meng tz0 dm更高级别离散数学mathematics paper 3 discrete.pdf
M11/5/MATHL/HP3/ENG/TZ0/DM
22117207
MATHEMATICS
HIGHERLEVEL
PAPER3–DISCRETEMATHEMATICS
Monday9May2011(morning)
1hour
INSTRUCTIONSTOCANDIDATES
Donotopenthiaminationpaperuntilinstructedtodoso.
Answerallthequestions.
Unlessotherwisestatedinthequestion,allnumericalanswersmustbegivenexactlyorcorrect
tothreesignicantgures.
2211-72074pages
©InternationalBaccalaureateOrganization2011
–2–M11/5/MATHL/HP3/ENG/TZ0/DM
Pleasestarteachquestiononanege.Fullmarksarenotnecessarilyawardedforacorrectanswerwith
noworking.Answersmustbesupportedbyworkingand/orexplanations.Inparticular,solutionsfound
fromagraphicdisplaycalculatorshouldbesupportedbysuitableworking,e.g.ifgraphsareusedtofind
asolution,youshouldsketchtheseaspartofyouranswer.Whereananswerisincorrect,somemarksmay
begivenforacorrectmethod,providedthisisshownbywrittenworking.Youarethereforeadvisedtoshow
allworking.
1.[ummark:13]
(a) UsetheEuclideanalgorithmtofindthegreatestcommondivisorofthenumbers
56and315.[4marks]
(b) (i) Findthegeneralsolutiontothediophantineequation56x+315y=21.
(ii) Henceorotherwisefindthesmallestpositivesolutiontothecongruence
315x≡21(modulo56).[9marks]
2.[ummark:7]
Thecompletegraph Hhasthefollowingcostadjacencymatrix.
ABCD