Question Number 63438 by ajfour last updated on 04/Jul/19 Commented by ajfour last updated on 04/Jul/19 $${Determine}\:\:{a}:{b}:{c}\:\:{if}\:{the} \\ $$$${inscribed}\:\bigtriangleup{ABC}\:{has}\:{constant} \\ $$$${perimeter}\:{but}\:{maximum}\:{area}. \\ $$ Commented by…
Question Number 128971 by Ahmed1hamouda last updated on 11/Jan/21 $${solve}\:{the}\:{following}\:{equation} \\ $$$$\left({x}\mathrm{tan}\left(\frac{{y}}{{x}}\right)−{y}\mathrm{sec}^{\mathrm{2}} \left(\frac{{y}}{{x}}\right)\right){dx}−{x}\mathrm{sec}^{\mathrm{2}} \left(\frac{{y}}{{x}}\right){dy}=\mathrm{0} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 63433 by minh2001 last updated on 04/Jul/19 $$\underset{\mathrm{1}} {\overset{{x}} {\int}}{x}^{\mathrm{2}} −\mathrm{3}{x}\sqrt{{x}}{dx}\:=\frac{−\mathrm{716}}{\mathrm{15}} \\ $$$${then}\:{calculate}\:\underset{{x}} {\overset{{x}+\mathrm{1}} {\int}}\frac{\mathrm{1}}{{x}+\mathrm{3}}{dx} \\ $$ Commented by MJS last updated on…
Question Number 63430 by rajesh4661kumar@gamil.com last updated on 04/Jul/19 Answered by MJS last updated on 04/Jul/19 $${y}=\mathrm{sin}^{−\mathrm{1}} \:\sqrt{{x}−\mathrm{1}} \\ $$$$\mathrm{1}\leqslant{x}\leqslant\mathrm{2}\:\Rightarrow\:\mathrm{0}\leqslant\sqrt{{x}−\mathrm{1}}\leqslant\mathrm{1}\:\Rightarrow \\ $$$$\Rightarrow\:\mathrm{0}\leqslant{y}\leqslant\frac{\pi}{\mathrm{2}} \\ $$ Terms…
Question Number 63428 by Rio Michael last updated on 04/Jul/19 $${It}\:{is}\:{given}\:{that}\:{S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}^{\mathrm{2}\:} −\mathrm{3}{r}−\mathrm{1}\right).\:{Use}\:{the}\:{the}\:{formulae} \\ $$$${of}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}\:\:} {and}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\:\:{to}\:{show}\:{that}\:{S}_{{n}} ={n}^{\mathrm{3}} . \\…
Question Number 63427 by Rio Michael last updated on 04/Jul/19 $$\left(\mathrm{1}\right)\:{A}\:{plane}\:{contains}\:{the}\:{lines}\:\frac{{x}+\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{4}−{y}}{\mathrm{2}}=\frac{{z}−\mathrm{2}}{\mathrm{3}}\:{and}\: \\ $$$${r}=\:\left(\mathrm{2}{i}+\mathrm{2}{j}\:+\:\mathrm{12}{k}\right)+{t}\left(−{i}+\mathrm{2}{j}\:+\mathrm{4}{k}\right).\:{find} \\ $$$$\left({a}\right)\:{the}\:{angle}\:{between}\:{these}\:{lines}. \\ $$$$\left({b}\right)\:{A}\:{cartesian}\:{equation}\:{of}\:{the}\:{plane}. \\ $$$$\left(\mathrm{2}\right)\:{Given}\:{the}\:{lines}\:\boldsymbol{{l}}_{\mathrm{1}} :\frac{{x}−\mathrm{10}}{\mathrm{3}}=\frac{{y}−\mathrm{1}}{\mathrm{1}}=\frac{{z}−\mathrm{9}}{\mathrm{4}}\:\:\boldsymbol{{l}}_{\mathrm{2}} :{r}=\left(−\mathrm{9}{j}+\mathrm{13}{k}\right)+\mu\left({i}+\mathrm{2}{j}−\mathrm{3}{k}\right) \\ $$$${where}\:\mu\:{is}\:{a}\:{parameter};\:\boldsymbol{{l}}_{\mathrm{3}} :\frac{{x}+\mathrm{10}}{\mathrm{4}}=\frac{{y}+\mathrm{5}}{\mathrm{3}}=\frac{{z}+\mathrm{4}}{\mathrm{1}}. \\…
Question Number 63426 by Rio Michael last updated on 04/Jul/19 $${show}\:{that}\: \\ $$$$\left({a}\right)\:{cos}\left[\mathrm{2}{cos}^{−\mathrm{1}} \left({x}\right)\:+{sin}^{−\mathrm{1}} \left({x}\right)\right]=\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\: \\ $$$$\left({b}\right)\:\frac{{sin}\alpha\:+\:{sin}\beta}{{cos}\alpha−{cos}\beta}={cot}\left(\frac{\beta−\alpha}{\mathrm{2}}\right) \\ $$$$\left({c}\right)\:\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{3}}+{p}\right)\approxeq\:\mathrm{1}−\sqrt{\mathrm{3}}\:{if}\:{p}\:{is}\:{small}\:{enough}\:{to}\:{neglect}\:{p}^{\mathrm{2}} . \\ $$$$\left({d}\right)\:{if}\:\theta\:=\frac{\mathrm{1}}{\mathrm{2}}{sin}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right),\:{show}\:{that}\:{sin}\theta−{cos}\theta\:=\:\pm\frac{\mathrm{1}}{\mathrm{2}} \\…
Question Number 63425 by Rio Michael last updated on 04/Jul/19 $${The}\:{probability}\:{that}\:{a}\:{vaccinated}\:{person}\left({V}\right)\:{contracts}\:{a}\:{disease} \\ $$$${is}\:\frac{\mathrm{1}}{\mathrm{20}}.\:{For}\:{a}\:{person}\:{vaccinated}\left({V}\:'\right)\:,\:{the}\:{probability}\:{of}\:{contracting} \\ $$$${a}\:{disease}\:{is}\:\frac{\mathrm{5}}{\mathrm{6}}.\:{In}\:{a}\:{certain}\:{town}\:\mathrm{90\%}{of}\:{thepopulation}\:{has} \\ $$$${been}\:{vaccinated}\:{against}\:{a}\:{disease}.\:{A}\:{person}\:{is}\:{selected}\:{at} \\ $$$${random}\:{from}\:{the}\:{town},{find}\:{the}\:{probability}\:{that}: \\ $$$$\left({a}\right)\:{he}\:{has}\:{the}\:{disease}, \\ $$$$\left({b}\right)\:{he}\:{is}\:{vaccinated}\:{or}\:{he}\:{has}\:{the}\:{disease}. \\ $$$${sir}\:{Forkum}\:{Michael}…
Question Number 63424 by Rio Michael last updated on 04/Jul/19 $${A}\:{colony}\:{of}\:{bacteria}\:{if}\:{left}\:{undisturbed}\:{will}\:{grow}\:{at}\:{a}\:{rate} \\ $$$${proportional}\:{to}\:{the}\:{number}\:{of}\:{bacteria},\:{P}\:{present}\:{at}\:{time},{t}. \\ $$$${However},{a}\:{toxic}\:{substance}\:{is}\:{being}\:{added}\:{slowly}\:{such}\:{that} \\ $$$${at}\:{time}\:{t},\:{the}\:{bacteria}\:{also}\:{die}\:{at}\:{the}\:{rate}\:\mu{Pt}\:{where}\:\mu\:{is} \\ $$$${a}\:{positive}\:{constant}. \\ $$$$\left({a}\right)\:\:{Show}\:{that}\:{at}\:{time}\:{t}\:{the}\:{rate}\:{of}\:{growth}\:{of}\:{the}\:{bacteria}\:{in} \\ $$$${the}\:{colony}\:{is}\:{governed}\:{by}\:{the}\:{differential}\:{equation} \\ $$$$\:\frac{{dP}}{{dt}}=\:\left({k}−\mu{t}\right){p}\:{where}\:{k}\:{is}\:{apositive}\:{constant}.…
Question Number 63423 by minh2001 last updated on 04/Jul/19 $${Prove}\:{that}\: \\ $$$$\frac{{sin}^{\mathrm{2}} \left(\mathrm{36}°\right)}{{cos}^{\mathrm{2}} \left(\mathrm{72}°\right)}+{sin}^{\mathrm{2}} \left(\mathrm{72}°\right){cos}\left(\mathrm{36}°\right)=\frac{\mathrm{45}+\mathrm{11}\sqrt{\mathrm{5}}}{\mathrm{16}} \\ $$ Commented by Tony Lin last updated on 04/Jul/19…