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Question Number 61297 Answers: 0 Comments: 0

A container is 50% full of water at triple point phase . It′s Isolated and subjected to space system defining no gravity acting on the particles , Which state of matter is now more dominant , solid , liquid , or gas ? Calculate the intermolecular distances between simultaneous two distinct states of water.

$$\mathrm{A}\:\mathrm{container}\:\mathrm{is}\:\mathrm{50\%}\:\mathrm{full}\:\mathrm{of}\:\mathrm{water}\:\mathrm{at}\:\mathrm{triple}\:\mathrm{point} \\ $$$$\mathrm{phase}\:.\:\mathrm{It}'\mathrm{s}\:\mathrm{Isolated}\:\mathrm{and}\:\mathrm{subjected}\:\mathrm{to}\:\mathrm{space} \\ $$$$\mathrm{system}\:\mathrm{defining}\:\mathrm{no}\:\mathrm{gravity}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{particles}\:,\:\mathrm{Which}\:\mathrm{state}\:\mathrm{of}\:\mathrm{matter}\:\mathrm{is}\:\mathrm{now} \\ $$$$\mathrm{more}\:\mathrm{dominant}\:,\:\mathrm{solid}\:,\:\mathrm{liquid}\:,\:\mathrm{or}\:\mathrm{gas}\:? \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{intermolecular}\:\mathrm{distances} \\ $$$$\mathrm{between}\:\mathrm{simultaneous}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{states} \\ $$$$\mathrm{of}\:\mathrm{water}.\: \\ $$

Question Number 61284 Answers: 0 Comments: 0

(998^(999) × 999^(998) × 2019^(2019) ) mod (1000) = ?

$$\left(\mathrm{998}^{\mathrm{999}} \:×\:\mathrm{999}^{\mathrm{998}} \:×\:\mathrm{2019}^{\mathrm{2019}} \right)\:\:{mod}\:\left(\mathrm{1000}\right)\:\:=\:\:? \\ $$

Question Number 61283 Answers: 1 Comments: 0

(x+y)(x^2 +y^2 )(x^3 + y^3 ) = 2 (x^4 +y^4 )(x^6 +y^6 )(x^8 + y^8 ) = 4 (x^3 + y^3 )(x^5 + y^5 )(x^7 + y^7 ) = 6 (x^4 + y^4 )(x^5 + y^5 )(x^9 + y^9 )(x^(10) + y^(10) ) = ?

$$\left({x}+{y}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\left({x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \right)\:\:=\:\:\mathrm{2} \\ $$$$\left({x}^{\mathrm{4}} +{y}^{\mathrm{4}} \right)\left({x}^{\mathrm{6}} +{y}^{\mathrm{6}} \right)\left({x}^{\mathrm{8}} \:+\:{y}^{\mathrm{8}} \right)\:\:=\:\:\mathrm{4} \\ $$$$\left({x}^{\mathrm{3}} +\:{y}^{\mathrm{3}} \right)\left({x}^{\mathrm{5}} +\:{y}^{\mathrm{5}} \right)\left({x}^{\mathrm{7}} \:+\:{y}^{\mathrm{7}} \right)\:\:=\:\:\mathrm{6} \\ $$$$\left({x}^{\mathrm{4}} \:+\:{y}^{\mathrm{4}} \right)\left({x}^{\mathrm{5}} \:+\:{y}^{\mathrm{5}} \right)\left({x}^{\mathrm{9}} \:+\:{y}^{\mathrm{9}} \right)\left({x}^{\mathrm{10}} \:+\:{y}^{\mathrm{10}} \right)\:\:=\:\:? \\ $$

Question Number 61274 Answers: 1 Comments: 0

If p , x_1 ,x_2 ,...x_i and q,y_1 ,y_2 ,...y_i form two infinite arithmetic sequences with common difference a and b respectively , then find the locus of the point ( α , β ) where α = (1/n) Σ_(i=1) ^n x_i and β= (1/n) Σ_(i=1) ^n y_(i .)

$${If}\:{p}\:,\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...{x}_{{i}} \:{and}\:{q},{y}_{\mathrm{1}} ,{y}_{\mathrm{2}} ,...{y}_{{i}} \:{form}\:{two}\: \\ $$$${infinite}\:{arithmetic}\:{sequences}\:{with}\:{common}\: \\ $$$${difference}\:\:{a}\:{and}\:{b}\:{respectively}\:, \\ $$$${then}\:{find}\:{the}\:{locus}\:{of}\:{the}\:{point}\:\left(\:\alpha\:,\:\beta\:\right)\: \\ $$$${where}\:\alpha\:=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {x}_{{i}} \:{and}\:\beta=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {y}_{{i}\:.} \\ $$

Question Number 61272 Answers: 0 Comments: 0

Question Number 61273 Answers: 2 Comments: 2

Suppose α ,β,γ,δ are real numbers such that α+β+γ+δ = α^7 +β^7 +γ^7 +δ^7 =0 Prove that α(α+β)(α+γ)(α+δ)=0

$${Suppose}\:\alpha\:,\beta,\gamma,\delta\:{are}\:{real}\:{numbers} \\ $$$${such}\:{that}\:\alpha+\beta+\gamma+\delta\:=\:\alpha^{\mathrm{7}} +\beta^{\mathrm{7}} +\gamma^{\mathrm{7}} +\delta^{\mathrm{7}} =\mathrm{0} \\ $$$${Prove}\:{that}\:\alpha\left(\alpha+\beta\right)\left(\alpha+\gamma\right)\left(\alpha+\delta\right)=\mathrm{0} \\ $$

Question Number 61270 Answers: 2 Comments: 0

Question Number 61269 Answers: 3 Comments: 0

Let p(x) be a quadratic polynomial such that for distinct α and β , p(α) = α and p(β) =β prove that α and β are roots of p[p(x)]−x=0 Find the remaining roots .

$${Let}\:{p}\left({x}\right)\:{be}\:{a}\:{quadratic}\:{polynomial}\:{such} \\ $$$${that}\:{for}\:{distinct}\:\alpha\:{and}\:\beta\:, \\ $$$${p}\left(\alpha\right)\:=\:\alpha\:{and}\:{p}\left(\beta\right)\:=\beta \\ $$$${prove}\:{that}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\:\:{p}\left[{p}\left({x}\right)\right]−{x}=\mathrm{0}\: \\ $$$${Find}\:{the}\:{remaining}\:{roots}\:. \\ $$

Question Number 61268 Answers: 0 Comments: 0

Let a,b,c,d,e ≥ −1 and a+b+c+d+e=5 Find the maximum and minimum value of S =(a+b)(b+c)(c+d)(d+e)(e+a)

$${Let}\:{a},{b},{c},{d},{e}\:\geqslant\:−\mathrm{1}\:{and}\:{a}+{b}+{c}+{d}+{e}=\mathrm{5} \\ $$$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${value}\:{of}\:{S}\:=\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{d}\right)\left({d}+{e}\right)\left({e}+{a}\right) \\ $$

Question Number 61267 Answers: 0 Comments: 0

Question Number 61261 Answers: 1 Comments: 0

for polynomial p(x),the value of p(3) is −2.which of the following must be true about p(x)? (a)x−5 is the factor of p(x) (b)x−2is the factor of p(x) (c)x+2 is the factor of p(x) (d)the reminder when p(x) is divide d by x−3 is −2

$${for}\:{polynomial}\:{p}\left({x}\right),{the}\:{value}\:{of}\: \\ $$$${p}\left(\mathrm{3}\right)\:{is}\:−\mathrm{2}.{which}\:{of}\:{the}\:{following}\: \\ $$$${must}\:{be}\:{true}\:{about}\:{p}\left({x}\right)? \\ $$$$\left({a}\right){x}−\mathrm{5}\:{is}\:{the}\:{factor}\:{of}\:{p}\left({x}\right) \\ $$$$\left({b}\right){x}−\mathrm{2}{is}\:{the}\:{factor}\:{of}\:{p}\left({x}\right) \\ $$$$\left({c}\right){x}+\mathrm{2}\:{is}\:{the}\:{factor}\:{of}\:{p}\left({x}\right) \\ $$$$\left({d}\right){the}\:{reminder}\:{when}\:\:{p}\left({x}\right)\:{is}\:{divide} \\ $$$${d}\:{by}\:{x}−\mathrm{3}\:{is}\:−\mathrm{2} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 61258 Answers: 1 Comments: 2

Calculate, using cartesian coodinates, the following integrals: 1) ∫∫_D dxdy being D={ (x,y)∈R^2 /0≤x≤(1/2),y+x≤1,y≥0} 2) ∫∫_D x^3 ydxdy being D={(x,y)∈R^2 /0≤x≤(1/2),y+x≤1,y≥0} 3) ∫∫_D (x/y)dxdy being D={(x,y)∈R^2 /xy≤16,x≥y,x−6≤y,x≥0,y≥1} Help please!

$$\boldsymbol{{C}}{alculate},\:{using}\:{cartesian}\:{coodinates},\:{the}\:{following} \\ $$$${integrals}: \\ $$$$ \\ $$$$\left.\mathrm{1}\right)\:\int\int_{{D}} {dxdy}\:\:{being}\:\:{D}=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{0}\leqslant{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}},{y}+{x}\leqslant\mathrm{1},{y}\geqslant\mathrm{0}\right\} \\ $$$$\left.\mathrm{2}\right)\:\int\int_{{D}} {x}^{\mathrm{3}} {ydxdy}\:\:{being}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{0}\leqslant{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}},{y}+{x}\leqslant\mathrm{1},{y}\geqslant\mathrm{0}\right\} \\ $$$$\left.\mathrm{3}\right)\:\int\int_{{D}} \frac{{x}}{{y}}{dxdy}\:\:{being}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{xy}\leqslant\mathrm{16},{x}\geqslant{y},{x}−\mathrm{6}\leqslant{y},{x}\geqslant\mathrm{0},{y}\geqslant\mathrm{1}\right\} \\ $$$$ \\ $$$${Help}\:\:{please}! \\ $$

Question Number 61241 Answers: 5 Comments: 0

Question Number 61240 Answers: 1 Comments: 3

∫ ((x^(2 ) − 4)/((x^2 + 4)^2 )) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{2}\:} −\:\mathrm{4}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 61237 Answers: 0 Comments: 0

Question Number 61235 Answers: 0 Comments: 6

Question Number 61232 Answers: 0 Comments: 3

let U_n =∫_1 ^(+∞) (([nx]−[(n−1)x])/x^3 ) dx with n≥1 1) find U_n interms of n 2) find lim_(n→+∞) U_n 3) study the serie Σ_(n=1) ^∞ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\left[{nx}\right]−\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\:\:{with}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 61229 Answers: 1 Comments: 0

let f_n (a) =∫_0 ^a x^n (√(a^2 −x^2 ))dx with a>0 1) determine a explicit form of f(a) 2) let g_n (a) =f^′ (a) give g_n (a) at form of integral and give its value 3) find the value of ∫_0 ^2 x^3 (√(4−x^2 ))dx and ∫_0 ^(√3) x^4 (√(3−x^2 ))dx

$${let}\:{f}_{{n}} \left({a}\right)\:=\int_{\mathrm{0}} ^{{a}} \:{x}^{{n}} \sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}_{{n}} \left({a}\right)\:={f}^{'} \left({a}\right)\:\:\:{give}\:{g}_{{n}} \left({a}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{give}\:{its} \\ $$$${value}\: \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \:{x}^{\mathrm{3}} \sqrt{\mathrm{4}−{x}^{\mathrm{2}} }{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} {x}^{\mathrm{4}} \sqrt{\mathrm{3}−{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 61215 Answers: 2 Comments: 0