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Te Tauanga me te Whakatauira, Kaupae 3, 201290644M Te whakaoti whārite
9.30 i te ata Rāapa 21 Whiringa-ā-rangi 2012 Whiwhinga: Whā
Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu (NSN) kei tō pepa whakauru ki te tau kei runga ake nei.
Me whakautu e koe ngā pātai KATOA kei roto i te pukapuka nei.
Whakaaturia ngā mahinga KATOA.
Me mātua riro mai i a koe te pukaiti o ngā Tikanga Tātai me ngā Papatau L3–STATMF.
Ki te hiahia koe ki ētahi atu wāhi hei tuhituhi whakautu, whakamahia te (ngā) whārangi kei muri i te pukapuka nei, ka āta tohu ai i ngā tau pātai.
Tirohia mehemea kei roto nei ngā whārangi 2 – 27 e raupapa tika ana, ā, kāore hoki he whārangi wātea.
HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE HEI TE MUTUNGA O TE WHAKAMĀTAUTAU.
906445
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© Mana Tohu Mātauranga o Aotearoa, 2012. Pūmau te mana.Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.
MĀ TE KAIMĀKA ANAKE Paearu Paetae
Paetae Paetae Kaiaka Paetae KairangiTe whakaoti whārite. Te whakaoti rapanga e whai wāhi
mai ana tēnei mea te whārite.Te tātari, te whakamāori rānei i ngā hua ka puta, i te hātepe rānei ka whāia hei whakaoti whārite, hei whakaoti rānei i ngā rapanga o te tikanga rorohiko mō te kauwhata rārangi.
Whakakaotanga o te tairanga mahinga
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Kia 55 meneti hei whakautu i ngā pātai o tēnei pukapuka.
PĀTAI TUATAHI
(a) Kawhakareretētahikamupenewakarererangiingāmomowakarererangieruaiwaengaingātāonenuierua.KakawengāwakarererangiX30ingāpāhihie30,ā,e45menetiteroaotererenga.KakawengāwakarererangiX15ingāpāhihie15,ā,e60menetiteroaotererenga.
Itētahirānoa,kawhakareretekamupenewakarererangiitemōrahipāhihie900.Eaikingātureotemanatūrererangimōngāhaorarereongākaihautūwakarererangi,kataeaetekamupenewakarererangitewhakareremōtemōrahiote45haoraiiarā.
Mēnākamahiatemōrahiongātāpuitangaiiarerenga,kapāngāaukatiewhaiakeneikitēneitūāhua:
30x+15y ≤900 0.75x+1.0y ≤45 x ≥0 y ≥0
inakoxtemahaongārerengakamahiaetewakarererangiX30,ā,koytemahaongārerengakamahiaetewakarererangiX15.
Ingārāekaweanangāwakarererangikatoaitetokomahamōrahiongāpāhihi,kawhakaaturiatehuamoniatekamupenewakarererangimaiitekawepāhihiingārerengamātewhāriteP=330x+195y.
Tātaitiatemōrahihuamonikataeapeaiiarāetekamupenewakarererangimaiitekawepāhihi.
Kahomaitetukutukukeiraroheiāwhinaiakoekitewhakautuitepātai.
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You are advised to spend 55 minutes answering the questions in this booklet.
QUESTION ONE
(a) Anairlinefliestwodifferenttypesofaircraftbetweentwocities.X30aircraftcarry30passengersandtake45minutestocompletetheflight.X15aircraftcarry15passengersandtake60minutestocompletetheflight.
Onanygivenday,theairlinewillflyamaximumof900passengers.Underaviationauthorityregulationsforpilotflyinghours,theairlineisallowedtoflyforamaximumof45hoursperday.
Assumingthemaximumnumberofbookingsismadeoneachflight,thefollowingconstraintsapplytothissituation:
30x+15y ≤900 0.75x+1.0y ≤45 x ≥0 y ≥0
where xisthenumberofflightsmadebytheX30aircraftandyisthenumberofflightsmadebytheX15aircraft.
Onadaywithallaircraftcarryingthemaximumnumberofpassengers,theairline’sprofitfromcarryingpassengersontheflightsisgivenbytheequationP=330x+195y.
Calculatetheairline’spotentialmaximumdailyprofitfromcarryingpassengers.
Thegridbelowisprovidedtohelpyoutoanswerthequestion.
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(b) KawhakamahiahokiteX30heikaweingāikamengāhuarākaumataiwaengaingātāonenuierua.Katākaitiangāhuaingākātenemeterōrahiote0.1m3.E32kgtetaumahaongākāteneikamata.E20kgtetaumahaongākātenehuarākaumata.
Ewāteaanaitepurimangawakarererangiterōrahie9m3mōngākātene.Kataeatekawetaeatukitetaumahamōrahie2400kgongākātene.
Kautainaetekamupenewakarererangiteutuote$37mōiakāteneika,mete$22mōiakātenehuarākau.
Whakamahiangātikangarorohikomōtekauwhatarārangiheiwhakatauitemahakātenearotaumōiamomohuaheiwhakamōrahiitemoniwhiwhikitekamupenewakarererangimōtekawengaika,huarākauhoki.
Kahomaitetukutukukeiraroheiāwhinakitewhakautuitepātai.
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(c) Kauruatutekamupenewakarererangikitētahikirimanaheikaweitemōkitoote10kāteneoteikamata,metemōkitoote20kāteneotehuarākaumataiiarā.
Tātaihiatehuamoniwhiwhimōrahihōuatekamupenewakarererangimōtekawehua.
Parahautiatōwhakautukingātātaingamengāwhakaaroarorau.
(b) TheX30isalsousedtocarryfreshfishandfruitbetweenthetwocities.Theproduceispackagedincartonswithavolumeof0.1m3.Freshfishcartonsweigh32kg.Freshfruitcartonsweigh20kg.
Thecargoholdoftheaircrafthasspacefor9m3ofcartons.Itcancarryamaximumof2400kgofthecartons.
Theairlinecharges$37percartonoffishand$22percartonoffruit.
Uselinearprogrammingtechniquestodeterminetheoptimalnumberofcartonsofeachtypeofproducetomaximisetheairline’sfreightincomefromfishandfruit.
Thegridbelowisprovidedtohelpyoutoanswerthequestion.
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(c) Theairlineentersintoacontracttocarryaminimumof10cartonsoffreshfishandaminimumof20cartonsoffreshfruitdaily.
Calculatetheairline’snewmaximumfreightincome.
Justifyyouranswerwithcalculationsandreasoning.
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PĀTAI TUARUA
(a) Keitekauwhataoraroewhakaaturiaanatewhārite =y x x2 cos enohoaitexheitātoro(radian).
5 x
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Mātewhakamahiitetikangaweheruamengāuaratīmataote2mete5,mahiakiaruangāwhitiauauheikimiiteuaraāwhiwhiotepūtakeotewhārite =x x2 cos 0 etakotoanaiwaengaiēneiuaraerua.
QUESTION TWO
(a) Thegraphshownbelowhastheequation =y x x2 cos ,wherexisinradians.
5 x
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Usingthebisectionmethodwithstartingvalues2and5,performtwoiterationstofindanapproximatevalueoftherootoftheequation =x x2 cos 0 thatliesbetweenthesetwovalues.
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(b) WhakamahiatetikangaNewton-Raphsonheikimiotinga,kiatikakitematiwhaiirae2,kitewhāritee0.2x sin x=0(xā-tātoro)etakotoanaiwaengaix=1mex=20.
= +yx
x xdd
0.2e sin e cosx x0.2 0.2
Tuhipoka: Mēnā ko y = e0.2x sin x,
kātahi ko
(b) UsetheNewton-Raphsonmethodtofind,correctto2decimalplaces,asolutionoftheequatione0.2xsinx=0(xinradians)thatliesbetweenx=1andx=20.
=
= +
y xyx
x x
Note: If e sin ,
then dd
0.2e sin e cos
x
x x
0.2
0.2 0.2
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(c) Mātewhakamahiitekauwhataotētahipāngay = f (x)keiraro,whakaatumāngātikangaāhuahanga MEtewhakamāramaā-tuhi,kapēheaetaeaaitewhakamahitetikangaNewton-Raphsonheikimiiteāwhiwhitangamōtepūtakeotewhāritef (x)=0kapāiwaengaote5mete10.
KōrerotiaētahihuakinoitewhakamahiitetikangaNewton-Raphsonheiwhakaāwhiwhiitēneipūtake.
5 10 15x
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(c) Usingthegraphofafunctiony = f (x)below,showgeometricallyANDwithawrittenexplanationhowtheNewton-Raphsonmethodcouldbeusedtofindanapproximationtotherootoftheequationf (x)=0thatoccursbetween5and10.
StateanydisadvantagesofusingtheNewton-Raphsonmethodtoapproximatethisroot.
5 10 15x
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PĀTAI TUATORU
(a) KotētahiongāwakarererangiitirawakawhakarerehiaeAirNewZealandkoteBombardierQ300.E50ngāpāhihikataeatekawe.Tērātētahirerengakī,maiiNgāmotukiTeWhanganui-a-Tara,meēneimomoutuhaere.
Momo utu haere Utu mō ia pāhihiKapotūru $83
PenapenaAtamai $132HangoreRawa $205
MēnākatohuaaitetokomahaongāpāhihiihokotīkitiKapotūru,etohuanaabitetokomahaongāpāhihiihokotīkitiPenapenaAtamai,ā,etohuanaacitetokomahaongāpāhihiihokotīkitiHangoreRawa,katohutēneipūnahaongāwhāriteingāmōhiohiomōtēneirerenga.
a + b + c=5083a+132b+205c=71822a = c
WhakaotiatēneipūnahawhāriteheikimitokohiangāpāhihiotererengaihokotīkitiKapotūru.
QUESTION THREE
(a) OneofthesmallestaircraftthatAirNewZealandfliesistheBombardierQ300.Itseats50passengers.OneparticularflightfromNewPlymouthtoWellington,whichwasfull,hadthefollowingfares.
Type of fare Cost per passengerGrabaseat $83SmartSaver $132FlexiPlus $205
IfarepresentsthenumberofpassengerswhoboughtaGrabaseatfare,brepresentsthenumberofpassengerswhoboughtaSmartSaverfare,andcrepresentsthenumberofpassengerswhoboughtaFlexiPlusfare,thenthefollowingsystemofequationsrepresentsinformationaboutthisflight.
a + b + c=5083a+132b+205c=71822a = c
Solvethissystemofequationstofindouthowmanypassengers,ontheflight,paidaGrabaseatfare.
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(b) KawhakahaereaAirNewZealandingāmomowakarereranginuietoru,eāheianatereremaiiTāmakiMakauraukiVancouver:koteBoeing747-400,teBoeing777-300meteBoeing777-200.
Etorungāmomotīkitiewāteaanamōtēneirerenga:
• pakihimatua• whakamoamoamatua• whakamoamoa
Keitepapatauewhaiakeneingāmōhiohioepāanakingāmoniwhiwhimōiarerengamaiingāhokotīkiti.
Momo waka rererangi
Momo tīkiti / tokomaha o ngā pāhihi Moni whiwhi ina kī te waka
rererangiPakihi matua Whakamoamoa
matuaWhakamoamoa
747-400 46 39 294 $824 903777-300 44 44 244 $751 960777-200 26 36 242 $620 142
Kimihiateutumōiamomotīkitimātewhakatūitētahipūnahawhārite.
(b) AirNewZealandoperatesthreetypesoflargeaircraftthatarecapableofflyingfromAucklandtoVancouver:theBoeing747-400,theBoeing777-300andtheBoeing777-200.
Threetypesofticketsareavailableforthisflight:
• businesspremier• premiumeconomy• economy
Thefollowingtableprovidesinformationabouttheincomeforeachflightfromticketsales.
Type of aircraft
Type of ticket / number of passengers Income when aircraft is fullBusiness
premierPremium economy
Economy
747-400 46 39 294 $824903777-300 44 44 244 $751960777-200 26 36 242 $620142
Findthecostofeachtypeofticketbysettingupasystemofequations.
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(c) Hemaioroorongāwhāritetukutahiewhaiakenei.
5p–7q+18=015p=21q–7
Kīmaiheahaai,ā,whiriwhiriahokingāuaraoAmeBitepūnahawhāritekahomaikiraro,kiapāaitēneimomomaiorooroōritekingāpapaāhuahangaetohuaanaengāwhāriteewhaiakenei.
4x+2y–10z =17 15z–6x–3y =32 5y–7 =Ax + Bz
(c) Thefollowingsimultaneousequationsareinconsistent.
5p–7q+18=015p=21q–7
Statewhy,andfindthevaluesofAandBinthesystemofequationsgivenbelow,sothatthissametypeofinconsistencyappliestothegeometricplanesrepresentedbythefollowingequations.
4x+2y–10z =17 15z–6x–3y =32 5y–7 =Ax + Bz
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KeiraroneitētahitukutukuanōkataeaekoetewhakamahiheiāwhinakitewhakautuitePātaiTuatahi(a).
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BelowisasparegridthatyoucanusetohelpyouanswerQuestionOne(a).
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KeiraroneitētahitukutukuanōkataeaekoetewhakamahiheiāwhinakitewhakautuitePātaiTuatahi(b).
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BelowisasparegridthatyoucanusetohelpyouanswerQuestionOne(b).
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He puka anō mēnā ka hiahiatia.Tuhia te (ngā) tau pātai mēnā e hāngai ana.
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QUESTION NUMBER
Extra paper if required.Write the question number(s) if applicable.
Level 3 Statistics and Modelling, 201290644 Solve equations
9.30 am Wednesday 21 November 2012 Credits: Four
Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.
You should attempt ALL the questions in this booklet.
Show ALL working.
Make sure that you have the Formulae and Tables Booklet L3–STATF.
If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.
Check that this booklet has pages 2 – 27 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
© New Zealand Qualifications Authority, 2012. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
ASSESSOR’S USE ONLY Achievement Criteria
Achievement Achievement with Merit Achievement with ExcellenceSolve equations. Solve problems involving
equations.Analyse or interpret the outcome or the process used to solve equations or linear programming problems.
Overall level of performance
English translation of the wording on the front cover
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