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AllQuestion and Answers: Page 204

Question Number 201906 Answers: 2 Comments: 0

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Question Number 201864 Answers: 0 Comments: 2

(ct^2 −(1/(ct^2 ))+(√2))^2 =c^2 (t^4 +(1/t^4 )) Find t=f(c).

$$\left({ct}^{\mathrm{2}} −\frac{\mathrm{1}}{{ct}^{\mathrm{2}} }+\sqrt{\mathrm{2}}\right)^{\mathrm{2}} ={c}^{\mathrm{2}} \left({t}^{\mathrm{4}} +\frac{\mathrm{1}}{{t}^{\mathrm{4}} }\right) \\ $$$${Find}\:\:{t}={f}\left({c}\right). \\ $$

Question Number 201860 Answers: 1 Comments: 0

𝚺_(n=1) ^∞ (((−1)^n H_n )/(n+1))=??

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

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))≥(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 −3x^3 +2x−7=0 , find the value of Π_(cyc) (a^3 −1)

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

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

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

Question Number 201848 Answers: 1 Comments: 0

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

Question Number 201839 Answers: 0 Comments: 1

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y′′′−y′′+y′=sec(t),−(π/2)<t<(π/2)

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

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