Question Number 13591 by FilupS last updated on 21/May/17 $$\mathrm{for}: \\ $$$${x}^{\mathrm{2}} +\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{it}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}: \\ $$$${r}=\mathrm{2sin}\left(\theta\right) \\ $$ Commented by FilupS last updated…
Question Number 144663 by mathdanisur last updated on 27/Jun/21 $${x}\in\left(\mathrm{0};\pi\right)\:{and}\:\left({a};{b}\right)\:{real}\:{numbers}\:{fixed}. \\ $$$${Find}\:{the}\:{range}\:{of}\:{function}: \\ $$$${g}\left({x}\right)=\:\frac{\left(\mathrm{1}+{a}^{\mathrm{2}} +{cot}^{\mathrm{2}} {x}\right)\centerdot\left(\mathrm{1}+{b}^{\mathrm{2}} +{cot}^{\mathrm{2}} {x}\right)}{\mathrm{1}\:+\:{cot}^{\mathrm{2}} {x}} \\ $$ Answered by mindispower last…
Question Number 144645 by mathdanisur last updated on 27/Jun/21 $$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left(\frac{\mathrm{1}\:+\:{tanx}}{\mathrm{1}\:+\:{sinx}}\right)^{\frac{\mathrm{1}}{\boldsymbol{{sinx}}}} =\:? \\ $$ Answered by liberty last updated on 27/Jun/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}\right)^{\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}}…
Question Number 144634 by loveineq last updated on 27/Jun/21 $$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:\left({a}+{b}\right)\left({b}+{c}\right)\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} +\mathrm{2}{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\frac{\mathrm{2}{b}\left({c}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{c}+{a}}\:\geqslant\:\mathrm{6} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} +\mathrm{3}{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\frac{\mathrm{3}{b}\left({c}^{\mathrm{3}} +{a}^{\mathrm{3}} \right)}{{c}+{a}}\:\geqslant\:\mathrm{8} \\…
Question Number 144629 by mathdanisur last updated on 27/Jun/21 $${if}\:\:{f}^{\mathrm{2}} \left(\mathrm{2}{x}-\mathrm{1}\right)-\mathrm{10}{f}\left(\mathrm{3}{x}-\mathrm{2}\right)+\mathrm{25}=\mathrm{0} \\ $$$${find}\:\:{f}\:^{'} \left(\mathrm{1}\right)+{f}\left(\mathrm{1}\right)=? \\ $$ Answered by liberty last updated on 27/Jun/21 $$\left[\mathrm{f}\left(\mathrm{2x}−\mathrm{1}\right)\right]^{\mathrm{2}} −\mathrm{10f}\left(\mathrm{3x}−\mathrm{2}\right)+\mathrm{25}=\mathrm{0}…
Question Number 79089 by TawaTawa last updated on 22/Jan/20 Commented by mr W last updated on 22/Jan/20 $${set}\:\mathrm{2}: \\ $$$${standard}\:{deviation}\:\sigma_{\mathrm{2}} =\mathrm{2}\sigma_{\mathrm{1}} =\mathrm{2}×\mathrm{3}.\mathrm{5}=\mathrm{7} \\ $$$${variance}\:\sigma_{\mathrm{2}} ^{\mathrm{2}}…
Question Number 79085 by key of knowledge last updated on 22/Jan/20 $$\alpha\:\mathrm{and}\:\beta\:\left(\mathrm{or}\:\mathrm{more}\right)\:\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)−\mathrm{k}=\mathrm{0} \\ $$$$\mathrm{and}\:\:\alpha\beta<\mathrm{0}\:\:\:\:\:\:\:\:\Rightarrow\mathrm{k}\in? \\ $$ Commented by mr W…
Question Number 144609 by mathdanisur last updated on 26/Jun/21 $${How}\:{many}\:{digits}\:{doest}\:{the}\:{number} \\ $$$$\mathrm{2021}^{\mathrm{2022}} \:\:{have}.? \\ $$ Answered by MJS_new last updated on 27/Jun/21 $$\mathrm{log}_{\mathrm{10}} \:\mathrm{2021}^{\mathrm{2022}} \:=\mathrm{2022log}_{\mathrm{10}}…
Question Number 144600 by mathdanisur last updated on 26/Jun/21 $${if}\:{x},{y},{z}>\mathrm{0}\:;\:{xy}+{yz}+{zx}=\mathrm{1}\:{prove}\:{that}: \\ $$$${xyz}\:+\:\sqrt[{\mathrm{3}}]{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)\left(\mathrm{1}+{y}^{\mathrm{3}} \right)\left(\mathrm{1}+{z}^{\mathrm{3}} \right)}\:\geqslant\:\mathrm{1} \\ $$ Answered by mindispower last updated on 27/Jun/21 $${x},{y},{z}>\mathrm{0}…
Question Number 144603 by loveineq last updated on 26/Jun/21 $$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:\left({a}+{b}\right)\left({b}+{c}\right)\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}{a}+{b}\right)\left({a}+{b}\right)+\left({b}+\mathrm{2}{c}\right)\left({b}+{c}\right)\:\geqslant\:\mathrm{8}+\frac{\mathrm{1}}{\mathrm{2}}\left({a}+\mathrm{2}{b}+{c}\right)\left({c}+{a}\right)\:\:\:\:\:\:\: \\ $$$$\mathrm{Determine}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}.\:\:\: \\ $$ Answered by mindispower last updated on 27/Jun/21 $$\left(\mathrm{2}{a}+{b}\right)\left({a}+{b}\right)+\left({b}+\mathrm{2}{c}\right)\left({b}+{c}\right) \\…