PAPER2:纸2
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MATHEMATICALTRIPOSPartIA
Friday1June20011.30to4.30
PAPER2
Beforeyoubeginreadtheseinstructionscarefully.
EachquestioninSectionIIcarriestwicethecreditofeachquestioninSectionI.You
mayattemptallfourquestionsinSectionIandatmostfivequestionsfromSection
II.InSectionIInomorethanthreequestionsoneachcoursemaybeattempted.
Completeanswersarepreferredtofragments.
Writeononesideofthepaperonlyandbegineachansweronaseparatesheet.
Writelegibly;otherwiseyouplaceyourselfatagravedisadvantage.
Attheendoftheexamination:
Tieupyouranswersintwobundles,markedBandFaccordingtothecodeletter
affixedtoeachquestion.Attachabluecoversheettoeachbundle;writethecodein
theboxmarked‘SECTION’onthecoversheet.DonottieupquestionsfromSection
IandSectionIIinseparatebundles.
Youmustalsocompleteagreenmastercoversheetlistingallthequestionsattempted
byyou.
Everycoversheetmustbearyourexaminationnumberanddesknumber.2
SECTIONI
1BDifferentialEquations
Findthesolutionto
dy(x)
dx+tanh(x)y(x)=H(x),
intherange−∞
definedby
H(x)=0x<0
1x>0.
Sketchthesolution.
2BDifferentialEquations
Thefunctiony(x)satisfiestheinhomogeneoussecond-orderlineardifferential
equation
y−y−2y=18xe−x.
Findthesolutionthatsatisfiestheconditionsthaty(0)=1andy(x)isboundedasx→∞.
3FProbability
ThefollowingproblemisknownasBertrand’sparadox.Achordhasbeenchosen
atrandominacircleofradiusr.Findtheprobabilitythatitislongerthanthesideof
theequilateraltriangleinscribedinthecircle.Considerthreedifferentcases:
a)themiddlepointofthechordisdistributeduniformlyinsidethecircle,
b)thetwoendpointsofthechordareindependentanduniformlydistributedover
thecircumference,
c)thedistancebetweenthemiddlepointofthechordandthecentreofthecircle
isuniformlydistributedovertheinterval[0,r].
[Hint:drawingdiagramsmayhelpconsiderably.]
Paper23
4FProbability
TheRuritanianauthoritiesdecidedtopardonandreleaseoneoutofthreeremaining
inmates,A,BandC,keptinstrictisolationinthenotoriousAlkazafprison.Theinmates
knowthis,butcan’tguesswhoamongthemistheluckyone;thewaitingisagonising.A
sympathetic,butcorrupted,prisonguardapproachesAandofferstoname,inexchange
forafee,anotherinmate(notA)whoisdoomedtostay.Hesays:“Thisreducesyour
chancestoremainherefrom2/3to1/2:willitmakeyoufeelbetter?”Ahesitatesbut
thenacceptstheoffer;theguardnamesB.
AssumethatindeedBwillnotbereleased.Determinetheconditionalprobability
PAremainsBnamed=P(A&Bremain)P(Bnamed)
andthuschecktheguard’sclaim,inthreecases:
a)whentheguardiscompletelyunbiased(i.e.,namesanyofBandCwith
probability1/2ifthepairB,Cistoremainjailed),
b)ifhehatesBandwouldcertainlynamehimifBistoremainjailed,
c)ifhehatesCandwouldcertainlynamehimifCistoremainjailed.
Paper2[TURNOVER4
SECTIONII
5BDifferentialEquations
Therealsequenceyk,k=1,2,...satisfiesthedifferenceequation
yk+2−yk+1+yk=0.
Showthatthegeneralsolutioncanbewritten
yk=acosπk
3+bsinπk
3,
whereaandbarearbitraryrealconstants.
Nowletyksatisfy
yk+2−yk+1+yk=1
k+2.(∗)
Showthataparticularsolutionof(∗)canbewrittenintheform
yk=k
n=1ank−n+1,
where
an+2−an+1+an=0,n≥1,
anda1=1,a2=1.
Hence,findthegeneralsolutionto(∗).
Paper25
6BDifferentialEquations
Thefunctiony(x)satisfiesthelinearequation
y(x)+p(x)y(x)+q(x)y(x)=0.
TheWronskian,W(x),oftwoindependentsolutionsdenotedy1(x)andy2(x)isdefined
tobe
W(x)=y1y2y1y2.
Lety1(x)begiven.Inthiscase,showthattheexpressionforW(x)canbe
interpretedasafirst-orderinhomogeneousdifferentialequationfory2(x).Hence,by
explicitderivation,showthaty2(x)maybeexpressedas
y2(x)=y1(x)x
x0W(t)
y1(t)2dt,(∗)
wheretherˆoleofx0shouldbebrieflyelucidated.
ShowthatW(x)satisfies
dW(x)
dx+p(x)W(x)=0.
Verifythaty1(x)=1−xisasolutionof
xy(x)−(1−x2)y(x)−(1+x)y(x)=0.(†)
Hence,using(∗)withx0=0andexpandingtheintegrandinpowersofttoordert3,find
thefirstthreenon-zerotermsinthepowerseriesexpansionforasolution,y2(x),of(†)
thatisindependentofy1(x)andsatisfiesy2(0)=0,y2(0)=1.
Paper2[TURNOVER6
7BDifferentialEquations
Considerthelinearsystem
˙z+Az=h,(∗)
where
z(t)=x(t)
y(t)
,A=1+a−2
1−1+a
,h(t)=2cost
cost−sint
,
wherez(t)isrealandaisarealconstant,a≥0.
Finda(complex)eigenvector,e,ofAanditscorresponding(complex)eigenvalue,
l.Showthatthesecondeigenvectorandcorrespondingeigenvaluearerespectively¯eand¯l,wherethebaroverthesymbolssignifiescomplexconjugation.Henceexplainhowthe
generalsolutionto(∗)canbewrittenas
z(t)=α(t)e+¯α(t)¯e,
whereα(t)iscomplex.
Writedownadifferentialequationforα(t)andhence,fora>0,deducethesolution
to(∗)whichsatisfiestheinitialconditionz(0)=0¯.
Isthelinearsystemresonant?
Bytakingthelimita→0ofthesolutionalreadyfounddeducethesolution
satisfyingz(0)=0¯whena=0.
8BDifferentialEquations
Carnivoroushuntersofpopulationhpreyonvegetariansofpopulationp.Inthe
absenceofhuntersthepreywillincreaseinnumberuntiltheirpopulationislimitedby
theavailabilityoffood.Intheabsenceofpreythehunterswilleventuallydieout.The