Matrices AT

Transcription

1 VCE Further Mathematics Matrices AT Part A Outcome 1 Define and explain key concepts and apply related mathematical techniques and models in routine contexts. Outcome 2 Select and apply the mathematical concepts, models and techniques in a range of contexts of increasing complexity. Outcome 3 Select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches. Problem Solving Task: Matrices This task will be marked out of a TOTAL of 56 marks This task has two parts: Part A uses matrices to solve problems involving transition situations. (27 marks) Part B covers basic knowledge and simple operations as well as using matrices to represent data and solve problems. (30 marks) You will have 80 minutes to complete each part. You can access your bound reference and an approved CAS calculator. Answer in the space provided. 1 P a g e

2 PART A 26 MARKS A round robin tournament of Rock Paper Scissors was held between a computer (C) and four other participants: Doug, Ellie, Fong, and Grace. The results of the 10 games are shown in this dominance matrix. 1. Explain why, in a game of win or loss without draws, a dominance matrix cannot be symmetric. Multiply this dominance matrix by a summing matrix to give the one step order of dominance between the 5 competitors. 1+1=2 marks Show, and complete, a matrix calculation to determine the overall order of dominance, using one and two step dominances. State the most and least dominant players = 3 marks 2 P a g e

3 Long term analysis of the playing form of footballers shows that: 72% of players who play well one week, play well the following week. 25% of players who play well one week, play reasonably the following week. 3% of players who play well one week, play poorly the following week. 22% of players who play reasonably one week, play well the following week. 55% of players who play reasonably one week, play reasonably the following week. 23% of players who play reasonably one week, play poorly the following week. 8% of players who play poorly one week, play well the following week. 40% of players who play poorly one week, play reasonably the following week. 52% of players who play poorly one week, play poorly the following week. This scenario involves 18 players who each play every game in a 21 game season. 2. Give a transition matrix, T, that shows the weekly changes of how well people play. In the first game of the season, all 18 players play well. Express this as an initial state matrix, S 0. 3 P a g e

4 Use a matrix calculation to determine the number of players who play well, reasonably, and poorly, in the second game of the season. Round your answer to whole numbers = 2 marks (d) Calculate (to whole numbers) the number of players playing well, reasonably and poorly in: (i) the third game of the season. (ii) the 10 th game of the season = 3 marks (e) (i) In which game does the team reach a consistent number of players playing well, reasonably and poorly for the rest of the season? Give the number of players in each state in that game. (ii) How does this change if all players played poorly in the first game? = 3 marks 4 P a g e

5 A radio station aims to play 300 songs each week. The songs are designated as high rotation (played more than once per day), medium rotation (played once per day) and low rotation (played less than once per day). Each week the station changes the list by moving some songs up or down the list, and by adding some new hit songs to high rotation and deleting some older songs from the list. The transition matrix for songs follows the pattern shown in this matrix. This is interpreted as: for this week s songs on high rotation 28% are on high rotation next week, 15% are moved to medium rotation, 32% are moved to low rotation and 25% are deleted from the play list. 3. Interpret the 4 th column of this transition matrix. In the first week of a new year, there are 20 songs on high rotation, 120 songs on medium rotation and 160 songs on low rotation. Put this information into an initial state matrix that can be used with the transition matrix to determine future states. Using the transition matrix determine the number of songs on each type of rotation in the following week, and the number deleted. Round your answers to units = 2 marks 5 P a g e

6 (d) The eventual outcome of this transition process, with songs deleted every week, is that there will eventually be no songs left to play. To illustrate this trend, determine the number of songs on each rotation, and the total number deleted, in the 20 th week = 2 marks (e) A recurrence relation, S n+1 = T S n + A can be applied from the start of the new year to ensure that the number of songs on each type of rotation is maintained. Calculate the value of the matrix A. Round values to units. An alternative rotation system is suggested to the radio station, using a transition matrix as follows. 4. Describe how this triangular matrix changes the way in which songs move through the rotation categories, compared with using matrix T. 6 P a g e

7 Given that S 0 remains the same as in 3, determine the number of songs in each category in week 2 using F as the transition matrix = 2 marks Determine the matrix B that would have to be applied in the recurrence relation S n+1 = S n F + B to maintain a constant number of songs on each rotation each week. Total: 27 marks 7 P a g e

8 VCE Further Mathematics Matrices AT Part B Outcome 1 Define and explain key concepts and apply related mathematical techniques and models in routine contexts. Outcome 2 Select and apply the mathematical concepts, models and techniques in a range of contexts of increasing complexity. Outcome 3 Select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches. Task Problem Solving Task: Matrices This task will be marked out of a TOTAL of 56 marks This task has two parts: Part A uses matrices to solve problems involving transition situations. (27 marks) Part B covers basic knowledge and simple operations as well as using matrices to represent data and solve problems. (30 marks) You will have 80 minutes to complete each part. You can access your bound reference and an approved CAS calculator. Answer in the space provided. 1 P a g e

9 PART B 30 MARKS For the given matrices shown below, answer questions 1 5. You can detach this page to assist answering the questions on the next page. 2 P a g e

10 1. Write down the order of Matrix C 5. Where possible, perform the following matrix operations. 2. Give the names of the matrices that are: Square G + M Symmetric 2N C Binary J + 5G 3. Give the names of any Identity matrices (d) PC Permutation matrices (e) DE 4. Identify the matrix that is C T (f) O 2 G -1 (g) (F + J) T each, 5 (g) 2 marks = 16 marks 3 P a g e

11 A café sells coffee in three sizes, standard (S) medium (M) and large (L). Sales for one week are shown in this matrix. 6. Pre-multiplying matrix C by the summing matrix [ ] provides what information? Post-multiplying matrix C by the summing matrix shown here, provides what information? Sales for a second week are shown in matrix D. Calculate C + D and interpret the information = 3 marks 4 P a g e

12 The takings from the sale of coffees for the first three days of the first week as shown in matrix C are given in matrix R. (d) Construct and then solve a matrix equation which can be used to calculate the individual prices of each size of coffee = 3 marks The coffee shop has one barista (Pip) and four people who wait on tables and deliver orders to the barista. As occasionally happens, they are not all on speaking terms with each other. The communication network between Pip and the four wait staff, Quintin, Rachel, Shelley and Tam, is shown here. This network shows that Pip will talk to Quinton, Rachel and Tam, but not to Shelley. However, Shelley and Quinton will talk to Pip. Quinton and Pip will talk with each other. And so on. 5 P a g e

13 7. Show this communication network as a matrix. Interpret the values on the leading diagonal of your matrix What is the least number of communication steps required for Rachel to send a message to Pip and get a reply? (d) Use a matrix method to determine the number of one and two step communications between each worker. Leave your answer as a 5 5 matrix = 2 marks (e) Interpret the values on the leading diagonal of your answer to (d). Total: 30 marks 6 P a g e