Question Number 84651 by ajfour last updated on 14/Mar/20 $${A}\:{person}\:{stands}\:{in}\:{the}\:{diagonal} \\ $$$${produced}\:{of}\:{the}\:{square}\:{base}\:{of}\:{a} \\ $$$${church}\:{tower},\:{at}\:{a}\:{distance}\:\mathrm{2}{a} \\ $$$${from}\:{it},\:{and}\:{observes}\:{the}\:{angle} \\ $$$${of}\:{elevation}\:{of}\:{each}\:{of}\:{the}\:{two} \\ $$$${outer}\:{corners}\:{of}\:{the}\:{top}\:{of}\:{the} \\ $$$${tower}\:{to}\:{be}\:\mathrm{30}°,\:{while}\:{that}\:{of}\:{the} \\ $$$${nearest}\:{corner}\:{is}\:\mathrm{45}°.\:{Find}\:{the} \\…
Question Number 150187 by mathdanisur last updated on 10/Aug/21 $$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{5}\:\:\Rightarrow\:\:\mathrm{f}^{\:−\mathrm{1}} \left(\mathrm{2}\right)\:=\:? \\ $$ Commented by amin96 last updated on 10/Aug/21 $$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}=\mathrm{2}\:\:\Rightarrow\:\:\:{x}=\pm\frac{\mathrm{3}{i}}{\:\sqrt{\mathrm{2}}}\:\:\: \\ $$$${f}\left(\pm\frac{\mathrm{3}{i}}{\:\sqrt{\mathrm{2}}}\right)=\mathrm{2}\:\:\:\:\Rightarrow\:\:{f}^{−\mathrm{1}}…
Question Number 150186 by mathdanisur last updated on 10/Aug/21 $$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{sin}^{\mathrm{4}} \left(\mathrm{3x}\right)\:\:\Rightarrow\:\:{f}\:^{'} \left(\frac{\pi}{\mathrm{12}}\right)\:=\:? \\ $$ Answered by puissant last updated on 10/Aug/21 $$\mathrm{f}'\left(\mathrm{x}\right)=\mathrm{4}×\mathrm{3cos}\left(\mathrm{3x}\right)\mathrm{sin}^{\mathrm{3}} \left(\mathrm{3x}\right) \\ $$$$\Rightarrow\:\mathrm{f}'\left(\mathrm{x}\right)=\mathrm{12cos}\left(\mathrm{3x}\right)\mathrm{sin}^{\mathrm{3}}…
Question Number 150181 by liberty last updated on 10/Aug/21 $$\:\left(\mathrm{x}−\mathrm{6}\right)^{\mathrm{3}} +\left(\mathrm{x}−\mathrm{5}\right)^{\mathrm{3}} +\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{3}} =\mathrm{3}\left(\mathrm{x}−\mathrm{6}\right)\left(\mathrm{x}−\mathrm{5}\right)\left(\mathrm{x}−\mathrm{4}\right) \\ $$$$\mathrm{x}=? \\ $$ Answered by Rasheed.Sindhi last updated on 13/Aug/21 $$\:\left(\mathrm{x}−\mathrm{6}\right)^{\mathrm{3}}…
Question Number 150180 by jlewis last updated on 10/Aug/21 $$\mathrm{show}\:\mathrm{that}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{Sin}\:\mathrm{x}/\mathrm{x}\:=\mathrm{1} \\ $$$$ \\ $$ Answered by liberty last updated on 10/Aug/21 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\right)=\frac{\mathrm{xcos}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} }\: \\ $$…
Question Number 150177 by mr W last updated on 10/Aug/21 Commented by mr W last updated on 10/Aug/21 $${an}\:{other}\:{way}\:{to}\:{solve}\:{Q}\mathrm{149894} \\ $$ Commented by mr W…
Question Number 84641 by M±th+et£s last updated on 14/Mar/20 Commented by M±th+et£s last updated on 14/Mar/20 $${a}\:{square},\:{a}\:{circle}\:{and}\:{tow}\:{semicircles} \\ $$$${the}\:{are}\:{of}\:{the}\:{square}\:{is}\:\mathrm{4}.{what}\:{is}\:{the} \\ $$$${length}\:{of}\:{the}\:{blue}\:{lenght} \\ $$ Answered by…
Question Number 19104 by Tinkutara last updated on 04/Aug/17 $$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{FO}\:\mathrm{intersects} \\ $$$$\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\Delta\mathrm{FXY},\:\mathrm{second}\:\mathrm{time} \\…
Question Number 19101 by Tinkutara last updated on 04/Aug/17 $$\mathrm{A}\:\mathrm{polynomial}\:{f}\left({x}\right)\:\mathrm{with}\:\mathrm{rational} \\ $$$$\mathrm{coefficients}\:\mathrm{leaves}\:\mathrm{remainder}\:\mathrm{15},\:\mathrm{when} \\ $$$$\mathrm{divided}\:\mathrm{by}\:{x}\:−\:\mathrm{3}\:\mathrm{and}\:\mathrm{remainder}\:\mathrm{2}{x}\:+\:\mathrm{1}, \\ $$$$\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} .\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left({x}\:−\:\mathrm{3}\right)\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} . \\ $$ Commented…
Question Number 84637 by Rio Michael last updated on 14/Mar/20 $$\mathrm{prove}\:\mathrm{that}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{x}}\right)^{{x}} \:={e} \\ $$ Commented by ajfour last updated on 14/Mar/20 $${prove}\:{that}\:\:\mathrm{sin}\:\theta=\frac{{p}}{{h}}\:. \\ $$…