Question Number 114758 by bobhans last updated on 21/Sep/20 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\mathrm{4}{n}^{\mathrm{2}} \left(\mathrm{10}{n}−\mathrm{6}\right)\left(\mathrm{10}{n}−\mathrm{4}\right)}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{10}{n}−\mathrm{1}\right)\left(\mathrm{10}{n}+\mathrm{1}\right)}\:=? \\ $$ Answered by Olaf last updated on 21/Sep/20 $${u}_{{n}} \:=\:\frac{\mathrm{4}{n}^{\mathrm{2}}…
Question Number 49220 by malwaan last updated on 04/Dec/18 $$\left(\mathrm{1}+\mathrm{x}−\mathrm{2x}^{\mathrm{2}} \:\right)^{\mathrm{8}} =? \\ $$ Commented by Abdo msup. last updated on 04/Dec/18 $${S}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{8}} \:{C}_{\mathrm{8}}…
Question Number 114739 by mr W last updated on 20/Sep/20 $${find}\:{the}\:{largest}\:{and}\:{smallest} \\ $$$${coefficient}\:{in}\:\left(\mathrm{4}+\mathrm{3}{x}\right)^{−\mathrm{5}} . \\ $$ Answered by mr W last updated on 20/Sep/20 $$\left(\mathrm{4}+\mathrm{3}{x}\right)^{−\mathrm{5}}…
Question Number 180272 by Shrinava last updated on 09/Nov/22 $$\mathrm{In}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{following}\:\mathrm{identity} \\ $$$$\mathrm{is}\:\mathrm{true}:\:\:\:\mathrm{6R}^{\mathrm{2}} \:=\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:\:\:,\:\:\:\mathrm{a}\:\neq\:\mathrm{c}. \\ $$$$\mathrm{Ptove}\:\mathrm{that}\:\mathrm{Euler}'\mathrm{s}\:\mathrm{line}\:\mathrm{is}\:\mathrm{antiparallel}\:\mathrm{to}\:\:\mathrm{AC}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 180273 by Shrinava last updated on 09/Nov/22 $$\mathrm{In}\:\mathrm{triangle}\:\:\mathrm{ABC}\:\:\mathrm{with}\:\mathrm{angles}\:\:\alpha\:,\:\beta\:,\:\gamma \\ $$$$\mathrm{correspondently}\:,\:\mathrm{Euler}'\mathrm{s}\:\mathrm{line}\:\mathrm{interescts} \\ $$$$\mathrm{BC}\:\:\mathrm{at}\:\mathrm{point}\:\:\mathrm{P}.\:\mathrm{Ite}'\mathrm{s}\:\mathrm{put}\:\:\delta\:\:\mathrm{is}\:\mathrm{angle} \\ $$$$\mathrm{between}\:\mathrm{Euler}'\mathrm{s}\:\mathrm{line}\:\mathrm{and}\:\:\mathrm{BC}\:\left(\angle\mathrm{BPH}\right). \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{true} \\ $$$$\mathrm{tan}\:\delta\:=\:\frac{\mathrm{2}\:\mathrm{cos}\:\beta\:\mathrm{cos}\:\gamma\:−\:\mathrm{cos}\:\alpha}{\mathrm{sin}\:\left(\beta\:−\:\gamma\right)} \\ $$ Terms of Service…
Question Number 180274 by Ar Brandon last updated on 09/Nov/22 $$\mathrm{Solve}\:\mathrm{in}\:\mathbb{C}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{z}^{\mathrm{4}} +\left(\mathrm{7}−{i}\right){z}^{\mathrm{3}} +\left(\mathrm{12}−\mathrm{15}{i}\right){z}^{\mathrm{2}} +\left(\mathrm{4}+\mathrm{4}{i}\right){z}+\mathrm{16}+\mathrm{192}{i}=\mathrm{0} \\ $$$$\mathrm{Knowing}\:\mathrm{that}\:\mathrm{it}\:\mathrm{has}\:\mathrm{one}\:\mathrm{real}\:\mathrm{root}\:\mathrm{and}\:\mathrm{a}\:\mathrm{purely}\:\mathrm{imaginary}\:\mathrm{root} \\ $$$$\mathrm{of}\:\mathrm{equal}\:\mathrm{magnitude}. \\ $$ Answered by Frix…
Question Number 49202 by rahul 19 last updated on 04/Dec/18 $$\left.\mathrm{1}\right)\:{If}\:\omega\:{is}\:{an}\:{imaginary}\:{fifth}\:{root}\:{of} \\ $$$${unity},\:{then}\:{find}\:{value}\:{of}\: \\ $$$$\mathrm{log}\:_{\mathrm{2}} \:\mid\mathrm{1}+\omega+\omega^{\mathrm{2}} +\omega^{\mathrm{3}} −\frac{\mathrm{1}}{\omega}\mid\:? \\ $$$$\left.\mathrm{2}\right)\:{Find}\:{value}\:{of}\:: \\ $$$$\left({i}+\sqrt{\mathrm{3}}\right)^{\mathrm{100}} +\left({i}−\sqrt{\mathrm{3}}\right)^{\mathrm{100}} +\mathrm{2}^{\mathrm{100}} \:?…
Question Number 180268 by Noorzai last updated on 09/Nov/22 Answered by Ar Brandon last updated on 09/Nov/22 $$\mathrm{Q121680} \\ $$ Answered by LEKOUMA last updated…
Question Number 49200 by rahul 19 last updated on 04/Dec/18 $$\left.\mathrm{1}\right){Find}\:{the}\:{area}\:{of}\:{the}\:{triangle}\:{formed} \\ $$$${by}\:{roots}\:{of}\:{cubic}\:{equation} \\ $$$$\left({z}+\alpha{b}\right)^{\mathrm{3}} =\alpha^{\mathrm{3}_{} } \:\:\left(\alpha\neq\mathrm{0}\right). \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{Find}\:{product}\:{of}\:{all}\:{possible}\:{values} \\ $$$${of}\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}{i}}{\mathrm{2}}\right)^{\frac{\mathrm{3}}{\mathrm{4}}} \:.…
Question Number 49188 by behi83417@gmail.com last updated on 04/Dec/18 $${solve}\:{for}\:{x},{y},{z}\in{R}. \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{yz}}=\mathrm{1} \\ $$$$\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{xz}}=\mathrm{2} \\ $$$$\boldsymbol{\mathrm{z}}^{\mathrm{2}} +\boldsymbol{\mathrm{xy}}=\mathrm{3} \\ $$ Commented by tanmay.chaudhury50@gmail.com last…