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Question Number 216560    Answers: 0   Comments: 0

An object of mass M, initially at rest at the coordinate origin, explodes into three parts. Fragment A has a mass M/2, and fragments B and C have a mass M/4 each. After the explosion, fragment A moves in the +X direction at 10 m/s and fragment B moves in the +Y direction at 8 m/s. Find the direction and speed of fragment C

An object of mass M, initially at rest at the coordinate origin, explodes into three parts. Fragment A has a mass M/2, and fragments B and C have a mass M/4 each. After the explosion, fragment A moves in the +X direction at 10 m/s and fragment B moves in the +Y direction at 8 m/s. Find the direction and speed of fragment C

Question Number 216542    Answers: 0   Comments: 0

if y=cosu then prove that y′=sinu∙u′ by newton′s formula.

$${if}\:{y}={cosu}\:{then}\:{prove}\:{that}\:{y}'={sinu}\centerdot{u}' \\ $$$${by}\:{newton}'{s}\:{formula}. \\ $$

Question Number 216538    Answers: 0   Comments: 1

Find all integer solutions of 3^m =2n^2 +1. I only found m=1, 2, 5 by computer from m=1 to m=30000. Is there any greater solutions?

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{3}^{{m}} =\mathrm{2}{n}^{\mathrm{2}} +\mathrm{1}. \\ $$$$ \\ $$$${I}\:{only}\:{found}\:{m}=\mathrm{1},\:\mathrm{2},\:\mathrm{5}\:{by}\:{computer} \\ $$$${from}\:{m}=\mathrm{1}\:{to}\:{m}=\mathrm{30000}. \\ $$$${Is}\:{there}\:{any}\:{greater}\:{solutions}? \\ $$

Question Number 216537    Answers: 0   Comments: 0

using first principle solve y=((x+2)/( (√x)+2))

$$\mathrm{using}\:\mathrm{first}\:\mathrm{principle}\:\mathrm{solve} \\ $$$$\mathrm{y}=\frac{\mathrm{x}+\mathrm{2}}{\:\sqrt{\mathrm{x}}+\mathrm{2}} \\ $$

Question Number 216534    Answers: 0   Comments: 24

Question Number 216572    Answers: 0   Comments: 0

Question Number 216532    Answers: 0   Comments: 0

Let f :R_+ →R such as f(x)=f(x)+f(y) 1) Prove that f is derivable iff f is derivable at x=1. 2) Prove that if so, f(x)=Log_a x) where a is positive value to precise

$${Let}\:{f}\::\mathbb{R}_{+} \rightarrow\mathbb{R}\:{such}\:{as}\:{f}\left({x}\right)={f}\left({x}\right)+{f}\left({y}\right) \\ $$$$\left.\mathrm{1}\right)\:{Prove}\:{that}\:\:{f}\:{is}\:{derivable}\:{iff}\:\: \\ $$$${f}\:{is}\:{derivable}\:{at}\:{x}=\mathrm{1}. \\ $$$$\left.\mathrm{2}\left.\right)\:{Prove}\:{that}\:{if}\:{so},\:{f}\left({x}\right)={Log}_{{a}} {x}\right)\: \\ $$$${where}\:{a}\:{is}\:{positive}\:{value}\:{to}\:{precise} \\ $$

Question Number 216526    Answers: 1   Comments: 0

Question Number 216525    Answers: 2   Comments: 0

Question Number 216515    Answers: 2   Comments: 0

Question Number 216513    Answers: 0   Comments: 0

Prove or disprove that: If p=(√(Σ_(k=0) ^n 3^k )) (n>0) is an integer, then p is prime.

$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}: \\ $$$$\mathrm{If}\:{p}=\sqrt{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{3}^{{k}} }\:\left({n}>\mathrm{0}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer},\:\mathrm{then}\:{p}\:\mathrm{is}\:\mathrm{prime}. \\ $$

Question Number 216507    Answers: 0   Comments: 1

Question Number 216493    Answers: 1   Comments: 0

Res_(z=c) {f(z)}=(1/(2πi)) ∮_( C) f(z)dz Res_(z=1) {((z^(21) +z^2 +z+1)/((z−1)^3 ))}=(1/(2πi)) ∮_( C) ((z^(21) +z^2 +z+1)/((z−1)^3 ))dz (1/(2πi)) ∮_( C) (((z^(21) +z^2 +z+1)/((z−1)^2 ))/(z−1))dz=lim_(z→1) ((z^(21) +z^2 +z+1)/((z−1)^2 )) L′hosiptal :) lim_(z→1) ((21z^(20) +2z+1)/(2(z−1))) and... Twice!! lim_(z→1) ((420z^(19) +2)/2)=211 ∴Res_(z=1) {f(z)}=211 ★Caution★ f(α)′′=′′(1/(2πi)) ∮_( C) ((f(z))/(z−α)) dz Why did I use big quotes for this equation?? because the conditions for establshing this equation are that path C must be a simple closed curve and there must be no singularity in path C

$$\mathrm{Res}_{{z}={c}} \left\{{f}\left({z}\right)\right\}=\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:\mathrm{C}} \:{f}\left({z}\right)\mathrm{d}{z} \\ $$$$\mathrm{Res}_{{z}=\mathrm{1}} \left\{\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{3}} }\right\}=\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{3}} }\mathrm{d}{z} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\:\frac{\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{2}} }}{{z}−\mathrm{1}}\mathrm{d}{z}=\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{L}'\mathrm{hosiptal}\::\right) \\ $$$$\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{21}{z}^{\mathrm{20}} +\mathrm{2}{z}+\mathrm{1}}{\mathrm{2}\left({z}−\mathrm{1}\right)}\:\:\mathrm{and}...\:\mathrm{Twice}!! \\ $$$$\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{420}{z}^{\mathrm{19}} +\mathrm{2}}{\mathrm{2}}=\mathrm{211} \\ $$$$\therefore\mathrm{Res}_{{z}=\mathrm{1}} \left\{{f}\left({z}\right)\right\}=\mathrm{211} \\ $$$$\bigstar\mathrm{Caution}\bigstar \\ $$$${f}\left(\alpha\right)''=''\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\:\frac{{f}\left({z}\right)}{{z}−\alpha}\:\mathrm{d}{z}\: \\ $$$$\mathrm{Why}\:\mathrm{did}\:\mathrm{I}\:\mathrm{use}\:\mathrm{big}\:\mathrm{quotes}\:\mathrm{for}\:\mathrm{this}\: \\ $$$$\mathrm{equation}?? \\ $$$$\mathrm{because}\:\mathrm{the}\:\mathrm{conditions}\:\mathrm{for}\:\mathrm{establshing} \\ $$$$\mathrm{this}\:\mathrm{equation}\:\mathrm{are}\:\mathrm{that}\:\mathrm{path}\:{C}\: \\ $$$$\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{curve} \\ $$$$\mathrm{and}\:\mathrm{there}\:\mathrm{must}\:\mathrm{be}\:\mathrm{no}\:\mathrm{singularity} \\ $$$$\mathrm{in}\:\mathrm{path}\:\mathrm{C} \\ $$

Question Number 216491    Answers: 2   Comments: 0

Question Number 216489    Answers: 1   Comments: 1

find residuo ((x^(21) +x^2 +x+1)/((x−1)^3 ))

$${find}\:\:{residuo} \\ $$$$\:\:\:\:\:\:\:\:\frac{{x}^{\mathrm{21}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 216487    Answers: 1   Comments: 2

Find the value of ω^7 + ω^8 + ω^(12) where ω is omega function.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\omega^{\mathrm{7}} \:\:+\:\:\omega^{\mathrm{8}} \:\:+\:\:\omega^{\mathrm{12}} \:\:\mathrm{where} \\ $$$$\omega\:\:\mathrm{is}\:\mathrm{omega}\:\mathrm{function}. \\ $$

Question Number 216486    Answers: 1   Comments: 0

∫_( 0) ^( 1) x(√(x ((x ((x ((x ...))^(1/5) ))^(1/4) ))^(1/3) )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}\sqrt{\mathrm{x}\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:\sqrt[{\mathrm{4}}]{\mathrm{x}\:\:\sqrt[{\mathrm{5}}]{\mathrm{x}\:...}}}}\:\:\mathrm{dx} \\ $$

Question Number 216485    Answers: 2   Comments: 0

Solve for x in: i^x = 2

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:\:\mathrm{in}:\:\:\:\mathrm{i}^{\mathrm{x}} \:\:=\:\:\mathrm{2} \\ $$

Question Number 216478    Answers: 1   Comments: 1

Question Number 216477    Answers: 2   Comments: 0

Question Number 216471    Answers: 1   Comments: 3

Question Number 216445    Answers: 2   Comments: 2

Prove that Γ((1/2)) = (√π)

$$\mathrm{Prove}\:\mathrm{that}\:\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\:=\:\:\sqrt{\pi} \\ $$

Question Number 216437    Answers: 2   Comments: 0

Question Number 216454    Answers: 0   Comments: 7

Reponse a l exercice N8: Reponses par ordre:(1,2,3,4,5,6) imsge 1 imsge 2 image 3 imsge 5 imsge 4 imsge 6

$$\mathrm{Reponse}\:\mathrm{a}\:\:\mathrm{l}\:\mathrm{exercice}\:\:\mathrm{N8}: \\ $$$$\mathrm{Reponses}\:\mathrm{par}\:\mathrm{ordre}:\left(\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\right) \\ $$$$\boldsymbol{\mathrm{imsge}}\:\mathrm{1}\: \\ $$$$\boldsymbol{\mathrm{imsge}}\:\mathrm{2} \\ $$$$\boldsymbol{\mathrm{image}}\:\mathrm{3} \\ $$$$\boldsymbol{\mathrm{imsge}}\:\mathrm{5} \\ $$$$\boldsymbol{\mathrm{imsge}}\:\mathrm{4} \\ $$$$\boldsymbol{\mathrm{imsge}}\:\mathrm{6} \\ $$

Question Number 216425    Answers: 1   Comments: 1

Question Number 216421    Answers: 1   Comments: 0

If asinθ + bcosθ = acosecθ + bsecθ then prove that each term is equal to (a^(2/3) − b^(2/3) )(√(a^(2/3) + b^(2/3) )).

$$\mathrm{If}\:{a}\mathrm{sin}\theta\:+\:{b}\mathrm{cos}\theta\:=\:{a}\mathrm{cosec}\theta\:+\:{b}\mathrm{sec}\theta\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{each}\:\mathrm{term}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left({a}^{\frac{\mathrm{2}}{\mathrm{3}}} \:−\:{b}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)\sqrt{{a}^{\frac{\mathrm{2}}{\mathrm{3}}} \:+\:{b}^{\frac{\mathrm{2}}{\mathrm{3}}} }. \\ $$

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