Question Number 117116 by mathocean1 last updated on 10/Oct/20 $${x}\:,\:{y}\:,{z}\:,\:{t}\:\in\:\mathbb{Z}. \\ $$$${x}\:{and}\:{y}\:{are}\:{x}\:{are}\:{respectively}\:{the} \\ $$$${divisor}\:{of}\:{y}\:{and}\:{t}. \\ $$$${Justify}\:{the}\:{existence}\:{of}\:{k}\:\in\:\mathbb{Z}\:{such} \\ $$$${that}\:{yt}={xzk}. \\ $$$${Deduct}\:{that}\:{x}^{{m}\:} {is}\:{a}\:{divisor}\:{of}\:{y}^{{m}} \\ $$$${where}\:{m}\:\in\:\mathbb{N}. \\ $$…
Question Number 117007 by TANMAY PANACEA last updated on 08/Oct/20 $${find}\:{unit}\:{digit}\:{of} \\ $$$$\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!+\mathrm{5}!+….+\mathrm{100}! \\ $$ Answered by mindispower last updated on 08/Oct/20 $${if}\:{n}\geqslant\mathrm{5}\:{n}!=\mathrm{0}\left(\mathrm{10}\right) \\ $$$$\mathrm{1}+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!=\mathrm{24}+\mathrm{6}+\mathrm{4}+\mathrm{1}=\mathrm{35}…
Question Number 116611 by gaminghanzo12 last updated on 05/Oct/20 Commented by Dwaipayan Shikari last updated on 05/Oct/20 $$\mathrm{Q}\:\mathrm{116516} \\ $$ Answered by TANMAY PANACEA last…
Question Number 51028 by gunawan last updated on 23/Dec/18 $${f}\left({z}\right)=\sqrt{{r}}\:{e}^{{i}\frac{\theta}{\mathrm{2}}} \\ $$$${f}'\left({z}\right)=…? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18 $${df}=\left(\frac{\partial{f}}{\partial{r}}\right)_{\theta} {dr}+\left(\frac{\partial{f}}{\partial\theta}\right)_{{r}} {d}\theta \\…
Question Number 50970 by peter frank last updated on 22/Dec/18 $${Given}\:\mathrm{3}−\mathrm{2}{i}\:{and}\:\mathrm{1}+{i} \\ $$$${are}\:{the}\:{two}\:{of}\:{roots}\:{of} \\ $$$${the}\:{equation} \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{3}} +{dx}+{e} \\ $$$${find}\:{the}\:{values}\:{of} \\ $$$${a},{b},{c},{d}\:{and}\:{e} \\…
Question Number 50890 by peter frank last updated on 21/Dec/18 $${Determine}\:{whether} \\ $$$${the}\:{following}\:\:{is}\:{true}\:{for}\:{all} \\ $$$${value}\:{of}\:{x} \\ $$$$\mathrm{0}\leqslant\frac{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\leqslant\frac{\mathrm{4}}{\mathrm{3}} \\ $$ Answered by MJS last…
Question Number 50849 by peter frank last updated on 21/Dec/18 $$\mathrm{solve}\:{for}\:{z}\:\:{in}\:{the}\:{form}\:\:{x}+{iy}\: \\ $$$${if}\:{tanz}=\mathrm{0}.\mathrm{5}\: \\ $$ Answered by ajfour last updated on 21/Dec/18 $${e}^{{iz}} \:=\:\mathrm{cos}\:{z}+{i}\mathrm{sin}\:{z} \\…
Question Number 116359 by bemath last updated on 03/Oct/20 $$\mathrm{If}\:\mathrm{0}\:<\:\theta\:<\:\frac{\pi}{\mathrm{4}}\:\mathrm{such}\:\mathrm{that}\:\mathrm{cosec}\:\theta−\mathrm{sec}\:\theta=\frac{\sqrt{\mathrm{13}}}{\mathrm{6}} \\ $$$$\mathrm{then}\:\mathrm{cot}\:\theta−\mathrm{tan}\:\theta\:\mathrm{equals}\:\mathrm{to}\:\_\_ \\ $$ Answered by bobhans last updated on 03/Oct/20 $$\Rightarrow\:\mathrm{let}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:=\:\mathrm{r}\:.\:\mathrm{Then}\:\left(\frac{\mathrm{1}}{\mathrm{sin}\:\theta}\:−\frac{\mathrm{1}}{\mathrm{cos}\:\theta}\right)^{\mathrm{2}} =\:\frac{\mathrm{13}}{\mathrm{36}} \\ $$$$\Rightarrow\:\frac{\left(\mathrm{cos}\:\theta−\mathrm{sin}\:\theta\right)^{\mathrm{2}}…
Question Number 50806 by Tawa1 last updated on 20/Dec/18 $$\mathrm{Determine}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{roots}\:\mathrm{of}\:\:−\:\mathrm{16}\:,\:\:\mathrm{giving}\:\mathrm{the}\:\mathrm{results}\:\mathrm{in}\:\mathrm{polar} \\ $$$$\mathrm{form}\:\mathrm{and}\:\mathrm{in}\:\mathrm{exponential}\:\mathrm{form} \\ $$$$\boldsymbol{\mathrm{Answers}}:\:\:\:\:\:\sqrt{\mathrm{2}}\:\left(\mathrm{1}\:+\:\boldsymbol{\mathrm{j}}\right)\:,\:\:\sqrt{\mathrm{2}}\:\left(−\:\mathrm{1}\:+\:\boldsymbol{\mathrm{j}}\right)\:,\:\:\:\:\:\sqrt{\mathrm{2}}\:\left(−\:\mathrm{1}\:−\:\boldsymbol{\mathrm{j}}\right),\:\:\:\:\sqrt{\mathrm{2}}\left(\mathrm{1}\:−\:\boldsymbol{\mathrm{j}}\right) \\ $$ Answered by mr W last updated on 20/Dec/18 $${x}={r}\left(\mathrm{cos}\:\theta+{j}\:\mathrm{sin}\:\theta\right)…
Question Number 181841 by mr W last updated on 01/Dec/22 $${what}\:{is}\:{larger},\:\sqrt{\mathrm{11}}+\sqrt{\mathrm{13}}\:{or}\:\mathrm{7}? \\ $$ Answered by hmr last updated on 01/Dec/22 $$ \\ $$$${assume}\:{that}:\: \\ $$$$\sqrt{\mathrm{11}}\:+\:\sqrt{\mathrm{13}\:}\:<\:\mathrm{7}…