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AlgebraQuestion and Answers: Page 295

Question Number 74742 Answers: 2 Comments: 1

If α and β are the roots of x^2 − x + 1 = 0, Find α^(23) + β^(23) without demoivre′s theorem.

$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:+\:\mathrm{1}\:\:\:=\:\:\mathrm{0},\:\: \\ $$$$\mathrm{Find}\:\:\:\:\:\:\:\:\:\alpha^{\mathrm{23}} \:+\:\beta^{\mathrm{23}} \:\:\:\:\:\mathrm{without}\:\mathrm{demoivre}'\mathrm{s}\:\mathrm{theorem}. \\ $$

Question Number 74716 Answers: 1 Comments: 0

Question Number 74712 Answers: 0 Comments: 2

Question Number 74663 Answers: 1 Comments: 0

If x^x y^y z^z = c show that at x = y = z (∂^2 z/(∂x∂y)) = − (x log ex)^(−1)

$$\mathrm{If}\:\:\:\:\mathrm{x}^{\mathrm{x}} \:\mathrm{y}^{\mathrm{y}} \:\mathrm{z}^{\mathrm{z}} \:\:\:=\:\:\:\mathrm{c}\:\:\:\:\:\:\:\mathrm{show}\:\mathrm{that}\:\mathrm{at}\:\:\:\:\:\mathrm{x}\:\:=\:\:\mathrm{y}\:\:=\:\:\mathrm{z} \\ $$$$\:\:\:\:\:\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}\partial\mathrm{y}}\:\:\:=\:\:\:−\:\left(\mathrm{x}\:\mathrm{log}\:\mathrm{ex}\right)^{−\mathrm{1}} \\ $$

Question Number 74604 Answers: 0 Comments: 3

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Question Number 74599 Answers: 0 Comments: 0

Hello,verry Nice day let U_n =E((((3+(√(17)))/2))^n ),n∈N^∗ show that U_n ≡n(2)

$$\mathrm{Hello},\mathrm{verry}\:\mathrm{Nice}\:\mathrm{day}\: \\ $$$$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\mathrm{E}\left(\left(\frac{\mathrm{3}+\sqrt{\mathrm{17}}}{\mathrm{2}}\right)^{\mathrm{n}} \right),\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{U}_{\mathrm{n}} \equiv\mathrm{n}\left(\mathrm{2}\right) \\ $$

Question Number 74615 Answers: 2 Comments: 0

Question Number 74614 Answers: 1 Comments: 0

Question Number 74611 Answers: 1 Comments: 0

Question Number 74609 Answers: 0 Comments: 1

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Question Number 74554 Answers: 1 Comments: 0

Find the superimum of the set {(n^2 /2^n )}

$$\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{superimum}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{set}}\:\left\{\frac{\boldsymbol{{n}}^{\mathrm{2}} }{\mathrm{2}^{\boldsymbol{{n}}} }\right\} \\ $$

Question Number 74412 Answers: 0 Comments: 0

a,b,c are given real constants. p,q,r,t are unknowns from which we can choose values of two of them (non zero) and have to determine the other two (non zero), obeying two equations given below p^2 +q(1+bq)t^2 +q(ap+br)t +r(1+bq)t+r(ap+br) = 0 pt+q(a+cq)t^2 +r(a+cq)t +cqrt+cr^2 = 0 Can this be done solving a quadratic eq. and none higher..?

$${a},{b},{c}\:{are}\:{given}\:{real}\:{constants}. \\ $$$${p},{q},{r},{t}\:{are}\:{unknowns}\:{from}\:{which} \\ $$$${we}\:{can}\:{choose}\:{values}\:{of}\:{two}\:{of} \\ $$$${them}\:\left({non}\:{zero}\right)\:{and}\:{have}\:{to}\: \\ $$$${determine}\:{the}\:{other}\:{two}\:\left({non}\:{zero}\right), \\ $$$${obeying}\:\:{two}\:{equations}\:{given}\:{below} \\ $$$$\:{p}^{\mathrm{2}} +{q}\left(\mathrm{1}+{bq}\right){t}^{\mathrm{2}} +{q}\left({ap}+{br}\right){t} \\ $$$$\:\:\:\:\:+{r}\left(\mathrm{1}+{bq}\right){t}+{r}\left({ap}+{br}\right)\:=\:\mathrm{0} \\ $$$${pt}+{q}\left({a}+{cq}\right){t}^{\mathrm{2}} +{r}\left({a}+{cq}\right){t} \\ $$$$\:\:\:\:+{cqrt}+{cr}^{\mathrm{2}} =\:\mathrm{0} \\ $$$${Can}\:{this}\:{be}\:{done}\:{solving}\:{a} \\ $$$${quadratic}\:{eq}.\:{and}\:{none}\:{higher}..? \\ $$

Question Number 74329 Answers: 1 Comments: 1

Solve : ax+by=r bx−ay=s

$${Solve}\:: \\ $$$${ax}+{by}={r} \\ $$$${bx}−{ay}={s} \\ $$

Question Number 74266 Answers: 0 Comments: 0

please kindly help me with the solutions to these question?very urgent please (1) if dy=x^3 dx. find the equation of y in terms of x if the curve passes through (1,1) 2) Given that the volume v(t) of cell at a time t changes according to ((dV(t))/dt)= sin t, with v(t)=4. find v(t) 3) Given (dP/dt) + 3P = 0. determine P (t) if p(0)= 4 4) Radium decomposes at a rate proportion to the amount present. if the half−life of the radium is 1000 years. what is the percentage lost in 100 years? 5) Calculate the pressure of a gas after phase transition at 171°K from 101.3kPa pressure at 472°K taking R = 0.1886kJ\kgK and L = 35.73kg

$${please}\:{kindly}\:{help}\:{me}\:{with}\:{the}\:{solutions}\:{to}\:{these}\:{question}?{very}\:{urgent}\:{please}\:\left(\mathrm{1}\right)\:{if}\:{dy}={x}^{\mathrm{3}} {dx}.\:{find}\:{the}\:{equation}\:{of}\:{y}\:{in}\:{terms}\:{of}\:{x} \\ $$$$\:{if}\:{the}\:{curve}\:{passes}\:{through}\:\left(\mathrm{1},\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{Given}\:{that}\:{the}\:{volume}\:{v}\left({t}\right)\:{of}\:{cell}\:{at}\:{a}\:{time}\:{t}\:{changes}\:{according}\:{to} \\ $$$$\frac{{dV}\left({t}\right)}{{dt}}=\:{sin}\:{t},\:{with}\:{v}\left({t}\right)=\mathrm{4}.\:{find}\:{v}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{Given}\:\frac{{dP}}{{dt}}\:+\:\mathrm{3}{P}\:=\:\mathrm{0}.\:{determine}\:{P}\:\left({t}\right)\: \\ $$$${if}\:{p}\left(\mathrm{0}\right)=\:\mathrm{4} \\ $$$$\left.\mathrm{4}\right)\:{Radium}\:{decomposes}\:{at}\:{a}\:{rate}\:{proportion}\:{to}\:{the}\:{amount}\: \\ $$$${present}.\:{if}\:{the}\:{half}−{life}\:{of}\:{the}\:{radium}\:{is} \\ $$$$\mathrm{1000}\:{years}.\:{what}\:{is}\:{the}\:{percentage}\:{lost}\:{in}\:\mathrm{100}\:{years}? \\ $$$$\left.\mathrm{5}\right)\:{Calculate}\:{the}\:{pressure}\:{of}\:{a}\:{gas}\:{after}\:{phase} \\ $$$${transition}\:{at}\:\mathrm{171}°{K}\:{from}\:\mathrm{101}.\mathrm{3}{kPa} \\ $$$${pressure}\:{at}\:\mathrm{472}°{K}\:{taking}\:{R}\:=\:\mathrm{0}.\mathrm{1886}{kJ}\backslash{kgK}\:{and}\:{L}\:=\:\mathrm{35}.\mathrm{73}{kg} \\ $$

Question Number 74213 Answers: 2 Comments: 1

Question Number 74209 Answers: 1 Comments: 0

Question Number 74196 Answers: 0 Comments: 1

solve for x: 4^x +6^x =9^x

$${solve}\:{for}\:{x}: \\ $$$$\mathrm{4}^{{x}} +\mathrm{6}^{{x}} =\mathrm{9}^{{x}} \\ $$

Question Number 74164 Answers: 2 Comments: 0

factorize 6x^2 −11xy −10y^2

$${factorize} \\ $$$$\mathrm{6}{x}^{\mathrm{2}} \:−\mathrm{11}{xy}\:−\mathrm{10}{y}^{\mathrm{2}} \\ $$

Question Number 74168 Answers: 3 Comments: 1

Question Number 74148 Answers: 1 Comments: 0

{ ((−x(√3)+2my(√2)=((√3)/3))),((2mx−3y(√6)=1)) :} help me solve it.

$$\begin{cases}{−{x}\sqrt{\mathrm{3}}+\mathrm{2}{my}\sqrt{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}}\\{\mathrm{2}{mx}−\mathrm{3}{y}\sqrt{\mathrm{6}}=\mathrm{1}}\end{cases}\:\:\:\:\:\: \\ $$$$ \\ $$$${help}\:{me}\:{solve}\:{it}. \\ $$

Question Number 74129 Answers: 1 Comments: 0

2C_4 ^n = 35C_3 ^(n/2) ⇒ n = ?

$$\mathrm{2}\boldsymbol{{C}}_{\mathrm{4}} ^{\boldsymbol{{n}}} \:=\:\mathrm{35}\boldsymbol{{C}}_{\mathrm{3}} ^{\frac{\boldsymbol{{n}}}{\mathrm{2}}} \: \\ $$$$\Rightarrow\:\boldsymbol{{n}}\:=\:? \\ $$

Question Number 74121 Answers: 0 Comments: 1

Factor the polynomial ((c/2))x^2 +(b−((3c)/2))x+(c−b+a)

$$\mathrm{Factor}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\left(\frac{{c}}{\mathrm{2}}\right){x}^{\mathrm{2}} +\left({b}−\frac{\mathrm{3}{c}}{\mathrm{2}}\right){x}+\left({c}−{b}+{a}\right) \\ $$

Question Number 74111 Answers: 1 Comments: 1

Question Number 74109 Answers: 1 Comments: 3

Question Number 74063 Answers: 0 Comments: 0

Question Number 74024 Answers: 1 Comments: 1

{ ((h^2 +y^2 +(k−z)^2 =s^2 )),((a^2 +(b−y)^2 +z^2 =s^2 )),((ah+y(y−b)+z(z−k)=0)),((((h+a)/2)+yz−(b−y)(k−z)=1)),((b+a(k−z)+hz=1)),((k+h(b−y)+ay=1)) :} Find s_(min) or at least express s=f(y) or g(z).

$$\begin{cases}{{h}^{\mathrm{2}} +{y}^{\mathrm{2}} +\left({k}−{z}\right)^{\mathrm{2}} ={s}^{\mathrm{2}} }\\{{a}^{\mathrm{2}} +\left({b}−{y}\right)^{\mathrm{2}} +{z}^{\mathrm{2}} ={s}^{\mathrm{2}} }\\{{ah}+{y}\left({y}−{b}\right)+{z}\left({z}−{k}\right)=\mathrm{0}}\\{\frac{{h}+{a}}{\mathrm{2}}+{yz}−\left({b}−{y}\right)\left({k}−{z}\right)=\mathrm{1}}\\{{b}+{a}\left({k}−{z}\right)+{hz}=\mathrm{1}}\\{{k}+{h}\left({b}−{y}\right)+{ay}=\mathrm{1}}\end{cases} \\ $$$${Find}\:\:{s}_{{min}} \:{or}\:{at}\:{least}\:{express} \\ $$$$\:{s}={f}\left({y}\right)\:{or}\:{g}\left({z}\right). \\ $$

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