Question Number 17158 by Tinkutara last updated on 01/Jul/17 $$\mathrm{Please}\:\mathrm{solve}\:\mathrm{Q}.\:\mathrm{16069}.\:\mathrm{Ask}\:\mathrm{from}\:\mathrm{me}\:\mathrm{the} \\ $$$$\mathrm{solution}\:\mathrm{if}\:\mathrm{needed}\:\mathrm{and}\:\mathrm{please}\:\mathrm{explain}\:\mathrm{it}. \\ $$ Commented by mrW1 last updated on 01/Jul/17 $$\mathrm{one}\:\mathrm{can}\:\mathrm{prove} \\ $$$$\mathrm{case}\:\mathrm{1}:\:\mathrm{AB}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{CD}\:\mathrm{and}\:\mathrm{AD}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{BC}: \\…
Question Number 148231 by qaz last updated on 26/Jul/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{dx}} =? \\ $$ Commented by abdullahalholanymath last updated on 26/Jul/21 $$?????? \\ $$…
Question Number 17153 by Tinkutara last updated on 01/Jul/17 $$\mathrm{If}\:{m},\:{n}\:\in\:{N}\left({n}\:>\:{m}\right),\:\mathrm{then}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${n}\mid\mathrm{sin}\:{x}\mid\:=\:{m}\mid\mathrm{sin}\:{x}\mid\:\mathrm{in}\:\left[\mathrm{0},\:\mathrm{2}\pi\right]\:\mathrm{is} \\ $$ Answered by mrW1 last updated on 01/Jul/17 $${n}\mid\mathrm{sin}\:{x}\mid\:=\:{m}\mid\mathrm{sin}\:{x}\mid \\…
Question Number 148226 by mathdanisur last updated on 26/Jul/21 Answered by mitica last updated on 26/Jul/21 $$\Sigma\frac{\mathrm{1}}{{x}\left({px}+\mathrm{1}\right)}=\Sigma\frac{\left(\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} }{{p}+\frac{\mathrm{1}}{{x}}}\geqslant \\ $$$$\frac{\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)^{\mathrm{2}} }{{p}+{q}+{r}+\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}}=\frac{\mathrm{9}}{{p}+{q}+{r}+\mathrm{3}} \\ $$ Commented by…
Question Number 17152 by Tinkutara last updated on 01/Jul/17 $$\mathrm{If}\:\mathrm{sin}{A}\:=\:\mathrm{sin}{B}\:\mathrm{and}\:\mathrm{cos}{A}\:=\:\mathrm{cos}{B},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{A}\:=\:{B}\:+\:{n}\pi,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{2}\right)\:{A}\:=\:{B}\:−\:{n}\pi,\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{3}\right)\:{A}\:=\:\mathrm{2}{n}\pi\:+\:{B},\:{n}\:\in\:{I} \\ $$$$\left(\mathrm{4}\right)\:{A}\:=\:{n}\pi\:−\:{B},\:{n}\:\in\:{I} \\ $$ Answered by ajfour last updated…
Question Number 148221 by 0731619 last updated on 26/Jul/21 Answered by Olaf_Thorendsen last updated on 26/Jul/21 $$\mathrm{N}\left({x}\right)\:=\:\mathrm{tan}^{\mathrm{2}} \left(\mathrm{tan}{x}\right)−\mathrm{tan}^{\mathrm{2}} {x} \\ $$$$\mathrm{N}\left({x}\right)\:\underset{\mathrm{0}} {\sim}\:\mathrm{tan}^{\mathrm{2}} \left({x}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}+\frac{\mathrm{2}{x}^{\mathrm{5}} }{\mathrm{15}}\right)−\left({x}+\frac{{x}^{\mathrm{3}}…
Question Number 17151 by Tinkutara last updated on 01/Jul/17 $$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{cos}^{\mathrm{2}} \theta\:−\:\mathrm{2cos}\theta\:=\:\mathrm{4sin}\theta\:−\:\mathrm{sin2}\theta\:\mathrm{where} \\ $$$$\theta\:\in\:\left[\mathrm{0},\:\pi\right]\:\mathrm{is} \\ $$ Answered by ajfour last updated on 01/Jul/17 $$\mathrm{4sin}\:\theta−\mathrm{2sin}\:\theta\mathrm{cos}\:\theta+\mathrm{2cos}\:\theta−\mathrm{cos}\:^{\mathrm{2}}…
Question Number 82687 by M±th+et£s last updated on 23/Feb/20 Commented by jagoll last updated on 23/Feb/20 $${what}\:{is}\:\frac{{d}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{dx}^{\frac{\mathrm{1}}{\mathrm{2}}} }\:? \\ $$ Commented by JDamian last…
Question Number 17150 by Tinkutara last updated on 01/Jul/17 $$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{cos}\:{x}\:+\:\mathrm{3}\:\mathrm{cos}\:\left(\mathrm{2}{Kx}\right)\:=\:\mathrm{4} \\ $$$$\mathrm{has}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{solution},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\frac{{P}}{{P}\:+\:\mathrm{1}},\:{P}\:\neq\:−\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number}\:\mathrm{whose} \\ $$$$\mathrm{rational}\:\mathrm{approximation}\:\mathrm{does}\:\mathrm{not} \\ $$$$\mathrm{exceed}\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:{K}\:\mathrm{is}\:\mathrm{irrational}\:\mathrm{number} \\…
Question Number 148222 by BHOOPENDRA last updated on 26/Jul/21 Answered by iloveisrael last updated on 26/Jul/21 $$\mathrm{v}_{\mathrm{3}} =\lambda\mathrm{v}_{\mathrm{1}} +\alpha\mathrm{v}_{\mathrm{2}} \\ $$$$\begin{pmatrix}{−\mathrm{3}}\\{\:\:\:\:\mathrm{4}}\\{\:\:\:\:\mathrm{7}}\end{pmatrix}\:=\:\begin{pmatrix}{\:\:\lambda}\\{\:\:\mathrm{0}}\\{−\lambda}\end{pmatrix}\:+\begin{pmatrix}{\:\:\mathrm{0}}\\{\mathrm{2}\alpha}\\{\mathrm{2}\alpha}\end{pmatrix} \\ $$$$\:\begin{cases}{\lambda=−\mathrm{3}}\\{\alpha=\mathrm{2}}\end{cases}\Rightarrow\mathrm{v}_{\mathrm{3}} =−\mathrm{3v}_{\mathrm{1}} +\mathrm{2v}_{\mathrm{2}}…