Te Tauanga me te Whakatauira, Kaupae 3, 2012 · 2017. 5. 30. · Te Tauanga me te Whakatauira, Kaupae 3, 2012 90644M Te whakaoti whārite 9.30 i te ata Rāapa 21 Whiringa-ā-rangi

See back cover for an English translation of this cover

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

3SUPERVISOR’S USE ONLY

9 0 6 4 4 M

© 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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Te Tauanga me te Whakatauira 90644M, 2012

MĀ TE KAIMĀKA

ANAKE

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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Statistics and Modelling 90644, 2012

ASSESSOR’S USE ONLY

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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

y

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Mātewhakamahiitetikangaweheruamengāuaratīmataote2mete5,mahiakiaruangāwhitiauauheikimiiteuaraāwhiwhiotepūtakeotewhārite =x x2 cos 0 etakotoanaiwaengaiēneiuaraerua.

QUESTION TWO

(a) Thegraphshownbelowhastheequation =y x x2 cos ,wherexisinradians.

5 x

y

–5

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15

20

–10

–15

–20

0 10 15

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

y

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(c) Usingthegraphofafunctiony = f (x)below,showgeometricallyANDwithawrittenexplanationhowtheNewton-Raphsonmethodcouldbeusedtofindanapproximationtotherootoftheequationf (x)=0thatoccursbetween5and10.

StateanydisadvantagesofusingtheNewton-Raphsonmethodtoapproximatethisroot.

5 10 15x

y

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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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KeiraroneitētahitukutukuanōkataeaekoetewhakamahiheiāwhinakitewhakautuitePātaiTuatahi(b).

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ANAKETAU PĀTAI

He puka anō mēnā ka hiahiatia.Tuhia te (ngā) tau pātai mēnā e hāngai ana.

27



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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Te Tauanga me te Whakatauira, Kaupae 3, 2012 · 2017. 5. 30. · Te Tauanga me te Whakatauira, Kaupae 3, 2012 90644M Te whakaoti whārite 9.30 i te ata Rāapa 21 Whiringa-ā-rangi