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AllQuestion and Answers: Page 1927

Question Number 8720    Answers: 1   Comments: 5

Question Number 8757    Answers: 0   Comments: 1

A balloon is inflated such that every point expands at a units/second. An ant runs from one point A to another point B. If the ant moves b units/second, what will influence if or not the ant will ever reach point B?

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{is}\:\mathrm{inflated}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every} \\ $$$$\mathrm{point}\:\mathrm{expands}\:\mathrm{at}\:{a}\:\mathrm{units}/\mathrm{second}. \\ $$$$ \\ $$$$\mathrm{An}\:\mathrm{ant}\:\mathrm{runs}\:\mathrm{from}\:\mathrm{one}\:\mathrm{point}\:\boldsymbol{{A}}\:\mathrm{to}\:\mathrm{another} \\ $$$$\mathrm{point}\:\boldsymbol{{B}}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{moves}\:{b}\:\mathrm{units}/\mathrm{second}, \\ $$$$\mathrm{what}\:\mathrm{will}\:\mathrm{influence}\:\mathrm{if}\:\mathrm{or}\:\mathrm{not}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{will} \\ $$$$\mathrm{ever}\:\mathrm{reach}\:\mathrm{point}\:\boldsymbol{{B}}? \\ $$

Question Number 8707    Answers: 1   Comments: 0

Find an integer x that satisfies the equation x^5 −101x^3 −999x^2 +100900=0

$${Find}\:{an}\:{integer}\:{x}\:{that}\:{satisfies}\:{the}\:{equation}\: \\ $$$${x}^{\mathrm{5}} −\mathrm{101}{x}^{\mathrm{3}} −\mathrm{999}{x}^{\mathrm{2}} +\mathrm{100900}=\mathrm{0} \\ $$

Question Number 8704    Answers: 1   Comments: 0

Solving for A. U(z) = U_b +((2A)/(h+1))(ρ×g×sin(α))^n [H^(n+1) −(H−Z)^(n+1) ]

$${Solving}\:{for}\:{A}. \\ $$$${U}\left({z}\right)\:=\:{U}_{{b}} +\frac{\mathrm{2}{A}}{{h}+\mathrm{1}}\left(\rho×{g}×{sin}\left(\alpha\right)\right)^{{n}} \left[{H}^{{n}+\mathrm{1}} −\left({H}−{Z}\right)^{{n}+\mathrm{1}} \right] \\ $$

Question Number 8695    Answers: 2   Comments: 0

solving for B? U_b = U_s − (2/(n+1))(((ρ×g×sinα)/B))^n H^(n+1)

$${solving}\:{for}\:{B}? \\ $$$${U}_{{b}} \:=\:{U}_{{s}} \:−\:\frac{\mathrm{2}}{{n}+\mathrm{1}}\left(\frac{\rho×{g}×{sin}\alpha}{{B}}\right)^{{n}} {H}^{{n}+\mathrm{1}} \\ $$

Question Number 8691    Answers: 1   Comments: 1

∫_0 ^( ∞) x^(−ln(x)) dx

$$\int_{\mathrm{0}} ^{\:\infty} {x}^{−\mathrm{ln}\left({x}\right)} {dx} \\ $$

Question Number 8718    Answers: 0   Comments: 2

(((x−5)^(x^2 −11×−26) +(x−5)^(x^2 −171×+26) )/(x^2 +3x−203))=x . fine x.

$$\frac{\left({x}−\mathrm{5}\right)^{{x}^{\mathrm{2}} −\mathrm{11}×−\mathrm{26}} +\left({x}−\mathrm{5}\right)^{{x}^{\mathrm{2}} −\mathrm{171}×+\mathrm{26}} }{{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{203}}={x}\:.\: \\ $$$${fine}\:{x}. \\ $$$$ \\ $$

Question Number 8685    Answers: 0   Comments: 0

Divide a circle in two equal parts by drawing an arc.

$$\mathrm{Divide}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{parts} \\ $$$$\mathrm{by}\:\mathrm{drawing}\:\mathrm{an}\:\mathrm{arc}. \\ $$

Question Number 8683    Answers: 0   Comments: 3

Question Number 8681    Answers: 0   Comments: 0

Question Number 8678    Answers: 0   Comments: 8

Question Number 8672    Answers: 1   Comments: 0

∫_0 ^∞ x.e^(−x^2 ) dx evaluate above expression.

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\mathrm{x}.\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \:\mathrm{dx} \\ $$$$\mathrm{evaluate}\:\mathrm{above}\:\mathrm{expression}. \\ $$

Question Number 8674    Answers: 1   Comments: 0

Question Number 8667    Answers: 1   Comments: 0

Question Number 8663    Answers: 1   Comments: 0

if I=a(1−(r/(100)))^n make n the subject of formular

$${if}\:{I}={a}\left(\mathrm{1}−\frac{{r}}{\mathrm{100}}\right)^{{n}} \\ $$$${make}\:{n}\:{the}\:{subject}\:{of}\:{formular} \\ $$$$ \\ $$

Question Number 8662    Answers: 1   Comments: 0

If the roots of 2x^2 +7x+5=0 are the reciprocal roots of ax^2 +bx+c=0, then a−c = ______.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{5}=\mathrm{0}\:\mathrm{are}\:\mathrm{the}\: \\ $$$$\mathrm{reciprocal}\:\mathrm{roots}\:\mathrm{of}\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0}, \\ $$$$\mathrm{then}\:{a}−{c}\:=\:\_\_\_\_\_\_. \\ $$

Question Number 8661    Answers: 1   Comments: 0

p_n =n^(th) prime number p_1 =2, p_2 =3, p_3 = 5, ... Does the following converge: Σ_(i=1) ^∞ (p_i /p_(i+1) ) Prove/disprove

$${p}_{{n}} ={n}^{{th}} \:\mathrm{prime}\:\mathrm{number} \\ $$$${p}_{\mathrm{1}} =\mathrm{2},\:{p}_{\mathrm{2}} =\mathrm{3},\:{p}_{\mathrm{3}} =\:\mathrm{5},\:... \\ $$$$ \\ $$$$\mathrm{Does}\:\mathrm{the}\:\mathrm{following}\:\mathrm{converge}: \\ $$$$\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{p}_{{i}} }{{p}_{{i}+\mathrm{1}} } \\ $$$$\mathrm{Prove}/\mathrm{disprove} \\ $$

Question Number 8653    Answers: 1   Comments: 0

Question Number 8650    Answers: 0   Comments: 3

Question Number 8647    Answers: 0   Comments: 1

Question Number 8644    Answers: 0   Comments: 4

Write an expression involving e and π which when evaluated gives result 1.

$$\mathrm{Write}\:\mathrm{an}\:\mathrm{expression}\:\mathrm{involving} \\ $$$${e}\:\mathrm{and}\:\pi\:\mathrm{which}\:\mathrm{when}\:\mathrm{evaluated} \\ $$$$\mathrm{gives}\:\mathrm{result}\:\mathrm{1}. \\ $$

Question Number 8635    Answers: 1   Comments: 0

Find the value of x 5^(√x) − 5^x − 7 = 100

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$$$\mathrm{5}^{\sqrt{\mathrm{x}}} \:−\:\mathrm{5}^{\mathrm{x}} \:−\:\mathrm{7}\:=\:\mathrm{100} \\ $$

Question Number 8633    Answers: 1   Comments: 2

Find lim_(n→∞) (n^p /x^n ) where p∈Z^+ .

$$\mathrm{Find}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{n}^{\mathrm{p}} }{\mathrm{x}^{\mathrm{n}} }\:\:\:\mathrm{where}\:\mathrm{p}\in\mathbb{Z}^{+} . \\ $$

Question Number 8632    Answers: 1   Comments: 0

evaluate ∫_0 ^3 ∫_0 ^4 1+x dy dx

$${evaluate} \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\mathrm{1}+{x}\:{dy}\:{dx} \\ $$$$\: \\ $$

Question Number 8628    Answers: 1   Comments: 0

is it always satisfying? A=lim[n→∞]∫f(n,x)dx B=∫lim[n→∞]f(n,x)dx A=B?? please show counter example checking (1) f(n,x)=x^n ,x[0→1] A=lim_(n→∞) ∫_0 ^1 x^n dx=lim_(n→∞) (1/(n+1))x^(n+1) =0 B=∫_0 ^1 lim_(n→∞) x^n dx=∫_0 ^1 0dx=0 so A=B (2) f(n,x)=(1+(x/n))^n A=lim_(n→∞) ∫(1+(x/n))^n dx =lim_(n→∞) (n/((n+1)))(1+(x/n))^(n+1) =e^x B=∫lim_(n→∞) (1+(x/n))^n dx =∫e^x dx =e^x so A=B . . .

$${is}\:{it}\:{always}\:{satisfying}? \\ $$$$\boldsymbol{{A}}=\mathrm{lim}\left[{n}\rightarrow\infty\right]\int{f}\left({n},{x}\right){dx} \\ $$$$\boldsymbol{{B}}=\int\mathrm{lim}\left[{n}\rightarrow\infty\right]{f}\left({n},{x}\right){dx} \\ $$$${A}={B}?? \\ $$$${please}\:{show}\:{counter}\:{example} \\ $$$$ \\ $$$$ \\ $$$${checking} \\ $$$$\left(\mathrm{1}\right) \\ $$$${f}\left({n},{x}\right)={x}^{{n}} \:\:,{x}\left[\mathrm{0}\rightarrow\mathrm{1}\right] \\ $$$${A}={lim}_{{n}\rightarrow\infty} \int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {dx}={lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}+\mathrm{1}}{x}^{{n}+\mathrm{1}} =\mathrm{0} \\ $$$${B}=\int_{\mathrm{0}} ^{\mathrm{1}} {lim}_{{n}\rightarrow\infty} {x}^{{n}} {dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{0}{dx}=\mathrm{0} \\ $$$${so}\:\:{A}={B} \\ $$$$ \\ $$$$\left(\mathrm{2}\right) \\ $$$${f}\left({n},{x}\right)=\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} \\ $$$${A}={lim}_{{n}\rightarrow\infty} \int\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} {dx} \\ $$$$\:\:={lim}_{{n}\rightarrow\infty} \frac{{n}}{\left({n}+\mathrm{1}\right)}\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}+\mathrm{1}} \\ $$$$\:\:={e}^{{x}} \\ $$$${B}=\int{lim}_{{n}\rightarrow\infty} \left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} {dx} \\ $$$$\:\:=\int{e}^{{x}} {dx} \\ $$$$\:\:={e}^{{x}} \\ $$$${so}\:\:{A}={B} \\ $$$$ \\ $$$$.\:.\:. \\ $$

Question Number 8618    Answers: 1   Comments: 1

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