Question Number 70898 by Henri Boucatchou last updated on 09/Oct/19 $$\boldsymbol{{Solve}}\::\: \\ $$$$\left.\mathrm{1}.\right)\:\:\sqrt{\boldsymbol{{x}}−\mathrm{2}}\:+\:\sqrt{\mathrm{4}−\boldsymbol{{x}}}\:=\:\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{{x}}+\mathrm{11} \\ $$$$\left.\mathrm{2}.\right)\:\:\boldsymbol{{x}}^{\mathrm{4}} \:+\:\boldsymbol{{x}}^{\mathrm{3}} \:−\:\mathrm{2}\boldsymbol{{ax}}^{\mathrm{2}} \:−\:\boldsymbol{{ax}}\:+\:\boldsymbol{{a}}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$ Answered by MJS…
Question Number 5357 by Junaid Mirza last updated on 11/May/16 Commented by Yozzii last updated on 11/May/16 $${Let}\:{the}\:{lowest}\:{position}\:{of}\:{the}\:{bob}\:{be} \\ $$$${that}\:{where}\:{its}\:{gravitational}\:{potential} \\ $$$${energy}\:{is}\:{zero}.\:{Then},\:{by}\:{the}\:{law}\:{of} \\ $$$${conservation}\:{of}\:{energy}\:{for}\:{the}\:{bob}, \\…
Question Number 136425 by Ar Brandon last updated on 21/Mar/21 $$\mathrm{If}\:\alpha>\mathrm{0}\:\mathrm{and}\:\beta>\mathrm{0},\:\mathrm{prove} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\left(\alpha\mathrm{x}\right)}{\beta^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=\frac{\pi}{\mathrm{2}\beta}\mathrm{ln}\left(\alpha\beta\right) \\ $$ Commented by Dwaipayan Shikari last updated…
Question Number 70891 by Henri Boucatchou last updated on 09/Oct/19 $$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{x}}\left(\boldsymbol{{x}}−\mathrm{1}\right)\left(\boldsymbol{{x}}−\mathrm{2}\right)….\left(\boldsymbol{{x}}−\mathrm{10}\right) \\ $$$$\boldsymbol{{f}}\:'\left(\mathrm{0}\right)\:=\:? \\ $$ Commented by kaivan.ahmadi last updated on 09/Oct/19 $${let}\:{g}\left({x}\right)=\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)…\left({x}−\mathrm{10}\right)\Rightarrow{f}\left({x}\right)={xg}\left({x}\right) \\ $$$$\Rightarrow{f}'\left({x}\right)={g}\left({x}\right)+{xg}'\left({x}\right)\Rightarrow…
Question Number 5354 by Rasheed Soomro last updated on 11/May/16 Commented by Rasheed Soomro last updated on 11/May/16 $$\mathrm{ABCD}\:\mathrm{and}\:\mathrm{EFGH}\:\mathrm{are}\:\mathrm{squares}\:\mathrm{in} \\ $$$$\mathrm{above}\:\mathrm{figure}.\mathrm{The}\:\mathrm{centres}\:\mathrm{of}\:\mathrm{the}\:\:\mathrm{arcs}\: \\ $$$$\mathrm{are}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the}\:\mathrm{external}\:\mathrm{square}. \\ $$$$\mathrm{If}\:\:\mathrm{EF}=\mathrm{t}\:\mathrm{then}\:\mathrm{AB}=?…
Question Number 70886 by Omer Alattas last updated on 09/Oct/19 Commented by mathmax by abdo last updated on 10/Oct/19 $$\mathrm{1}−{cosu}\:\sim\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:{and}\:{sinu}\:\sim\:{u}\:{if}\:{u}\in{V}\left(\mathrm{0}\right)\:\Rightarrow \\ $$$$\mathrm{1}−{cos}\left(\frac{\pi{t}}{\mathrm{3}}\right)\sim\frac{\left(\frac{\pi{t}}{\mathrm{3}}\right)^{\mathrm{2}} }{\mathrm{2}}\:=\pi^{\mathrm{2}} \frac{{t}^{\mathrm{2}}…
Question Number 5351 by sanusihammed last updated on 10/May/16 $${A}\:{cylindrical}\:{iron}\:{rod}\:\mathrm{8}{cm}\:{and}\:\mathrm{6}{cm}\:{in}\:{diameter}\:{stands}\:{in}\:{a} \\ $$$${cylindrical}\:{tin}\:{of}\:\mathrm{12}{cm}\:{in}\:{diameter}.\:{Water}\:{is}\:{poured}\:{into}\:{the} \\ $$$${tin}\:{until}\:{it}\:{depth}\:{is}\:\mathrm{8}{cm}.\:{How}\:{far}\:{would}\:{the}\:{level}\:{drop}\:{when}\:{the}\: \\ $$$${rod}\:{is}\:{removed}\:? \\ $$ Commented by Yozzii last updated on 11/May/16…
Question Number 136420 by Dwaipayan Shikari last updated on 21/Mar/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right)\theta}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{\theta}{\mathrm{2}}{log}\left(\mathrm{2}\right)+{sin}\theta\frac{{log}\left({sin}\theta\right)}{\mathrm{4}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \theta\right)+\frac{{sin}\theta}{\mathrm{16}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \theta\right) \\ $$$${Prove}\:{or}\:{disprove}\:\:\:\:\:\mathrm{0}<\theta<\pi \\ $$ Answered…
Question Number 70887 by naka3546 last updated on 09/Oct/19 Commented by ajfour last updated on 09/Oct/19 Commented by ajfour last updated on 09/Oct/19 $${s}\mathrm{sin}\:\alpha={r}\mathrm{sin}\:\gamma \\…
Question Number 5350 by sanusihammed last updated on 09/May/16 $${Suppose}\:{the}\:{radius}\:{of}\:{the}\:{earth}\:{is}\:\mathrm{6400}{km}.\:{the}\:{acceleration}\: \\ $$$${of}\:{a}\:{person}\:{at}\:{latitude}\:\mathrm{60}°\:{due}\:{to}\:{the}\:{earth}\:{rotation}\:{is}\:? \\ $$$$ \\ $$$$\left({a}\right)\:\mathrm{0}.\mathrm{034}{m}/{s}^{\mathrm{2}} \\ $$$$\left({b}\right)\:\mathrm{232}.\mathrm{7}{m}/{s}^{\mathrm{2}} \\ $$$$\left({c}\right)\:\mathrm{0}.\mathrm{0169}{m}/{s}^{\mathrm{2}} \\ $$$$\left({d}\right)\:\mathrm{465}.\mathrm{4}{m}/{s}^{\mathrm{2}} \\ $$$$ \\…