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Question Number 212499 by efronzo1 last updated on 15/Oct/24 $$\:\:\mathrm{Given}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{and}\:\mathrm{d}\:\mathrm{are}\:\mathrm{reals}\: \\ $$$$\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{10}}\\{\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} =\mathrm{10}\:}\\{\mathrm{ab}+\mathrm{cd}=\mathrm{0}}\end{cases} \\ $$$$\:\:\mathrm{Find}\:\mathrm{ac}\:+\:\mathrm{bd}. \\ $$ Answered by ajfour…
Question Number 212492 by MrGaster last updated on 15/Oct/24 $$\mathrm{Given}\:\mathrm{that}\:{x}\:\in\:\mathbb{R}\left(\right. \\ $$$$\:\left(\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{real}\:\mathrm{numbers}\right)\mathrm{d} \\ $$$$\mathrm{an}\:{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\left(\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{naturalu}\right. \\ $$$$\left.\mathrm{nmbers}\right)\:\mathrm{prove}\:\mathrm{that}\:: \\ $$$$\mid\mathrm{cos}\:{x}\mid+\mid\mathrm{cos}\:\mathrm{2}{x}\mid+\mid\mathrm{cos}\:\mathrm{3}{x}\mid+\ldots+\mid\mathrm{cos}\left({n}+\mathrm{1}\right){x}\mid\geq\frac{{n}}{\mathrm{2}} \\ $$ Terms of Service…
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Question Number 212477 by Durganand last updated on 14/Oct/24 Answered by mehdee7396 last updated on 14/Oct/24 $$\frac{{sin}\mathrm{2}{acosa}−{sinacos}\mathrm{2}{a}}{{sin}\mathrm{2}{asina}} \\ $$$$=\frac{{sina}}{{sin}\mathrm{2}{asina}}=\frac{\mathrm{1}}{{sin}\mathrm{2}{a}}\:\:\checkmark \\ $$ Answered by A5T last…
Question Number 212479 by hardmath last updated on 14/Oct/24 $$\begin{cases}{\mathrm{2x}\:+\:\mathrm{3y}\:−\:\mathrm{z}\:=\:\mathrm{7}}\\{\mathrm{4x}\:−\:\mathrm{y}\:+\:\mathrm{2z}\:=\:\mathrm{1}}\\{−\mathrm{x}\:+\:\mathrm{5y}\:+\:\mathrm{3z}\:=\:\mathrm{14}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\mathrm{x},\mathrm{y},\mathrm{z}\:=\:? \\ $$ Answered by A5T last updated on 14/Oct/24 $$\mathrm{3}\left({ii}\right)+\left({i}\right)\Rightarrow\mathrm{14}{x}+\mathrm{5}{z}=\mathrm{10}…\left({iv}\right) \\ $$$$\mathrm{5}\left({ii}\right)+\left({iii}\right)\Rightarrow\mathrm{19}{x}+\mathrm{13}{z}=\mathrm{19}…\left({v}\right) \\ $$$$\mathrm{13}\left({iv}\right)−\mathrm{5}\left({v}\right)\Rightarrow\mathrm{87}{x}=\mathrm{35}\Rightarrow{x}=\frac{\mathrm{35}}{\mathrm{87}}\Rightarrow{z}=\frac{\mathrm{76}}{\mathrm{87}}\Rightarrow{y}=\frac{\mathrm{205}}{\mathrm{87}} \\…
Question Number 212470 by Spillover last updated on 14/Oct/24 Answered by Ar Brandon last updated on 14/Oct/24 $$\Omega=\int_{\mathrm{0}} ^{\infty} \frac{\left(\mathrm{1}−{x}\right)\mathrm{ln}{x}}{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$$$\:\:\:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{\mathrm{1}−{x}^{\mathrm{6}}…
Question Number 212464 by liuxinnan last updated on 14/Oct/24 Commented by liuxinnan last updated on 14/Oct/24 $${is}\:{it}\:{right} \\ $$ Answered by MrGaster last updated on…
Question Number 212467 by Hanuda354 last updated on 14/Oct/24 $$\mathrm{Use}\:\:\mathrm{the}\:\:\mathrm{integration}\:\:\mathrm{by}\:\:\mathrm{parts}. \\ $$$$ \\ $$$$\int\:\frac{\mathrm{2}}{\:\sqrt{{r}^{\mathrm{2}} −\mathrm{4}}}\:{dr}\: \\ $$ Answered by mehdee7396 last updated on 14/Oct/24 $${r}=\mathrm{2}{coshx}\Rightarrow{dr}=\mathrm{2}{sinhxdx}…
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