Te Tauanga me te Whakatauira, Kaupae 3, 2012

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1 See back cover for an English translation of this cover 90644M SUPERVISOR S USE ONLY Te Tauanga me te Whakatauira, Kaupae 3, M 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. MĀ TE KAIMĀKA ANAKE Paearu Paetae Te whakaoti whārite. Paetae Paetae Kaiaka Paetae Kairangi Te whakaoti rapanga e whai wāhi mai ana tēnei mea te whārite. Whakakaotanga o te tairanga mahinga 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. Mana Tohu Mātauranga o Aotearoa, 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.

2 2 Kia 55 meneti hei whakautu i ngā pātai o tēnei pukapuka. MĀ TE KAIMĀKA ANAKE PĀTAI TUATAHI (a) Ka whakarere tētahi kamupene waka rererangi i ngā momo waka rererangi e rua i waenga i ngā tāone nui e rua. Ka kawe ngā waka rererangi X30 i ngā pāhihi e 30, ā, e 45 meneti te roa o te rerenga. Ka kawe ngā waka rererangi X15 i ngā pāhihi e 15, ā, e 60 meneti te roa o te rerenga. I tētahi rā noa, ka whakarere te kamupene waka rererangi i te mōrahi pāhihi e 900. E ai ki ngā ture o te manatū rererangi mō ngā haora rere o ngā kaihautū waka rererangi, ka taea e te kamupene waka rererangi te whakarere mō te mōrahi o te 45 haora i ia rā. Mēnā ka mahia te mōrahi o ngā tāpuitanga i ia rerenga, ka pā ngā aukati e whai ake nei ki tēnei tūāhua: 30x + 15y x + 1.0y 45 x 0 y 0 ina ko x te maha o ngā rerenga ka mahia e te waka rererangi X30, ā, ko y te maha o ngā rerenga ka mahia e te waka rererangi X15. I ngā rā e kawe ana ngā waka rererangi katoa i te tokomaha mōrahi o ngā pāhihi, ka whakaaturia te huamoni a te kamupene waka rererangi mai i te kawe pāhihi i ngā rerenga mā te whārite P = 330x + 195y. Tātaitia te mōrahi huamoni ka taea pea i ia rā e te kamupene waka rererangi mai i te kawe pāhihi. Ka homai te tukutuku kei raro hei āwhina i a koe ki te whakautu i te pātai. He tukutuku anō kei te whārangi Te Tauanga me te Whakatauira 90644M, 2012

3 3 MĀ TE KAIMĀKA ANAKE Te Tauanga me te Whakatauira 90644M, 2012

4 4 You are advised to spend 55 minutes answering the questions in this booklet. ASSESSOR S USE ONLY QUESTION ONE (a) An airline flies two different types of aircraft between two cities. X30 aircraft carry 30 passengers and take 45 minutes to complete the flight. X15 aircraft carry 15 passengers and take 60 minutes to complete the flight. On any given day, the airline will fly a maximum of 900 passengers. Under aviation authority regulations for pilot flying hours, the airline is allowed to fly for a maximum of 45 hours per day. Assuming the maximum number of bookings is made on each flight, the following constraints apply to this situation: 30x + 15y x + 1.0y 45 x 0 y 0 where x is the number of flights made by the X30 aircraft and y is the number of flights made by the X15 aircraft. On a day with all aircraft carrying the maximum number of passengers, the airline s profit from carrying passengers on the flights is given by the equation P = 330x + 195y. Calculate the airline s potential maximum daily profit from carrying passengers. The grid below is provided to help you to answer the question. There is a spare grid on page Statistics and Modelling 90644, 2012

5 5 ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

6 6 (b) Ka whakamahia hoki te X30 hei kawe i ngā ika me ngā huarākau mata i waenga i ngā tāone nui e rua. Ka tākaitia ngā hua i ngā kātene me te rōrahi o te 0.1 m 3. E 32 kg te taumaha o ngā kātene ika mata. E 20 kg te taumaha o ngā kātene huarākau mata. E wātea ana i te purimanga waka rererangi te rōrahi e 9 m 3 mō ngā kātene. Ka taea te kawe tae atu ki te taumaha mōrahi e 2400 kg o ngā kātene. Ka utaina e te kamupene waka rererangi te utu o te $37 mō ia kātene ika, me te $22 mō ia kātene huarākau. MĀ TE KAIMĀKA ANAKE Whakamahia ngā tikanga rorohiko mō te kauwhata rārangi hei whakatau i te maha kātene arotau mō ia momo hua hei whakamōrahi i te moni whiwhi ki te kamupene waka rererangi mō te kawenga ika, huarākau hoki. Ka homai te tukutuku kei raro hei āwhina ki te whakautu i te pātai. He tukutuku anō kei te whārangi Te Tauanga me te Whakatauira 90644M, 2012

7 7 MĀ TE KAIMĀKA ANAKE (c) Ka uru atu te kamupene waka rererangi ki tētahi kirimana hei kawe i te mōkito o te 10 kātene o te ika mata, me te mōkito o te 20 kātene o te huarākau mata i ia rā. Tātaihia te huamoni whiwhi mōrahi hōu a te kamupene waka rererangi mō te kawe hua. Parahautia tō whakautu ki ngā tātainga me ngā whakaaro arorau. Te Tauanga me te Whakatauira 90644M, 2012

8 (b) The X30 is also used to carry fresh fish and fruit between the two cities. The produce is packaged in cartons with a volume of 0.1 m 3. Fresh fish cartons weigh 32 kg. Fresh fruit cartons weigh 20 kg. The cargo hold of the aircraft has space for 9 m 3 of cartons. It can carry a maximum of 2400 kg of the cartons. The airline charges $37 per carton of fish and $22 per carton of fruit. 8 ASSESSOR S USE ONLY Use linear programming techniques to determine the optimal number of cartons of each type of produce to maximise the airline s freight income from fish and fruit. The grid below is provided to help you to answer the question. There is a spare grid on page Statistics and Modelling 90644, 2012

9 9 ASSESSOR S USE ONLY (c) The airline enters into a contract to carry a minimum of 10 cartons of fresh fish and a minimum of 20 cartons of fresh fruit daily. Calculate the airline s new maximum freight income. Justify your answer with calculations and reasoning. Statistics and Modelling 90644, 2012

10 10 PĀTAI TUARUA MĀ TE KAIMĀKA ANAKE (a) Kei te kauwhata o raro e whakaaturia ana te whārite y = 2xcos x e noho ai te x hei tātoro (radian). y x Mā te whakamahi i te tikanga weherua me ngā uara tīmata o te 2 me te 5, mahia kia rua ngā whitiauau hei kimi i te uara āwhiwhi o te pūtake o te whārite 2xcos x = 0 e takoto ana i waenga i ēnei uara e rua. Te Tauanga me te Whakatauira 90644M, 2012

11 11 QUESTION TWO ASSESSOR S USE ONLY (a) The graph shown below has the equation y = 2xcos x, where x is in radians y x Using the bisection method with starting values 2 and 5, perform two iterations to find an approximate value of the root of the equation 2xcos x = 0 that lies between these two values. Statistics and Modelling 90644, 2012

12 12 (b) Whakamahia te tikanga Newton-Raphson hei kimi otinga, kia tika ki te mati whaiira e 2, ki te whārite e 0.2x sin x = 0 (x ā-tātoro) e takoto ana i waenga i x = 1 me x = 20. MĀ TE KAIMĀKA ANAKE Tuhipoka: Mēnā ko y = e 0.2x sin x, dy 0.2x 0.2x kātahi ko = 0.2e sin x+ e cos x dx Te Tauanga me te Whakatauira 90644M, 2012

13 (b) Use the Newton-Raphson method to find, correct to 2 decimal places, a solution of the equation e 0.2x sin x = 0 (x in radians) that lies between x = 1 and x = x Note:If y = e sin x, y then d 0.2x 0.2x = 0.2e sin x+ e cos x dx 13 ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

14 14 (c) Mā te whakamahi i te kauwhata o tētahi pānga y = f (x) kei raro, whakaatu mā ngā tikanga āhuahanga ME te whakamārama ā-tuhi, ka pēhea e taea ai te whakamahi te tikanga Newton Raphson hei kimi i te āwhiwhitanga mō te pūtake o te whārite f (x) = 0 ka pā i waenga o te 5 me te 10. Kōrerotia ētahi huakino i te whakamahi i te tikanga Newton-Raphson hei whakaāwhiwhi i tēnei pūtake. MĀ TE KAIMĀKA ANAKE 15 y x Te Tauanga me te Whakatauira 90644M, 2012

15 (c) Using the graph of a function y = f (x) below, show geometrically AND with a written explanation how the Newton-Raphson method could be used to find an approximation to the root of the equation f (x) = 0 that occurs between 5 and 10. State any disadvantages of using the Newton-Raphson method to approximate this root. 15 ASSESSOR S USE ONLY 15 y x Statistics and Modelling 90644, 2012

16 16 PĀTAI TUATORU (a) Ko tētahi o ngā waka rererangi iti rawa ka whakarerehia e Air New Zealand ko te Bombardier Q300. E 50 ngā pāhihi ka taea te kawe. Tērā tētahi rerenga kī, mai i Ngāmotu ki Te Whanganui-a-Tara, me ēnei momo utu haere. MĀ TE KAIMĀKA ANAKE Momo utu haere Utu mō ia pāhihi Kapotūru $83 Penapena Atamai $132 Hangore Rawa $205 Mēnā ka tohu a a i te tokomaha o ngā pāhihi i hoko tīkiti Kapotūru, e tohu ana a b i te tokomaha o ngā pāhihi i hoko tīkiti Penapena Atamai, ā, e tohu ana a c i te tokomaha o ngā pāhihi i hoko tīkiti Hangore Rawa, ka tohu tēnei pūnaha o ngā whārite i ngā mōhiohio mō tēnei rerenga. a + b + c = 50 83a + 132b + 205c = a = c Whakaotia tēnei pūnaha whārite hei kimi tokohia ngā pāhihi o te rerenga i hoko tīkiti Kapotūru. Te Tauanga me te Whakatauira 90644M, 2012

17 17 QUESTION THREE ASSESSOR S USE ONLY (a) One of the smallest aircraft that Air New Zealand flies is the Bombardier Q300. It seats 50 passengers. One particular flight from New Plymouth to Wellington, which was full, had the following fares. Type of fare Cost per passenger Grabaseat $83 Smart Saver $132 Flexi Plus $205 If a represents the number of passengers who bought a Grabaseat fare, b represents the number of passengers who bought a Smart Saver fare, and c represents the number of passengers who bought a Flexi Plus fare, then the following system of equations represents information about this flight. a + b + c = 50 83a + 132b + 205c = a = c Solve this system of equations to find out how many passengers, on the flight, paid a Grabaseat fare. Statistics and Modelling 90644, 2012

18 18 (b) Ka whakahaere a Air New Zealand i ngā momo waka rererangi nui e toru, e āhei ana te rere mai i Tāmaki Makaurau ki Vancouver: ko te Boeing , te Boeing me te Boeing E toru ngā momo tīkiti e wātea ana mō tēnei rerenga: pakihi matua whakamoamoa matua whakamoamoa Kei te papatau e whai ake nei ngā mōhiohio e pā ana ki ngā moni whiwhi mō ia rerenga mai i ngā hoko tīkiti. Momo waka rererangi Pakihi matua Momo tīkiti / tokomaha o ngā pāhihi Whakamoamoa matua Whakamoamoa Moni whiwhi ina kī te waka rererangi $ $ $ MĀ TE KAIMĀKA ANAKE Kimihia te utu mō ia momo tīkiti mā te whakatū i tētahi pūnaha whārite. Te Tauanga me te Whakatauira 90644M, 2012

19 (b) Air New Zealand operates three types of large aircraft that are capable of flying from Auckland to Vancouver: the Boeing , the Boeing and the Boeing Three types of tickets are available for this flight: business premier premium economy economy The following table provides information about the income for each flight from ticket sales. Type of aircraft Type of ticket / number of passengers Business premier 19 Premium economy Economy Income when aircraft is full $ $ $ ASSESSOR S USE ONLY Find the cost of each type of ticket by setting up a system of equations. Statistics and Modelling 90644, 2012

20 20 (c) He maiorooro ngā whārite tukutahi e whai ake nei. 5p 7q + 18 = 0 15p = 21q 7 MĀ TE KAIMĀKA ANAKE Kī mai he aha ai, ā, whiriwhiria hoki ngā uara o A me B i te pūnaha whārite ka homai ki raro, kia pā ai tēnei momo maiorooro ōrite ki ngā papa āhuahanga e tohua ana e ngā whārite e whai ake nei. 4x + 2y 10z = 17 15z 6x 3y = 32 5y 7 = Ax + Bz Te Tauanga me te Whakatauira 90644M, 2012

21 (c) The following simultaneous equations are inconsistent. 5p 7q + 18 = 0 15p = 21q 7 State why, and find the values of A and B in the system of equations given below, so that this same type of inconsistency applies to the geometric planes represented by the following equations. 4x + 2y 10z = 17 15z 6x 3y = 32 5y 7 = Ax + Bz 21 ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

22 22 Kei raro nei tētahi tukutuku anō ka taea e koe te whakamahi hei āwhina ki te whakautu i te Pātai Tuatahi (a). MĀ TE KAIMĀKA ANAKE Te Tauanga me te Whakatauira 90644M, 2012

23 23 Below is a spare grid that you can use to help you answer Question One (a). ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

24 24 Kei raro nei tētahi tukutuku anō ka taea e koe te whakamahi hei āwhina ki te whakautu i te Pātai Tuatahi (b). MĀ TE KAIMĀKA ANAKE Te Tauanga me te Whakatauira 90644M, 2012

25 25 Below is a spare grid that you can use to help you answer Question One (b). ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

26 26 TAU PĀTAI He puka anō mēnā ka hiahiatia. Tuhia te (ngā) tau pātai mēnā e hāngai ana. MĀ TE KAIMĀKA ANAKE Te Tauanga me te Whakatauira 90644M, 2012

27 27 QUESTION NUMBER Extra paper if required. Write the question number(s) if applicable. ASSESSOR S USE ONLY Statistics and Modelling 90644, 2012

28 English translation of the wording on the front cover Level 3 Statistics and Modelling, Solve equations 9.30 am Wednesday 21 November 2012 Credits: Four 90644M 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. ASSESSOR S USE ONLY Achievement Criteria Solve equations. Achievement Achievement with Merit Achievement with Excellence Solve problems involving equations. Overall level of performance Analyse or interpret the outcome or the process used to solve equations or linear programming problems. New Zealand Qualifications Authority, All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

Matawhenua, Kaupae 1, 2012

1 See back cover for an English translation of this cover 91010MR Matawhenua, Kaupae 1, 2012 91010M Te whakahāngai i ngā huatau me ngā pūkenga matawhenua māmā hei whakaatu i te māramatanga ki tētahi taiao