mathematics stage 2c 2d calc assumed exam 2013

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Western Australian Certificate of EducationExamination, 2013

Question/Answer Booklet

Copyright School Curriculum and Standards Authority 2013

*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 candidateStandard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction fluid/tape, 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 your responsibility to ensure that you do not have any unauthorised notes or other items of a non-personal nature in the examination room. If you have any unauthorised material with you, hand it to the supervisor before reading any further.

Number of additional answer booklets used(if applicable):

Ref: 13-082

CALCULATOR-ASSUMEDMATHEMATICS 2C/2D 2

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Instructions to candidates

1. The rules for the conduct of Western Australian external examinations are detailed in the Year 12 Information Handbook 2013. Sitting this examination implies that you agree to abide by these rules.

2. Write your answers in this Question/Answer Booklet.

3. You must be careful to confine your response to the specific question asked and to follow any instructions that are specified to a particular question.

4. 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 the number of the question that you are continuing to answer at the top of the page.

5. Show all your working clearly. Your working should be in sufficient detail to allow your answers to be checked readily and for marks to be awarded for reasoning. Incorrect answers given without supporting reasoning cannot be allocated any marks. For any question or part question worth more than two marks, valid working or justification is required to receive full marks. If you repeat any question, ensure that you cancel the answer you do not wish to have marked.

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

7. The Formula Sheet is not handed in with your Question/Answer Booklet.

Structure of this paper

SectionNumber of questions available

Number of questions to be answered

Working time

(minutes)

Marks available

Percentage of exam

Section One: Calculator-free 6 6 50 50 33

13

Section Two:Calculator-assumed 13 13 100 100 66

23

Total 100

CALCULATOR-ASSUMED 3 MATHEMATICS 2C/2D

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Section Two: Calculator-assumed (100 Marks)

This section has 13 questions. Answer all questions. Write your answers in the spaces provided.

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 the number of the question that you are continuing to answer at the top of the page.

Suggested working time: 100 minutes.

Question 7 (5 marks)

The Venn diagram below shows two events, A and B, and some probabilities.

Determine

(a) P(A B). (2 marks)

(b) P(A B). (1 mark)

(c) P( ). (1 mark)

(d) P(A|B). (1 mark)


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Income tax rates for 201213 for both Australian citizens and foreign residents working in Australia are given in the tables below. The rates apply from 1 July 2012.

Australian citizens

Taxable income Tax on this income0$18 200 Nil$18 201$37 000 19c for each $1 over $18 200$37 001$80 000 $3572 plus 32.5c for each $1 over $37 000$80 001$180 000 $17 547 plus 37c for each $1 over $80 000$180 001 and over $54 547 plus 45c for each $1 over $180 000

Foreign residents

Taxable income Tax on this income0$80 000 32.5c for each $1$80 001$180 000 $26 000 plus 37c for each $1 over $80 000$180 001 and over $63 000 plus 45c for each $1 over $180 000

(a) Calculate the tax payable by an Australian citizen with a taxable income of $144 280 in the 201213 financial year. (3 marks)


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(b) Calculate the taxable income of a foreign resident who paid $43 530.60 in tax in the 201213 financial year. (3 marks)


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An event organiser is concerned about the number of people attending concerts without tickets. In a recent music festival, 6000 tickets were sold but the festival security company reported that, in a sample of 56 people, four people had no ticket. Assume that all 6000 ticket holders attended the concert.

(a) Using a capture-recapture method, estimate the number of people who attended the festival. (3 marks)

(b) Estimate the number of people in attendance who did not have a ticket. (1 mark)


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Peta was investigating prime numbers and made the conjecture:

For every two-digit prime number p, p 13, at least one of p 10 or p + 10 is prime.

(a) Test Petas conjecture with three different two-digit prime numbers p 13. (3 marks)

(b) What conclusion can you draw about Petas conjecture, on the basis of your results from part (a)? (1 mark)


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The two-way table below shows some of the characteristics of the residents of an over-50s lifestyle village. Residents are classified according to age, gender and source of income.

Male FemaleUnder

65 65 and

overUnder

65 65 and

over TotalWholly self-funded 22 48

Pension dependent 18 42Total 28 16 53 122

(a) Complete the table. (3 marks)

(b) How many residents depend on a pension? (1 mark)

(c) If a resident is selected at random, what is the probability that the resident is

(i) male? (1 mark)

(ii) at least 65 years old? (1 mark)

(iii) pension dependent, given that they are female and aged under 65? (2 marks)


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(d) Use the information from part (a) to complete the following Venn diagram. (4 marks)


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The network below represents a rail transport network connecting the central business district S to seven metropolitan centres A, B, C, G and a port P. The number on each arc represents the time, in minutes, that a train takes to travel between the centres.

(a) What is the fastest route from the central business district S to the port P? To obtain full marks, numbers must be added to the above diagram to show that an appropriate method has been used. (4 marks)


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(b) State the time required by the route found in part (a). (1 mark)

(c) Following a storm the rail link DG is flooded and not available. What effect, if any, will this have on the fastest rail link connecting the central business district and the port? Justify your answer. (2 marks)


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The diagram below represents a product distribution network linking a production centre A to a central distribution centre H. The products are moved from A through a number of intermediate centres, B, C, D, E, F and G. The number on each arc represents the maximum number of units of product that can be moved per day along the network segment.

(a) What is the maximum amount of product, in units per day, that can be moved from production centre A to the central distribution centre H? Show systematic working to allow your solution to be checked. (5 marks)


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(b) What effect, if any, would there be on the maximum daily flow of product from A to H if the capacity of the link DG was increased to 45 units per day? Justify your answer. (2 marks)


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The following table shows details of the projected population for Western Australia to 2056, at which time it is estimated that the total population will be 4 293 000, of which 78.2% will be living in the capital city.

Units 2016 2021 2026 2031 2036 2041 2046 2051 2056

Total population 000 2527 2765 3000 3231 3453 3669 3880 4088 4293Population aged 014 years % 19.1 19.0 18.5 18.1 17.6 17.3 17.2 17.1 17.0Population aged 1564 years % 66.5 65.1 64.1 63.3 62.8 62.4 62.1 61.4 60.8Population aged 65 years and over % 14.4 15.9 17.4 18.6 19.6 20.3 20.7 21.5 22.2Population aged 85 years and over % 1.8 1.9 2.2 2.5 3.2 3.7 4.0 4.3 4.6Median age of total population years 37.5 38.3 39.2 40.0 40.7 41.1 41.4 41.8 42.1Population living in the capital city % 74.6 75.1 75.6 76.0 76.5 76.9 77.4 77.8 78.2

(a) What is the projected median age for people in Western Australia in 2046? (1 mark)

(b) What is the projected percentage of people in Western Australia aged 1564 years in 2021? (1 mark)

(c) What is the projected population of people in Western Australia aged 1564 years in 2021? (2 marks)


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(d) What is the estimated percentage increase in total population from 2016 to 2056? (3 marks)

(e) The government is considering a plan to encourage people to move away from the capital city. Describe how this would affect the projected median age reported in this table. (2 marks)


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The diagram shows the graphs of y = 2(0.5)x and y = 4 x.

(a) Use the graphs given to estimate the solutions to the equation 2(0.5)x = 4 x. Show on your graph where you found the solutions. (3 marks)


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(b) On the diagram, sketch the graph of y = 0.5(x + 2)(x 3)2. (3 marks)

(c) Use the graphs to determine the number of solutions to the equation 0.5(x + 2)(x 3)2 = 4 x. (1 mark)

(d) Use your calculator to determine the coordinates of the point(s) of intersection of the two functions y = 2(0.5)x and y = 4 x. Express your answer(s) correct to two decimal places. (4 marks)


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The diagram below (not drawn to scale) is a survey plan of a mining lease for an area of land ABCD. All dimensions are in metres (m) and angles are in degrees.

(a) To develop the mine a road needs to be constructed along the line segment AC. Using trigonometry, calculate the length of this road to the nearest metre. (4 marks)

(b) If the road along AC is costed to the nearest metre at a rate of $1500 per metre, what is the total cost of constructing the road? (1 mark)


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(c) Using trigonometry, determine the size of ABC. (3 marks)

(d) Using trigonometry, determine the area of the lease contained in triangle ADC. (2 marks)


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Quentin recorded and graphed official quarterly interest rates over four years beginning in September 2008. A table showing part of his data is given below.

Quarter Month and year Interest rate1 September 2008 7.02%2 December 2008 4.35%3 March 2009 3.25%4 June 2009 3.00%5 September 2009 3.00%

15 March 2012 4.25%16 June 2012 3.54%

A diagram of his plot of the entire data is given below.


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Using his calculator Quentin determined that the quarterly interest rate as a percentage could be modelled by the cubic function y = 0.0104x3 + 0.275x2 2.04x + 7.9.

(a) Sketch Quentins function on the diagram on the opposite page. (4 marks)

(b) Use Quentins function to determine the interest rate in September 2012. (2 marks)

(c) Use the sketch of Quentins function to estimate the greatest difference between the actual interest rate and the rate given by Quentins function. (1 mark)

(d) Why would Quentins function have been inaccurate in predicting interest rates in 2013? (1 mark)


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Eddy and Eileen have decided to invest $16 000 for their eight-year old son Gary. He is to be given the money on his twentieth birthday, in twelve years time. They have chosen a savings fund that pays interest at the rate of 4.8% per annum compounded quarterly.

The following table shows the value of the investment at three-monthly intervals during the first two years.

Time Value of investmentAt start $16 000After 3 months $16 000 1.012 = $16 192After 6 months $16 192 1.012 = $16 000 1.0122 = $16 386.30After 9 months $16 386.30 1.012 = $16 000 1.0123 = $16 582.94After 1 year $16 582.94 1.012 = $16 000 1.0124 = $16 781.94After 15 months $16 781.94 1.012 = $16 000 1.0125 = $16 983.32After 18 months $16 983.32 1.012 = $16 000 1.0126 = $17 187.12After 21 months $17 187.12 1.012 = $16 000 1.0127 = $17 393.36After 2 years $17 393.36 1.012 = $16 000 1.0128 = $17 602.08

(a) The value of the investment at the end of each three months (quarter) can be written as:Value = A(1.012)n where, n is the number of quarters.

(i) What is the value of A? (1 mark)

(ii) What is the value of n after two years? (1 mark)

(iii) What is the quarterly interest rate? (1 mark)


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(b) Determine how much Gary will receive on his twentieth birthday. (3 marks)

(c) Write a recursive rule for the value of the investment after n quarters. (2 marks)


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Louise and Andrea were tossing a fair coin and investigating run size. They had defined run size as the number of times the toss resulted in the same outcome (either heads or tails) before changing.

Examples: Five tosses resulting in HHHTT represents a run of size 3, followed by a run of size 2. Six tosses resulting in TTTHHT represents a run of size 3, followed by a run of size 2,

then a run of size 1.

(a) Write an example in which 10 coins are tossed and there is a run of size 3, followed by a run of size 5, then a run of size 2. (1 mark)

(b) The result from 20 tosses of a coin are shown below.

HHHTTHHTTTHHHTHTHHTT (1 mark)

How many run sizes of 3 are there?

(c) Louise and Andrea had tossed a fair coin 100 times in their investigation and recorded the number of times each run size occurred. The results are given in the table below.

Run size 1 2 3 4 5 6Frequency 18 9 9 5 1 2

(i) What was the largest run size? (1 mark)

(ii) In the process of calculating the mean size of a run, Louise divided by 100 while Andrea divided by 44. Who was correct? Explain. (2 marks)


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(d) Using the information in part (c), determine

(i) the mean size of a run. (1 mark)

(ii) the median size of a run. (1 mark)

(iii) the standard deviation of the run size. (1 mark)

(e) Louise and Andrea wanted to collect more data and decided it would be quicker to run a simulation using a spreadsheet on their calculators. What calculator function would be useful to generate numbers to represent the outcomes of the coin toss? (1 mark)

End of questions


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Additional working space

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Question number:

Published by the School Curriculum and Standards Authority of Western Australia27 Walters Drive

OSBORNE PARK WA 6017

This document apart from any third party copyright material contained in it may be freely copied, or communicated on an intranet, for non-commercial purposes in educational institutions, provided that it is not changed and that the School Curriculum and Standards Authority is acknowledged as the copyright owner, and that the Authoritys moral rights are not infringed.

Copying or communication for any other purpose can be done only within the terms of the Copyright Act 1968 or with prior written permission of the School Curriculum and Standards Authority. Copying or communication of any third party copyright material can be done only within the terms of the Copyright Act 1968 or with permission of the copyright owners.

Any content in this document that has been derived from the Australian Curriculum may be used under the terms of the Creative Commons Attribution-NonCommercial 3.0 Australia licence.

ACKNOWLEDGEMENTS

Section Two Question 8 Australian Taxation Office. (2012). Income tax rates [Tables].

Retrieved April 21, 2013, from www.ato.gov.au/content/12333.htm Question 14 Australian Bureau of Statistics. (2012). Population projections [Table].

Retrieved April 21, 2013, from www.abs.gov.au/AUSSTATS/[email protected]/DetailsPage/4102.0Sep%202012. Used under the Creative Commons Attribution 2.5 Australia licence.

Question 17 Data source: Reserve Bank of Australia. (2013). Interest rates.

Retrieved August 5, 2013, from www.rba.gov.au/statistics/tables/index.html#share_mkts

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mathematics stage 2c 2d calc assumed exam 2013

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