Reading Time: 15 minutes Writing Time: 1 hour 30 minutes Letter Student Number: Structure of Book Section A - Core Section B - Modules Number of questions Number of questions to be answered Number of marks 9 9 36 Number of modules Number of modules to be answered Number of marks 4 2 24 Total 60 Students are to write in blue or black pen. Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, one bound reference, one approved technology (calculator or software) and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. For approved computer-based CAS, full functionality may be used. Students are NOT permitted to bring into the examination room: blank sheets of paper and/or correction fluid/tape. Materials Supplied Question and answer book of 33 pages. Formula sheet. Working space is provided throughout the book. Instructions Write your name in the space provided above on this page. All written responses must be in English. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
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SECTION A Core Answer all questions in the spaces provided. Instructions for Section A You need not give numerical answers as decimals unless instructed to do so. Alternative forms may include, for example, π, surds or fractions. In Recursion and financial modelling, all answers should be rounded to the nearest cent unless otherwise instructed. Unless otherwise indicated, the diagrams in this book are not drawn to scale. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 1
Data Analysis QUESTION 1 (3 marks) The distribution of average number of shots at goal per game (AV SPG) for the top 40 soccer players in the European leagues is shown in the histogram below: a. What is the percentage of players in this group who average more than five shots at goal per game? b. Describe the shape of the distribution of the average shots at goal per game data for these players. c. What is the modal class interval for this data? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 2
QUESTION 2 (5 marks) A dot plot is shown below of the total number of goals scored so far this season for the top 40 players in the European leagues. a. What type of data is the number of goals scored? Some of the summary statistics for the number of goals scored data are in the table below. Minimum Q 1 Median Q 3 Maximum 1 5 20.5 30 b. Add the median value for the number of goals scored to the table above. c. Explain why there are no outliers in this data. Include appropriate calculations that support your answer. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 3
d. Construct a boxplot of the number of goals scored data on the grid below. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 4
QUESTION 3 (3 marks) The percentaged segmented bar charts shown below compare the percentage of time that two teams in the English Premier league, Manchester City (Man City) and Newcastle United (New United), have the ball in their possession on average during a game. Manchester City is currently on top of the ladder and Newcastle United are sitting in 10 th position. a. Do the percentaged segmented bar charts support the contention that a higher ladder position is associated with a greater time that a team has the ball in their possession? Explain your answer quoting appropriate statistics. 2 marks b. A soccer coach tells his team that, if they have the ball in their possession for a greater percentage of time, it follows that they will have a higher ladder position. Can this conclusion be drawn from the percentage segmented bar charts above? Explain your answer. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 5
QUESTION 4 (7 marks) A scatterplot is shown below of the total number of goals scored this season against the average number of shots per game for the top 40 players in the European leagues: The equation of the least squares regression line, correct to 2 decimal places, for this data is Total Number of Goals Scored 0.12 4.81 Average Shots per Game The value of the coefficient of determination is 0.5993, correct to 4 decimal places. a. Describe the relationship between total number of goals scored and average shots per game in terms of strength, direction and form. b. Add the least squares regression line to the scatterplot shown above. ANSWER ON GRAPH c. Interpret the slope of the least squares regression line for this relationship. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 6
d. Interpret the coefficient of determination for this relationship. e. Calulate the residual of total goals scored for the player who averaged 4.5 shots per game. Give your answer correct to one decimal place. 2 marks The residual plot for the relationship between number of goals scored and average number of shots per game is shown below: f. Explain what the residual plot indicates about the relationship between number of goals scored and the average shots per game. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 7
QUESTION 5 (4 marks) The brand value of most English Premier League (EPL) teams has been steadily increasing over the last few years. The estimated brand value in millions of US dollars of the Everton soccer club over the years 2011 through to 2017 is listed below. Year Estimated Brand Value in millions of US dollars 2011 61 2012 79 2013 78 2014 121 2015 228 2016 279 2017 344 The relationship between brand value in $millions and year is best described using a logarithmic transformation of the brand value in $millions axis. a. Determine the equation of the least squares regression line that results from this transformation. Write the gradient and vertical intercept values correct to three significant figures. 2 marks log( brand value in $millions ) year b. Predict the brand value of Everton soccer club in millions of US dollars in the year 2020, using the equation from part a. Give your answer correct to the nearest million dollars. c. Explain why this prediction may not be reliable. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 8
QUESTION 6 (2 marks) The time series plot below shows the average attendance at soccer games played in the Italian premier league during the ten years from 1996 through to 2005 inclusive. Smooth the time series plot using the three point moving median method. Mark each smoothed point with a cross (x). 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 9
Recursion and Financial Modelling QUESTION 7 (3 marks) Harry, who is a soccer player, purchases an investment property worth $400 000. The property is depreciated at 2.5% per annum on a flat rate basis. a. By what amount is the property depreciated each year if it is depreciated at 2.5% of $400 000 each year? b. Write a recurrence relation, in terms of P n+1 and P n, that gives the book value of Harry s property after n years. c. Determine the book value of Harry s property after 11 years. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 10
QUESTION 8 (4 marks) Tim is another player who has taken out a loan to purchase a home. The balance of Tim s loan after n months is given by the recurrence relation L 0 = 520 000 L n+1 = 1.0035 L n 2800 a. Use the recurrence relation to show how the balance of Tim s loan after two months would be determined. 2 marks b. Tim wants to know how much interest he will pay during the fifth year of his loan. Determine the total interest paid on Tim s loan during the fifth year. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 11
QUESTION 9 (5 marks) Mile is another soccer player who is concerned that his soccer career will not last forever and that he must save for his future. He has savings of $50 000, but he has committed to saving an additional $24 000 every year. He will invest his money in an annuity investment at 3.87% per annum compounding annually and add $24 000 to the account at the end of each year when the interest is calculated. a. How much money would Mile have in his account after 2 years? Mile knows that he will need at least $650 000 in his account to support his lifestyle after soccer. b. How long would it take before Mile has at least $650 000? Give your answer correct to the nearest year. Mile has been told that he would be better to compound monthly at the same rate and add $2000 per month to his investment. c. How much sooner would Mile reach his goal of at least $650 000? Give your answer correct to the nearest month. 2 marks d. Explain why Mile would achieve his saving goal sooner if he compounds more often. END OF SECTION A The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 12
SECTION B Modules Instructions for Section B Select two modules and answer all questions within the selected modules. You need not give numerical answers as decimals unless instructed to do so. Alternative forms may include, for example, π, surds or fractions. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Contents Page Module 1 Matrices... 16 Module 2 Networks and Decision Mathematics... 20 Module 3 Geometry and Measurement... 26 Module 4 Graphs and Relations... 32 The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 13
Module 1: Matrices QUESTION 1 (3 marks) Hannah runs a canteen at the local junior soccer club. The canteen sells sushi (S), noodles (N) and tacos (T). The number of each item sold during games 1, 2 and 3 is shown in the matrix A below: S N T 12 17 14 A = [ 24 32 35] 29 22 39 The element in row i and column j of matrix A is a ij. Game 1 Game 2 Game 3 a. What is the meaning of element a 23? Sushi is sold for $3, noodles are sold for $1.50 and tacos are sold for $5. 3 The matrix B = [ 0]. 5 b. Calculate the matrix product AB. c. Explain the information contained in the matrix product AB. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 14
QUESTION 2 (4 marks) At the junior soccer club the coach is conducting trials to determine who will play keeper. Four players, Alana (A), Beatrice (B), Carol (C) and Dana (D), take part in a round robin competition, where each player competes against each other player exactly once. In each trial there is a winner and a loser. A win is indicated by a 1 in the matrix, K, below: loser A B C D K = winner A B C D 0 1 1 0 0 0 1 1 [ ] 0 0 0 1 1 0 0 0 a. Explain why the shaded elements in the matrix, K, are all zero. The coach considers the sum of K and K 2 when deciding who will be keeper. The player with the greatest overall dominance will be selected. b. i. What does the sum of the elements in the second row of K 2 mean? ii. Who will be keeper for the soccer club? In the junior boys team, Eli (E), Finn (F), George (G) and Harry (H) competed in a similar round robin tournament to decide who will play keeper. The matrix J 2 shows the two step dominances each player has over each other player: J 2 = winner E F G H loser E F G H 0 1 1 0 1 0 0 1 [ ] 0 2 0 1 0 0 1 0 c. Who did Finn defeat in the keeper trials? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 15
QUESTION 3 (5 marks) The soccer club have players in a number of different age levels, junior (J), intermediate (I) and seniors (S). At the beginning of each season some players move up in age group, some remain in the same age group and some players leave (L). The changes can be modelled using the transition matrix: this season J I S L T = 0.8 0 0 0 0.1 0.6 0 0 [ ] 0 0.2 0.7 0 0.1 0.2 0.3 1 J I S L next season a. Explain, in terms of the context of this problem, why the transition matrix T is a lower triangular matrix. At the start of the 2016 season, there were 50 senior players, and 100 players in each of the intermediate and junior levels. This is represented by the matrix, 200 200 S 2016 = [ ] 100 0 J I S L b. At the start of the 2017 season, there were 140 intermediate players. Show the calculation that produces an element of 140 in the second row of S 2017. c. At the start of which year s season would there be fewer than 40 players in the senior team, if this pattern was to continue? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 16
The club wants to keep up its player numbers, so after the 2017 season they recruit new players so that the numbers remain steady at the same numbers for each of the junior, intermediate and senior teams. d. How many new players would they need to recruit at each of the junior, intermediate and senior level so that the numbers at each level remain at the 2017 numbers? 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 17
Module 2: Networks and decision mathematics QUESTION 1 (3 marks) A soccer coach has arranged a training schedule based around a number of points on the soccer pitch. These are shown on the diagram below as points A, B, C, D, E, F, G, H and I. The players must run along the lines of the pitch between each training point. a. Simone runs along from point A to F to E to D to E. Give a name to the route that Simone takes. Caroline wants to start and finish at point A and go to every training point on the pitch without repeating any training point. This is not possible unless she misses one training point. b. Which training point would Caroline need to miss? If a network was drawn with the training points as vertices and the lines on the pitch as edges, there would be a number of loops on the network. c. List all vertices that would have one or more loops at that vertex. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 18
QUESTION 2 (3 marks) The men s senior soccer team have five players that they have not yet allocated positions for. A bipartite graph is shown below that shows the players and the positions that they could play: a. One player is required for each of the five positions. Complete the table below of who should play each position. Goalkeeper Sweeper Forward Midfielder Defender b. Adam has indicated that he could also play Forward. Does this change the allocations made in part a? Explain your answer. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 19
Three umpires, Anna, Bronte and Carla, will each attend each of the games played one Saturday. The umpires will be allocated so that the minimum distance is travelled overall. The distances, in km, that each umpire lives from each of the Senior, Reserves and Junior games is listed in Table 1 below: Table 1 Senior Game Reserves Game Junior Game Anna 11 23 13 Bronte 12 25 14 Carla 15 26 9 Table 2 shows the same information after the Hungarian Algorithm has been applied: Table 2 Senior Game Reserves Game Junior Game Anna 0 0 2 Bronte 0 1 2 Carla 6 5 0 c. In Table 2 there is a zero in the row for Carla. When all values in the table are considered, what conclusion about minimum total distance can be made from this zero? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 20
QUESTION 3 (6 marks) A working bee is to be held at the soccer club. The activity network for one of the projects is shown below. The duration of each activity is also given in hours: a. Which activity is predecessor to the greatest number of other activities? b. List the critical path for this project. Two activities can be delayed for a longer time than any other activities without increasing the critical time for this project. c. Name the two acitivities that have the largest float time. d. What is the latest finishing time for activity C? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 21
The project needs to be finished earlier. Five activities, G, H, I, J or K could be reduced if an extra person was available at the working bee for each activity. The reduction in time per activity would be two hours for each activity that had an additional person. e. What is the minimum time in which the project could now be completed? f. What is the minimum number of additional people required to complete the project in the reduced time? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 22
Module 3: Geometry and Measurement QUESTION 1 (4 marks) A soccer club has a goal that is shaped as a trapezoidal prism. The dimensions of the goal are shown below: A diagram of one end of the goal is shown below: a. Show that the length AB is 2.56 m correct to two decimal places. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 23
The net on the goal needs replacing. The net does not cover the front or the base of the goal. b. Determine the area of netting that is required for replacement. Give your answer in square metres, correct to one decimal place. 2 marks c. A soccer player stands 40 metres directly in front of the goal and kicks the ball off the ground as shown in the diagram below. Assuming that the path of the ball is a straight line, what is the maximum angle of elevation of the ball that would result in a goal? Give your answer correct to the nearest degree. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 24
QUESTION 2 (4 marks) a. A soccer player must travel from Perth, Western Australia (32⁰ S, 116⁰ E) to Beijing, China (40⁰ N, 116⁰ E). Assume that the radius of the Earth is 6400 km. Find the shortest great circle distance between Perth and Beijing. Round your answer correct to the nearest kilometre. b. The player then needs to fly from Beijing (40⁰ N, 116⁰ E) to Boulder, Colorado, USA (40⁰ N, 105⁰ W). The flight to Boulder takes 12 hours and 15 minutes. Assume that 15⁰ of longitude equates to a one hour time difference. He leaves Beijing at 7.10 am on Monday 5 th of November, 2018. On what day and at what time will he arrive in Boulder, Colorado? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 25
c. Assume that the radius of the Earth is 6400 km. What is the shortest distance from Beijing (40⁰ N, 116⁰ E) to Boulder, Colorado, USA (40⁰ N, 105⁰ W)? Give your answer correct to the nearest kilometre. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 26
QUESTION 3 (4 marks) The soccer pitch is 90 metres wide and 120 metres long. It is watered using sprinklers that are centred at points A, B and C as shown. Point C is at the midpoint of the sideline. The sprinkler at point C waters the shaded sector shown above with a radius of 90 metres. a. Show that the area watered by the sprinkler placed at point C is 4763 m 2 correct to the nearest square metre. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 27
The sprinklers at A and B also water the soccer pitch. They both spray over a sector with a radius of 60 metres. A diagram is shown below, where the shaded areas are watered by more than one sprinkler: b. What area of the soccer pitch is watered by more than one sprinkler? Give your answer in square metres, correct to the nearest whole square metre. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 28
Module 4: Graphs and Relations QUESTION 1 (2 marks) The graph below shows the fee structure at a local soccer club, where the fees paid depend on the age of the player at the start of the calendar year. a. What is the fee if a player is 10 years old at the start of the calendar year? One family pay $1750 in fees in 2018. They have a number of children who play soccer with the club, including one child aged seven, one child aged 12 and one child aged 14. Their other children who play soccer are aged 16 or more. b. How many children from this family who are 16 years old or more play soccer with the club? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 29
QUESTION 2 (6 marks) The soccer club have club shirts that they sell to their members for a set amount. The graph of the revenue against n, the number of shirts sold, is shown on the graph below: The revenue equation is R = 75n where 75 is the gradient of the line. a. Show that the gradient of the revenue line is 75. The shirts are made at a local supplier. The cost, C, to the club for n shirts is given by the equation C = 1000 + 55n. b. Add the line C = 1000 + 55n to the graph above. ANSWER ON GRAPH c. Write an equation for the profit, P, for the sale of shirts at this club. The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 30
d. What is the minimum number of shirts the club need to sell in order to make a profit? The club negotiate a different arrangement with their supplier of shirts. From the next season they will pay an upfront fee of $800 and then the first 100 shirts will cost $60 each. After the first 100, the cost per additional shirt will drop to $40 each. This can be modelled using the relationship 60n + a, 0 < n 100 C = { bn + c, n > 100 e. Determine the values of a, b and c. 2 marks The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 31
QUESTION 3 (4 marks) The soccer club need to take a team and some officials to a game. They will travel by sharing vehicles, where the number of utes is x and the number of cars is y. A number of constraints apply to this situation: Each ute can only take 2 people and each car can take 5 people. At least 15 people need to go to the game. Only 10 people are able to drive, so a maximum of 10 vehicles can go to the game. 2x + 5y 15 x + y 10 There are a maximum of 7 utes available to go to the game. x 7 There are a maximum of 8 cars available to go to the game. y 8 A graph showing the lines produced by each of these inequalities: a. Shade the region that is defined by this set of inequalities. ANSWER ON GRAPH b. If four utes are driven to the game, what is the minimum number of cars that could also go? The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 32
There is also a large amount of equipment required at the game, so there now must be at least two utes, forming a new constraint inequality. The total number of vehicles going to the game must also be minimised. The new constraint must also be considered. The objective function is N = x + y. c. How many of each vehicle should be going to the game if N is to be minimised? Give all possible solutions to this problem. 2 marks END OF QUESTION AND ANSWER BOOK The School For Excellence 2018 Units 3 & 4 Further Mathematics Written Examination 2 Page 33