Mathematical Methods 2017 Sample paper

Transcription

1 South Australian Certificate of Education The external assessment requirements of this subject are listed on page 20. Question Booklet 2 Mathematical Methods 2017 Sample paper 2 Part 2 (Questions 11 to 16) 78 marks Answer all questions in Part 2 Write your answers in this question booklet You may write on page 19 if you need more space Allow approximately 90 minutes Approved calculators may be used complete the box below SACE Board of South Australia 2017 SEQ Copy the information from your SACE label here FIGURES CHECK LETTER BIN Graphics calculator 1. Brand Model 2. Brand Model For office use only Supervisor check Re-marked

2 PART 2 (Questions 11 to 16) (78 marks) Question 11 A 12-hour music festival started at 12 noon and finished at 12 midnight. The rate d E at which dt patrons entered the venue over the 12 hours is modelled by the function de 04. t 220te 15 for 0t 12, dt where t is the number of hours after 12 noon and d E is the number of patrons entering the venue per hour. dt Before 12 noon there were no patrons inside the venue. The graph of y d E is shown below: dt 220 y y E d dt t page 2 of 20

3 (a) Find the rate at which patrons entered the venue at 12 noon. (b) Use your calculator to find the time when the rate at which patrons entered the venue was at a maximum. (c) The model implies that patrons continuously entered the venue. Identify the characteristic of the model that reflects this. E (d) (i) On the graph of y d, provide a representation of the number of patrons that d t the model implies entered the venue between 1 pm and 4 pm. (ii) Calculate the number of patrons that the model implies entered the venue between 1 pm and 4 pm. page 3 of 20 PLEASE TURN OVER

4 The rate d L at which patrons left the venue over the 12 hours is modelled by the function dt dl 015. t 50te for 0t 12. dt (e) On the axes on page 2, sketch the graph of y L d L. Label your graph y d d t d t. (3 marks) (f ) Find the time, to the nearest half-hour, when the rate at which patrons entered the venue was the same as the rate at which patrons left the venue. (g) Given that 12 0 de dl dt 296, state the meaning of the value 296 in the context of the dt dt number of patrons inside the venue. page 4 of 20

5 Question 12 A manufacturer produces nails and sells them in 500-gram packets. For several years, records have shown that the weights of the nails have a distribution with a mean and standard deviation of = 6.7 g and = 0.17 g. (a) As part of the manufacturer s ongoing quality assurance program, the nails produced in 1 week are collected and randomly placed in groups of 25 nails. The weight W of each individual nail is recorded and the sum of weights S 25 of each group of 25 nails is also recorded. Two histograms are shown below. One histogram illustrates the distribution of W and the other histogram illustrates the distribution of S 25. Histogram A Histogram B Identify which histogram, A or B, illustrates the distribution of W. Histogram Explain your answer. page 5 of 20 PLEASE TURN OVER

6 (b) (i) Let S 73 be the sum of weights of a random sample of 73 nails. According to the central limit theorem, the distribution of S 73 will be approximately normal, provided that the sample size is sufficiently large. Is the sample size of 73 sufficiently large in this case? Explain your answer, referring to part (a) as appropriate. (ii) Calculate the mean of S 73. (iii) State the standard deviation of S 73. (iv) Find an approximate value for PS page 6 of 20

7 (c) The nails are sold in packets that have been filled by a machine. The machine stops filling a packet when the packet weighs 500 g or more. (i) Using your answer to part (b)(iii), state how likely it is that the machine will stop filling a packet when the packet contains 73 nails or fewer. Explain your answer. (ii) The label on each packet states, Contains at least 74 nails. Is this statement reasonable? Justify your answer. page 7 of 20 PLEASE TURN OVER

8 Question 13 Let f x 1 for x 0. x (a) (i) Find f ( x ). (ii) Find the equation of the tangent to the graph of y f x at x 1. (iii) Find the x-intercept of the tangent to the graph of y f x at x 1. The graph of y f x are also shown. y 3 is shown below. The tangents to the graph of y f x at x 2 and x x page 8 of 20

9 (b) (i) The tangent to the graph of y f x 1 3 y x Find the x-intercept of this tangent. at x 4 has the equation at x 6 has the equation (ii) The tangent to the graph of y f x 1 3 y x Find the x-intercept of this tangent. (c) Consider the tangent to the graph of y f x at x a. Let x a be the x-intercept of this tangent. (i) On the basis of your answers to parts (a) and (b), make a conjecture describing how x a is related to a. page 9 of 20 PLEASE TURN OVER

10 (ii) Prove or disprove the conjecture that you made in part (c)(i). (5 marks) page 10 of 20

11 Question 14 (a) State the properties that qualify a function as being suitable to be a probability density function. The continuous random variable X has the probability density function f, where 168x 1 x f x x 1. elsewhere (b) On the axes below, sketch the graph of y f x. y x page 11 of 20 PLEASE TURN OVER

12 (c) What is the probability that X is less than 1 2? (d) Calculate the mean of X. (e) Calculate the standard deviation of X. page 12 of 20

13 (f ) For this probability density function, the median value of X is the value of the random variable x for which PX x (i) Show that 168x ( 1x) 168x 336x 168x. (ii) Hence find the median value of X. (3 marks) (g) What feature of the graph of y f x median of X? reflects the fact that the mean of X is less than the page 13 of 20 PLEASE TURN OVER

14 Question 15 (a) A seed wholesaler is selling a large quantity of seeds. To determine the proportion of these seeds that will germinate and grow into seedlings, the wholesaler plants a random sample of 100 seeds and finds that 87 germinate and grow into seedlings. (i) Calculate a 95% confidence interval for the proportion of seeds that, when planted, will germinate and grow into seedlings. (ii) On the basis of the confidence interval that you calculated in part (a)(i), the wholesaler claims that at least 80% of the seeds will germinate and grow into seedlings. Explain why it is reasonable for the wholesaler to make this claim. Assume, now, that 80% of the seeds will germinate and grow into seedlings. The manager of a plant nursery has purchased these seeds and is planning to germinate them in trays consisting of six pots (as shown in the photograph). Only trays with one or more seedlings in all six pots can be sold. The manager wants to make sure that more than 90% of trays can be sold. tray with six pots (b) The manager decides to plant two seeds in every pot. (i) Calculate the probability that, if two seeds are planted in a single pot, at least one of the seeds will germinate and grow into a seedling. page 14 of 20

15 (ii) Using your answer to part (b)(i), calculate the probability that, if two seeds are planted in each of the six pots in each tray, at least one of the seeds in each pot will germinate and grow into a seedling. (c) If three seeds are planted in each of the six pots in each tray, is it likely that more than 90% of trays could be sold? Show calculations to support your answer. (4 marks) page 15 of 20 PLEASE TURN OVER

16 Question 16 In drag races, cars are initially stationary on the starting line and then travel 400 metres in a straight line in as short a time as possible. Car A (shown in the photograph) competes in drag races. The design features of Car A are considered in the development of a mathematical model to predict the speed (v A in metres per second) of the car t seconds after it leaves the starting line. The model assumes that all the components of Car A Source: work optimally and that the driver attempts to finish in as short a time as possible. The model is useful for predictions only for the first 7 seconds after the car leaves the starting line. The model is shown below: va e t for 0t 7. (a) Find the maximum speed of Car A according to this model. 2 (b) (i) Calculate v t, correct to the nearest whole number. 0 A d (ii) Interpret your answer to part (b)(i) in the context of Car A competing in a drag race. page 16 of 20

17 (c) Complete the following table according to the model on page 16. t (seconds) distance (metres) travelled by Car A after t seconds 0 The photograph shows Car B, a drag-racing car under construction. The design features and the same assumptions that apply to Car A are considered in the development of a new mathematical model to predict the speed (v B in metres per second) of Car B, t seconds after it leaves the starting line. The model for Car B is shown below: vb e t for 0t 7. (d) Complete the following table according to the model above. t (seconds) distance (metres) travelled by Car B after t seconds 0 (e) If Car A races against Car B, which car would win the race, according to the models? Car Explain your answer. page 17 of 20 PLEASE TURN OVER

18 (f) (i) Find d y dt 1 2t if y ln ( e k), where k is a positive real constant. 2 (ii) Hence show that e 2t 2t 4. 9 d t 49ln e tc. (3 marks) (iii) Hence determine the time Car A takes to travel the 400 metres, correct to two decimal places. (3 marks) page 18 of 20

19 You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. Question 13(c)(ii) continued ). page 19 of 20 PLEASE TURN OVER

20 2017 SAMPLE MATHEMATICAL METHODS PAPER The purpose of this sample paper is to show the structure of the Mathematical Methods examination and the style of questions that may be used. The following extract is from the 2017 subject outline for Mathematical Methods: EXTERNAL ASSESSMENT Assessment Type 3: Examination (30%) Students undertake a 3-hour external examination. The examination is based on the key questions and key concepts in the six topics. The considerations for developing teaching and learning strategies are provided as a guide only, although applications described under this heading may provide contexts for examination questions. The examination consists of a range of problems, some focusing on knowledge and routine skills and applications, and others focusing on analysis and interpretation. Some problems may require students to interrelate their knowledge, skills, and understanding from more than one topic. Students provide explanations and arguments, and use correct mathematical notation, terminology, and representations throughout the examination. A formula sheet is included in the examination booklet. Students may take two unfolded A4 sheets (four sides) of handwritten notes into the examination room. Students may have access to approved electronic technology during the external examination. However, students need to be discerning in their use of electronic technology to fi nd solutions to questions/problems in examinations. All specifi c features of the assessment design criteria for this subject may be assessed in the external examination. Source: Mathematical Methods 2017 Subject Outline Stage 2, p 44, on the SACE website, page 20 of 20 end of question booklet