Mathematics A Thursday 29 October 2015 Paper Two Question and response book

Transcription

1 2015 Senior External Examination Mathematics A Thursday 29 October 2015 Paper Two Question and response book 1:15 pm to 4:25 pm Time allowed Perusal time: 10 minutes Working time: 3 hours Examination materials provided Paper Two Question and response book Paper Two Resource book Equipment allowed QCAA-approved equipment ruler (metric, parallel or rolling) protractor drawing compass set squares templates (without formulas) non-programmable calculator graphing calculator Not allowed: Calculators with computer algebra system (CAS) functionality. Directions Do not write in this book during perusal time. Paper Two has four extended-response questions. Attempt all questions. Candidate use Print your candidate number here 1 5 Attach barcode here Number of books used Supervisor use only Supervisor s initials QCAA use only Marker number Assessment Paper Two assesses the following assessment criteria: Knowledge and procedures Modelling and problem solving (MP) Communication and justification (CJ) Assessment standards are at the end of this book. After the examination session The supervisor will collect this book when you leave. For all Queensland schools

2

3 Paper Two has four extended-response questions. Attempt all questions. Write your responses in the spaces provided. Show full working in all responses. Partial credit can only be awarded if working is shown. Question 1 a. An investment of $1200 for one year earns $37.80 simple interest. Calculate the rate of interest. b. A machine that makes boxes costs $ Its value depreciates by 5 cents for every box it makes. Each year it makes boxes. Determine the depreciated value of this machine at the end of two years. 1

4 c. A bank has three different types of savings accounts, as described in the table. Type A Type B Type C Account service fees per month: if minimum monthly balance stays at or above $350 Nil $5.00 Nil if balance drops below $350 $3.00 $5.00 Nil Number of fee-free transactions per month 10 Unlimited 6 Fee per transaction over the free limit 80 cents Nil 80 cents i. Cameron has a Type A account. In September his minimum balance was $362 and he made thirteen transactions. Calculate the fee he was charged for September. ii. In any month, Jess normally has between $200 and $300 in the bank. She usually makes about eight transactions each month. Explain why Jess would choose a Type C account at this bank. (MP) 2

5 d. Peter has 3000 shares with a market value of $4.20 each. The par value of the shares is $2 and they have been paying a dividend of 7% p.a. Peter can get 5.5% p.a. interest from his bank for a term deposit. Determine if Peter would do better to keep the shares or sell them and put his money in the bank. Show full working to justify your response. Ignore any brokerage fees paid on the sale of the shares or any bank fees. 3

6 Question 2 a. The observation deck of Sydney Tower is 250 m above the ground level of the Sydney Football Stadium. The angle of depression of the Sydney Football Stadium from the deck of Sydney Tower is 6. Sydney Tower Not to scale 250 m Sydney Football Stadium Calculate the horizontal distance between the bottom of Sydney Tower and Sydney Football Stadium. b. A boy flying a kite has let out 100 m of kite string. The kite is directly above the park entrance which is 85 m away. Not to scale 100 m 85 m Calculate the angle of elevation of the kite. 4

7 c. Yummee Cheesecakes come in two sizes as shown. Each cheesecake is the same thickness. The smaller cake will serve eight people. Not to scale 20 cm 30 cm How many serves of the same quantity are in the larger cheesecake? (MP) 5

8 d. Johannesburg in South Africa has latitude and longitude coordinates of 27 S, 28 E and Brisbane has coordinates of 27 S, 153 E. A flight to Johannesburg is scheduled to leave Brisbane on Tuesday 10 November 2015 at 6.30 pm and is listed to take 19 hours 40 minutes. What will be the local day, date and expected time of arrival of this flight in Johannesburg? 6

9 e. A goal defence in netball is 190 cm tall and can extend their reach by 50 cm. The goal shooter is 175 cm tall and reaches up another 40 cm to shoot. The goal defence can lean in at 20 and hold their body and arms straight to defend the shot. A successful block occurs when the goal defence covers half of the ball. If the netball has an 11 cm radius, determine if the goal defence will be successful in defending the shot if the rules state the goal defence s nearest foot must be 91 cm from the shooter s nearest foot. List two limitations of this mathematical model. Not to scale Goal shooter Goal defence (MP) 7

10 Question 3 a. The following drawing shows a scale plan for a timber deck to be added to a house timber decking bed 1 Drawn to scale balcony First floor plan If the length of the timber decking is 5200 mm as shown on the plan, determine the scale used. b. The bookcase below has been constructed using five pieces of timber, each 300 mm by 1200 mm. To strengthen the structure, a brace is to be attached along the diagonal at the back. Calculate the length of bracing required. Not to scale 8

11 c. A builder takes measurements along the string line as shown below. Determine if the corners are square. Not to scale 1200 mm 270 mm 720 mm 660 mm 1600 mm 2000 mm 9

12 d. The gable roof as shown below has a pitch of 26 and an overhang of 600 mm. Each sheet of roofing material has an effective width of 762 mm (this allows for overlap). It is cut to the required lengths and costs $24.13 per metre. Determine if a quote of $15000 is sufficient for the cost of the roofing material. Justify your decision with complete mathematical reasoning. F 600 mm overhang 26 B E G A 16 m D C 25 m Not to scale (MP) 10

13 Question 4 a. The costs, in $, of connecting various locations on a school campus with computer cable are given in the table below. B C D E A B C D 2500 i. Draw a network to represent this situation, showing the cost of connection along each arc. ii. Using the minimum spanning tree, find the least cost of connecting the cable

14 b. The activities and their completion times (in hours) that are needed to complete a project are shown in the network below. For this project, the minimum time to complete the whole project would be increased if one of the activities was delayed. S, 3 Not to scale P, 4 V, 3 start Q, 5 T, 6 X, 8 finish R, 12 U, 1 W, 4 Y, 3 Name an activity that would delay the project... c. The following table shows nine activities and their immediate predecessors for a project. The duration of each activity is not yet known. Immediate Activity predecessors A B C D A E B F C G D, E H F I G, H i. Use the information in the table above to complete the network below by including activities G, H and I. A D B E C ii. F There is only one critical path for this project. How many non-critical activities are there? 12

15 d. A hot food caravan has food continuously cooked so that the speed of sales in the service area is not affected. The time taken to serve the food by one employee is shown in the table below. The exchange of money averages an additional 10 seconds per customer. Food Time (seconds) Hot chips 15 Chiko roll 5 Hamburger 10 Hot dog 5 With no queue, Customer A arrives and orders a Chiko roll. Customer B arrives 10 seconds after customer A and orders hot chips and a hamburger. Five seconds after B arrives, Customer C arrives and wishes to purchase 3 hot dogs. Use the table below to show the progress of the queue and servicing for the first 60 seconds. Time (seconds) Customer arrival Customer served Queue Queue length

16 e. Historical figures show that, on average, customers join the queue of a busy supermarket at the rate of approximately 220 every hour and each checkout operator can serve on average 7 customers every 5 minutes. Determine the smallest number of servers required to ensure that there is no queue. Discuss one strength and one limitation of this model. (MP) End of Paper Two 14

17 Additional page for responses (if required) Question

18 Assessment standards from the Mathematics A Senior External Syllabus 2006 Criterion Standard A Standard B Standard C Standard D Standard E Knowledge and procedures The overall quality of a candidate s achievement across the full range within the contexts of application, technology and complexity, and across topics, consistently demonstrates: accurate recall, selection and use of definitions and rules use of technology recall and selection of procedures, and their accurate and proficient use. The overall quality of a candidate s achievement across a range within the contexts of application, technology and complexity, and across topics, generally demonstrates: accurate recall, selection and use of definitions and rules use of technology recall and selection of procedures, and their accurate use. The overall quality of a candidate s achievement in the contexts of application, technology and complexity, generally demonstrates: accurate recall and use of basic definitions and rules use of some technology accurate use of basic procedures. The overall quality of a candidate s achievement in the contexts of application, technology and complexity, sometimes demonstrates: accurate recall and use of some definitions and rules use of some technology. The overall quality of a candidate s achievement rarely demonstrates knowledge and use of procedures. Modelling and problem solving (MP) The overall quality of a candidate s achievement across the full range within each context, and across topics generally demonstrates mathematical thinking which includes: interpreting, clarifying and analysing a range of situations, and identifying variables selecting and using effective strategies informed decision making and sometimes demonstrates mathematical thinking which includes: selecting and using procedures to solve a wide range of problems initiative in exploring the problem recognising strengths and limitations of models. The overall quality of a candidate s achievement across a range within each context, and across topics, generally demonstrates mathematical thinking which includes: interpreting, clarifying and analysing a range of situations, and identifying variables selecting and using strategies and sometimes demonstrates mathematical thinking which includes: selecting and using procedures required to solve a range of problems informed decision making. The overall quality of a candidate s achievement demonstrates mathematical thinking which includes: interpreting and clarifying a range of situations selecting strategies and/or procedures. The overall quality of a candidate s achievement demonstrates mathematical thinking which includes following basic procedures and/or using strategies. The overall quality of a candidate s achievement rarely demonstrates mathematical thinking which includes following basic procedures and/or using strategies. 16

19 (continued) Criterion Standard A Standard B Standard C Standard D Standard E Communication and justification (CJ) The overall quality of a candidate s achievement across the full range within each context consistently demonstrates: accurate use of mathematical terms and symbols accurate use of language organisation of information into various forms suitable for a given use use of mathematical reasoning to develop logical arguments in support of conclusions, results and/ or decisions justification of procedures. The overall quality of a candidate s achievement across a range within each context generally demonstrates: accurate use of mathematical terms and symbols accurate use of language organisation of information into various forms suitable for a given use use of mathematical reasoning to develop simple logical arguments in support of conclusions, results and/ or decisions. The overall quality of a candidate s achievement in some contexts generally demonstrates: accurate use of basic mathematical terms and symbols accurate use of basic language organisation of information into various forms use of some mathematical reasoning to develop simple logical arguments. The overall quality of a candidate s achievement sometimes demonstrates evidence of the use of the basic conventions of language and mathematics. The overall quality of a candidate s achievement rarely demonstrates use of the basic conventions of language or mathematics. 17

20 The State of Queensland (Queensland Curriculum and Assessment Authority) 2015 Copyright enquiries should be made to: Manager Publishing Unit Queensland Curriculum & Assessment Authority PO Box 307, Spring Hill QLD 4004 Australia Level 7, 154 Melbourne Street, South Brisbane T F