Question Number 191498 by MATHEMATICSAM last updated on 24/Apr/23 $$\mathrm{If}\:\mathrm{A}\left(\frac{\mathrm{2}{c}}{{a}}\:,\:\frac{{c}}{{b}}\right),\:\mathrm{B}\left(\frac{{c}}{{a}}\:,\:\mathrm{0}\right)\:\mathrm{and}\:\mathrm{C}\left(\frac{\mathrm{1}\:+\:{c}}{{a}}\:,\:\frac{\mathrm{1}}{{b}}\right) \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{points},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that},\: \\ $$$$\mathrm{i}.\:\:\frac{\left(\mathrm{AB}\right)^{\mathrm{2}} \:+\:\left(\mathrm{BC}\right)^{\mathrm{2}} }{\left(\mathrm{CA}\right)^{\mathrm{2}} }\:=\:\frac{{c}^{\mathrm{2}} \:+\:\mathrm{1}}{\left({c}\:−\:\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{ii}.\:\left(\mathrm{AB}\right)^{\mathrm{2}} \:+\:\left(\mathrm{BC}\right)^{\mathrm{2}} \:−\:\left(\mathrm{AC}\right)^{\mathrm{2}} \:=\:\frac{\mathrm{2}{c}\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}}…
Question Number 60422 by Tawa1 last updated on 20/May/19 Commented by MJS last updated on 20/May/19 $$\mathrm{what}'\mathrm{s}\:\mathrm{a}\:“\mathrm{lidless}\:\mathrm{box}''? \\ $$$$\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:“\mathrm{ends}''\:\mathrm{of}\:\mathrm{a}\:\mathrm{box}? \\ $$ Commented by Tawa1 last…
Question Number 60423 by Tawa1 last updated on 20/May/19 Answered by MJS last updated on 20/May/19 $$\begin{pmatrix}{\mathrm{20}}\\{\angle\mathrm{30}°}\end{pmatrix}\:=\begin{pmatrix}{\mathrm{10}\sqrt{\mathrm{3}}}\\{\mathrm{10}}\end{pmatrix} \\ $$$$\begin{pmatrix}{\mathrm{8}}\\{\angle−\mathrm{50}°}\end{pmatrix}\:=\begin{pmatrix}{\mathrm{8sin}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\\{−\mathrm{8cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\end{pmatrix} \\ $$$$\begin{pmatrix}{\mathrm{10}\sqrt{\mathrm{3}}}\\{\mathrm{10}}\end{pmatrix}\:+\begin{pmatrix}{\mathrm{8sin}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\\{−\mathrm{8cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\end{pmatrix}\:\approx\begin{pmatrix}{\mathrm{22}.\mathrm{46}}\\{\mathrm{3}.\mathrm{87}}\end{pmatrix} \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{this}\:\mathrm{vector}\:\mathrm{is}\:\mathrm{22}.\mathrm{79} \\ $$$$\mathrm{the}\:\mathrm{direction}\:\mathrm{of}\:\mathrm{this}\:\mathrm{vector}\:\mathrm{is}\:\mathrm{E}\:\mathrm{9}.\mathrm{78}°\:\mathrm{N}…
Question Number 60421 by Tawa1 last updated on 20/May/19 Answered by MJS last updated on 20/May/19 $$\eta=\frac{\pi}{\mathrm{8}}×\frac{{pr}^{\mathrm{4}} }{{LQ}} \\ $$$$\eta_{{min}} =\frac{\pi}{\mathrm{8}}×\frac{.\mathrm{97}{p}×\left(.\mathrm{98}{r}\right)^{\mathrm{4}} }{\mathrm{1}.\mathrm{01}{L}×\mathrm{1}.\mathrm{005}{Q}}\approx.\mathrm{881}\frac{\pi{pr}^{\mathrm{4}} }{\mathrm{8}{LQ}} \\ $$$$\eta_{{max}}…
Question Number 60410 by tanmay last updated on 20/May/19 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{{x}^{\mathrm{2}} }{{tanxsinx}}\right]\:\left[.\right]={grestest}\:{integer}\:{function} \\ $$ Commented by kaivan.ahmadi last updated on 20/May/19 $${if}\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}} \\ $$$${sinx}<{x}<{tgx}\Rightarrow{xsinx}<{x}^{\mathrm{2}} <{xtgx}\Rightarrow…
Question Number 191483 by Mastermind last updated on 24/Apr/23 $$\mathrm{Compute}\:\mathrm{the}\:\mathrm{min}−\mathrm{max}\:\mathrm{polynomial} \\ $$$$\mathrm{q}_{\mathrm{1}} ^{\ast} \left(\mathrm{x}\right)\:\mathrm{to}\:\mathrm{e}^{\mathrm{x}} \:\mathrm{on}\:\mathrm{interval}\:\left[−\mathrm{1},\:\mathrm{1}\right]. \\ $$$$ \\ $$$$\mathrm{Anyone}?? \\ $$ Terms of Service Privacy…
Question Number 125939 by Dwaipayan Shikari last updated on 15/Dec/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{cos}\mathrm{2}{x}−{tanx}.{cot}\left({tanx}\right)}{{sin}\mathrm{2}{x}−{tan}\left({tanx}\right){log}\left({cos}^{\mathrm{2}} {x}\right)}{dx} \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 125884 by Dwaipayan Shikari last updated on 14/Dec/20 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)^{{n}} \begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}}{\left(\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\left(\sqrt{\mathrm{5}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\right)^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} +\left(\sqrt{\mathrm{5}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\right)^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} \right)\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 60337 by arcana last updated on 20/May/19 $$\mathrm{15},\mathrm{25},\mathrm{42},…? \\ $$ Commented by MJS last updated on 20/May/19 $$\mathrm{only}\:\mathrm{one}\:\mathrm{of}\:\infty\:\mathrm{solutions}: \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}−{y}=\mathrm{0} \\ $$$${a}+{b}+{c}−\mathrm{15}=\mathrm{0}…
Question Number 125857 by Dwaipayan Shikari last updated on 14/Dec/20 $$\mathrm{1}+\mathrm{4}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{7}} +\mathrm{7}\left(\frac{\mathrm{1}.\mathrm{3}}{\mathrm{2}.\mathrm{4}}\right)^{\mathrm{7}} +\mathrm{10}\left(\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}}{\mathrm{2}.\mathrm{4}.\mathrm{6}}\right)^{\mathrm{7}} +… \\ $$ Answered by Olaf last updated on 14/Dec/20 $$\mathrm{S}\:=\:\mathrm{1}+\underset{{n}=\mathrm{1}} {\overset{\infty}…