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AlgebraQuestion and Answers: Page 238

Question Number 116221 Answers: 3 Comments: 0

Given α,β and ϕ are the roots of x^3 −px^2 +qx−pq = 0 . Find the value of (α/β)+(β/α)+(β/ϕ)+(ϕ/β)+(α/ϕ)+(ϕ/α)=?

$$\mathrm{Given}\:\alpha,\beta\:\mathrm{and}\:\varphi\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{pq}\:=\:\mathrm{0}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\varphi}+\frac{\varphi}{\beta}+\frac{\alpha}{\varphi}+\frac{\varphi}{\alpha}=? \\ $$

Question Number 116192 Answers: 2 Comments: 1

Question Number 116173 Answers: 0 Comments: 2

How many ways can the letters of the word DENKENMATHEMATICAL be arranged if no same letters must be together

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\:\mathrm{DENKENMATHEMATICAL} \\ $$$$\mathrm{be}\:\mathrm{arranged}\:\mathrm{if}\:\mathrm{no}\:\mathrm{same}\:\mathrm{letters}\:\mathrm{must}\:\mathrm{be}\:\mathrm{together} \\ $$

Question Number 116163 Answers: 1 Comments: 0

I need a general rule for factorising 1) a^n +b^n 2) a^n −b^n when n is even and when n is odd.

$$\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{general}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{factorising} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} \\ $$$$\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}. \\ $$

Question Number 116138 Answers: 3 Comments: 1

If α and β are the roots of the quadratic equation x^2 −10x+2=0 and α >β, find: (i) (1/β)−(1/α) (ii)α^3 −β^3

$$\mathrm{If}\:\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{2}=\mathrm{0}\:\mathrm{and}\:\alpha\:>\beta,\:\mathrm{find}: \\ $$$$\left(\mathrm{i}\right)\:\frac{\mathrm{1}}{\beta}−\frac{\mathrm{1}}{\alpha} \\ $$$$\left(\mathrm{ii}\right)\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} \\ $$

Question Number 116226 Answers: 2 Comments: 1

(1/(2+(√2))) +(1/(3(√2)+2(√3))) +(1/(4(√3)+3(√4)))+...+(1/(100(√(99))+99(√(100))))

$$\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}}\:+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{4}}}+...+\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{99}}+\mathrm{99}\sqrt{\mathrm{100}}} \\ $$

Question Number 116092 Answers: 1 Comments: 2

Solve for x and y if x^y =36

$${Solve}\:{for}\:{x}\:{and}\:{y}\:{if} \\ $$$${x}^{{y}} =\mathrm{36} \\ $$

Question Number 116091 Answers: 2 Comments: 0

Is there a formular to tell how many times a digit occur in an interval. e.g. How many times digits 2 occur between 1 − 100

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{formular}\:\mathrm{to}\:\mathrm{tell}\:\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{a}\:\mathrm{digit}\:\mathrm{occur}\:\mathrm{in}\:\mathrm{an}\:\mathrm{interval}. \\ $$$$ \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{digits}\:\:\mathrm{2}\:\:\mathrm{occur}\:\mathrm{between}\:\:\mathrm{1}\:−\:\mathrm{100} \\ $$

Question Number 116059 Answers: 2 Comments: 0

Σ_(n=1) ^∞ ((n!)/3^(n+1) )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}!}{\mathrm{3}^{{n}+\mathrm{1}} } \\ $$

Question Number 116057 Answers: 2 Comments: 0

Σ_(n=2) ^∞ (3/(3n+1))=?

$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{3}}{\mathrm{3}{n}+\mathrm{1}}=? \\ $$

Question Number 116056 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (5^n /(n!))=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{5}^{{n}} }{{n}!}=? \\ $$

Question Number 115986 Answers: 1 Comments: 0

Question Number 115916 Answers: 1 Comments: 0

Question Number 115909 Answers: 1 Comments: 0

solve the system of equations x+((3x−y)/(x^2 +y^2 ))=3 , y−((x+3y)/(x^2 +y^2 ))=0

$${solve}\:{the}\:{system}\:{of}\:{equations} \\ $$$${x}+\frac{\mathrm{3}{x}−{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{3}\:,\:{y}−\frac{{x}+\mathrm{3}{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{0} \\ $$

Question Number 115908 Answers: 1 Comments: 2

what is the cofficient of x^2 (1+x)(1+2x)(1+4x)......(1+2^n x)

$${what}\:{is}\:{the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{4}{x}\right)......\left(\mathrm{1}+\mathrm{2}^{{n}} {x}\right) \\ $$

Question Number 115906 Answers: 1 Comments: 0

find all pairs of integers (x,y) such that x^3 +y^3 =(x+y)^2

$${find}\:{all}\:{pairs}\:{of}\:{integers}\:\left({x},{y}\right)\:{such}\:{that} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\left({x}+{y}\right)^{\mathrm{2}} \\ $$

Question Number 115897 Answers: 2 Comments: 1

Given a_n = (√(1 + (1 − (1/n))^2 )) + (√(1 + (1 + (1/n))^2 )) The value of Σ_(n=1) ^(2015) ((4/a_n )) is ...

$$\mathrm{Given} \\ $$$${a}_{{n}} \:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{2015}} {\sum}}\left(\frac{\mathrm{4}}{{a}_{{n}} }\right)\:\mathrm{is}\:... \\ $$

Question Number 115858 Answers: 2 Comments: 0

What are all ordered pairs of real number (x,y) for which 5^(y−x) (x+y) = 1 and (x+y)^(x−y) = 5

$${What}\:{are}\:{all}\:{ordered}\:{pairs}\:{of}\:{real} \\ $$$${number}\:\left({x},{y}\right)\:{for}\:{which}\: \\ $$$$\mathrm{5}^{{y}−{x}} \:\left({x}+{y}\right)\:=\:\mathrm{1}\:{and}\:\left({x}+{y}\right)^{{x}−{y}} \:=\:\mathrm{5} \\ $$

Question Number 115857 Answers: 1 Comments: 0

What are all real values of p for which the inequality −3<((x^2 +px−2)/(x^2 −x+1))<2 is satisfied by all real values of x

$${What}\:{are}\:{all}\:{real}\:{values}\:{of}\:{p}\:{for} \\ $$$${which}\:{the}\:{inequality}\: \\ $$$$−\mathrm{3}<\frac{{x}^{\mathrm{2}} +{px}−\mathrm{2}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}<\mathrm{2}\:{is}\:{satisfied}\: \\ $$$${by}\:{all}\:{real}\:{values}\:{of}\:{x} \\ $$

Question Number 115871 Answers: 1 Comments: 1

Question Number 115815 Answers: 1 Comments: 0

Montrer que ∀(a,b,c)∈(R_+ ^∗ )^3 (1/(a^2 +bc))+(1/(b^2 +ac))+(1/(c^2 +ab))≤(1/2)((1/(ab))+(1/(bc))+(1/(ac)))

$$\mathrm{Montrer}\:\mathrm{que}\:\:\forall\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left(\mathbb{R}_{+} ^{\ast} \right)^{\mathrm{3}} \\ $$$$\frac{\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{ac}}+\frac{\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{ab}}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{ab}}+\frac{\mathrm{1}}{\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{ac}}\right) \\ $$

Question Number 115773 Answers: 0 Comments: 0

((1−x)/( (√(x^2 −2x+5)))) = (3/5)

$$\:\frac{\mathrm{1}−{x}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}}\:=\:\frac{\mathrm{3}}{\mathrm{5}} \\ $$

Question Number 115564 Answers: 1 Comments: 0

Question Number 115544 Answers: 1 Comments: 1

Given x^2 +12(√x) = 5 then x+2(√x) ?

$${Given}\:{x}^{\mathrm{2}} +\mathrm{12}\sqrt{{x}}\:=\:\mathrm{5} \\ $$$${then}\:{x}+\mathrm{2}\sqrt{{x}}\:? \\ $$

Question Number 115484 Answers: 1 Comments: 3

find the value of ((111)/(1+1+1)) , ((222)/(2+2+2)) , ((333)/(3+3+3)), ((444)/(4+4+4)) ((555)/(5+5+5)), ((666)/(6+6+6)) , ((777)/(7+7+7)) , ((888)/(8+8+8)) ((999)/(9+9+9))

$${find}\:{the}\:{value}\:{of}\: \\ $$$$\frac{\mathrm{111}}{\mathrm{1}+\mathrm{1}+\mathrm{1}}\:,\:\frac{\mathrm{222}}{\mathrm{2}+\mathrm{2}+\mathrm{2}}\:,\:\frac{\mathrm{333}}{\mathrm{3}+\mathrm{3}+\mathrm{3}},\:\frac{\mathrm{444}}{\mathrm{4}+\mathrm{4}+\mathrm{4}} \\ $$$$\frac{\mathrm{555}}{\mathrm{5}+\mathrm{5}+\mathrm{5}},\:\frac{\mathrm{666}}{\mathrm{6}+\mathrm{6}+\mathrm{6}}\:,\:\frac{\mathrm{777}}{\mathrm{7}+\mathrm{7}+\mathrm{7}}\:,\:\frac{\mathrm{888}}{\mathrm{8}+\mathrm{8}+\mathrm{8}} \\ $$$$\frac{\mathrm{999}}{\mathrm{9}+\mathrm{9}+\mathrm{9}} \\ $$

Question Number 115436 Answers: 1 Comments: 0

(√((a)^(1/3) +(b)^(1/3) ))=(1/( (√(b−a×s^3 ))))(((−s^2 ×(a^2 )^(1/3) )/2)+s((ab))^(1/3) +(b^2 )^(1/3) ) what is s?

$$\sqrt{\sqrt[{\mathrm{3}}]{{a}}+\sqrt[{\mathrm{3}}]{{b}}}=\frac{\mathrm{1}}{\:\sqrt{{b}−{a}×{s}^{\mathrm{3}} }}\left(\frac{−{s}^{\mathrm{2}} ×\sqrt[{\mathrm{3}}]{{a}^{\mathrm{2}} }}{\mathrm{2}}+{s}\sqrt[{\mathrm{3}}]{{ab}}+\sqrt[{\mathrm{3}}]{{b}^{\mathrm{2}} }\right) \\ $$$${what}\:{is}\:{s}? \\ $$

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