Question Number 8847 by tawakalitu last updated on 31/Oct/16 Commented by Rasheed Soomro last updated on 01/Nov/16 $$\mathrm{Let}\:\mathrm{r}\:\mathrm{be}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{in}\:\mathrm{each}\:\mathrm{case} \\ $$$$\mathrm{N}\:\mathrm{is}\:\mathrm{GCD}\:\mathrm{of}\:\mathrm{1305}−\mathrm{r},\mathrm{4665}−\mathrm{r}\:\mathrm{and}\:\mathrm{6905}−\mathrm{r} \\ $$$$\mathrm{Let}\:\mathrm{1305}−\mathrm{r}=\mathrm{AN},\:\mathrm{4665}−\mathrm{r}=\mathrm{BN}\:\:\mathrm{and}\:\:\mathrm{6905}−\mathrm{r}=\mathrm{CN} \\ $$$$\mathrm{AN}+\mathrm{r}=\mathrm{1305}\:,\:\mathrm{BN}+\mathrm{r}=\mathrm{4665}\:\:\mathrm{and}\:\:\mathrm{CN}+\mathrm{r}=\mathrm{6905} \\…
Question Number 74383 by aliesam last updated on 23/Nov/19 Commented by mathmax by abdo last updated on 23/Nov/19 $${A}\left({x}\right)=\frac{\mathrm{16}\sqrt{{x}−\sqrt{{x}}}−\mathrm{3}\sqrt{\mathrm{2}}{x}−\mathrm{4}\sqrt{\mathrm{2}}}{\mathrm{16}\left({x}−\mathrm{4}\right)^{\mathrm{2}} }{let}\:{use}\:{hospital}\:{theorem}\:\:{let}\:{f}\left({x}\right)=\mathrm{16}\sqrt{{x}−\sqrt{{x}}}\:−\mathrm{3}\sqrt{\mathrm{2}}{x}−\mathrm{4}\sqrt{\mathrm{2}} \\ $$$${and}\:{g}\left({x}\right)=\mathrm{16}\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:\:{we}\:{have}\:{f}^{'} \left({x}\right)=\mathrm{16}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}}{\mathrm{2}\sqrt{{x}−\sqrt{{x}}}}\:−\mathrm{3}\sqrt{\mathrm{2}} \\…
Question Number 8846 by Rasheed Soomro last updated on 31/Oct/16 $$\mathrm{Let}\:\mathrm{by}\:\left(\mathrm{a}_{\mathrm{1}} ,\mathrm{a}_{\mathrm{2}} ,…\mathrm{a}_{\mathrm{n}} \right)\:\mathrm{we}\:\mathrm{mean}\:\mathrm{LCM} \\ $$$$\mathrm{of}\:\:\mathrm{a}_{\mathrm{1}} ,\mathrm{a}_{\mathrm{2}} ,…\mathrm{a}_{\mathrm{n}} \:,\mathrm{where}\:\mathrm{a}_{\mathrm{i}} \in\mathbb{N}. \\ $$$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}\:\left(\:\left(\mathrm{a},\mathrm{b}\right),\left(\mathrm{b},\mathrm{c}\right)\:\:\right)=\left(\mathrm{a},\mathrm{b},\mathrm{c}\right). \\ $$ Answered…
Question Number 8845 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{figure}\:\mathrm{out}\:\mathrm{this}\: \\ $$$$\mathrm{Quantitative}\:\mathrm{reasoning} \\ $$$$\mathrm{How}\:\mathrm{did}\:\mathrm{they}\:\mathrm{get}\:\mathrm{this}\:\mathrm{answers}. \\ $$$$ \\ $$$$\mathrm{2},\mathrm{613},\mathrm{400}\:=\:\mathrm{3} \\ $$$$\mathrm{2},\mathrm{451},\mathrm{100}\:=\:\mathrm{1} \\ $$$$\mathrm{2},\mathrm{541},\mathrm{100}\:=\:\mathrm{2} \\ $$$$\mathrm{3},\mathrm{000},\mathrm{001}\:=\:\mathrm{1} \\…
Question Number 139918 by mathsuji last updated on 02/May/21 $$\mathrm{1}+{cos}\mathrm{2}°+{cos}\mathrm{4}°+{cos}\mathrm{6}°+…+{cos}\mathrm{88}° \\ $$ Answered by mr W last updated on 02/May/21 $$\underset{{k}=\mathrm{0}} {\overset{\mathrm{44}} {\sum}}\left(\mathrm{cos}\:\frac{\mathrm{2}{k}\pi}{\mathrm{180}}+{i}\:\mathrm{sin}\:\frac{\mathrm{2}{k}\pi}{\mathrm{180}}\right)=\underset{{k}=\mathrm{0}} {\overset{\mathrm{44}} {\sum}}{e}^{\frac{\mathrm{2}{k}\pi}{\mathrm{180}}{i}}…
Question Number 139912 by mnjuly1970 last updated on 02/May/21 $$ \\ $$$$\:\:\:\:\:{prove}\:{that}::\: \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\Omega}\::=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:={log}\left(\varphi\right) \\ $$$$\:\:\:\:\:\:\:\:\varphi:={golden}\:{ratio}\:… \\ $$$$ \\ $$ Answered by…
Question Number 8839 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Prove}\:\mathrm{that}.\: \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\mathrm{n}\right)^{\mathrm{1}/\mathrm{n}} \:=\:\mathrm{e} \\ $$ Answered by FilupSmith last updated on 31/Oct/16 $${L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:{n}\right)^{\mathrm{1}/{n}}…
Question Number 8838 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Show}\:\mathrm{that}\::\:\:\mathrm{e}^{\mathrm{i}\pi\:+\:\mathrm{1}} \:=\:\mathrm{0} \\ $$ Answered by FilupSmith last updated on 31/Oct/16 $$\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:{e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$$${e}^{{ix}} =\mathrm{cos}\left({x}\right)+{i}\mathrm{sin}\left({x}\right)…
Question Number 139911 by mathdanisur last updated on 02/May/21 $$\underset{\:−\mathrm{1}} {\overset{\:\mathrm{1}} {\int}}\frac{\mathrm{1}+\boldsymbol{{z}}^{\mathrm{2}} }{\mathrm{1}+\boldsymbol{{z}}^{\mathrm{4}} }\:{d}\boldsymbol{{z}}=? \\ $$ Answered by Dwaipayan Shikari last updated on 02/May/21 $$\int_{−\mathrm{1}}…
Question Number 74371 by liki last updated on 23/Nov/19 Commented by liki last updated on 23/Nov/19 $$….{plz}\:{help}\:{me}\:\mathrm{16}{a}\:,,{emergerce} \\ $$ Answered by MJS last updated on…