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Question Number 63689 Answers: 0 Comments: 3

Show that if a∣b then an∣bn

$${Show}\:{that}\:\:{if}\:\:{a}\mid{b}\:\:{then}\:{an}\mid{bn} \\ $$

Question Number 63684 Answers: 0 Comments: 0

cot 118

$$\mathrm{cot}\:\mathrm{118} \\ $$

Question Number 63552 Answers: 1 Comments: 1

Calculate ∫_0 ^(1/2) x(√(x^2 +1)) dx+∫_(1/2) ^1 x^2 (√(x^3 +1)) dx+∫_1 ^2 x^3 (√(x^4 +1)) dx+∫_2 ^3 x^4 (√(x^5 +1 ))dx+...+∫_(78) ^(79) x^(80) (√(x^(81) +1)) dx+∫_(79) ^(80) x^(81) (√(x^(82) +1)) dx usingΣ_(n=2) ^(80) ∫_(n−1) ^n x^(n+1) (√(x^(n+2) +1))dx

$${Calculate}\:\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\mathrm{2}}} {\int}}{x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}+\underset{\frac{\mathrm{1}}{\mathrm{2}}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}+\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\:{dx}+\underset{\mathrm{2}} {\overset{\mathrm{3}} {\int}}{x}^{\mathrm{4}} \sqrt{{x}^{\mathrm{5}} +\mathrm{1}\:}{dx}+...+\underset{\mathrm{78}} {\overset{\mathrm{79}} {\int}}{x}^{\mathrm{80}} \sqrt{{x}^{\mathrm{81}} +\mathrm{1}}\:{dx}+\underset{\mathrm{79}} {\overset{\mathrm{80}} {\int}}{x}^{\mathrm{81}} \sqrt{{x}^{\mathrm{82}} +\mathrm{1}}\:{dx} \\ $$$${using}\underset{{n}=\mathrm{2}} {\overset{\mathrm{80}} {\sum}}\underset{{n}−\mathrm{1}} {\overset{{n}} {\int}}{x}^{{n}+\mathrm{1}} \sqrt{{x}^{{n}+\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 63534 Answers: 1 Comments: 0

find the set of values of x for which y is real if y=(((x−2)(x−1))/(x+2)) , x≠−2, x∈R

$${find}\:{the}\:{set}\:{of}\:{values}\:{of}\:{x}\:{for}\:{which}\:{y}\:{is}\:{real}\:{if}\: \\ $$$$\:{y}=\frac{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{1}\right)}{{x}+\mathrm{2}}\:,\:{x}\neq−\mathrm{2},\:{x}\in\mathbb{R} \\ $$

Question Number 63532 Answers: 1 Comments: 0

prove that there exist unique intergers p and s sucb that a = bp + s with −((∣b∣)/2)< s ≤((∣b∣)/2) hence find p and s given that a=49 and b=26

$${prove}\:{that}\:{there}\:{exist}\:{unique}\:{intergers}\:{p}\:{and}\:{s}\:{sucb}\:{that} \\ $$$${a}\:=\:{bp}\:+\:{s}\:{with}\:−\frac{\mid{b}\mid}{\mathrm{2}}<\:{s}\:\leqslant\frac{\mid{b}\mid}{\mathrm{2}} \\ $$$${hence}\:{find}\:{p}\:{and}\:{s}\:{given}\:{that}\:{a}=\mathrm{49}\:{and}\:{b}=\mathrm{26} \\ $$

Question Number 63517 Answers: 1 Comments: 0

Given that ∣z−6∣=2∣z+6−9i∣, a) Use algebra to show that the locus of z is a circle, stating its center and its radius. b) sketch the locus z on an argand diagram.

$$\mathrm{Given}\:\mathrm{that}\:\:\mid{z}−\mathrm{6}\mid=\mathrm{2}\mid{z}+\mathrm{6}−\mathrm{9}{i}\mid, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{Use}\:\mathrm{algebra}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}, \\ $$$$\mathrm{stating}\:\mathrm{its}\:\mathrm{center}\:\mathrm{and}\:\mathrm{its}\:\mathrm{radius}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{locus}\:{z}\:\mathrm{on}\:\mathrm{an}\:\mathrm{argand}\:\mathrm{diagram}. \\ $$

Question Number 63507 Answers: 1 Comments: 0

let U_n =∫_(1/n) ^1 ((√(x^2 +x+1)) −(√(x^2 −x+1)))dx (n>0) 1)calculate lim_(n→+∞) U_n 2) find nature of Σ U_n

$${let}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\right){dx}\:\:\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{nature}\:{of}\:\:\Sigma\:{U}_{{n}} \\ $$

Question Number 63499 Answers: 1 Comments: 0

given that a∣b, show that −a∣b.

$${given}\:{that}\:\:\:{a}\mid{b},\:{show}\:{that}\:−{a}\mid{b}. \\ $$

Question Number 63447 Answers: 1 Comments: 2

How to calculate ⌈(n) using gamma function ∀n∈R

$${How}\:{to}\:{calculate}\:\lceil\left({n}\right)\: \\ $$$${using}\:{gamma}\:{function} \\ $$$$\forall{n}\in{R} \\ $$

Question Number 63428 Answers: 0 Comments: 2

It is given that S_n =Σ_(r=1) ^n (3r^(2 ) −3r−1). Use the the formulae of Σ_(r=1) ^n r^(2 ) and Σ_(r=1) ^n r to show that S_n =n^3 . sir Forkum Michael

$${It}\:{is}\:{given}\:{that}\:{S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}^{\mathrm{2}\:} −\mathrm{3}{r}−\mathrm{1}\right).\:{Use}\:{the}\:{the}\:{formulae} \\ $$$${of}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}\:\:} {and}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\:\:{to}\:{show}\:{that}\:{S}_{{n}} ={n}^{\mathrm{3}} . \\ $$$${sir}\:{Forkum}\:{Michael} \\ $$

Question Number 63425 Answers: 0 Comments: 0

The probability that a vaccinated person(V) contracts a disease is (1/(20)). For a person vaccinated(V ′) , the probability of contracting a disease is (5/6). In a certain town 90%of thepopulation has been vaccinated against a disease. A person is selected at random from the town,find the probability that: (a) he has the disease, (b) he is vaccinated or he has the disease. sir Forkum Michael

$${The}\:{probability}\:{that}\:{a}\:{vaccinated}\:{person}\left({V}\right)\:{contracts}\:{a}\:{disease} \\ $$$${is}\:\frac{\mathrm{1}}{\mathrm{20}}.\:{For}\:{a}\:{person}\:{vaccinated}\left({V}\:'\right)\:,\:{the}\:{probability}\:{of}\:{contracting} \\ $$$${a}\:{disease}\:{is}\:\frac{\mathrm{5}}{\mathrm{6}}.\:{In}\:{a}\:{certain}\:{town}\:\mathrm{90\%}{of}\:{thepopulation}\:{has} \\ $$$${been}\:{vaccinated}\:{against}\:{a}\:{disease}.\:{A}\:{person}\:{is}\:{selected}\:{at} \\ $$$${random}\:{from}\:{the}\:{town},{find}\:{the}\:{probability}\:{that}: \\ $$$$\left({a}\right)\:{he}\:{has}\:{the}\:{disease}, \\ $$$$\left({b}\right)\:{he}\:{is}\:{vaccinated}\:{or}\:{he}\:{has}\:{the}\:{disease}. \\ $$$${sir}\:{Forkum}\:{Michael} \\ $$

Question Number 63424 Answers: 0 Comments: 0

A colony of bacteria if left undisturbed will grow at a rate proportional to the number of bacteria, P present at time,t. However,a toxic substance is being added slowly such that at time t, the bacteria also die at the rate μPt where μ is a positive constant. (a) Show that at time t the rate of growth of the bacteria in the colony is governed by the differential equation (dP/dt)= (k−μt)p where k is apositive constant. when t=0, (dP/dt)=2P and when t=1, (dP/dt)=((19)/(10))P (b) show that (dP/dt)= (1/(10))(20−t)P. Sir Forkum Michael.

$${A}\:{colony}\:{of}\:{bacteria}\:{if}\:{left}\:{undisturbed}\:{will}\:{grow}\:{at}\:{a}\:{rate} \\ $$$${proportional}\:{to}\:{the}\:{number}\:{of}\:{bacteria},\:{P}\:{present}\:{at}\:{time},{t}. \\ $$$${However},{a}\:{toxic}\:{substance}\:{is}\:{being}\:{added}\:{slowly}\:{such}\:{that} \\ $$$${at}\:{time}\:{t},\:{the}\:{bacteria}\:{also}\:{die}\:{at}\:{the}\:{rate}\:\mu{Pt}\:{where}\:\mu\:{is} \\ $$$${a}\:{positive}\:{constant}. \\ $$$$\left({a}\right)\:\:{Show}\:{that}\:{at}\:{time}\:{t}\:{the}\:{rate}\:{of}\:{growth}\:{of}\:{the}\:{bacteria}\:{in} \\ $$$${the}\:{colony}\:{is}\:{governed}\:{by}\:{the}\:{differential}\:{equation} \\ $$$$\:\frac{{dP}}{{dt}}=\:\left({k}−\mu{t}\right){p}\:{where}\:{k}\:{is}\:{apositive}\:{constant}. \\ $$$${when}\:{t}=\mathrm{0},\:\frac{{dP}}{{dt}}=\mathrm{2}{P}\:{and}\:{when}\:{t}=\mathrm{1},\:\frac{{dP}}{{dt}}=\frac{\mathrm{19}}{\mathrm{10}}{P} \\ $$$$\left({b}\right)\:{show}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dP}}{{dt}}=\:\frac{\mathrm{1}}{\mathrm{10}}\left(\mathrm{20}−{t}\right){P}. \\ $$$$\:{Sir}\:{Forkum}\:{Michael}. \\ $$

Question Number 63296 Answers: 1 Comments: 1

A random Variable Y has probability function P, defined by P(y) = { (((y^2 /k) , y= 1,2,3)),(((((y−7)^2 )/k) , y= 4,5,6)),((0 , otherwise.)) :} Find (i) The value of the constant k. (ii) the mean and varriance of Y. (iii) The variance R, where R= 2Y −3.

$${A}\:{random}\:{Variable}\:{Y}\:{has}\:{probability}\:{function}\:{P},\:{defined}\:{by} \\ $$$$\:{P}\left({y}\right)\:=\:\begin{cases}{\frac{{y}^{\mathrm{2}} }{{k}}\:,\:{y}=\:\mathrm{1},\mathrm{2},\mathrm{3}}\\{\frac{\left({y}−\mathrm{7}\right)^{\mathrm{2}} }{{k}}\:,\:{y}=\:\mathrm{4},\mathrm{5},\mathrm{6}}\\{\mathrm{0}\:\:\:\:,\:{otherwise}.}\end{cases} \\ $$$${Find}\: \\ $$$$\left({i}\right)\:{The}\:{value}\:{of}\:{the}\:{constant}\:{k}. \\ $$$$\left({ii}\right)\:{the}\:{mean}\:{and}\:{varriance}\:{of}\:{Y}. \\ $$$$\left({iii}\right)\:{The}\:{variance}\:{R},\:{where}\:{R}=\:\mathrm{2}{Y}\:−\mathrm{3}. \\ $$

Question Number 63267 Answers: 0 Comments: 3

lim_(n→∞) (((n^3 + 1)/(n^3 − 1)))^(2n − n^3 )

$$\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\left(\frac{\mathrm{n}^{\mathrm{3}} \:+\:\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \:−\:\mathrm{1}}\right)^{\mathrm{2n}\:−\:\mathrm{n}^{\mathrm{3}} } \\ $$

Question Number 63288 Answers: 1 Comments: 0

Question Number 62907 Answers: 1 Comments: 1

∫(√(((1+x)/(1−x)) ))(1+x) dx

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

Question Number 62880 Answers: 0 Comments: 1

when f(E^c ) is equal to (f(E))^c

$${when}\:\:{f}\left({E}^{{c}} \right)\:{is}\:{equal}\:{to}\:\left({f}\left({E}\right)\right)^{{c}} \\ $$

Question Number 62874 Answers: 0 Comments: 3

lim_(x→0) ((sin(6x))/(tan(5x))) how to solve this w/o L′hospital′s rule?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\left(\mathrm{6}{x}\right)}{{tan}\left(\mathrm{5}{x}\right)} \\ $$$$ \\ $$$${how}\:\:{to}\:\:{solve}\:\:{this}\:\:{w}/{o}\:\:{L}'{hospital}'{s}\:\:{rule}? \\ $$

Question Number 62712 Answers: 0 Comments: 0

$$ \\ $$

Question Number 62750 Answers: 0 Comments: 1

An element X has RAM of 88g.when a current of 0.5A was passed through fused chloride of X for 32minutes and 10sec. 0.44g of X was deposited at the cathode (a)number of faraday? (b)write formular of X ions (c)write the formular of OH

$${An}\:{element}\:{X}\:{has}\:{RAM} \\ $$$${of}\:\mathrm{88}{g}.{when}\:{a}\:{current} \\ $$$${of}\:\mathrm{0}.\mathrm{5}{A}\:{was}\:{passed}\:{through} \\ $$$${fused}\:{chloride}\:{of}\:{X}\:{for} \\ $$$$\mathrm{32}{minutes}\:{and}\:\mathrm{10}{sec}. \\ $$$$\mathrm{0}.\mathrm{44}{g}\:{of}\:{X}\:{was}\:{deposited} \\ $$$${at}\:{the}\:{cathode} \\ $$$$\left({a}\right){number}\:{of}\:{faraday}? \\ $$$$\left({b}\right){write}\:{formular}\:{of}\: \\ $$$${X}\:{ions} \\ $$$$\left({c}\right){write}\:{the}\:{formular}\:{of}\:{OH} \\ $$

Question Number 62413 Answers: 0 Comments: 0

calculate W_n = ∫_0 ^(π/2) cos^n xdx ( n from N) and J_n =∫_0 ^(π/2) sin^n xdx

$${calculate}\:\:{W}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{{n}} {xdx}\:\:\:\left(\:{n}\:{from}\:{N}\right)\:{and}\:{J}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}^{{n}} {xdx} \\ $$

Question Number 62410 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−t) ln(t) dt =−γ ( γ is the constant of euler)

$$\:\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}} {ln}\left({t}\right)\:{dt}\:=−\gamma\:\:\:\:\:\:\:\left(\:\:\gamma\:{is}\:{the}\:{constant}\:{of}\:{euler}\right) \\ $$

Question Number 62395 Answers: 0 Comments: 1

The Most Beautiful Equation for me is: e^(iπ) +1=0 INCREDIBLE! #Euler′sIdentity

$$\mathrm{The}\:\mathrm{Most}\:\mathrm{Beautiful}\:\mathrm{Equation} \\ $$$$\mathrm{for}\:\mathrm{me}\:\mathrm{is}: \\ $$$$\mathrm{e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{INCREDIBLE}! \\ $$$$#\mathrm{Euler}'\mathrm{sIdentity} \\ $$

Question Number 62341 Answers: 2 Comments: 3

How many real root does the equation x^8 − x^7 + 2x^6 − 2x^5 + 3x^4 − 3x^3 + 4x^2 − 4x + (5/2) = 0 has