2016 MATHEMATICAL METHODS

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1 2016 MATHEMATICAL METHODS External Examination 2016 FOR OFFICE USE ONLY SUPERVISOR CHECK ATTACH SACE REGISTRATION NUMBER LABEL TO THIS BOX Graphics calculator Brand Model Computer software RE-MARKED Friday 18 November: 9 am Time: 3 hours Examination material: one 35-page question booklet one SACE registration number label Pages: 35 Questions: 15 Approved dictionaries, notes, calculators, and computer software may be used. Instructions to Students 1. You will have 10 minutes to read the paper. You must not write in your question booklet or use a calculator during this reading time but you may make notes on the scribbling paper provided. 2. Answer all parts of Questions 1 to 15 in the spaces provided in this question booklet. There is no need to fill all the space provided. You may write on pages 21 and 32 if you need more space, making sure to label each answer clearly. 3. The total mark is 153. The allocation of marks is shown below: Question Marks Appropriate steps of logic and correct answers are required for full marks. 5. Show all working in this booklet. (You are strongly advised not to use scribbling paper. Work that you consider incorrect should be crossed out with a single line.) 6. Use only black or blue pens for all work other than graphs and diagrams, for which you may use a sharp dark pencil. 7. State all answers correct to three significant figures, unless otherwise stated or as appropriate. 8. Diagrams, where given, are not necessarily drawn to scale. 9. The list of mathematical formulae is on page 33. You may remove the page from this booklet before the examination begins. 10. Complete the box on the top right-hand side of this page with information about the electronic technology you are using in this examination. 11. Attach your SACE registration number label to the box at the top of this page. SACE Board of South Australia 2016

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3 QUESTION 1 (a) Write the equation ln M 3ln x4ln 2 without logarithms. (3 marks) (b) Write the equation ln M. 3x2 in the form M k a x (3 marks) page 3 of 35 PLEASE TURN OVER

4 QUESTION 2 a Let D b b a, E , and F (a) Determine DE. (b) Given that a and b are integers, solve for a and b if DE F. page 4 of 35

5 QUESTION 3 Find d y without simplifying when: dx (a) y x x4. 3 x (b) y e 2 2 x. (3 marks) (c) y 2 ln x 3. page 5 of 35 PLEASE TURN OVER

6 QUESTION 4 The glycogen level (in grams) in the muscles of a male athlete competing in a long-distance running race was measured every 15 minutes; the results are shown in the table below. Time (t minutes) Muscle glycogen level (M grams) (a) Find the equation, in the form M mt c, for the linear model that best represents the results. (b) State the r 2 value and describe the strength and direction of the linear relationship between time and the muscle glycogen level of the male athlete. (3 marks) (c) (i) Use the equation for the linear model found in part (a) to predict the muscle glycogen level of the male athlete at 90 minutes. (ii) Find the residual at t 90 minutes. page 6 of 35

7 (iii) Use the residual found in part (c)(ii) to determine whether the model overestimates or underestimates the muscle glycogen level of the male athlete at 90 minutes. (d) Use the equation for the linear model found in part (a) to find the time when the muscle glycogen level of the male athlete was 350 grams. The glycogen level in the muscles of a female athlete who was running in the same race was also measured and can be modelled by the function F t. (e) Find the time when the muscle glycogen level of the male athlete was the same as the muscle glycogen level of the female athlete. (f ) Which athlete s muscle glycogen level decreased at the fastest rate? Explain your answer. page 7 of 35 PLEASE TURN OVER

8 QUESTION 5 Residents living near a suburban oval were asked by the local council whether they were in favour of night performances being held at the oval. Of the 250 replies, 120 were in favour of night performances being held at the oval. (a) (i) Calculate the sample proportion of residents who were in favour of night performances being held at the oval. (ii) Use the sample proportion calculated in part (a)(i) to determine the 95% confidence interval for the population proportion. (iii) Interpret the meaning of this 95% confidence interval. (iv) State the width of this 95% confidence interval. page 8 of 35

9 (b) Does your answer to part (a)(ii) support the claim: The majority of residents living near this suburban oval are in favour of night performances being held at the oval? (c) Determine the smallest sample size necessary to ensure the width of the 95% confidence interval for the percentage of residents who are in favour of night performances being held at the oval is no greater than 4%. (3 marks) page 9 of 35 PLEASE TURN OVER

10 QUESTION 6 (a) On the set of axes below, draw and clearly label the graphs of y7x and y3. y 10 9 y = 2x x (3 marks) (b) On the set of axes above, shade the feasible region that is defined by the following constraints: x 0 y 0 y2x1 y7 x y 3. (c) State the coordinates of the vertices of the feasible region. page 10 of 35

11 (d) Find the coordinates of the point that maximises the objective function P3x4 y over the feasible region. (e) Determine an objective function that has more than one optimal solution, including the point identified in part (d). page 11 of 35 PLEASE TURN OVER

12 QUESTION 7 Every 2 months, for a particular species of frog: each female frog lays 50 eggs before dying 20% of the eggs survive to become tadpoles 10% of the tadpoles survive to become female frogs. This two-month cycle has been summarised in the following matrix: Breeding rate L Survival rate of eggs Survival rate of tadpoles Mr Smith stocks his pond with tadpoles and frogs. He does not buy any eggs, but he does buy 20 tadpoles and 5 female frogs. (a) Represent the initial population of Mr Smith s pond as a 3 1matrix, X 0 eggs tadpoles. frogs (b) (i) Calculate LX 0. (ii) What does matrix LX 0 represent? page 12 of 35

13 (c) Calculate L 2 X 0. (d) Calculate L 3 X 0. (e) Using your answers from parts (b), (c), and (d), explain what happens to the numbers of eggs, tadpoles, and female frogs in Mr Smith s pond over the long term. page 13 of 35 PLEASE TURN OVER

14 QUESTION 8 An unbiased spinner has five equal sectors. Two of the sectors are black, and three of the sectors are white. (a) The spinner is spun once. Determine the probability that it stops on a black sector. (b) The spinner is spun 10 times. Determine the probability that it stops on a black sector: (i) exactly six times. (ii) at least six times. page 14 of 35

15 (iii) more than three times and at most eight times. (3 marks) (c) The spinner is spun 1000 times. (i) Find the mean and standard deviation for the number of times it stops on a black sector. (ii) Use a normal approximation to find the probability that it stops on a black sector fewer than 420 times. Show your working. page 15 of 35 PLEASE TURN OVER

16 QUESTION 9 Scientists studying a colony of geckos have noticed an increase in the population. The population can be modelled by the function Pt e 02. t where t is time (in months) since the population was first measured. (a) Find the initial population of geckos. (b) Find the population of geckos 5 months after the first measurement. (c) (i) Solve for t if P t 80. (ii) Interpret your answer to part (c)(i). page 16 of 35

17 (d) (i) Find Pt, and express it in the form where k is a constant. k Pt t e 1 49e t 2 (3 marks) (ii) Hence or otherwise, determine the rate of increase in the population of geckos after 10 months. page 17 of 35 PLEASE TURN OVER

18 QUESTION 10 Jacob owns a rabbit. He feeds his rabbit a combination of two brands of rabbit food: Bunnifill and Rabbitmeal. Let x be the number of grams of Bunnifill eaten by the rabbit per week. Let y be the number of grams of Rabbitmeal eaten by the rabbit per week. Bunnifill contains 2 units of fat, 2 units of protein, and 2 units of vitamins per gram of product. Rabbitmeal contains 3 units of fat, 1 unit of protein, and 6 units of vitamins per gram of product. (a) The vet recommends that a rabbit should eat no more than 240 units of fat per week. Write this as a constraint in terms of x and y. The other weekly dietary constraints recommended for a rabbit are: 2x y100 x3y90 x 0 y 0. (b) Interpret the meaning of the constraint 2x y 100 in the context of the question. page 18 of 35

19 (c) On the set of axes below, graph all the constraints and shade the feasible region. y x (4 marks) (d) State the vertices of the feasible region. page 19 of 35 PLEASE TURN OVER

20 Jacob purchases Bunnifill and Rabbitmeal from Lamani Pet Store. The objective function for the profit P (in dollars) made by Lamani Pet Store from the sale of Bunnifill and Rabbitmeal is expressed as P006. xky. Lamani Pet Store makes a profit of $6.60 for the sale of a packet containing 50 grams of Bunnifill and 40 grams of Rabbitmeal. (e) (i) Determine the value of k. (ii) Interpret your answer to part (e)(i). (iii) Using your answers to part (d) and to part (e)(i), calculate the maximum weekly profit that Lamani Pet Store can make from the sale of Bunnifill and Rabbitmeal to Jacob. page 20 of 35

21 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 12(e) continued ). page 21 of 35 PLEASE TURN OVER

22 QUESTION 11 Let f x3 2. x (a) Calculate f 1. (b) Show that f 5 3h 1 h 1. h (c) Find the average rate of change of f x3 2 from x1 to x1h. x (3 marks) page 22 of 35

23 (d) Hence or otherwise, determine the slope of the tangent to the curve y 3 2 at x 1. x (e) Determine the equation of the tangent to the curve y 3 2 at x 1. x page 23 of 35 PLEASE TURN OVER

24 QUESTION 12 A fishing boat has a maximum capacity of 30 passengers. The cost per passenger to hire the boat is shown on the graph below, where C is the cost per passenger in dollars and p is the number of passengers. 500 C The linear model connecting ln C and ln p is given by ln C ln p. p (a) Using the laws of logarithms, clearly show that the equation for the power model connecting C and p is C 500. p (3 marks) (b) (i) Find the value of C 16. page 24 of 35

25 (ii) What does your answer to part (b)(i) tell you about the cost of hiring a fishing boat? (c) Using the equation for the power model in part (a), determine the minimum number of passengers who could hire the boat if the cost per passenger was no more than $120. (d) Find the total cost of hiring the boat for 25 passengers. (e) Find the average rate of decrease in the cost per passenger to hire the boat, when between 16 and 25 passengers hire the boat. (3 marks) page 25 of 35 PLEASE TURN OVER

26 QUESTION 13 (a) If Z is the standard normal distribution, determine the value of k if PZ k (b) It has been found that the delay times for Select Airways flights are normally distributed with a mean of 15 minutes. (i) If 12% of Select Airways flights have a delay time of 20 minutes or more, use your answer to part (a) to show that the standard deviation for the delay times for Select Airways flights is approximately 4.27 minutes. (ii) On the normal distribution curve below, shade the area that represents the 12% of Select Airways flights that have a delay time of 20 minutes or more. 15 (c) Find the probability that a Select Airways flight will have a delay time of more than 15 minutes but less than 20 minutes. page 26 of 35

27 (d) In a sample of 50 Select Airways flights, approximately how many will have a delay time of less than 10 minutes? The management at Select Airways recorded the delay time of randomly selected flights. Let D 100 be the distribution of the average delay times for samples of 100 flights. (e) Find the mean and the standard deviation of D 100. (f ) (i) Determine the value of P 14 D (ii) What does your answer to part (f )(i) suggest about the delay times of Select Airways flights? page 27 of 35 PLEASE TURN OVER

28 QUESTION 14 Every Sunday Annie tries to visit either her parents or her grandparents. The pattern of her visits is recorded in the following transition table. Next Sunday Parents Grandparents Neither Parents This Sunday Grandparents Neither (a) State the 3 3 transition matrix, V, for Annie s Sunday visits. (b) (i) Find V 2. (ii) Interpret the values found in row 3 of V 2. page 28 of 35

29 (c) (i) Find V 16. (ii) Interpret your answer to part (c)(i). (iii) In the subsequent 52 weeks, on how many Sundays would you expect Annie to visit her grandparents? page 29 of 35 PLEASE TURN OVER

30 QUESTION 15 The body temperature of a patient, C (degrees Celsius), was monitored for 8 hours after medication was administered. The patient s temperature over time, t (hours), can be modelled by Ct te 06. t, where 0t 8. The graph of C t is shown below. 41 C ( Celsius) t (hours) (a) Find the value of C t (b) (i) Solve for t if te 39. (ii) What does your answer to part (b)(i) tell you about the patient? page 30 of 35

31 06. t (c) (i) Show that C t 4e t. (ii) Determine the value of t such that Ct 0. (d) (i) On the following set of axes, graph Ct. C'(t) t (hours) (3 marks) (ii) State the values of t such that Ct 0. (iii) What does your answer to part (d)(ii) suggest about the patient? page 31 of 35 PLEASE TURN OVER

32 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 15(d)(iii) continued ). page 32 of 35

33 You may remove this page from the booklet by tearing along the perforations so that you can refer to it while you write your answers. LIST OF MATHEMATICAL FORMULAE FOR USE IN STAGE 2 MATHEMATICAL METHODS Standardised Normal Distribution A measurement scale X is transformed into a standard scale Z using the formula X Z where is the population mean and is the standard deviation for the population distribution. Binomial Probability PX kc p 1 p n k n k k where p is the probability of a success in one trial and the possible values of X are k 01,, n and C n k n nn 1n k1. nk k k Condence Interval Mean A 95% con dence interval for the mean of a normal population with standard deviation, based on a simple random sample of size n with sample mean x, is x 196. x n n For suitably large samples, an approximate 95% con dence interval can be obtained by using the sample standard deviation s in place of. Binomial Mean and Standard Deviation The mean and standard deviation of a binomial count X and a proportion of successes p X n are X np p p p1 p X np1 p p n where p is the probability of a success in one trial. Sample Size Mean The sample size n required to obtain a 95% con dence interval of width w for the mean of a normal population with standard deviation is n 1.96 w. 2 2 Derivatives f x y x n e kx ln x log e x y f x d dx nx n1 ke kx 1 x Condence Interval Population Proportion An approximate 95% con dence interval for the population proportion p, based on a large simple random sample of size n with sample proportion p X n, is p1 p p1 p p 196. p p1.96. n n Sample Size Proportion The sample size n required to obtain an approximate 95% con dence interval of approximate width w for a proportion is n w 2 p 1 p where p is a given preliminary value for the proportion. Properties of Derivatives d dx f x g x f x g x d dx f x g x f x g x d dx x kfx d dx f x g x f x g x f x g x d dx f g x f gxgx Laws of Logarithms log Alog Blog AB A log Alog Blog B n log A nlog A page 33 of 35 PLEASE TURN OVER

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DRAFT. Mathematical Methods 2017 Sample paper. Question Booklet 1

DRAFT. Mathematical Methods 2017 Sample paper. Question Booklet 1 1 South Australian Certificate of Education Mathematical Methods 017 Sample paper Question Booklet 1 Part 1 Questions 1 to 10 Answer all questions in Part 1 Write your answers in this question booklet