PLACE YOUR CANDIDATE … · Question 3 (approximately 8 minutes) 20 elite long-jump athletes were involved in a sports performance investigation. It was found that the distance of

© Copyright for part(s) of this examination may be held by individuals and/or organisations other than the Office of Tasmanian Assessment, Standards and Certification.

PLACE YOUR CANDIDATE

LABEL HERE

Pages: 16

Questions: 4

Attachment: Information sheet

Tasmanian Certificate of Education

External Assessment 2019

GENERAL MATHEMATICS (MTG315115)

PART 1 – Bivariate Data Analysis

Time Allowed: 36 minutes

On the basis of your performance in this examination, the examiners will provide a result on the following

criterion taken from the course document:

Criterion 4 Demonstrate knowledge and understanding of bivariate data analysis.

Candidate Instructions

1. You MUST make sure that your responses to the questions in this examination paper will show

your achievement in the criteria being assessed.

2. Answer ALL questions. Answers must be written in the spaces provided on the examination paper.

3. You should make sure you answer all parts within each question so that the criterion can be

assessed.

4. This examination is 3 hours in length. It is recommended that you spend approximately 36 minutes

in total answering the questions in this booklet.

5. The External Examination Information Sheet for General Mathematics can be used throughout the

examination. No other written material is allowed into the examination.

6. All written responses must be in English.

Section Total: /36

MTG315115 Page 2 of 16

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Additional Instructions for Candidates

Logical and mathematical presentation of answers and the statement of the arguments leading to your answer will be considered when assessing this part.

You are expected to provide a calculator approved by the Office of Tasmanian Assessment, Standards and Certification.

For questions worth 1 mark, whilst no workings are required, markers may consider appropriate step(s) taken to come to an answer.

For questions worth 2 or more marks, markers will look at the presentation of answer(s) and at the argument(s) leading to the answer(s).

For questions worth 3 or more marks, you are required to show relevant working.

Spare diagrams and grids have been provided in the back of the booklet for you to use if required.

If you use any of these spare diagrams and/or grids you MUST indicate you have done so in your answer to that question.


Question 1 (approximately 6 minutes) The town of Winston is divided by the Spring River. A developer proposes a new factory sited on the North side of the town. The town council conducts a survey of local residents to see which Winston residents are in favour of the idea of the new factory. The results are tabulated below. (a) Complete the following table to show the percentages of North and South

Winston residents that are in favour of or against the new factory. (2 marks)

Question 1 continues.

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North Winston

South Winston

Proposed

factory site

Number of residents

North Winston

South Winston

Total

In favour of the factory 70 620 690

Against the factory 290 105 395

Total 360 725 1085

Grand total

Percent of residents

North Winston

South Winston

In favour of the factory

Against the factory

Total 100% 100%


Question 1 (continued) (b) Display the percentage information using a segmented column chart. (2 marks) (c) Use the data to write a sentence against the proposed factory. (1 mark)

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(d) Use the data to write a sentence in support of the factory. (1 mark)

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Against the factory

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Question 2 (approximately 10 minutes)

The fuel consumption of a small truck is being tested. The data and graph below show how fuel consumption changes if the truck is loaded with different amounts. (a) Find the linear relationship between the fuel consumption of the truck (F) and the weight

of the load it carries (w). (3 marks)

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(b) What is the intercept with the vertical axis of the graph and what does it represent? (2 marks)

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(c) What is the correlation coefficient? What can you conclude from it? (2 marks)

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0 1 2 Weight of load (tonne)

Fuel consumption (litres per 100 km)

10

11

12

13

14

15

Weight of load (tonne)

0.4 0.9 1.6 1.8 2.4

Fuel consumption (litres per 100 km)

10.4 11.4 13.2 13.5 14.7

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Question 2 (continued)

(d) Use your equation to predict the load carried by the truck if its fuel consumption was 16 litres per 100 km. (2 marks)

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(e) Why are predictions which use extrapolated data unreliable? (1 mark)

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20 elite long-jump athletes were involved in a sports performance investigation. It was found that the distance of an athlete’s long jump (‘D’, measured in m) was related to the speed of their run-up (‘r’, measured in m/s) according to the equation:

D = 0.59 r + 1.35 A plot of residuals to this model is presented below.

(a) What is a ‘residuals plot’? (2 marks)

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(b) The residuals plot is missing one point. The missing data belongs to an athlete who had a run-up speed of 8.2 m/s and a long jump of 6.3 m. Find the residual for this athlete and plot it on the graph. (4 marks)

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6 8 9 10 7

0.1

0.2

0.3

0

- 0.1

- 0.2

- 0.3

Run-up speed (m/s)

Residual (m)



(c) What is the size of the largest residual and why is this relevant? (2 marks)

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Question 4 (approximately 12 minutes) My BBQ Heaven is a shop that specialises in selling barbeques (BBQs). The table and graph shows the actual number of barbeques sold each financial quarter as well as some seasonally adjusted data. (Note: Financial quarters are: Q1 July – Sept, Q2 Oct - Dec etc.)

(a) What is the seasonal pattern? Why do you think this occurs? (1 mark)

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The seasonal index for each quarter, Q1 – Q3 is given in the table below.

(b) Find the Quarter 4 index and write a sentence explaining what it means. (2 marks)

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Year 1 Year 2 Year 3

Financial Quarter Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Quarter number 1 2 3 4 5 6 7 8 9 10 11 12

Actual BBQs sold 30 81 75 40 27 88 78 42 35 89 82 46

De-seasonalised BBQs sold

58 56 57 55 52 61 59 59

1 2 3 4 5 6 7 8 9 10 11 12

50

100

BB

Q S

ale

s

Actual data De-seasonalised

Quarter number

Quarter

Q1 Q2 Q3 Q4

Index 0.52 1.45 1.32

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(c) Use the seasonal indices to de-seasonalise the last four pieces of data in the first table.

(2 marks)

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(d) Complete the graph of de-seasonalised data. (1 mark)

(e) Which was the most successful quarter for sales on a de-seasonalised basis? Explain what your answer means. (2 marks)

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The equation of the de-seasonalised trend line is:

De-seasonalised Number sold = 0.67 Q + 54.4

(f) What does the gradient of this line imply? (1 mark)

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(g) Predict the number of barbeques that My BBQ Heaven will sell in the second quarter of year 5. (3 marks)

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Spare Diagrams

Spare table for question 1 (b)

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Against the factory

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Spare Diagrams

Spare graph for question 3

Spare grid and graph for question 4

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1 2 3 4 5 6 7 8 9 10 11 12

50

100

BB

Q S

ale

s

Actual data De-seasonalised

6 8 9 10 7

0.1

0.2

0.3

0

- 0.1

- 0.2

- 0.3

Run-up speed m/s

Residual (m)

Quarter number

Year 1 Year 2 Year 3

Financial Quarter Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

Quarter number 1 2 3 4 5 6 7 8 9 10 11 12

Actual BBQs sold 30 81 75 40 27 88 78 42 35 89 82 46

De-seasonalised BBQs sold

58 56 57 55 52 61 59 59

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This examination paper and any materials associated with this examination (including answer booklets, cover sheets, rough note

paper, or information sheets) remain the property of the Office of Tasmanian Assessment, Standards and Certification (TASC).



LABEL HERE

Pages: 12

Questions: 3

Attachment: Information Sheet




PART 2 – Growth and Decay in Sequences




Criterion 5 Demonstrate knowledge and understanding of growth and decay in sequences.






assessed.






Section Total: /36


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Spare graph grids have been provided in the back of the booklet for you to use if required.

If you use any of these spare graph grids you MUST indicate you have done so in your answer to that question.


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Question 5 (approximately 7 minutes) The number of seals in an offshore island colony is recorded below. (a) What feature of the data indicates that the population numbers are reducing

exponentially? (1 mark)

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(b) Find the common ratio between terms. (1 mark)

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(c) What is the sequence rule? (1 mark)

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(d) How many seals will there be in year 6? (2 marks)

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(e) During which year will the recorded population be less than 40 seals? (2 marks)

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Year Population

1 4000

2 2800

3 1960


Question 6 (approximately 14 minutes) Upon completing a building design course, Millie applies for work and receives job offers from two different building companies. INTEGRITY HOMES offer her: A starting salary of $64 000 per year with an annual

increase of $ 2 200 each year for 8 years. STAUNCH-BUILT offer her: A starting salary of $50 000 per year with an annual

increase of 12% of the previous year’s salary for 8 years. (a) Detail Millie’s salary over the first 4 years of working for each company by completing the

tables below. (2 marks) (b) What type of sequence (arithmetic, geometric or neither) is represented in each case?

How can you tell? (2 marks)

Integrity Homes: .......................................................................................................

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Staunch-built: ..........................................................................................................

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(c) What is the sequence rule for the salaries offered by each company? (2 marks)

Integrity Homes: ......................................................................................................

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Staunch-built: ..........................................................................................................

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INTEGRITY HOMES

Year 1 Year 2 Year 3 Year 4

$ 64 000

STAUNCH-BUILT

Year 1 Year 2 Year 3 Year 4

$ 50 000



(d) Find the amount earned in the 8th year of working for each company. (2 marks)

Integrity Homes: ...............................................................................................................

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(e) Find the total earned over 8 years of working for each company. (4 marks)

Integrity Homes: ...............................................................................................................

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(f) Based on salary alone, give one reason why Millie might choose each company. (2 marks)

Integrity Homes: ....................................................................................................

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Staunch-built: ........................................................................................................

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A breeder raises mice for sale to a pet shop.

The natural growth rate of a population of mice is 20% per week.

The breeder starts with 130 mice.

Every week the breeder sells 35 mice to the pet shop.

(a) Write a difference (recurrence) equation which models the situation. (2 marks)

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(b) Model the difference (recurrence) equation on your calculator and then complete the table below which shows the number of mice that the breeder has at the end of each week. (2 marks)

Week 0 1 2 3 4 5

Mice 130

(c) Plot a graph of population numbers. (1 mark)


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1 2 3 4 5 weeks 0

100

200

Mice

Time



(d) When will the breeder be unable to make his weekly sale to the pet shop? (1 mark)

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(e) Rewrite your difference (recurrence) equation to model the situation if, instead, the supplier sells 10 mice each week. (1 mark)

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(f) Plot (on the same axes) a graph of the remodelled situation. (1 mark)

(g) How many mice can the supplier sell each week if he wishes to maintain his population at a steady number? (2 marks)

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(h) Write this in the form of a difference (recurrence) equation and plot it on the graph.

(2 marks)

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(i) The supplier says that the business is only worthwhile if he can sustainably sell 35 mice each week. Suggest a way that this might be possible. Include a difference (recurrence) equation in your answer. (3 marks)

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Spare Diagrams


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1 2 3 4 5 weeks 0

100

200

Mice

Time

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LABEL HERE

Pages: 12

Questions: 4





PART 3 – Finance




Criterion 6 Demonstrate knowledge and understanding of standard financial models.






assessed.






Section Total: /36


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Ravi took out a ‘pay day loan’ of $850. Over a period of 8 weeks he repaid the lender a total of $892. Find the flat rate (p.a.) of interest on the loan. (3 marks)

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$2 500 is invested for a period of 3 years at 5.8% p.a. interest compounded monthly.

(a) How much interest does the investment earn? (3 marks)

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(b) What is the effective interest rate on the investment? (2 marks)

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(c) Explain what effective interest means, making reference to the investment above. (2 marks)

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Question 10 (approximately 14 minutes) Marcia has just purchased a new work car costing $42 000. Her business plan involves replacing the car every 5 years. The car’s value will depreciate by 15% p.a. based on the reduced balance. (a) Complete the following depreciation table which shows the car’s value over a 5 year

period. (3 marks)

(b) Confirm the value after 5 years by using an appropriate formula. (1 mark)

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(c) Marcia is already planning for her next car. How much should Marcia try to save over 5 years given that inflation is expected to maintain a 4% p.a. rise in car prices, and that she can sell her old car for its depreciated value? (3 marks)

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Year Value at start of

year ($) Depreciation ($)

Value at end of year ($)

1 42 000

2

3

4

5



(d) In preparation for the next car purchase, Marcia starts a savings fund by depositing $400 per month into an account that pays 4.2% p.a. (compounding monthly).

How much will be in the account after 5 years? (3 marks)

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(e) How much should she have deposited into the account every month if she wished to fully cover the renewal costs of the car? (2 marks)

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(f) Given that Marcia keeps her monthly deposit at $400, use a difference equation to model the balance of her savings fund. (2 marks)

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Question 11 (Approximately 12 minutes)

Noah and Sara take out a housing loan with Austral Loans. They borrow $300 000 at a fixed interest rate of 5.85% p.a. compounded fortnightly with fortnightly payments made over 25 years.

(a) Use an appropriate formula to show that the repayment will be $878.95 (3 marks)

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After 10 years, Noah and Sara find that Federation Bank is offering lower interest rates. They consider refinancing their loan. (This means that they will borrow money from Federation Bank to pay off their loan to Austral Loans.)

(b) How much of the loan is still outstanding after 10 years? (3 marks)

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Noah and Sara borrow the funds required from Federation Bank at 4.80% p.a. compounded fortnightly. They also decide to increase their fortnightly payments to $962.17.

(c) How long will it take to discharge their new loan? (3 marks)

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(d) How much will they save making the changes? (3 marks)

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LABEL HERE

Pages: 12

Questions: 5





PART 4 – Trigonometry




Criterion 7 Demonstrate knowledge and understanding of applications of trigonometry.






assessed.






Section Total: /36


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For questions worth 3 or more marks, you are required to show relevant working. Spare diagrams have been provided in the back of the booklet for you to use if required.

If you use either of these spare diagrams you MUST indicate you have done so in your answer to that question.



(a) Find the area of the triangle shown. (2 marks)

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(b) Find the area of the triangle shown. (3 marks)

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10 cm

17 cm

19 cm

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18 m

20 m 75o



A radio-controlled helicopter is being flown at a consistent height above level ground. Its operator notices that its angle of elevation is 70o. The helicopter moves 120 metres directly away from the operator until the angle of elevation has become 26o.

(a) Detail the above information on the diagram below. (1 mark) (b) Find the distance that the helicopter is from its operator upon the second observation.

(4 marks)

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Two yachts leave Port (X) at 12:00 noon. ‘Liberty’ sails at 20 knots on a bearing of S 20o W. ‘Faith’ sails at 12 knots on a bearing of 130o T. (a) Indicate the yachts’ bearings as angles on the diagram above. (1 mark)

(b) How far has each yacht sailed by 4:20 pm? (2 marks)

Liberty: ...................................................................................................................

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Faith: ......................................................................................................................

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(c) Find the distance separating the yachts at this time. (3 marks)

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Liberty

X

N

Faith



(d) Find the bearing of ‘Liberty’ from ‘Faith’. Write your answer to the nearest degree. (3 marks)

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A flight leaves Melbourne (38°S, 145°E) at 10:00 am Monday to fly to Honolulu (21oN, 158oW) and travels by the shortest possible route flying at a speed of 880 km/h.

(a) Find the distance travelled in kilometres. (4 marks)

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(b) Find the travel time involved in the flight. (2 marks)

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(c) What are the time zones of Melbourne and Honolulu relative to UTC? (2 marks)

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(d) Find the local estimated time of arrival (ETA) of the flight in Honolulu. (3 marks)

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Hobart (42°53’S 147°20’E) is 3718 km directly south of Port Moresby in Papua New Guinea.

(a) Find the angular separation, θ, of the two cities. Write your answer to nearest minute.

(3 marks)

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(b) Find the coordinates of Port Moresby. (3 marks)

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θ

S

Equator

Hobart

N

Port Moresby

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Equator

Port Moresby

Hobart

S

N

Cross Section View

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Spare diagram for question 13



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S

Equator

Hobart

N

Port Moresby

Equator

Port Moresby

Hobart

S

N

Cross Section View

Liberty

X

N

Faith

θ

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LABEL HERE

Pages: 16

Questions: 4





PART 5 – Graphs and Networks




Criterion 8 Demonstrate knowledge and understanding of graphs and networks.

Section Total: /36






assessed.







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Spare diagrams and grids have been provided in the back of the booklet for you to use if required.

If you use any of these spare diagrams and/or grids you MUST indicate you have done so in

your answer to that question.


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The graph below shows how cars flow from A to X through a network of city roads. The weight on

each edge of the graph represents the capacity of the road and is measured in cars per minute.

(a) Find the capacity of the cuts shown. Write them on the graph. (2 marks)

(b) Show other cuts on your graph and use them to find the maximum flow from A to X.

(3 marks)

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(c) It has been suggested that one of the roads: AB, CF or FX should be upgraded to

increase the overall flow of the network. Which of these roads should receive the

upgrade? (1 mark)

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(d) What is the maximum worthwhile upgrade to this road? Explain your answer.

(2 marks)

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Cap = Cap =



The map below details the paths and features of a botanical garden.

(a) A mathematician wishes to make a Hamiltonian

circuit around the garden. What do they mean? Draw an example of such a circuit on the graph.

......................................................................... ......................................................................... ......................................................................... ......................................................................... ......................................................................... (b) What would be meant by an Eulerian circuit of the garden? (1 mark)

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(c) Is an Eulerian circuit possible? How can you tell? (1 mark)

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Entrance

Fountain Rose

garden

Cafe

Tall trees

Salvias Bulbs

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(2 marks)



The management of the garden wishes to upgrade some of the paths to make sure that all features of the garden are wheelchair accessible. The weights on the graph below show the cost in thousands of dollars ($ x 1000) of upgrading each path to wheelchair standard.

(d) Show the paths which should be upgraded if all features are to be wheelchair accessible but with costs kept to the minimum.

(e) What is the minimum cost of the upgrade? (1 mark)

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Management insists that the path between the bulbs and the salvias is to be wheelchair accessible.

(f) Show the paths which should be upgraded for minimum total cost under these circumstances.

(g) How much extra will the upgrade cost given this condition?

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F

E

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B

T

C S

4.2

3.8

2.5 3 5.2

4.6

1.4

3

5 3.2 4

4 3.5

(1 mark)

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B

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4.2

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2.5 3 5.2

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3

5 3.2 4

4 3.5

(1 mark)

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The following tasks are involved in the building of a pedestrian suspension bridge.

A Survey site and draw plans B Excavate site C Build concrete footings and anchorages D Prefabricate towers E Make the cables F Pre-build deck G Lift towers into place and support with cables H Position decking I Build approach ramps J Landscaping and paths The project is scheduled using the graph below. The weights give the length of each activity in days. (a) Which of the tasks listed are the immediate predecessors of task G? (1 mark)

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A (12)

E (7)

G (4) H (4) D (10)

F (14)

X (0) B (2)

C (14) I (7)

Start Finish

J (5)

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approach ramp

main cable

hanging cable

anchorage

footing

deck

tower



(b) Complete the critical path analysis on the graph presented by detailing the EST and LFT

of each activity. (3 marks)

(c) State the critical path. (1 mark)

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(d) What is the shortest possible completion time for building the bridge? (1 mark)

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(e) What is the float associated with ‘making the cables’? What does this mean? (2 marks)

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(f) The project manager says: “Landscaping and paths can be done at any time after the site

has been surveyed and plans have been drawn.” Redraw the graph given this decision.

(No calculations are required). (3 marks)

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Restaurants are invited to send a team of four chefs to take part in a cooking competition. Each

chef will have to prepare one course of a four course meal. Each course will be judged and

given a score out of a maximum of 20, and the team’s total will be the sum of the four results.

In preparation for the competition, a restaurant owner asks his four chefs to each prepare all

four courses. He scores each course and tabulates the results as shown.

He wishes to use the Hungarian algorithm to allocate a meal course to each chef so that the

overall total score is the greatest that can be achieved.

(a) As this is a maximisation problem, start by preparing a matrix in which all the scores are

subtracted from the maximum possible score. (The first two entries have already been

completed.) (1 mark)

(b) Explain what the numbers in your matrix actually represent, and why this means that the

question has become a minimising problem. (2 marks)

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3 6

Scores out of 20

Alis

ha

No

ah

Ga

bb

i

Zeke

Appetiser 17 14 19 17

Entree 15 10 16 11

Main course 19 12 17 12

Dessert 14 15 13 14

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(c) Perform the Hungarian algorithm to find the best possible allocation. (3 marks)

(d) Complete the bipartite graph below to show which chef should be allocated to the

preparation of each course. (2 marks)

Alisha Appetiser

Noah Entree

Gabbi Main course

Zeke Dessert

(e) What is the likely total score of the team in the competition? (1 mark)

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Spare Diagrams

Spare graphs for question 17

Spare graph for question 18(a)

Spare graph for question 18(d) and (f)

F

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B

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C S

4.2

3.8

2.5 3 5.2

4.6

1.4

3

5 3.2 4

4 3.5

F

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B

T

C S

4.2

3.8

2.5 3 5.2

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3

5 3.2 4

4 3.5

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E

F

B

D

C X

Cap 1 = Cap 2 =

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Spare Diagrams


Spare grid for question 20

Spare graph for question 20 (d)

Alisha Appetiser

Noah Entree

Gabbi Main course

Zeke Dessert

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A (12)

E (7)

G (4) H (4) D (10)

F (14)

X (0) B (2)

C (14) I (7)

Start Finish

J (5)

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