Question Number 153708 by liberty last updated on 09/Sep/21 Answered by MJS_new last updated on 09/Sep/21 $$\left(\mathrm{1}\right)\:\mathrm{200}{x}+\mathrm{160}{y}=\mathrm{300} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{200}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }=\mathrm{160}\sqrt{\mathrm{1}−{y}^{\mathrm{2}} } \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:{y}=\frac{\mathrm{15}−\mathrm{10}{x}}{\mathrm{8}}…
Question Number 88170 by Ar Brandon last updated on 08/Apr/20 $${Prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {tcos}\:{n}\pi{tdt}=\frac{\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$ Commented by jagoll last updated…
Question Number 22635 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n},\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{20}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} . \\ $$$$ \\…
Question Number 153704 by mathdanisur last updated on 09/Sep/21 Commented by mathdanisur last updated on 09/Sep/21 $$\mathrm{p}_{\boldsymbol{\mathrm{k}}} \:=\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} +…+\mathrm{x}^{\mathrm{2021}} }\:=\:\frac{\mathrm{1}\:-\:\mathrm{x}}{\mathrm{1}\:-\:\mathrm{x}^{\boldsymbol{\mathrm{k}}+\mathrm{1}} } \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}\:-\:\mathrm{x}}{\mathrm{1}\:-\:\mathrm{x}^{\boldsymbol{\mathrm{k}}+\mathrm{1}} } \\…
Question Number 88169 by jagoll last updated on 08/Apr/20 $$\mathrm{find}\:\mathrm{Laplace}\:\mathrm{transform}\: \\ $$$$\mathrm{t}^{\mathrm{3}} .\:\mathrm{cos}\:\:\mathrm{4t} \\ $$ Commented by mathmax by abdo last updated on 08/Apr/20 $${L}\left({x}^{\mathrm{3}}…
Question Number 153696 by SANOGO last updated on 09/Sep/21 Answered by puissant last updated on 09/Sep/21 $${x}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{k}} \\ $$$${Calcul}\:{des}\:\mathrm{5}\:{premiers}\:{termes}.. \\ $$$$\rightarrow\:{x}_{\mathrm{1}}…
Question Number 88160 by john santu last updated on 08/Apr/20 $${solve}\::\:{x}^{\mathrm{2}} \:=\:\mathrm{3}{x}\:+\:\mathrm{6}{y}\:;\:{xy}\:=\:\mathrm{5}{x}\:+\:\mathrm{4}{y} \\ $$ Commented by john santu last updated on 08/Apr/20 $${y}\:=\:\frac{{x}^{\mathrm{2}} −\mathrm{3}{x}}{\mathrm{6}}\:\:;\:{y}\left({x}−\mathrm{4}\right)\:=\:\mathrm{5}{x} \\…
Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{30}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{1}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples} \\ $$$$\mathrm{that}\:\mathrm{each}\:\mathrm{of}\:\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\…
Question Number 153698 by mathocean1 last updated on 09/Sep/21 $${solve}\:{in}\:\mathbb{R} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} −{y}^{\mathrm{3}} =\mathrm{19}}\\{{xy}=\mathrm{6}}\end{cases} \\ $$ Answered by amin96 last updated on 09/Sep/21 $$\left({x}−{y}\right)\left({x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}}…
Question Number 22624 by Tinkutara last updated on 21/Oct/17 $$\mathrm{A}\:\mathrm{uniform}\:\mathrm{chain}\:\mathrm{of}\:\mathrm{length}\:\mathrm{L}\:\mathrm{and}\:\mathrm{mass} \\ $$$$\mathrm{M}\:\mathrm{is}\:\mathrm{lying}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth}\:\mathrm{table}\:\mathrm{and}\:\mathrm{one} \\ $$$$\mathrm{third}\:\mathrm{of}\:\mathrm{its}\:\mathrm{length}\:\mathrm{is}\:\mathrm{hanging}\:\mathrm{vertically} \\ $$$$\mathrm{down}\:\mathrm{over}\:\mathrm{the}\:\mathrm{edge}\:\mathrm{of}\:\mathrm{the}\:\mathrm{table}.\:\mathrm{If}\:\mathrm{g}\:\mathrm{is} \\ $$$$\mathrm{acceleration}\:\mathrm{due}\:\mathrm{to}\:\mathrm{gravity},\:\mathrm{calculate} \\ $$$$\mathrm{work}\:\mathrm{required}\:\mathrm{to}\:\mathrm{pull}\:\mathrm{the}\:\mathrm{hanging}\:\mathrm{part} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{table}. \\ $$ Answered…