Question Number 8352 by tawakalitu last updated on 09/Oct/16 $$\mathrm{make}\:\mathrm{t}\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formular}. \\ $$$$\mathrm{s}\:=\:\mathrm{ut}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{gt}^{\mathrm{2}} \:+\:\mathrm{3t}^{\mathrm{3}} \\ $$ Answered by fernandodantas1996 last updated on 13/Oct/16 $$\mathrm{s}\:=\:\mathrm{t}\left(\mathrm{u}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{gt}+\mathrm{3t}^{\mathrm{2}} \right)\:\Leftrightarrow\:\mathrm{t}\:=\:\frac{\mathrm{s}}{\mathrm{u}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{gt}\:+\:\mathrm{3t}^{\mathrm{2}} }…
Question Number 8341 by Rasheed Soomro last updated on 09/Oct/16 $$\mathrm{What}\:\mathrm{are}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{conditions} \\ $$$$\mathrm{that}\:\left(\mathrm{a}+\mathrm{ib}\right)^{\mathrm{n}} \:\mathrm{is}\:\mathrm{cyclic}\:\mathrm{for}\:\mathrm{an}\:\:\mathrm{n}\:\mathrm{not}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{0}? \\ $$ Answered by prakash jain last updated on 09/Oct/16 $$\mid{a}+{ib}\mid=\mathrm{1}\Rightarrow\sqrt{{a}^{\mathrm{2}}…
Question Number 139399 by mathsuji last updated on 26/Apr/21 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{{log}\left({sin}\left({x}/\mathrm{2}\right)\right)+{log}\left({cos}\left({x}/\mathrm{2}\right)\right)}{\:\sqrt{\mathrm{2}}\centerdot{cos}\left(\pi/\mathrm{4}−{x}/\mathrm{2}\right)}\:{dx} \\ $$ Answered by Ar Brandon last updated on 26/Apr/21 $$\mathrm{Let}\:{u}=\frac{\pi}{\mathrm{2}}−{x} \\ $$…
Question Number 139394 by EnterUsername last updated on 26/Apr/21 $$\mathrm{If}\:\mid{z}−{i}\mid\leqslant\mathrm{2}\:\mathrm{and}\:{z}_{\mathrm{0}} =\mathrm{5}+\mathrm{3}{i},\:\mathrm{then}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mid{z}_{\mathrm{0}} +{iz}\mid\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{7}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\sqrt{\mathrm{7}}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{5}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{9} \\ $$ Answered by mr W last updated on…
Question Number 8301 by Rasheed Soomro last updated on 06/Oct/16 $$\mathrm{Is}\:\:\left\{\:\left(\omega+\mathrm{i}\right)^{\mathrm{0}} ,\:\left(\omega+\mathrm{i}\right)^{\mathrm{1}} ,\:\left(\omega+\mathrm{i}\right)^{\mathrm{2}} ,\:….,\:\left(\omega+\mathrm{i}\right)^{\mathrm{n}} \:\right\} \\ $$$$\mathrm{cyclic}\:\mathrm{for}\:\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}? \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{such}\:\mathrm{n}\:\mathrm{if}\:\mathrm{it}\:\mathrm{exists}. \\ $$$$\omega\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{cuberoot}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{i}=\sqrt{−\mathrm{1}} \\ $$…
Question Number 8282 by tawakalitu last updated on 06/Oct/16 $$\mathrm{Find}\:\mathrm{x},\:\mathrm{y}\:\mathrm{in}\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{1}}\\{\mathrm{x}^{\mathrm{8}} \:+\:\mathrm{y}^{\mathrm{8}} \:=\:\mathrm{x}^{\mathrm{10}} \:+\:\mathrm{y}^{\mathrm{10}} }\end{cases} \\ $$ Commented by Rasheed Soomro last…
Question Number 8277 by Yozzias last updated on 05/Oct/16 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{one}\:\mathrm{representation}\:\mathrm{for}\:\pi\approx\mathrm{3}.\mathrm{14}… \\ $$$$\mathrm{is}\:\pi=\mathrm{12cos}^{−\mathrm{1}} \left[\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{1}/\mathrm{4}} \left(\mathrm{1}+\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2r}} {\prod}}\left(\frac{\mathrm{3}}{\mathrm{2}}−\mathrm{k}\right)}{\left(\mathrm{2r}\right)!}\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{r}} \right)\right]. \\ $$$$ \\ $$ Terms of…
Question Number 139319 by mathsuji last updated on 25/Apr/21 $${prove}\:{that}\:{are}\:{axactly}\:\mathrm{1729}\:{positive} \\ $$$${integer}\:{solutions}\:{to}\:{the}\:{below}\:{equation} \\ $$$$\mathrm{4}{x}^{\mathrm{4}} +\mathrm{3}{y}^{\mathrm{3}} +\mathrm{2}{z}^{\mathrm{2}} +{t}=\mathrm{4311} \\ $$ Commented by Rasheed.Sindhi last updated on…
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Question Number 8236 by Yozzias last updated on 03/Oct/16 $$\mathrm{Define}\:\mathrm{a}\:\mathrm{3}×\mathrm{3}\:\mathrm{matrix}\:\mathrm{whose}\:\mathrm{entries} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{first}\:\mathrm{9}\:\mathrm{positive}\:\mathrm{integers}. \\ $$$$\mathrm{Let}\:\mathrm{s}_{\mathrm{k}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{elements} \\ $$$$\mathrm{across}\:\mathrm{the}\:\mathrm{kth}\:\mathrm{row}.\:\mathrm{Is}\:\mathrm{there}\:\mathrm{such}\:\mathrm{a}\: \\ $$$$\mathrm{matrix}\:\mathrm{where}\:\mathrm{s}_{\mathrm{1}} \::\:\mathrm{s}_{\mathrm{2}} \::\:\mathrm{s}_{\mathrm{3}} \:=\:\mathrm{1}\::\:\mathrm{2}\::\:\mathrm{3}\:? \\ $$$$−−−−−−−−−−−−−−−−−−−− \\…