Mathematics B Monday 2 November 2015 Paper Two Question book

Transcription

1 2015 Senior External Examination Mathematics B Monday 2 November 2015 Paper Two Question book 1:15 pm to 4:25 pm Time allowed Perusal time: 10 minutes Working time: 3 hours Examination materials provided Paper Two Question book Paper Two Response book Equipment allowed QCAA-approved equipment ruler graduated in millimetres protractor graphics calculator additional calculator Equipment not allowed Directions Calculators with computer algebra system (CAS) functionality You may write in this book during perusal time. Paper Two has five questions. Attempt all questions. 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 Take this book when you leave. For all Queensland schools

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3 Paper Two has five questions. Attempt all questions. Each question assesses Knowledge and procedures, Modelling and problem solving (MP) or a combination of both. Communication and justification (CJ) will be assessed by an overall judgment of your responses to all questions. Write your responses in the response book. Show full working where necessary to meet the standards for each criterion. Question 1 a. Given that y = x x 2, use algebraic methods to calculate: i. the x and y intercepts ii. any stationary points and determine their nature iii. the least and greatest values of this function in the interval 1 x 4. b. A manufacturing company makes door hinges. The cost, in dollars, for making x hinges per hour is C x = x x x + 160where 0 x 150. i. Graph the cost function across the domain 0 x 150 using the graph paper provided in the response book. ii. For what production interval will the production costs decrease as the number of hinges increases? c. A surveyor estimates the area of a lake by using the Trapezoidal rule. The surveyor prepares the following diagram from an aerial photograph of the lake. The offsets taken from the line AB are equally spaced with AB = 400 metres. All other measurements in the diagram are given in metres. Not to scale A B If the depth of water in the lake is rising by 1 cm/hour, calculate the rate at which water is entering the lake. (MP) 1

4 Question 2 a. Determine: i. 2x dx ii. 6cos4x dx iii. e 3x 1 dx b. Determine b, showing full working dx = log x + 3 e b c. The region formed by the intersection of the line 2x y = 3 and the curve y = x 2 + 6x 5 is shown below. y B O A x i. Using a graphics calculator, determine the coordinates of the points of intersection, A and B. ii. Write an expression that could be used to determine the bounded area. d. A curve has the equation y = xlog e x. i. State the limitation that exists on x in this equation. ii. A normal to this curve is drawn at the point B on the curve where x = 1. The vertical line that passes through the turning point C of the curve meets the normal through B at point A. Using a graphics calculator, determine the area of the enclosed region BAC. (MP) 2

5 Question 3 a. In simplest form, express 135 in radians in terms of. b. The velocity of a particle at time, t seconds, is given by dx = 10 + sin t cm/s. dt i. If the particle has an initial displacement of 3 cm, calculate its displacement between t = 3 and t = 7 seconds. ii. Determine the initial acceleration of the particle. c. The area of the region bounded by the x axis, the curve y xe x 2 = and the line x = a is shown below. y O a x 2 2 i. d Given that e x xe 2 = -- x, calculate the value of a if the shaded area is 0.5. dx ii. As a approaches infinity, what is the effect on the calculated area? (MP) 3

6 Question 4 a. Without using a calculator, simplify: i. ii. 3x x x log log b. Without using a calculator, solve for x: i. ii. 2 3x 3 = 16 2 x log 10 5x + 2 = log 10 x 2 + log 10 6 c. A scientist records the level of smoke pollution, P micrograms per cubic metre ( g/m 3 ), against the minimum temperature, T ( C), over a one-week period. The data is displayed in the following table. Day Minimum temperature (T C) Pollution (P g/m 3 ) Monday Tuesday Wednesday Thursday Friday Saturday Sunday The linear equation, P = at + b, is used by the scientist to model the data. Use a graphics calculator to determine the values of the constants a and b. The scientist assumes the Saturday data to be an outlier and removes it. An exponential equation, P = ce dt, is then used to remodel the data. Use a graphics calculator to determine the values of the constants c and d. Compare and contrast the strengths and limitations of the linear and exponential equations in modelling the relationship between P and T. (MP) 4

7 Question 5 a. How long will it take for $5000 invested at 4% p.a. compounding half-yearly to grow to $6500? b. Sarah has taken out a $ car loan over five years at 10% p.a. compounding monthly with monthly repayments. i. Calculate Sarah s monthly repayment. ii. After one year, Sarah refinances her existing car loan. The outstanding balance is to be paid off over four years at 8.4% p.a. compounding monthly. Sarah believes the new monthly repayment will save her a lot of money. How reasonable is Sarah s claim? (MP) End of Paper Two 5

8 Assessment standards from the Mathematics B Senior External Syllabus 2006 Criterion Standard A Standard B Standard C Standard D Standard E Knowledge and procedures 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 accurate use of technology recall and selection of procedures and their accurate and proficient use effective transfer and application of mathematical procedures. 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 accurate use of technology recall and selection of procedures and their accurate use. in the contexts of Application, Technology and Complexity generally demonstrates: accurate recall and use of basic definitions and rules use of technology accurate recall, selection and use of basic procedures. in the contexts of Application, Technology and Complexity sometimes demonstrates: accurate recall and use of some definitions and rules use of technology use of basic procedures. rarely demonstrates knowledge and use of procedures. 6

9 (continued) Criterion Standard A Standard B Standard C Standard D Standard E Modelling and problem solving across the full range within each context, and across topics, generally demonstrates mathematical thinking which includes: interpreting, clarifying and analysing a range of situations identifying assumptions and variables selecting and using effective strategies selecting suitable procedures required to solve a range of problems and sometimes demonstrates mathematical thinking which includes: suitable synthesis of procedures and strategies to solve problems initiative and insight in exploring the problem identifying strengths and limitations of models. 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 assumptions and variables selecting and using effective strategies selecting suitable procedures required to solve a range of problems and sometimes demonstrates mathematical thinking which includes: suitable synthesis of procedures and strategies. demonstrates mathematical thinking which includes: interpreting and clarifying a range of situations selecting strategies and/or procedures required to solve problems. sometimes demonstrates mathematical thinking which includes: following basic procedures and/or using strategies. rarely demonstrates mathematical thinking which includes following basic procedures and/or using strategies. 7

10 (continued) Criterion Standard A Standard B Standard C Standard D Standard E Communication and justification 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 propositions justification of procedures recognition of the effects of assumptions evaluation of the validity of arguments. 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 propositions justification of procedures. in all contexts generally demonstrates: accurate use of basic mathematical terms and symbols accurate use of language organisation of information into various forms use of some mathematical reasoning to develop simple logical arguments. sometimes demonstrates evidence of the use of the basic conventions of language and mathematics and occasional use of mathematical reasoning. rarely demonstrates use of the basic conventions of language and mathematics. 8

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12 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