mathematics stage 2c 2d calc assumed exam 2012

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Western Australian Certicate of Education

Examination, 2012

Question/Answer Booklet

Copyright School Curriculum and Standards Authority 2012

*MAT2CD-S2*MAT2CD-S2

MATHEMATICS

2C/2DSection Two:

Calculator-assumed

Time allowed for this sectionReading time before commencing work: ten minutesWorking time for section: one hundred minutes

Materials required/recommended for this sectionTo be provided by the supervisorThis Question/Answer BookletFormula Sheet (retained from Section One)

To be provided by the candidate

Standard items: pens (blue/black preferred), pencils (including coloured), sharpener,correction tape/uid, eraser, ruler, highlighters

Special items: drawing instruments, templates, notes on two unfolded sheets of A4 paper,and up to three calculators approved for use in the WACE examinations

Important note to candidatesNo other items may be taken into the examination room. It is yourresponsibility to ensure thatyou do not have any unauthorised notes or other items of a non-personal nature in theexamination room. If you have any unauthorised material with you, hand it to the supervisorbeforereading any further.

Number of additionalanswer booklets used(if applicable):

Ref: 12-088


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CALCULATOR-ASSUMEDMATHEMATICS 2C/2D 2

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Section

Section One:Calculator-free

Section Two:Calculator-assumed

Number ofquestionsavailable

6

12

Number ofquestions tobe answered

6

12

Workingtime

(minutes)

50

100

Total

Marksavailable

50

100

150

Percentageof total exam

3313

6623

100

Instructions to candidates

1. The rules for the conduct of Western Australian external examinations are detailed inthe Year 12 Information Handbook 2012. Sitting this examination implies that youagree to abide by these rules.

2. Write your answers in the spaces provided in this Question/Answer Booklet. Spare pagesare included at the end of this booklet. They can be used for planning your responsesand/or as additional space if required to continue an answer. Planning: If you use the spare pages for planning, indicate this clearly at the top of

the page.

Continuing an answer: If you need to use the space to continue an answer, indicate inthe original answer space where the answer is continued, i.e. give the page number.Fill in the number of the question(s) that you are continuing to answer at the top of thepage.

3. Show all your working clearly.Your working should be in sufcient detail to allow youranswers to be checked readily and for marks to be awarded for reasoning. Incorrectanswers given without supporting reasoning cannot be allocated any marks. For anyquestion or part question worth more than two marks, valid working or justication is

required to receive full marks. If you repeat an answer to any question, ensure that youcancel the answer you do not wish to have marked.

4. It is recommended that you do not use pencil, except in diagrams.

5. The Formula Sheet is nothanded in with your Question/Answer Booklet.

Structure of this paper


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CALCULATOR-ASSUMED 3 MATHEMATICS 2C/2D

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Section Two: Calculator-assumed (100 Marks)

This section has twelve(12)questions. Answer allquestions. Write your answers in the spacesprovided.

Spare pages are included at the end of this booklet. They can be used for planning your

responses and/or as additional space if required to continue an answer. Planning: If you use the spare pages for planning, indicate this clearly at the top of the page. Continuing an answer: If you need to use the space to continue an answer, indicate in the

original answer space where the answer is continued, i.e. give the page number. Fill in thenumber of the question(s) that you are continuing to answer at the top of the page.

Suggested working time: 100 minutes.

Question 7 (4 marks)

Given C = 6.2 102 andD= 7.7 103, determine the value of:

(a) CD. (2 marks)

Give the answer in scientic notation.

(b) CD. (2 marks)

Give the answer to two (2)signicant gures.


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From 1 July 2011, these fees (including GST) were recommended by the Real Estate Institute ofTasmania.

Real Estate Fees/Commission in Tasmania

Selling price of each property Real estate fees

Does not exceed $10 000 10.73% with a minimum of $100

From $10 001 $50 000 $1073 plus 5.85% of excess over $10 000

From $50 001 $100 000 $3413 plus 4.10% of excess over $50 000

From $100 001 upward $5463 plus 3.88% of excess over $100 000

Jenny owns several properties in Tasmania.

(a) If Jenny sells a house for $320 000, how much does she have to pay in real estate fees?(3 marks)

(b) Jenny wants to reduce the number of properties she owns. She is trying to decidewhether to sell a house for $320 000 or sell a group of four apartments, for $80 000each. Which option will result in her paying the smaller amount of real estate fees andhow much less will she pay in fees with this option? (3 marks)


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Marine researchers were interested in estimating the population of sea lions in one of thebreeding colonies on the west coast of Western Australia. On one day, they observed andtagged 23 sea lions in the colony. After allowing time for these tagged sea lions to mix with theothers, they observed a sample of 20 sea lions and found that seven of them had tags.

(a) Use the capture-recapture technique to estimate the total population of the sea lioncolony. (2 marks)

Three days later, the marine researchers observed a sample of 24 sea lions, and found that ve

of these were tagged.

(b) Use this additional information to form a better estimate of the total population of the sealion colony. (2 marks)

The researchers involved in this study were challenged by other marine researchers, whoclaimed that the population estimate was inaccurate because some of the sea lions originallytagged had lost their tags.

(c) (i) If this was the case, would such a claim imply a higher or lower estimate of the

sea lion population than that calculated in Part (b)? (1 mark)

(ii) Justify your answer to Part (c)(i). (1 mark)


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The table below produced by the Australian Bureau of Statistics shows results compiled fromHousehold Use of Information Technology (HUIT) data collected by the Multipurpose HouseholdSurvey (MPHS) for 201011.

Household Use of Information Technology, Australia, 201011

Households

(in thousands)

Home internet accessHouseholds with home internet access

by frequency of access

Households

without

home

internet

access

Households

with home

internet

access

Every

day

At least

weekly

At least

monthly

Never/

Dont

know

New South Wales 568 2164 1666 411 43 44

Victoria 458 1684 1280 354 34 16

Queensland 355 1347 1037 269 40 1

South Australia 160 497 373 109 10 5

Western Australia 163 714 557 129 21 7

Tasmania 62 146 110 29 4 3

Northern Territory 14 51 39 9 2 1

Australian CapitalTerritory

16 122 98 21 1 2

Total households 1796 6725 5160 1331 155 79

Note: All data are rounded.

(a) Determine the total number of households surveyed. (2 marks)

A household is randomly selected from those surveyed.

(b) Determine the probability that the household had home internet access. (2 marks)


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(c) Determine the probability that the household was in Western Australia and accessed theinternet at home every day. (1 mark)

(d) Given that the household had internet access, determine the probability that it was inVictoria. (2 marks)


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A city is to host an economic forum to be attended by the leaders of a number of nations. Toensure the safety of the leaders a sophisticated communication network, linking a number

of control points, is to be set up. There are eight control pointsP1,P

2, ...,P

8and the costs, in

thousands of dollars, of establishing a direct link between points is given in the following table.

P1

P2

P3

P4

P5

P6

P7

P8

P1

- 25 18 16 5 12 19 8

P2

25 - 22 20 24 25 22 23

P3

18 22 - 10 15 16 20 21

P4

16 20 10 - 19 21 28 27

P5

5 24 15 19 - 15 20 10

P6

12 25 16 21 15 - 28 30

P7

19 22 20 28 20 28 - 18

P8

8 23 21 27 10 30 18 -

The city needs to establish a minimal cost network.

(a) Use Prims algorithm to determine the minimal cost. (4 marks)

(b) Represent the solution found in Part (a) as a network. (2 marks)

P1

P3

P2

P5

P4

P6

P7

P8


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(c) The planners realise that the cost of connecting the control pointsP1andP

2can be

reduced by $8000 by using a remote device. By how much does the use of the remotedevice reduce the minimum cost of constructing the network? Justify your solution.

(2 marks)


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Hanna used her calculator to determine a function that modelled the temperature T(in C) on

one day last summer. Her temperature function wasT= 0.01t3+ 0.266t2 0.957t+ 21.77. The

function applied from midnight (t= 0)to midnight (t= 24).

(a) On the axes below sketch the temperature function for the 24 hour period. (4 marks)

T(C)

t(hours)2 4 6 8 10 12 14 16 18 20 22 24

40

38

36

34

32

30

28

2624

22

20

18

16

14

12

10

8

6

42

t (hours)

T(C)

(b) What did Hannas function suggest the temperature was at 5.00 pm? (1 mark)


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(c) The actual maximum temperature of 35.1 C occurred at 1.00 pm. Comment on howaccurate her function was for modelling the maximum temperature. (2 marks)

(d) According to the function, for what percentage of the 24 hour period was the temperature

above 30 C? (2 marks)


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The graph ofy=x

2+ 8has been drawn on the axes below.

12

10

8

6

4

2

2

4

6

8

10

4 2 2 4 6

y

x

(a) Sketch the graph ofy= 1.5xon the axes given. (4 marks)

(b) Use the graph to estimate the solution of the equation 1.5x=x

2+ 8. Show on your graph

where you found the solution. (2 marks)


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(c) Use your calculator to determine the second solution to the equation 1.5x=x

2+ 8.

(2 marks)

(d) The graphs of y= 1.5xandy= 2x+ c are to have one point of intersection.

Which would be the better estimate of csuch that the graphs ofy= 1.5xandy= 2x+ c

have only one point of intersection: c = 3, c= 0, c= 3? Explain your answer. (3 marks)


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When travelling from Perth to London, passengers need to take two ights. Unfortunately,

the rst ight is sometimes delayed, and this can result in passengers missing their second,

connecting ight.

Of 570 passengers who travelled from Perth to London on two ights, 152 had their rst ightdelayed. Of these delayed passengers, 65 missed their second, connecting ight. A total of 96

passengers missed their second, connecting ight.

LetDrepresent the set of passengers with their rst ight delayed andMrepresent the set ofpassengers who missed their second, connecting ight.

(a) Complete the Venn diagram below to show this information. Complete all regions. (4 marks)

D MD M

(b) Determine: n(D M). (1 mark)

(c) If one of these passengers is selected at random, determine

(i) P(D|M). (2 marks)

(ii) P(DM). (2 marks)


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(d) Determine the proportion of passengers who missed their second, connecting ight, even

though their rst ight was not delayed. (2 marks)


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In a mining operation ore is moved from a central stockpile S to a loading station L throughan ore processing plant consisting of six processors A, B, C, D, E and F linked by a systemof conveyor belts. The network below displays the operation with the arcs representing theconveyor belts. The number on each arc represents the maximum amount of ore, in tonnes per

minute, that can be moved along that conveyor belt.

(a) What is the maximum amount of ore, in tonnes per minute, that can be moved fromthe stockpile S to the loading station L? Show systematic working to allow your solutionto be checked. (5 marks)


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(b) What effect, if any, would there be on the maximum ow of ore from S to L if the capacity

of the conveyor belt FE was increased by 15 tonnes per minute? Justify your answer. (2 marks)


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The estimated resident population of Western Australia at 30 June 2011, according to theAustralian Bureau of Statistics (ABS), has been grouped into equal-sized intervals and thesedata are shown in the table and frequency histogram below.

Estimated resident population forWestern Australia, at 30 June 2011

Age (years)

Frequency

(number of

persons)

014 453 747

1529 515 339

3044 502 452

4559 462 003

6074 286 1877589 114 436

90104 12 246

Total 2 346 410

(a) For these data, determine

(i) the modal class. (1 mark)

(ii) the largest possible range. (1 mark)

(iii) the mean. (1 mark)

(iv) the standard deviation. (1 mark)

(v) the median class. Justify your answer. (2 marks)

600 000

500 000

400 000

300 000

200 000

100 000

0

Frequency

7 22 37 52 67 82 97

Age (years)


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The estimated resident population of Tasmania at 30 June 2011, according to the ABS, hasbeen grouped into equal-sized intervals and these data are shown in the table and frequencyhistogram below. The mean age for this population is 39.48 years and the median class is 3044years.

Estimated resident population forTasmania, at 30 June 2011

Age (years)

Frequency

(number of

persons)

014 97 694

1529 96 220

3044 94 446

4559 107 789

6074 78 3377589 32 710

90104 12 246

Total 519 442

(b) Does Western Australia or Tasmania tend to have ages that are more spread out? Justifyyour choice. (2 marks)

(c) A typical resident, in terms of their age, is to be chosen to represent each state. Shouldthese typical residents be chosen from the same age intervals? Explain your answer.

(3 marks)

120 000

100 000

80 000

60 000

40 000

20 000

0

Frequency

7 22 37 52 67 82 97

Age (years)


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The Jones family is planning the back garden of their new house. The diagram below (not drawnto scale) shows the area. All dimensions are in metres and angles are in degrees.

D 8.2m C 8.4m

9.0m

9.0m

A 8.2m B

75

8.2 m

9.0 m

8.2 m

9.0 m

8.4 m

The Jones family decides to build a limestone wall (one block high) from A to C to partition off theplayground area ACD from the rest of the back garden.

(a) Using trigonometry, calculate the length of this wall. (3 marks)

(b) Limestone blocks come in 500 mm lengths. How many blocks will the Jones family needto buy? (2 marks)


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(c) The playground area is to be covered by a shade sail with poles at A, C and D. Usingtrigonometry, determine

(i) the size of the angle ACD. (3 marks)

(ii) the area of the playground ACD. (2 marks)


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Triangle PQR has a base of (21 2x)cm and perpendicular height ofxcm, as shown in thetriangle below.

21 2x

x

P

RQ

(a) Ifx= 4 ,calculate the area of the triangle. (2 marks)

(b) Determine an expression for the area function of the triangle in terms ofx, i.e. completethe following:

A(x)= (1 mark)

(c) Sketch the area function on the axes below. Label all signicant features clearly.

(4 marks)

A(x)

x


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(d) What is the maximum possible area of triangle PQR? (1 mark)

(e) Line AB has been drawn parallel with base QR of the triangle and passing throughpoint P.

21 2

x

A

Q R

P S B

(i) For the point S, on line AB, what is the maximum area of triangle QRS? (1 mark)

(ii) If point S could be moved anywhere along line AB, describe how the locationwould affect the area of triangle QRS. (1 mark)

End of questions


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Additional working space

Question number:


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Question number:


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Question number:


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Question number:


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ACKNOWLEDGEMENTS

This examination paper apart from any third party copyright material contained in it may be freely copied, or communicated on anintranet, for non-commercial purposes in educational institutions, provided that it is not changed and that the School Curriculum andStandards Authority is acknowledged as the copyright owner.

Copying or communication for any other purpose can be done only within the terms of the Copyright Act or with prior writtenpermission of the Authority. Copying or communication of any third party copyright material can be done only within the terms of the

Copyright Act or with permission of the copyright owners.

Section Two

Question 8 Data source: Independent Real Estate Consulting. (n.d.).Agents fees, realestate fees, real estate commission TAS, effective 1 July 2011. RetrievedMarch 13, 2012, from http://irec.com.au/index.php?c=4.

Question 10Data source:

Australian Bureau of Statistics. (2012, February 23). 8146.0Household use of information technology, Australia, 201011 [Excel

spreadsheet]. Retrieved March, 2012, from www.abs.gov.au. Licensedunder a Creative Commons Attribution 2.5 Australia licence.

Question 16 Data source:Australian Bureau of Statistics. (2011, December 19).31010DO002_201106 Australian demographic statistics, June 2011 [Excelspreadsheet]. Retrieved March 27, 2012, from abs.gov.au. Licensed undera Creative Commons Attribution 2.5 Australia licence.

mathematics stage 2c 2d calc assumed exam 2012

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