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Question Number 201864    Answers: 0   Comments: 2

(ct^2 βˆ’(1/(ct^2 ))+(√2))^2 =c^2 (t^4 +(1/t^4 )) Find t=f(c).

$$\left({ct}^{\mathrm{2}} βˆ’\frac{\mathrm{1}}{{ct}^{\mathrm{2}} }+\sqrt{\mathrm{2}}\right)^{\mathrm{2}} ={c}^{\mathrm{2}} \left({t}^{\mathrm{4}} +\frac{\mathrm{1}}{{t}^{\mathrm{4}} }\right) \\ $$$${Find}\:\:{t}={f}\left({c}\right). \\ $$

Question Number 201860    Answers: 1   Comments: 0

𝚺_(n=1) ^∞ (((βˆ’1)^n H_n )/(n+1))=??

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\sum}}}\frac{\left(βˆ’\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}+\mathrm{1}}=?? \\ $$

Question Number 201859    Answers: 0   Comments: 0

If xyz ∈R^+ , xyz=1 , prove that the following inequality holds: (x/(2x^5 +x+4))+(y/(2y^5 +y+4))+(z/(2z^5 +z+4))β‰₯(3/7). Solution please with an advice to get better at inequalities and which book would you recommend. Thanks in advance!

$$\mathrm{If}\:{xyz}\:\in\mathbb{R}^{+} \:,\:{xyz}=\mathrm{1}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inequality}\:\mathrm{holds}: \\ $$$$\frac{{x}}{\mathrm{2}{x}^{\mathrm{5}} +{x}+\mathrm{4}}+\frac{{y}}{\mathrm{2}{y}^{\mathrm{5}} +{y}+\mathrm{4}}+\frac{{z}}{\mathrm{2}{z}^{\mathrm{5}} +{z}+\mathrm{4}}\geqslant\frac{\mathrm{3}}{\mathrm{7}}. \\ $$$$\boldsymbol{\mathrm{Solution}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{advice}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{better}} \\ $$$$\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{inequalities}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{book}}\:\boldsymbol{\mathrm{would}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{recommend}}. \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{advance}}! \\ $$$$\: \\ $$

Question Number 201857    Answers: 1   Comments: 1

If a,b,c,d,e are thr roots of 2x^5 βˆ’3x^3 +2xβˆ’7=0 , find the value of Ξ _(cyc) (a^3 βˆ’1)

$$\:\:\mathrm{If}\:{a},{b},{c},{d},{e}\:\mathrm{are}\:\mathrm{thr}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{2x}^{\mathrm{5}} βˆ’\mathrm{3x}^{\mathrm{3}} +\mathrm{2x}βˆ’\mathrm{7}=\mathrm{0}\:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{cyc}} {\prod}\left({a}^{\mathrm{3}} βˆ’\mathrm{1}\right)\: \\ $$

Question Number 201854    Answers: 0   Comments: 3

Question Number 201852    Answers: 0   Comments: 0

Question Number 201853    Answers: 1   Comments: 0

Question Number 201850    Answers: 0   Comments: 0

Question Number 201897    Answers: 0   Comments: 0

Question: The congruence equation β€²β€² ( 5a +3 )x ≑^(3a + 4) 19 β€²β€² is given. Find the sum of digits of the smallest three βˆ’digit natural number ” a ” such that the assumed equation has no solution in ” Z ”.

$$ \\ $$$$\:\:\:\:\:\mathrm{Q}{uestion}:\:\:{The}\:{congruence} \\ $$$$\:\:\:\:\:\:\:{equation}\:\:\:''\:\:\:\:\left(\:\mathrm{5}{a}\:+\mathrm{3}\:\right){x}\:\:\overset{\mathrm{3}{a}\:+\:\mathrm{4}} {\equiv}\:\mathrm{19}\:\:\:''\:\:{is}\:{given}. \\ $$$$\:\:\:\:\:\:\:{Find}\:{the}\:{sum}\:{of}\:{digits}\:{of}\:\: \\ $$$$\:\:\:\:\:\:\:{the}\:{smallest}\:\:{three}\:βˆ’{digit}\:{natural}\:{number}\:\:''\:{a}\:''\: \\ $$$$\:\:\:\:\:\:\:{such}\:{that}\:{the}\:{assumed}\:{equation}\:{has}\: \\ $$$$\:\:\:\:\:\:\:\:\:{no}\:\:{solution}\:\:{in}\:\:\:''\:\:\:\mathbb{Z}\:\:''. \\ $$$$ \\ $$

Question Number 201848    Answers: 1   Comments: 0

Question Number 201846    Answers: 0   Comments: 1

$$\:\: \\ $$$$ \\ $$

Question Number 201839    Answers: 0   Comments: 1

Question Number 201837    Answers: 0   Comments: 0

yβ€²β€²β€²βˆ’yβ€²β€²+yβ€²=sec(t),βˆ’(Ο€/2)<t<(Ο€/2)

$${y}'''βˆ’{y}''+{y}'={sec}\left({t}\right),βˆ’\frac{\pi}{\mathrm{2}}<{t}<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 201982    Answers: 0   Comments: 0

Question Number 201977    Answers: 0   Comments: 1

Question Number 201980    Answers: 1   Comments: 0

Question Number 201979    Answers: 2   Comments: 1

Question Number 201972    Answers: 1   Comments: 0

Question Number 201969    Answers: 2   Comments: 0

A dice is cast twice, and the sum of the appearing numbers is 10. The probability that the number 5 has appeared at least once is.

$$\mathrm{A}\:\mathrm{dice}\:\mathrm{is}\:\mathrm{cast}\:\mathrm{twice},\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{appearing}\:\mathrm{numbers}\:\mathrm{is}\:\mathrm{10}. \\ $$$$\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\mathrm{5}\:\mathrm{has}\: \\ $$$$\mathrm{appeared}\:\mathrm{at}\:\mathrm{least}\:\mathrm{once}\:\mathrm{is}. \\ $$

Question Number 201966    Answers: 2   Comments: 0

solve ((x^2 +3x+2))^(1/3) (((x+1))^(1/3) βˆ’((x+2))^(1/3) )= 1

$$ \\ $$$$\:\:\:\:\:\:\:{solve}\: \\ $$$$\:\: \\ $$$$\:\:\:\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}}\:\left(\sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}\:βˆ’\sqrt[{\mathrm{3}}]{{x}+\mathrm{2}}\:\right)=\:\mathrm{1} \\ $$$$ \\ $$

Question Number 201965    Answers: 1   Comments: 0

mβˆ’h=2p p(mβˆ’h)=kβˆ’q mkβˆ’qh=(1/3) kβˆ’2q=1+ph Assume one find the rest! βœ“

$${m}βˆ’{h}=\mathrm{2}{p} \\ $$$${p}\left({m}βˆ’{h}\right)={k}βˆ’{q} \\ $$$${mk}βˆ’{qh}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${k}βˆ’\mathrm{2}{q}=\mathrm{1}+{ph} \\ $$$${Assume}\:{one}\:{find}\:{the}\:{rest}! \\ $$$$\checkmark \\ $$

Question Number 201832    Answers: 1   Comments: 0

Question Number 201831    Answers: 0   Comments: 0

Question Number 201829    Answers: 2   Comments: 0

shortest distance from (βˆ’6,0)to x^2 βˆ’y^2 +16=0

$${shortest}\:{distance}\:{from}\:\left(βˆ’\mathrm{6},\mathrm{0}\right){to}\:{x}^{\mathrm{2}} βˆ’{y}^{\mathrm{2}} +\mathrm{16}=\mathrm{0} \\ $$

Question Number 201822    Answers: 0   Comments: 4

Do Not Use sin(ΞΈ)∼θ (ΞΈ is small Enough) ΞΈ^ +(g/β„“)sin(ΞΈ)=0 yβ€²β€²(t)+(g/β„“) sin(y(t))=0 yβ€²β€²(t)yβ€²(t)+(g/β„“)sin(y(t))yβ€²(t)=0 yβ€²(t)yβ€²β€²(t)=(1/2)βˆ™((d )/dt)(yβ€²(t))^2 (g/β„“)sin(y(t))yβ€²(t)=βˆ’(g/β„“)βˆ™((d )/dt)cos(y(t)) ∴ ((d )/dt)[(1/2)(yβ€²(t))^2 βˆ’(g/β„“)cos(y(t))]=0 ∴(1/2)(yβ€²(t))^2 βˆ’(g/β„“)cos(y(t))=c_1 Const yβ€²β€²(t)+(g/β„“)sin(y(t))=0β†’(yβ€²(t))^2 βˆ’((2g)/β„“)cos(y(t))=c_1 and... I canβ€²t Sove Diff Equa..

$$\mathrm{Do}\:\mathrm{Not}\:\mathrm{Use}\:\mathrm{sin}\left(\theta\right)\sim\theta\:\left(\theta\:\:\mathrm{is}\:\mathrm{small}\:\mathrm{Enough}\right) \\ $$$$\ddot {\theta}+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left(\theta\right)=\mathrm{0} \\ $$$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\:\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0} \\ $$$${y}''\left({t}\right){y}'\left({t}\right)+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=\mathrm{0} \\ $$$${y}'\left({t}\right){y}''\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} \\ $$$$\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=βˆ’\frac{\mathrm{g}}{\ell}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\mathrm{cos}\left({y}\left({t}\right)\right) \\ $$$$\therefore\:\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left[\frac{\mathrm{1}}{\mathrm{2}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} βˆ’\frac{\mathrm{g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)\right]=\mathrm{0} \\ $$$$\therefore\frac{\mathrm{1}}{\mathrm{2}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} βˆ’\frac{\mathrm{g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)={c}_{\mathrm{1}} \:\:\boldsymbol{\mathrm{Const}} \\ $$$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0}\rightarrow\left({y}'\left({t}\right)\right)^{\mathrm{2}} βˆ’\frac{\mathrm{2g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)={c}_{\mathrm{1}} \\ $$$$\mathrm{and}...\:\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{Sove}\:\mathrm{Diff}\:\:\mathrm{Equa}.. \\ $$

Question Number 201820    Answers: 1   Comments: 0

((∣3x+1βˆ£βˆ’βˆ£x+2∣)/(3βˆ’βˆ£2x∣)) β‰₯ 0 find the solution set.

$$\:\:\:\frac{\mid\mathrm{3x}+\mathrm{1}\midβˆ’\mid\mathrm{x}+\mathrm{2}\mid}{\mathrm{3}βˆ’\mid\mathrm{2x}\mid}\:\geqslant\:\mathrm{0}\: \\ $$$$\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}. \\ $$

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