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Question Number 28852 Answers: 0 Comments: 0

Testing of a Bakelite sample by schering Bridge having a standard capacitor of 106pF , balance was obtained with a capacitance of 0.351 F in parallel with non - inductive resistance in the remaining arm of the bridge being 130 Ω. Determine the capacitance and the equivalent series resistance of the specimen and draw the circuit diagram.

$$\mathrm{Testing}\:\mathrm{of}\:\mathrm{a}\:\mathrm{Bakelite}\:\mathrm{sample}\:\mathrm{by}\:\mathrm{schering}\:\mathrm{Bridge}\:\mathrm{having}\:\mathrm{a}\:\mathrm{standard}\:\mathrm{capacitor} \\ $$$$\mathrm{of}\:\:\mathrm{106pF}\:,\:\:\mathrm{balance}\:\mathrm{was}\:\mathrm{obtained}\:\mathrm{with}\:\mathrm{a}\:\mathrm{capacitance}\:\mathrm{of}\:\:\:\mathrm{0}.\mathrm{351}\:\mathrm{F}\:\:\mathrm{in}\:\mathrm{parallel} \\ $$$$\mathrm{with}\:\mathrm{non}\:-\:\mathrm{inductive}\:\mathrm{resistance}\:\mathrm{in}\:\mathrm{the}\:\mathrm{remaining}\:\mathrm{arm}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bridge}\:\mathrm{being}\:\:\:\mathrm{130}\:\Omega. \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{capacitance}\:\mathrm{and}\:\mathrm{the}\:\mathrm{equivalent}\:\mathrm{series}\:\mathrm{resistance}\:\mathrm{of}\:\mathrm{the}\:\mathrm{specimen} \\ $$$$\mathrm{and}\:\mathrm{draw}\:\mathrm{the}\:\mathrm{circuit}\:\mathrm{diagram}. \\ $$

Question Number 28833 Answers: 0 Comments: 0

let give ϕ(x) =x ,ϕ 2π periodique even developp f at fourier series then find the value of Σ_(n=1) ^∞ (((−1)^n )/n^2 ) and Σ_(n=0) ^∞ (1/((2n+1)^2 )) .

$${let}\:{give}\:\varphi\left({x}\right)\:={x}\:,\varphi\:\mathrm{2}\pi\:{periodique}\:{even} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series}\:{then}\:{find}\:{the}\:{value}\:{of} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} }\:{and}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 28832 Answers: 0 Comments: 0

find the value of A_n = ∫_1 ^(+∞) (dt/(t^(n+1) (√(t−1)))) .withn∈N .

$${find}\:{the}\:{value}\:{of}\:\:{A}_{{n}} =\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{{n}+\mathrm{1}} \sqrt{{t}−\mathrm{1}}}\:.{withn}\in{N}\:. \\ $$

Question Number 28835 Answers: 1 Comments: 4

Question Number 28830 Answers: 0 Comments: 0

let give f(x)= ch(αx) and 2π periodic with α≠0 developp f at fourier series.

$${let}\:{give}\:{f}\left({x}\right)=\:{ch}\left(\alpha{x}\right)\:{and}\:\mathrm{2}\pi\:{periodic}\:{with}\:\alpha\neq\mathrm{0} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series}. \\ $$

Question Number 28828 Answers: 0 Comments: 0

find f(x)=∫_0 ^1 ln(1+xt^2 )dt with ∣x∣<1 .

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:{with}\:\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 28827 Answers: 0 Comments: 0

let give F(x)=∫_0 ^∞ ((arctan(1+x(1+t^2 )))/(1+t^2 ))dt and x>0 calculate (dF/dx)(x).

$${let}\:{give}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{and}\:{x}>\mathrm{0} \\ $$$${calculate}\:\frac{{dF}}{{dx}}\left({x}\right).\:\: \\ $$

Question Number 28826 Answers: 0 Comments: 0

let give f(x)= e^(−x) cosx and 2π periodic 1) developp f at fourier series 2) find the value of Σ_(n=−∞) ^(n=+∞) (((−1)^n )/(1+n^2 )) .

$${let}\:{give}\:{f}\left({x}\right)=\:{e}^{−{x}} \:{cosx}\:\:{and}\:\mathrm{2}\pi\:{periodic} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}\:{at}\:{fourier}\:{series} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=−\infty} ^{{n}=+\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{1}+{n}^{\mathrm{2}} }\:. \\ $$

Question Number 28825 Answers: 0 Comments: 0

let give f(x)= e^(−x) cosx prove that f(x)= Σ_(n=0) ^∞ ((((√2))^n )/(n!)) cos(((3nπ)/4)) x^n .

$${let}\:{give}\:{f}\left({x}\right)=\:{e}^{−{x}} \:{cosx}\:{prove}\:{that} \\ $$$${f}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(\sqrt{\mathrm{2}}\right)^{{n}} }{{n}!}\:{cos}\left(\frac{\mathrm{3}{n}\pi}{\mathrm{4}}\right)\:{x}^{{n}} \:\:. \\ $$

Question Number 28824 Answers: 0 Comments: 0

by using residus theorem find the value of A_n = ∫_0 ^∞ (dx/(1+x^n )) with n integr and n≥2.

$${by}\:{using}\:{residus}\:{theorem}\:{find}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}. \\ $$

Question Number 28823 Answers: 0 Comments: 0

find I = ∫_(−∞) ^(+∞) (((x−1)cosx)/(x^2 −2x+2))dx and J= ∫_(−∞) ^(+∞) (((x−1)sinx)/(x^2 −2x +2)) dx.

$${find}\:\:{I}\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{\left({x}−\mathrm{1}\right){cosx}}{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}}{dx}\:{and} \\ $$$${J}=\:\int_{−\infty} ^{+\infty} \:\:\frac{\left({x}−\mathrm{1}\right){sinx}}{{x}^{\mathrm{2}} −\mathrm{2}{x}\:+\mathrm{2}}\:{dx}. \\ $$

Question Number 28821 Answers: 0 Comments: 0

find the value of ∫_0 ^(2π) ((4 cos(4θ))/(5−4cosθ)) dθ .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{4}\:{cos}\left(\mathrm{4}\theta\right)}{\mathrm{5}−\mathrm{4}{cos}\theta}\:{d}\theta\:. \\ $$

Question Number 28820 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(1−(t^2 /4)))/t^2 )dt.

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−\frac{{t}^{\mathrm{2}} }{\mathrm{4}}\right)}{{t}^{\mathrm{2}} }{dt}. \\ $$

Question Number 28819 Answers: 0 Comments: 0

let give f(x)= ∫_0 ^1 ((ln(1−x^2 t^2 ))/t^2 )dt with ∣x∣<1 by using derivation under ∫ find the value of f(x).

$${let}\:{give}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${by}\:{using}\:{derivation}\:{under}\:\int\:\:{find}\:{the}\:{value}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 28818 Answers: 0 Comments: 0

prove that (π/(4cos(((πα)/2))))=Σ_(p=0) ^∞ (((2p+1)(−1)^p )/((2p+1)^2 −α^2 )) α ∈R−Z.

$${prove}\:{that}\:\:\frac{\pi}{\mathrm{4}{cos}\left(\frac{\pi\alpha}{\mathrm{2}}\right)}=\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(\mathrm{2}{p}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{p}} }{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} \:−\alpha^{\mathrm{2}} } \\ $$$$\alpha\:\in{R}−{Z}. \\ $$

Question Number 28817 Answers: 0 Comments: 0

let give f(x)=e^(iαx) 2π prriodic and α ∈R−Z developp f at fourier series.

$${let}\:{give}\:{f}\left({x}\right)={e}^{{i}\alpha{x}} \:\:\mathrm{2}\pi\:{prriodic}\:{and}\:\alpha\:\in{R}−{Z} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series}. \\ $$

Question Number 28816 Answers: 0 Comments: 0

find the value of I= ∫_0 ^π (dθ/(1+cos^4 θ)) .

$${find}\:{the}\:{value}\:{of}\:\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\mathrm{1}+{cos}^{\mathrm{4}} \theta}\:. \\ $$

Question Number 28815 Answers: 1 Comments: 0

find the value of ∫_0 ^π (dx/(2cos^2 x +sin^2 x)) .

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}}\:. \\ $$

Question Number 28814 Answers: 0 Comments: 2

find Σ_(n=1) ^∞ (1/(1^3 +2^3 +...+n^3 )) .

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} \:+\mathrm{2}^{\mathrm{3}} +...+{n}^{\mathrm{3}} }\:. \\ $$

Question Number 28813 Answers: 0 Comments: 0

let give F(t)=∫_0 ^∞ ((sin(x^2 ))/x^2 ) e^(−tx^2 ) dx with t>0 find (dF/dt)(t).

$${let}\:{give}\:{F}\left({t}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\:{e}^{−{tx}^{\mathrm{2}} } {dx}\:\:{with}\:{t}>\mathrm{0} \\ $$$${find}\:\:\frac{{dF}}{{dt}}\left({t}\right). \\ $$$$ \\ $$

Question Number 28812 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ (((1−e^(−x) )sinx)/x^2 )dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 28811 Answers: 0 Comments: 1

find ∫_0 ^∞ ln(1+e^(−xt) )dx with t>0 then give the value of ∫_0 ^∞ ln(1+e^(−x) )dx.

$${find}\:\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+{e}^{−{xt}} \right){dx}\:{with}\:{t}>\mathrm{0}\:{then}\:{give}\:{the}\:{value}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+{e}^{−{x}} \right){dx}. \\ $$

Question Number 28808 Answers: 2 Comments: 0

Question Number 28806 Answers: 1 Comments: 0

If n(A)=15 and n(B)=25, (a) What are the greatest and least values of n(AuB)? (b) What are the greatest and least value of n(AnB)? (c) Draw Venn diagrams to illustrate the four situations in (a) and (b) above

$$\mathrm{If}\:\mathrm{n}\left(\mathrm{A}\right)=\mathrm{15}\:\mathrm{and}\:\mathrm{n}\left(\mathrm{B}\right)=\mathrm{25},\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AuB}\right)? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AnB}\right)? \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{Venn}\:\mathrm{diagrams}\:\mathrm{to}\:\mathrm{illustrate}\:\mathrm{the}\:\mathrm{four}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{situations}\:\mathrm{in}\:\left(\mathrm{a}\right)\:\mathrm{and}\:\left(\mathrm{b}\right)\:\mathrm{above} \\ $$

Question Number 28805 Answers: 1 Comments: 0

In a competition, a school awarded medals in different categories. 36 medals in dance,12 in dramatics and 18 medals in music.If these medals went to total 45,and only 4 persons got medals in all three catogories.Using set notations, how many received in exactly two of these categories?

$${In}\:{a}\:{competition},\:{a}\:{school}\:{awarded} \\ $$$${medals}\:{in}\:{different}\:{categories}. \\ $$$$\mathrm{36}\:{medals}\:{in}\:{dance},\mathrm{12}\:{in}\:{dramatics} \\ $$$${and}\:\mathrm{18}\:{medals}\:{in}\:{music}.{If}\:{these} \\ $$$${medals}\:{went}\:{to}\:{total}\:\mathrm{45},{and}\:{only} \\ $$$$\mathrm{4}\:{persons}\:{got}\:{medals}\:{in}\:{all}\:{three} \\ $$$${catogories}.{Using}\:{set}\:{notations}, \\ $$$${how}\:{many}\:{received}\:{in}\:{exactly} \\ $$$${two}\:{of}\:{these}\:{categories}? \\ $$

Question Number 28779 Answers: 3 Comments: 1

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