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Question Number 138789 Answers: 0 Comments: 0

Question Number 138783 Answers: 0 Comments: 4

Solve for real numbees: { ((a^(16) +1=((a+b)/2))),((b^(16) +1=((b+c)/2))),((c^(16) +1=((c+x)/2))) :}

$${Solve}\:{for}\:{real}\:{numbees}: \\ $$$$\begin{cases}{{a}^{\mathrm{16}} +\mathrm{1}=\frac{{a}+{b}}{\mathrm{2}}}\\{{b}^{\mathrm{16}} +\mathrm{1}=\frac{{b}+{c}}{\mathrm{2}}}\\{{c}^{\mathrm{16}} +\mathrm{1}=\frac{{c}+{x}}{\mathrm{2}}}\end{cases} \\ $$

Question Number 138774 Answers: 1 Comments: 1

Question Number 138771 Answers: 2 Comments: 0

∫_0 ^(π/2) ln (1−tan^2 x+tan^4 x)dx=?

$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {ln}\:\left(\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{tan}\:^{\mathrm{4}} {x}\right){dx}=? \\ $$

Question Number 138767 Answers: 1 Comments: 0

lim_(x→0) ((tan (πcos^2 x))/(sin (2πsin^2 x))) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left(\pi\mathrm{cos}\:^{\mathrm{2}} {x}\right)}{\mathrm{sin}\:\left(\mathrm{2}\pi\mathrm{sin}\:^{\mathrm{2}} {x}\right)}\:=? \\ $$

Question Number 138766 Answers: 3 Comments: 1

If α and β are the roots of the equation 3x^2 +x+2=0, find the equation whose roots are (1/α^2 ) and (1/β^2 ) and show that 27α^4 =11α+10.

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} +{x}+\mathrm{2}=\mathrm{0},\:\mathrm{find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{whose} \\ $$$$\mathrm{roots}\:\mathrm{are}\:\:\frac{\mathrm{1}}{\alpha^{\mathrm{2}} }\:\:\mathrm{and}\:\:\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:\:\mathrm{and}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{27}\alpha^{\mathrm{4}} =\mathrm{11}\alpha+\mathrm{10}. \\ $$

Question Number 138765 Answers: 2 Comments: 1

Question Number 138764 Answers: 2 Comments: 0

Find the impedence of an RC circuit with R= 10Ω and C=10μF at an angular frequency of 21800rads^(−1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{impedence}\:\mathrm{of}\:\mathrm{an}\:\mathrm{RC}\:\mathrm{circuit} \\ $$$$\mathrm{with}\:\mathrm{R}=\:\mathrm{10}\Omega\:\mathrm{and}\:\mathrm{C}=\mathrm{10}\mu\mathrm{F}\:\mathrm{at}\:\mathrm{an}\: \\ $$$$\mathrm{angular}\:\mathrm{frequency}\:\mathrm{of}\:\mathrm{21800rads}^{−\mathrm{1}} \\ $$

Question Number 138763 Answers: 0 Comments: 0

Make arbitrary movements between the 7 people sitting around the table so that at least one of the people sitting next to you is not a Mathematician..

$${Make}\:{arbitrary}\:{movements}\:{between} \\ $$$${the}\:\mathrm{7}\:{people}\:{sitting}\:{around}\:{the}\:{table} \\ $$$${so}\:{that}\:{at}\:{least}\:{one}\:{of}\:{the}\:{people}\:{sitting} \\ $$$${next}\:{to}\:{you}\:{is}\:{not}\:{a}\:{Mathematician}.. \\ $$

Question Number 138758 Answers: 1 Comments: 0

Question Number 138756 Answers: 1 Comments: 0

show that 0 ≤ (∫_0 ^( 2) ((((2−x)^n e^x )/(n!))dx)) ≤ ((2^n (e^2 −1))/(n!))

$${show}\:{that} \\ $$$$\mathrm{0}\:\leqslant\:\left(\int_{\mathrm{0}} ^{\:\mathrm{2}} \left(\frac{\left(\mathrm{2}−{x}\right)^{{n}} {e}^{{x}} }{{n}!}{dx}\right)\right)\:\leqslant\:\frac{\mathrm{2}^{{n}} \left({e}^{\mathrm{2}} −\mathrm{1}\right)}{{n}!} \\ $$

Question Number 138748 Answers: 1 Comments: 0

A committee of 3 members is to be formed from 8 members. Find the number of committees that can be formed if two particular club members cannot both be in a committee

$$\mathrm{A}\:\mathrm{committee}\:\mathrm{of}\:\mathrm{3}\:\mathrm{members}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{8}\:\mathrm{members}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{committees}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{if}\:\mathrm{two}\:\mathrm{particular} \\ $$$$\mathrm{club}\:\mathrm{members}\:\mathrm{cannot}\:\mathrm{both}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\:\mathrm{committee} \\ $$

Question Number 138742 Answers: 2 Comments: 0

........ nice .... calculus... evaluate :: Ω=∫_0 ^( 1) ((x^(e^π −1) −x^(e^γ −1) )/(ln((x)^(1/3) )))dx=^? 3(π−γ)

$$\:\:\:\:\:\:\:\:\:\:\:........\:{nice}\:\:\:....\:\:\:{calculus}... \\ $$$$\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{{e}^{\pi} −\mathrm{1}} −{x}^{{e}^{\gamma} −\mathrm{1}} }{{ln}\left(\sqrt[{\mathrm{3}}]{{x}}\:\right)}{dx}\overset{?} {=}\mathrm{3}\left(\pi−\gamma\right) \\ $$

Question Number 138735 Answers: 2 Comments: 0

Question Number 138734 Answers: 0 Comments: 1

Question Number 138733 Answers: 0 Comments: 0

.....mathematical ....analysis..... suppose f :[a , b]→R is a function and α:[a , b]→^(α↗) R (α is an increasing function on [a , b]) meanwhile α is continuous at y_0 where a≤y_0 ≤b . defining f(x)= { (( 1 x=y_0 )),(( 0 x≠y_0 )) :} prove that : f∈ R (α) .... Hint: f∈R (α) ⇔ ∀ ε>0 ∃ P_ε ; U(P_ε ,f,α)−L(P_ε ,f,α)<ε Reimann criterion ....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:....{analysis}..... \\ $$$$\:\:{suppose}\:\:\:\:{f}\::\left[{a}\:,\:{b}\right]\rightarrow\mathbb{R}\:{is}\:{a}\:{function} \\ $$$$\:\:\:{and}\:\:\:\alpha:\left[{a}\:,\:{b}\right]\overset{\alpha\nearrow} {\rightarrow}\mathbb{R}\:\left(\alpha\:{is}\:{an}\:{increasing}\:{function}\right. \\ $$$$\left.\:{on}\:\left[{a}\:,\:{b}\right]\right)\:\:{meanwhile}\:\alpha\:{is}\:{continuous}\:{at}\:{y}_{\mathrm{0}} \: \\ $$$$\:\:{where}\:\:\:{a}\leqslant{y}_{\mathrm{0}} \leqslant{b}\:\:.\:{defining}\: \\ $$$$\:\:\:{f}\left({x}\right)=\begin{cases}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{x}={y}_{\mathrm{0}} }\\{\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:{x}\neq{y}_{\mathrm{0}} }\end{cases} \\ $$$$\:\:\:\:{prove}\:\:{that}\::\:{f}\in\:\mathscr{R}\:\left(\alpha\right)\:.... \\ $$$$\:\:\:\:{Hint}:\:{f}\in\mathscr{R}\:\left(\alpha\right)\:\Leftrightarrow\:\forall\:\epsilon>\mathrm{0}\:\exists\:{P}_{\epsilon} \:;\:{U}\left({P}_{\epsilon} ,{f},\alpha\right)−{L}\left({P}_{\epsilon} ,{f},\alpha\right)<\epsilon \\ $$$$\:\:\:\:{Reimann}\:\:{criterion}\:.... \\ $$

Question Number 138730 Answers: 0 Comments: 0

Question Number 138728 Answers: 0 Comments: 0

∫_0 ^∞ ((ln(1+cos x))/(1+e^x ))dx=0

$$\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}{\mathrm{1}+{e}^{{x}} }{dx}=\mathrm{0} \\ $$

Question Number 138723 Answers: 0 Comments: 2

........advanced... ... ...math...... prove that _∗^∗ :::: 𝛀=Σ_(k=0) ^∞ {(1/(16^k ))((4/(8k+1))−(2/(8k+4))−(1/(8k+5))−(1/(8k+6)))}=π ....Bailey−Borwein formula....

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:........{advanced}...\:...\:...{math}...... \\ $$$$\:{prove}\:{that}\:_{\ast} ^{\ast} \:\::::: \\ $$$$\:\:\:\boldsymbol{\Omega}=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{\mathrm{16}^{{k}} }\left(\frac{\mathrm{4}}{\mathrm{8}{k}+\mathrm{1}}−\frac{\mathrm{2}}{\mathrm{8}{k}+\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{8}{k}+\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{8}{k}+\mathrm{6}}\right)\right\}=\pi \\ $$$$\:\:\:\:\:\:\:\:\:....{Bailey}−{Borwein}\:{formula}.... \\ $$$$\:\:\: \\ $$

Question Number 138725 Answers: 1 Comments: 0

Solve for real numbers ((sin(sinx))/(sinx)) + ((cos(cosx))/(cosx)) = 1

$${Solve}\:{for}\:{real}\:{numbers} \\ $$$$\frac{{sin}\left({sinx}\right)}{{sinx}}\:+\:\frac{{cos}\left({cosx}\right)}{{cosx}}\:=\:\mathrm{1} \\ $$

Question Number 138716 Answers: 1 Comments: 0

∫_2 ^∞ (1/(x^2 lnx))dx converges or diverges?

$$\int_{\mathrm{2}} ^{\infty} \frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \mathrm{lnx}}\mathrm{dx}\: \\ $$$$\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges}? \\ $$

Question Number 138818 Answers: 0 Comments: 0