Mathematics B Monday 2 November 2015 Paper One Question book

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1 2015 Senior External Examination Mathematics B Monday 2 November 2015 Paper One Question book 9 am to 12:10 pm Time allowed Perusal time: 10 minutes Working time: 3 hours Examination materials provided Paper One Question book Paper One Resource book Paper One Response book Equipment allowed QCAA-approved equipment ruler graduated in millimetres protractor graphics calculator additional calculator Equipment not allowed Calculators with computer algebra system (CAS) functionality Directions You may write in this book during perusal time. Paper One has six questions. Attempt all questions. Assessment Paper One 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 One has six 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 fair six-sided die is rolled 11 times. The results are presented in the following histogram. Die roll outcomes 3 Frequency Scores a. For the histogram data, i. state the mode ii. calculate the mean and standard deviation iii. determine the five-number summary. b. Present the histogram data as a boxplot (box-and-whisker plot). c. The same die is rolled four more times. It is claimed that the mean has not changed. Comment on the validity of this claim. Justify your response. (MP) 1

4 Question 2 a. The following describes the relationship y = f x. f x = x x x 0 0 x 4 x 4 i. Sketch y = f x and explain why it represents a function. ii. Determine the domain and range of f x. iii. State, giving a reason, if this function is continuous. b. Describe the set of transformations required to convert y = x to y = ½ x Using the graph paper provided in the response book, graph the transformed function. c. The tank shown below leaks water through a small hole in its base. Water leaves the tank at a constant rate of 0.03 m 3 /minute m 1 m 0.5 m x m When filled to a depth of x metres, the volume of water in the tank, V m 3, is given by V = x 3 + 6x x 48 What is the limitation that exists on the value of x? Use a graphics calculator to determine the value of x when the volume is m 3. Given that dv dv dx = -----, determine the rate at which the depth of the water is falling when the volume dt dx dt is m 3. (MP) 2

5 Question 3 a. Prove that 3cos 2sin = 2cos + 1 cos + 1. b. Solve algebraically for x if 2cosx = 1 and 2 x 2. c. The table below gives the mean monthly maximum temperature, T ( C), for Brisbane. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec T C i. A sine function of the form T = a sin (bm + c) + d is used to model the data. Using a graphics calculator, find estimates correct to two decimal places for each of the constants a, b, c and d. Use Jan = 1, Feb = 2 etc., for the values of the variable, m. ii. Calculate the period of the function. iii. Using your model, T = a sin (bm + c) + d, calculate the average rate of change of the mean monthly maximum temperature across the period from March through June for Brisbane. d. The diagram below shows a wall BF held in an upright position by two timber supports, CD and AE, and a wire stay AF. F D E 2.9 A B 48 C AE = 2.4 m, EF = 1.8 m, AF = 3.7 m, DC = 2.9 m and angle BCD = 48. Determine FD, the distance that F is above D. (MP) 3

6 Question 4 a. Given that f x = 2x and gx = 3x 2 + x 4, find: 5 i. f 3 ii. the inverse function, f 1 x iii. the composite function, g f x. b. Using an algebraic method, find the points of intersection of y = 3x 2 + 4x 1 and y = x c. Given that the graph of y = h x has an x intercept at x = 1 and h 1 = 0 at the point A 1 4, find the values of a, b and c if hx = ax 3 + 3bx + c. (MP) Question 5 a. Show from first principles that the derivative of fx = 2x is 4x. b. dy Determine for each of the following: dx i. y = cos 2x + 3log e x ii. y = x iii. y = x + 2 x 5 c. The displacement, x metres, of a particle from its starting point, after t seconds, is given by the equation x = e t 4 t. i. Find the initial position of the particle. ii. Determine the velocity of the particle when t = 2 seconds. iii. Calculate when the particle is at rest. 4

7 d. A ceramic brick is made in the shape of a right prism. Its ends are equilateral triangles of side length x cm. The brick has a length of y cm and a volume of 1000 cm 3. i. Show that the surface area of the brick, A cm 2, is given by x 2 3 A = x 3 2 ii. Using calculus methods, determine the minimum surface area of the brick. (MP) Question 6 a. A box contains 15 green marbles and 5 black marbles. A marble is randomly selected from the box, its colour noted and then put back in the box. If X is the random variable that represents the number of times a black marble is selected from the box in 12 trials, calculate P 7 x 9. b. The arm lengths of 18-year-old women are normally distributed with a mean of 64 cm and standard deviation of 4 cm. Calculate the percentage of 18-year-old women whose arm lengths are less than 57 cm. c. A six-sided die is loaded so that the probability of rolling an even number is twice as likely as rolling an odd number. The die is rolled 300 times. The binomial variable X represents the number of times an even number appears on the uppermost face of the die when it is rolled. Using the normal approximation for a binomial event, determine PX 210. (MP) End of Paper One 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

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

Mathematics B Monday 2 November 2015 Paper Two Question book

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