{0} \\…
Question Number 192992 by gatocomcirrose last updated on 01/Jun/23 $$\begin{cases}{\mathrm{2x}+\mathrm{3y}\equiv\mathrm{1}\left(\mathrm{mod26}\right)}\\{\mathrm{7x}+\mathrm{8y}\equiv\mathrm{2}\left(\mathrm{mod26}\right)}\end{cases} \\ $$$$ \\ $$ Answered by MM42 last updated on 01/Jun/23 $$\mathrm{2}{x}+\mathrm{3}{y}=\mathrm{26}{k}+\mathrm{1}\:\:\&\:\mathrm{7}{x}+\mathrm{8}{y}=\mathrm{26}{k}'+\mathrm{2} \\ $$$$\Rightarrow\mathrm{5}{x}=\mathrm{26}{k}''−\mathrm{2}\Rightarrow{x}\overset{\mathrm{26}} {\equiv}\mathrm{10}\:\checkmark…
Question Number 127459 by liberty last updated on 30/Dec/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{{x}}}}\:\right)^{{x}^{\mathrm{2}} } \:=?\: \\ $$ Answered by john_santu last updated on 30/Dec/20 $$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{x}}}}\:=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}+\frac{\mathrm{x}}{\mathrm{2x}+\mathrm{1}}}\:=\:\mathrm{1}+\frac{\mathrm{1}}{\frac{\mathrm{2x}^{\mathrm{2}} +\mathrm{2x}}{\mathrm{2x}+\mathrm{1}}} \\…
Question Number 61922 by Tawa1 last updated on 11/Jun/19 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}:\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1}}{\mathrm{n}\:+\:\mathrm{2}}.\:\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}!} \\ $$ Commented by maxmathsup by imad last updated on 12/Jun/19…
Question Number 61921 by maxmathsup by imad last updated on 11/Jun/19 $${let}\:{A}\:=\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left(\:{x}^{\mathrm{2}} \:−\mathrm{2}{i}\right)}{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{extract}\:{Re}\left({A}\right)\:{and}\:{Im}\left({A}\right)\:{and}\:{determine}\:{its}\:{values}\:\:\:\left({i}^{\mathrm{2}} =−\mathrm{1}\right) \\ $$ Commented by…
Question Number 127454 by snipers237 last updated on 29/Dec/20 $$\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{k}!\left({n}−{k}\right)!}{{n}!}\:=\:\mathrm{1} \\ $$ Answered by Ar Brandon last updated on 29/Dec/20 $$\forall\mathrm{n}\geqslant\mathrm{4},\:\forall\mathrm{k}\in\left\{\mathrm{2},…,\mathrm{n}−\mathrm{2}\right\},\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{k}}…
Question Number 192988 by Mingma last updated on 01/Jun/23 Answered by Subhi last updated on 01/Jun/23 $${suppose}\:{line}\:=\:{l} \\ $$$${line}\:=\:\sqrt{{l}^{\mathrm{2}} +{l}^{\mathrm{2}} }\:=\:\sqrt{\mathrm{2}\:}\:{l} \\ $$$$\frac{\sqrt{\mathrm{2}}\:{l}}{{sin}\left(\mathrm{180}−\left(\mathrm{45}−{x}+{x}\right)\right)}=\frac{{l}}{{sin}\left(\mathrm{45}−{x}\right)} \\ $$$${sin}\left(\mathrm{45}−{x}\right)=\frac{{sin}\left(\mathrm{135}\right)}{\:\sqrt{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}}…
Question Number 127455 by snipers237 last updated on 29/Dec/20 $${Prove}\:{that}\:{for}\:{all}\:{n}\geqslant\mathrm{1}\: \\ $$$$\left.\mathrm{1}\left.\right){There}\:{exist}\:\:{a}_{{n}} \in\right]\mathrm{0},\mathrm{1}\left[\:{such}\:{as}\:\:\right. \\ $$$${sin}\left(\frac{\mathrm{1}}{{n}}\right)=\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{\mathrm{6}{n}^{\mathrm{3}} }{cos}\left(\frac{\mathrm{1}}{{n}}{a}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{10}}\: \\ $$ Terms of…
Question Number 192991 by otchereabdullai@gmail.com last updated on 01/Jun/23 $${the}\:{first},\:{third}\:{and}\:{sixth}\:{terms}\:{of}\:{a} \\ $$$${linear}\:{sequence}\:{are}\:{the}\:{first}\:{three}\: \\ $$$${terms}\:{of}\:{an}\:{exponential}\:{sequence}.\: \\ $$$${find}\:{the}\:{common}\:{ratio} \\ $$ Answered by MM42 last updated on 01/Jun/23…