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AllQuestion and Answers: Page 723

Question Number 141528 Answers: 0 Comments: 0

Question Number 141526 Answers: 1 Comments: 0

Question Number 141525 Answers: 1 Comments: 0

Let <x_n > be a sequence defined by x_(n+1) = (1/k)(x_n +(k/x_n )) ∀ n ∈ N Show that <x_n > converges to (√(k/(k−1))) x_1 >0 , k>1

$$\mathrm{Let}\:<\mathrm{x}_{\mathrm{n}} >\:\mathrm{be}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\mathrm{x}_{\mathrm{n}+\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{k}}\left(\mathrm{x}_{\mathrm{n}} +\frac{\mathrm{k}}{\mathrm{x}_{\mathrm{n}} }\right)\:\forall\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{Show}\:\mathrm{that}\:<\mathrm{x}_{\mathrm{n}} >\:\mathrm{converges}\:\mathrm{to}\:\sqrt{\frac{\mathrm{k}}{\mathrm{k}−\mathrm{1}}} \\ $$$$\mathrm{x}_{\mathrm{1}} >\mathrm{0}\:,\:\mathrm{k}>\mathrm{1} \\ $$

Question Number 141521 Answers: 1 Comments: 0

{ ((((x+y))^(1/3) +(√(x−3y))=4)),((2x+3y=17)) :} Find x and y.

$$\begin{cases}{\sqrt[{\mathrm{3}}]{\boldsymbol{{x}}+\boldsymbol{{y}}}+\sqrt{\boldsymbol{{x}}−\mathrm{3}\boldsymbol{{y}}}=\mathrm{4}}\\{\mathrm{2}\boldsymbol{{x}}+\mathrm{3}\boldsymbol{{y}}=\mathrm{17}}\end{cases} \\ $$$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}. \\ $$

Question Number 141513 Answers: 0 Comments: 0

1. 27^((1/(log_2 3)) + log_(125) 0,2) =? 2. if, α = (π/(12)) find, cos^2 α + 2sinαcosα − sin^2 α=?

$$\mathrm{1}.\:\:\mathrm{27}^{\frac{\mathrm{1}}{{log}_{\mathrm{2}} \mathrm{3}}\:+\:{log}_{\mathrm{125}} \mathrm{0},\mathrm{2}} =? \\ $$$$\mathrm{2}.\:\:{if},\:\:\alpha\:=\:\frac{\pi}{\mathrm{12}} \\ $$$${find},\:\:{cos}^{\mathrm{2}} \alpha\:+\:\mathrm{2}{sin}\alpha{cos}\alpha\:−\:{sin}^{\mathrm{2}} \alpha=? \\ $$

Question Number 141512 Answers: 0 Comments: 0

Σ_(n=1 ) ^∞ ((sin((((2n−1)π)/6)))/((2n−1)^2 )) = a.G a = ? (G:= catalan constant)

$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{{sin}\left(\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} }\:=\:{a}.{G} \\ $$$$\:\:{a}\:=\:?\:\:\:\:\:\:\left({G}:=\:{catalan}\:{constant}\right) \\ $$$$ \\ $$

Question Number 141509 Answers: 1 Comments: 0

lim_(θ→0) ((sin^3 3θ − 3θ^3 )/(θ^3 + θ^4 ))

$$\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}^{\mathrm{3}} \mathrm{3}\theta\:\:\:−\:\:\:\mathrm{3}\theta^{\mathrm{3}} }{\theta^{\mathrm{3}} \:\:+\:\:\theta^{\mathrm{4}} } \\ $$

Question Number 141507 Answers: 3 Comments: 0

∫ (dx/(sin^4 x + cos^4 x)) dx

$$\int\:\frac{\mathrm{dx}}{\mathrm{sin}^{\mathrm{4}} \mathrm{x}\:\:\:+\:\:\:\mathrm{cos}^{\mathrm{4}} \mathrm{x}}\:\:\mathrm{dx} \\ $$

Question Number 141501 Answers: 0 Comments: 0

Question Number 141499 Answers: 0 Comments: 0

Question Number 141496 Answers: 3 Comments: 0

∫((√(2+9x^2 ))/x)dx SOS SOS HELP

$$\int\frac{\sqrt{\mathrm{2}+\mathrm{9}{x}^{\mathrm{2}} }}{{x}}{dx} \\ $$$${SOS}\:{SOS}\:{HELP} \\ $$

Question Number 141494 Answers: 0 Comments: 0

(((0 −1)),((0 0)) )!= (((1 γ)),((0 1)) ) γ=Euler Mascheroni Constant What is the physical representation of a Matrix factorial?

$$\begin{pmatrix}{\mathrm{0}\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\mathrm{0}}\end{pmatrix}!=\begin{pmatrix}{\mathrm{1}\:\:\gamma}\\{\mathrm{0}\:\:\mathrm{1}}\end{pmatrix}\:\:\:\:\:\:\gamma=\boldsymbol{{E}}{uler}\:{Mascheroni}\:{Constant} \\ $$$${What}\:{is}\:{the}\:{physical}\:{representation}\:{of}\:{a}\:{Matrix}\:{factorial}? \\ $$

Question Number 141475 Answers: 2 Comments: 0

Find the smallest value of (√(x^2 +y^2 )) among all values of x & y satisfying 3x−y = 20

$$\:{Find}\:{the}\:{smallest}\:{value}\:{of}\: \\ $$$$\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\:{among}\:{all}\:{values}\:{of} \\ $$$$\:{x}\:\&\:{y}\:{satisfying}\:\mathrm{3}{x}−{y}\:=\:\mathrm{20}\: \\ $$

Question Number 141469 Answers: 3 Comments: 0

simplify (log_3 11)(log_(11) 13)(log_(13) 15)(log_(15) 27)(log_(27) 81) please i need help

$$\mathrm{simplify} \\ $$$$\left(\mathrm{log}_{\mathrm{3}} \mathrm{11}\right)\left(\mathrm{log}_{\mathrm{11}} \mathrm{13}\right)\left(\mathrm{log}_{\mathrm{13}} \mathrm{15}\right)\left(\mathrm{log}_{\mathrm{15}} \mathrm{27}\right)\left(\mathrm{log}_{\mathrm{27}} \mathrm{81}\right) \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{help} \\ $$

Question Number 141489 Answers: 2 Comments: 0

Question Number 141463 Answers: 1 Comments: 2

lim_(x→∞) x^2 ( (((x^3 +x)/(x^3 +1)))^(1/(7 )) −cos (1/x))?

$$\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} \left(\:\sqrt[{\mathrm{7}\:\:}]{\frac{{x}^{\mathrm{3}} +{x}}{{x}^{\mathrm{3}} +\mathrm{1}}}\:−\mathrm{cos}\:\frac{\mathrm{1}}{{x}}\right)? \\ $$

Question Number 141457 Answers: 6 Comments: 0

Question Number 141454 Answers: 0 Comments: 0

Let a,b,c ≥ 0 and a+b+c = 4. Prove that ((a^2 +ab+b^2 )/((a+b+4)^2 ))+((b^2 +bc+c^2 )/((b+c+4)^2 ))+((c^2 +ca+a^2 )/((c+a+4)^2 )) ≤ (1/2)

$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} }{\left({a}+{b}+\mathrm{4}\right)^{\mathrm{2}} }+\frac{{b}^{\mathrm{2}} +{bc}+{c}^{\mathrm{2}} }{\left({b}+{c}+\mathrm{4}\right)^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} +{ca}+{a}^{\mathrm{2}} }{\left({c}+{a}+\mathrm{4}\right)^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 141453 Answers: 3 Comments: 1

Question Number 141452 Answers: 1 Comments: 1

Evaluate the integral ∫sin(dx)

$${Evaluate}\:{the}\:{integral}\:\:\:\int{sin}\left({dx}\right) \\ $$

Question Number 141446 Answers: 1 Comments: 0

Let P be the point ((a/2)(t^2 +(1/t^2 )),a(t−(1/t))) Find the locus of P, as t varies.

$$\mathrm{Let}\:\mathrm{P}\:\mathrm{be}\:\mathrm{the}\:\mathrm{point}\:\left(\frac{{a}}{\mathrm{2}}\left({t}^{\mathrm{2}} +\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right),{a}\left({t}−\frac{\mathrm{1}}{{t}}\right)\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{P},\:\mathrm{as}\:{t}\:\mathrm{varies}. \\ $$

Question Number 141444 Answers: 0 Comments: 0

Question Number 141492 Answers: 0 Comments: 1

Question Number 141437 Answers: 0 Comments: 0

Question Number 141435 Answers: 1 Comments: 0

Find all the arraangment of all the letters of the word SYLLABUSES such that each word contains the word BUS.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{arraangment}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{letters} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\mathrm{SYLLABUSES}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{each}\:\mathrm{word}\:\mathrm{contains}\:\mathrm{the}\:\mathrm{word}\:\mathrm{BUS}. \\ $$

Question Number 141431 Answers: 2 Comments: 0

2∫_0 ^( 1) ∫_0 ^( 2) (x+2y)^8 dxdy

$$\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{2}} \left({x}+\mathrm{2}{y}\right)^{\mathrm{8}} \:{dxdy} \\ $$

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