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AlgebraQuestion and Answers: Page 32

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If the roots of the quadratic equation (a − b + c)x^2 + (c − b − a)x + 2(b − c) = 0 are real and equal then find (a/(b − c)) .

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation} \\ $$$$\left({a}\:−\:{b}\:+\:{c}\right){x}^{\mathrm{2}} \:+\:\left({c}\:−\:{b}\:−\:{a}\right){x}\:+\:\mathrm{2}\left({b}\:−\:{c}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{are}\:\mathrm{real}\:\mathrm{and}\:\mathrm{equal}\:\mathrm{then}\:\mathrm{find}\:\frac{{a}}{{b}\:−\:{c}}\:. \\ $$

Question Number 210324 Answers: 1 Comments: 2

rationalize the denominator: (1/(a+b^(1/3) +c^(1/3) ))

$$\mathrm{rationalize}\:\mathrm{the}\:\mathrm{denominator}: \\ $$$$\frac{\mathrm{1}}{{a}+{b}^{\mathrm{1}/\mathrm{3}} +{c}^{\mathrm{1}/\mathrm{3}} } \\ $$

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Let a_1 =1 a_2 =2^1 a_3 =3^((2^1 )) a_4 =4^((3^((2^1 )) )) find the last two digits of a_(23) and so on

$${Let}\:{a}_{\mathrm{1}} =\mathrm{1}\:\:\:{a}_{\mathrm{2}} =\mathrm{2}^{\mathrm{1}} \:\:\:\:{a}_{\mathrm{3}} =\mathrm{3}^{\left(\mathrm{2}^{\mathrm{1}} \right)} \:\:{a}_{\mathrm{4}} =\mathrm{4}^{\left(\mathrm{3}^{\left(\mathrm{2}^{\mathrm{1}} \right)} \right)} \\ $$$${find}\:{the}\:{last}\:{two}\:{digits}\:{of}\:{a}_{\mathrm{23}} \:{and}\:{so}\:{on} \\ $$

Question Number 210310 Answers: 2 Comments: 4

Let a be the unique real zero of x^3 +x+1. find the simplest possible way to write ((18)/((a^2 +a+1)^2 )) as polynomial expression in a with ratio coefficients

$${Let}\:{a}\:{be}\:{the}\:{unique}\:{real}\:{zero}\:{of}\:{x}^{\mathrm{3}} +{x}+\mathrm{1}. \\ $$$${find}\:{the}\:{simplest}\:{possible}\:{way}\:{to}\:{write}\: \\ $$$$\frac{\mathrm{18}}{\left({a}^{\mathrm{2}} +{a}+\mathrm{1}\right)^{\mathrm{2}} }\:\:{as}\:{polynomial}\:{expression}\:{in}\:\:{a} \\ $$$${with}\:{ratio}\:{coefficients} \\ $$

Question Number 210309 Answers: 0 Comments: 1

For what value of p does the series Σ_(n=1) ^∞ (e^n /((2+e^(2n) )^p )) converge

$${For}\:{what}\:{value}\:{of}\:{p}\:{does}\:{the}\:{series} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{e}^{{n}} }{\left(\mathrm{2}+{e}^{\mathrm{2}{n}} \right)^{{p}} }\:\:\:\:\:{converge} \\ $$

Question Number 210308 Answers: 3 Comments: 1

Evaluate ∫((2y^4 )/(y^3 −y^2 +y−1))dy

$${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{2}{y}^{\mathrm{4}} }{{y}^{\mathrm{3}} −{y}^{\mathrm{2}} +{y}−\mathrm{1}}{dy} \\ $$

Question Number 210307 Answers: 3 Comments: 0

∫((x^2 −1)/((x^2 +1)((√(1+x^4 )) )))

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }\:\right)} \\ $$$$ \\ $$

Question Number 210297 Answers: 0 Comments: 0

Prove the theorem. A non empty subset W of a vector space V(F) is the subset of V if and only if αW_1 +βW_2 ∈W ∀α,β ∈ F and W_1 ,W_2 ∈W

$${Prove}\:{the}\:{theorem}. \\ $$$${A}\:{non}\:{empty}\:{subset}\:{W}\:\:{of}\:{a}\:{vector}\:{space}\:{V}\left({F}\right) \\ $$$${is}\:{the}\:{subset}\:{of}\:{V}\:\:{if}\:\:{and}\:{only}\:{if} \\ $$$$\alpha{W}_{\mathrm{1}} +\beta{W}_{\mathrm{2}} \:\in{W}\:\:\forall\alpha,\beta\:\in\:{F}\:\:{and}\:{W}_{\mathrm{1}} ,{W}_{\mathrm{2}} \:\in{W} \\ $$

Question Number 210298 Answers: 0 Comments: 0

For the given system of simultaneous linear equation 2x_1 −2x_2 +3x_3 +4x_4 −x_5 =0 −x_3 −2x_4 +3x_5 =0 −x_1 +x_2 +2x_3 +5x_4 +2x_5 =0 x_1 −x_2 +2x_3 +3x_4 =0 (a)Write the augmented matrix and convert it into echelon form (b)Hence find all the solution

$${For}\:{the}\:{given}\:{system}\:{of}\:{simultaneous}\: \\ $$$${linear}\:{equation} \\ $$$$\mathrm{2}{x}_{\mathrm{1}} −\mathrm{2}{x}_{\mathrm{2}} +\mathrm{3}{x}_{\mathrm{3}} +\mathrm{4}{x}_{\mathrm{4}} −{x}_{\mathrm{5}} =\mathrm{0} \\ $$$$−{x}_{\mathrm{3}} −\mathrm{2}{x}_{\mathrm{4}} +\mathrm{3}{x}_{\mathrm{5}} =\mathrm{0} \\ $$$$−{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +\mathrm{2}{x}_{\mathrm{3}} +\mathrm{5}{x}_{\mathrm{4}} +\mathrm{2}{x}_{\mathrm{5}} =\mathrm{0} \\ $$$${x}_{\mathrm{1}} −{x}_{\mathrm{2}} +\mathrm{2}{x}_{\mathrm{3}} +\mathrm{3}{x}_{\mathrm{4}} =\mathrm{0} \\ $$$$\left({a}\right){Write}\:{the}\:{augmented}\:\:{matrix}\:{and}\:{convert} \\ $$$${it}\:{into}\:{echelon}\:{form} \\ $$$$\left({b}\right){Hence}\:{find}\:{all}\:{the}\:{solution} \\ $$$$ \\ $$

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