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Question Number 33789 Answers: 3 Comments: 0

If the equation 2x^2 +14x−15=0 is divided by (x−4), the remainder is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:\:\mathrm{2}{x}^{\mathrm{2}} +\mathrm{14}{x}−\mathrm{15}=\mathrm{0}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\left({x}−\mathrm{4}\right),\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is} \\ $$

Question Number 34652 Answers: 2 Comments: 2

a point charge q is placed at (0,a/2,0) . find flux due to point charge through a square sheet of side a in xz plane whose centre coincides with orign. ( do not use gauss law)

$${a}\:{point}\:{charge}\:{q}\:{is}\:{placed} \\ $$$${at}\:\left(\mathrm{0},{a}/\mathrm{2},\mathrm{0}\right)\:.\:{find}\:{flux} \\ $$$${due}\:{to}\:{point}\:{charge}\:{through} \\ $$$${a}\:{square}\:{sheet}\:{of}\:{side}\:{a}\:\: \\ $$$${in}\:{xz}\:{plane}\:{whose}\:{centre} \\ $$$${coincides}\:{with}\:{orign}. \\ $$$$\left(\:{do}\:{not}\:{use}\:{gauss}\:{law}\right) \\ $$

Question Number 33753 Answers: 1 Comments: 0

Question Number 33747 Answers: 0 Comments: 0

Calculate ∫_(−∞) ^(+∞) e^(−x^2 ) dx using Residue theorem

$${Calculate}\:\int_{−\infty} ^{+\infty} {e}^{−{x}^{\mathrm{2}} } {dx}\:\:{using}\:\:{Residue}\:{theorem} \\ $$

Question Number 33743 Answers: 0 Comments: 1

let p(x)=(1+x^2 )(1+x^4 )....(1+x^2^n ) with n integr 1) find the roots of p(x) 2) factorize p(x) inside C[x]

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \right)\:{with}\:{n}\:{integr} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$

Question Number 33744 Answers: 0 Comments: 1

let P_n (x)=(1+x^2 )(1+x^4 )....(1+x^2^n ) calculate lim_(n→+∞) ∫_0 ^x P_n (t)dt with 0<x<1 .

$${let}\:\:{P}_{{n}} \left({x}\right)=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \right) \\ $$$${calculate}\:\:{lim}_{{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{{x}} \:{P}_{{n}} \left({t}\right){dt}\:\:{with}\:\:\mathrm{0}<{x}<\mathrm{1}\:. \\ $$

Question Number 33737 Answers: 1 Comments: 3

find the value of ∫_0 ^∞ ((cos(xt))/((t^2 + x^2 )^2 )) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left({xt}\right)}{\left({t}^{\mathrm{2}} \:+\:{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 33736 Answers: 2 Comments: 1

find the value of ∫_(−∞) ^(+∞) (x^2 /((1+x +x^2 )^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 33735 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(2x)dx)/((x^2 +1)( 2x^2 +3))) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{cos}\left(\mathrm{2}{x}\right){dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\:\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}\right)}\:. \\ $$

Question Number 33733 Answers: 1 Comments: 2

Solve : (x−2) × [x] = {x} −1 . • [.]= greatest integer function • {.}= fractional part function.

$${Solve}\:: \\ $$$$\left({x}−\mathrm{2}\right)\:×\:\left[{x}\right]\:=\:\left\{{x}\right\}\:−\mathrm{1}\:. \\ $$$$\bullet\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$$$\bullet\:\left\{.\right\}=\:{fractional}\:{part}\:\:{function}. \\ $$

Question Number 33724 Answers: 2 Comments: 1

if three persons selected at random are stopped on a street,what is the probability that (i)all were born on a friday? (ii)two were born on a friday and the others on a thursday? (iii)none was born on a friday

Question Number 33719 Answers: 1 Comments: 1

simplify S_n (x) =(1+x^2 )(1+x^4 )....(1+x^2^n ) 2) find lim_(n→+∞) S_n (x) if ∣x∣<1 .

$${simplify}\:{S}_{{n}} \left({x}\right)\:=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \left({x}\right)\:{if}\:\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 33718 Answers: 0 Comments: 2

find Σ_(n=1) ^(+∞) arctan( ((ln(1+(1/(n+1))) −ln(1+(1/n)))/(1+(1 +(1/n))(1+(1/(n+1))))))

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{+\infty} {arctan}\left(\:\frac{{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)\:−{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)}{\mathrm{1}+\left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\left.{n}+\mathrm{1}\right)}\right.}\right) \\ $$

Question Number 33716 Answers: 1 Comments: 1

calculate Σ_(n=0) ^∞ arctan( ((e^(n+1) −e^n )/(1+e^(2n+1) ))) .

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:{arctan}\left(\:\frac{{e}^{{n}+\mathrm{1}} \:\:−{e}^{{n}} }{\mathrm{1}+{e}^{\mathrm{2}{n}+\mathrm{1}} }\right)\:. \\ $$

Question Number 33717 Answers: 1 Comments: 2

find the value of Σ_(n=0) ^∞ artan( (((√(n+1)) −(√n))/(1+(√(n^2 +n)))) )

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{artan}\left(\:\frac{\sqrt{{n}+\mathrm{1}}\:−\sqrt{{n}}}{\mathrm{1}+\sqrt{{n}^{\mathrm{2}} +{n}}}\:\right) \\ $$

Question Number 33713 Answers: 1 Comments: 4

find tbe value of Σ_(n=2) ^∞ ((n^2 −n+1)/((n−1)^2 (n+1)^2 )) .

$${find}\:{tbe}\:{value}\:{of}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{{n}^{\mathrm{2}} −{n}+\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33710 Answers: 0 Comments: 1

1) find the radius of convergence?for Σ_(n=1) ^∞ (x^n /(n(n+1)(n+2))) and calculate its sum 2) find the value of Σ_(n=1) ^∞ (((−1)^n )/(n 2^n (n+1)(n+2)))

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{radius}\:{of}\:{convergence}?{for} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{x}^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\:{and}\:{calculate}\:{its}\:{sum} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\:\mathrm{2}^{{n}} \left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$

Question Number 33709 Answers: 0 Comments: 1

find Σ_(n=0) ^∞ (n+1)x^(3n) 2) calculate Σ_(n=0) ^∞ ((n+1)/8^n ) .

$${find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\left({n}+\mathrm{1}\right){x}^{\mathrm{3}{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}+\mathrm{1}}{\mathrm{8}^{{n}} }\:. \\ $$

Question Number 33708 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (n+1)x^(3n) 2) calculate Σ_(n=0) ^∞ ((n+1)/8^n ) .

Question Number 33707 Answers: 0 Comments: 0

find the radius of convergence for Σ_(n≥2) ( ∫_(n−(1/2)) ^(n+(1/2)) (dx/(√(x^3 +x +1))))x^n .

$${find}\:{the}\:{radius}\:{of}\:{convergence}\:{for} \\ $$$$\sum_{{n}\geqslant\mathrm{2}} \left(\:\int_{{n}−\frac{\mathrm{1}}{\mathrm{2}}} ^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} \:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{3}} +{x}\:+\mathrm{1}}}\right){x}^{{n}} \:\:. \\ $$

Question Number 33705 Answers: 1 Comments: 1

let α>0 find the fourier transform of f(t) = e^(−a^2 t^2 )

$${let}\:\:\alpha>\mathrm{0}\:\:{find}\:{the}\:{fourier}\:{transform}\:{of} \\ $$$${f}\left({t}\right)\:=\:{e}^{−{a}^{\mathrm{2}} {t}^{\mathrm{2}} } \\ $$

Question Number 33704 Answers: 0 Comments: 1

let f(t) = (1/(a^2 +t^2 )) witha>0 give the fourier transformfor f .

$${let}\:{f}\left({t}\right)\:=\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:\:{witha}>\mathrm{0}\:{give}\:{the}\:{fourier} \\ $$$${transformfor}\:{f}\:. \\ $$$$ \\ $$

Question Number 33703 Answers: 0 Comments: 0

give ∫_0 ^∞ ((x e^(−x) )/(1 −e^(−2x) )) sin(πx)dx at form of serie.

$${give}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{x}\:{e}^{−{x}} }{\mathrm{1}\:−{e}^{−\mathrm{2}{x}} }\:{sin}\left(\pi{x}\right){dx}\:\:{at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 33702 Answers: 0 Comments: 1

let p(x) =a_0 +a_1 x +a_2 x^2 +...a_n x^n prove that a_k = ((p^((k)) (0))/(k!)) ∀ k ∈[[0,n]] .

$${let}\:{p}\left({x}\right)\:={a}_{\mathrm{0}} \:+{a}_{\mathrm{1}} {x}\:+{a}_{\mathrm{2}} {x}^{\mathrm{2}} \:+...{a}_{{n}} {x}^{{n}} \\ $$$${prove}\:{that}\:\:{a}_{{k}} =\:\frac{{p}^{\left({k}\right)} \left(\mathrm{0}\right)}{{k}!}\:\:\forall\:{k}\:\in\left[\left[\mathrm{0},{n}\right]\right]\:. \\ $$

Question Number 33701 Answers: 0 Comments: 0

let Σ f_n (x) with f_n (x) = ((sin(nx))/(n^2 (n+1))) and S its sum x∈[−π,π] prove that ∀(x,y)∈[−π,π]^2 x≠y ⇒∣S(x)−S(y)∣<∣x−y∣ .

$${let}\:\Sigma\:{f}_{{n}} \left({x}\right)\:{with}\:{f}_{{n}} \left({x}\right)\:=\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:\:{and}\:{S}\:{its}\:{sum} \\ $$$${x}\in\left[−\pi,\pi\right]\:{prove}\:{that}\:\forall\left({x},{y}\right)\in\left[−\pi,\pi\right]^{\mathrm{2}} \\ $$$${x}\neq{y}\:\Rightarrow\mid{S}\left({x}\right)−{S}\left({y}\right)\mid<\mid{x}−{y}\mid\:. \\ $$

Question Number 33699 Answers: 0 Comments: 3

let S(x)=Σ_(n=0) ^∞ f_n (x) with f_n (x)= (((−1)^n )/(n!(x+n))) x∈]0,+∞[ 1) prove that S id defined .calculate S(1) and prove that ∀x>0 xS(x) −S(x+1) =(1/e) 2) prove that S is C^∞ on R^(+∗) 3) prove that S(x) ∼ (1/x) (x→0^+ ) .

$${let}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:{f}_{{n}} \left({x}\right)\:\:{with}\:{f}_{{n}} \left({x}\right)=\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!\left({x}+{n}\right)} \\ $$$$\left.{x}\in\right]\mathrm{0},+\infty\left[\right. \\ $$$$\left.\mathrm{1}\right)\:\:{prove}\:{that}\:{S}\:{id}\:{defined}\:.{calculate}\:{S}\left(\mathrm{1}\right)\:{and} \\ $$$${prove}\:{that}\:\forall{x}>\mathrm{0}\:\:{xS}\left({x}\right)\:−{S}\left({x}+\mathrm{1}\right)\:=\frac{\mathrm{1}}{{e}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{S}\:{is}\:{C}^{\infty} \:{on}\:{R}^{+\ast} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{S}\left({x}\right)\:\sim\:\frac{\mathrm{1}}{{x}}\:\left({x}\rightarrow\mathrm{0}^{+} \right)\:. \\ $$

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