Victorian Certificate of Education 2023 General Mathematics PDF

Victorian Certificate of Education 2023 GENERAL MATHEMATICS Written examination 1 Friday 27 October 2023 Reading time: 2.00 pm to 2.15 pm (15 minutes)...

Victorian Certificate of Education 2023 GENERAL MATHEMATICS Written examination 1 Friday 27 October 2023 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.45 pm (1 hour 30 minutes) MULTIPLE-CHOICE QUESTION BOOK Structure of book Number of Number of questions Number of questions to be answered marks 40 40 40 Total 40 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 book of 24 pages Formula sheet Answer sheet for multiple-choice questions Working space is provided throughout the book. Instructions Check that your name and student number as printed on your answer sheet for multiple-choice questions are correct, and sign your name in the space provided to verify this. Unless otherwise indicated, the diagrams in this book are not drawn to scale. At the end of the examination You may keep this question book and the formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. © VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2023 2023 GENMATH 1 2 THIS PAGE IS BLANK  3 2023 GENMATH 1 Instructions Answer all questions in pencil on the answer sheet provided for multiple-choice questions. Choose the response that is correct for the question. A correct answer scores 1; an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Data analysis Use the following information to answer Questions 1 and 2. The dot plot below shows the times, in seconds, of 40 runners in the qualifying heats of their 800 m club championship. 134 135 136 137 138 139 140 141 142 143 144 145 146 time (seconds) Question 1 The median time, in seconds, of these runners is A. 135.5 B. 136 C. 136.5 D. 137 E. 137.5 Question 2 The shape of this distribution is best described as A. positively skewed with one or more possible outliers. B. positively skewed with no outliers. C. approximately symmetric with one or more possible outliers. D. approximately symmetric with no outliers. E. negatively skewed with one or more possible outliers. TURN OVER 2023 GENMATH 1 4 Question 3 Gemma’s favourite online word puzzle allows her 12 attempts to guess a mystery word. Her number of attempts for the last five days is displayed in the table below. Day Number of attempts 1 8 2 11 3 5 4 6 5 9 On day six, how many attempts can she make so that the mean number of attempts for these six days is exactly eight? A. 5 B. 6 C. 7 D. 8 E. 9 Question 4 The time spent by visitors in a museum is approximately normally distributed with a mean of 82 minutes and a standard deviation of 11 minutes. 2380 visitors are expected to visit the museum today. Using the 68–95–99.7% rule, the number of these visitors who are expected to spend between 60 and 104 minutes in the museum is A. 1128 B. 1618 C. 2256 D. 2261 E. 2373 Question 5 The heights of a group of Year 8 students have a mean of 163.56 cm and a standard deviation of 8.14 cm. One student’s height has a standardised z-score of −0.85. This student’s height, in centimetres, is closest to A. 155.4 B. 156.6 C. 162.7 D. 170.5 E. 171.7  5 2023 GENMATH 1 Question 6 The histogram below displays the distribution of prices, in dollars, of the cars for sale in a used-car yard. The histogram has a logarithm (base 10) scale. 6 4 frequency 2 0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 log10 (price) Six of the cars in the yard have the following prices: $2450 $3175 $4999 $8925 $10 250 $105 600 How many of the six car prices listed above are in the modal class interval? A. 1 B. 2 C. 3 D. 4 E. 6 TURN OVER 2023 GENMATH 1 6 Use the following information to answer Questions 7 and 8. A teacher analysed the class marks of 15 students who sat two tests. The test 1 mark and test 2 mark, all whole number values, are shown in the scatterplot below. A least squares line has been fitted to the scatterplot. 50 48 46 44 42 40 38 36 34 test 2 mark 32 30 28 26 24 22 20 18 16 14 14 16 18 20 22 24 26 28 30 32 34 36 38 test 1 mark Question 7 The equation of the least squares line is closest to A. test 2 mark = −6.83 + 1.55 × test 1 mark B. test 2 mark = 15.05 + 0.645 × test 1 mark C. test 2 mark = −6.78 + 0.645 × test 1 mark D. test 2 mark = 1.36 + 1.55 × test 1 mark E. test 2 mark = 6.83 + 1.55 × test 1 mark Question 8 The least squares line shows the predicted test 2 mark for each student based on their test 1 mark. The number of students whose actual test 2 mark was within two marks of that predicted by the line is A. 3 B. 4 C. 5 D. 6 E. 7  7 2023 GENMATH 1 Question 9 A least squares line can be used to model the birth rate (children per 1000 population) in a country from the average daily food energy intake (megajoules) in that country. When a least squares line is fitted to data from a selection of countries it is found that: for a country with an average daily food energy intake of 8.53 megajoules, the birth rate will be 32.2 children per 1000 population for a country with an average daily food energy intake of 14.9 megajoules, the birth rate will be 9.9 children per 1000 population. The slope of this least squares line is closest to A.   −4.7 B.   −3.5 C.   −0.29 D.    2.7 E.   25 Question 10 A study of Year 10 students shows that there is a negative association between the scores of topic tests and the time spent on social media. The coefficient of determination is 0.72 From this information it can be concluded that A. a decreased time spent on social media is associated with an increased topic test score. B. less time spent on social media causes an increase in topic test performance. C. an increased time spent on social media is associated with an increased topic test score. D. too much time spent on social media causes a reduction in topic test performance. E. a decreased time spent on social media is associated with a decreased topic test score. TURN OVER 2023 GENMATH 1 8 Use the following information to answer Questions 11 and 12. The table below shows the height, in metres, and the age, in years, for 11 plantation trees. A scatterplot displaying this data is also shown. age height (years) (m) 11 10 9.5 10 8 8.0 9 13 9.7 8 9 9.1 7 11 9.4 height (m) 6 5 14 9.8 4 6 6.0 3 4 3.5 2 12 9.6 3 4 5 6 7 8 9 10 11 12 13 14 15 7 7.8 age (years) 5 4.0 Question 11 A reciprocal transformation applied to the variable age can be used to linearise the scatterplot. 1 With as the explanatory variable, the equation of the least squares line fitted to the linearised data is age closest to 1 A. height = −13.04 + 40.22 × age 1 B. height = −10.74 + 8.30 × age 1 C. height = 2.14 + 0.63 × age 1 D. height = 13.04 − 40.22 × age 1 E. height = 16.56 − 22.47 × age  9 2023 GENMATH 1 Question 12 The scatterplot can also be linearised using a logarithm (base 10) transformation applied to the variable age. The equation of the least squares line is height = −3.8 + 12.6 × log10(age) Using this equation, the age, in years, of a tree with a height of 8.52 m is closest to A. 7.9 B. 8.9 C. 9.1 D. 9.5 E. 9.9 TURN OVER 2023 GENMATH 1 10 Use the following information to answer Questions 13 and 14. The following graph shows a selection of winning times, in seconds, for the women’s 800 m track event from various athletic events worldwide. The graph shows one winning time for each calendar year from 2000 to 2022. 119 118 117 winning time (seconds) 116 115 114 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 year Data: https://www.worldathletics.org/records Question 13 The time series is smoothed using seven-median smoothing. The smoothed value for the winning time in 2006, in seconds, is closest to A. 116.0 B. 116.4 C. 116.8 D. 117.2 E. 117.6 Question 14 The median winning time, in seconds, for all the calendar years from 2000 to 2022 is closest to A. 116.8 B. 117.2 C. 117.6 D. 118.0 E. 118.3  11 2023 GENMATH 1 Question 15 The number of visitors to a public library each day for 10 consecutive days was recorded. These results are shown in the table below. Day number 1 2 3 4 5 6 7 8 9 10 Number of visitors 337 317 313 335 322 335 322 338 302 349 The eight-mean smoothed number of visitors with centring for day number 6 is A. 323 B. 324 C. 325 D. 326 E. 327 Question 16 The number of visitors each month to a zoo is seasonal. To correct the number of visitors in January for seasonality, the actual number of visitors, to the nearest percent, is increased by 35%. The seasonal index for that month is closest to A. 0.61 B. 0.65 C. 0.69 D. 0.74 E. 0.77 TURN OVER 2023 GENMATH 1 12 Recursion and financial modelling Question 17 A sequence of numbers is generated by the recurrence relation shown below. T0 = 5,    Tn + 1 = −Tn The value of T2 is A. −10 B.   −5 C.   0 D.    5 E.   10 Use the following information to answer Questions 18 and 19. Gus purchases a coffee machine for $15 000 and depreciates its value using the unit cost method. The rate of depreciation is $0.04 per cup of coffee made. A recurrence relation that models the year-to-year value Gn , in dollars, of the machine is G0 = 15000,    Gn +1 = Gn − 1314 Question 18 A rule for Gn , the value of the machine after n years is A. Gn = 15 000 − 0.04n B. Gn = 15 000 + 0.04n C. Gn = 15 000 − 1314n D. Gn = 1314 − 0.04n E. Gn = 1314 + 0.04n Question 19 The number of cups made by the machine per year is A.   1314 B. 13 686 C. 15 000 D. 31 536 E. 32 850  13 2023 GENMATH 1 Use the following information to answer Questions 20 and 21. For taxation purposes, Audrey depreciates the value of her $3000 computer over a four-year period. At the end of the four years, the value of the computer is $600. Question 20 If Audrey uses flat rate depreciation, the depreciation rate, per annum is A. 10% B. 15% C. 20% D. 25% E. 33% Question 21 If Audrey uses reducing balance depreciation, the depreciation rate, per annum is closest to A. 10% B. 15% C. 20% D. 25% E. 33% Question 22 Timmy took out a reducing balance loan of $500 000, with interest calculated monthly. The balance of the loan, in dollars, after n months, Tn , can be modelled by the recurrence relation T0 = 500 000,     Tn + 1 = 1.00325Tn − 2611.65 A final repayment that will fully repay the loan to the nearest cent is A. $2605.65 B. $2609.18 C. $2611.65 D. $2614.12 E. $2615.81 TURN OVER 2023 GENMATH 1 14 Question 23 Tavi took out a loan of $20 000, with interest compounding quarterly. She makes quarterly repayments of $653.65. The graph below represents the balance in dollars of Tavi’s loan at the end of each quarter of the first year of the loan. balance 20000 (0, 20 000) (1, 19527.56) 19000 (2, 19 050.83) (3, 18569.79) 18000 (4, 18 084.39) quarters 0 1 2 3 4 The effective interest rate for the first year of Tavi’s loan is closest to A. 3.62% B. 3.65% C. 3.66% D. 3.67% E. 3.68% Question 24 The following recurrence relation models the value, Pn , of a perpetuity after n time periods. P0 = a,   Pn + 1 = RPn − d The value of R can be found by calculating A. a + d a+d B. a a+d C. d a+d D. 1+ a a+d E. 1+ d  15 2023 GENMATH 1 Matrices Question 25 The daily maximum temperature at a regional town for two weeks is displayed in the table below. Monday Tuesday Wednesday Thursday Friday Saturday Sunday Week 1 20 °C 17 °C 23 °C 20 °C 18 °C 19 °C 30 °C Week 2 29 °C 27 °C 28 °C 21 °C 20 °C 20 °C 22 °C This information can also be represented by matrix M, shown below.  20 17 23 20 18 19 30  M    29 27 28 21 20 20 22  Element m21 indicates that A. the temperature was 29 °C on Monday in week 2. B. the temperature was 17 °C on Tuesday in week 1. C. the lowest temperature for these two weeks was 17 °C. D. the highest temperature for these two weeks was 29 °C. E. week 2 had a higher average maximum temperature than week 1. Question 26 Matrix P is a permutation matrix and matrix Q is a column matrix. 1 0 0 0 0 t  0 0 1 0 0  e    P  0 0 0 1 0 Q  a     0 1 0 0 0 m 0 0 0 0 1   s      When Q is multiplied by P, which three letters change position? A. t, e, a B. e, a, m C. a, m, s D. m, s, t E. e, a, s TURN OVER 2023 GENMATH 1 16 Question 27 The following transition matrix, T, models the movement of a species of bird around three different locations, M, N and O from one day to the next. this day M N O 1 9 3 0 M 10    T  1 1 1 N next day 3 10    1 0 0 O  3  Which one of the following statements best represents what will occur in the long term? A. No birds will remain at location M. B. No birds will remain at location N. C. All of the birds will end up at location M. D. All of the birds will end up at location O. E. An equal number of birds will be at all three locations.  17 2023 GENMATH 1 Question 28 Four table tennis teams played in a round-robin tournament. Each team played each other team once and there were no draws. The overall ranking of each team at the end of the tournament, based on number of wins, is shown in the table below. First Unicorns (U) Second Vampires (V) Third Scorpions (S) Fourth Titans (T) A dominance matrix can display the results of each game, where a ‘1’ in the matrix shows that the team named in that row defeated the team named in that column. The dominance matrix for this tournament could be A. loser B. loser S T U V S T U V S 0 1 1 1 S 0 0 1 1 T 0 0 1 1  T 0 0 0 0  winner winner U 0 0 0 1 U 1 1 0 1     V 0 0 0 0 V 1 1 0 0 C. loser D. loser S T U V S T U V S 0 1 0 0 S 0 1 0 0 T 0 0 1 0 T 0 0 0 0  winner winner U 1 0 0 1 U 1 1 0 1     V 1 1 0 0 V 1 1 0 0 E. loser S T U V S 0 1 0 0 T 0 0 0 0  winner U 1 1 1 1   V 0 1 1 0 TURN OVER 2023 GENMATH 1 18 Question 29 Matrix K is a 3 × 2 matrix. The elements of K are determined by the rule kij = (i − j)2. Matrix K is A. 0 1 2  B. 0 1 4  1 0 1 1 0 1      C.  0 1 D. 0 1 1 0  1 0       4 1   2 1  E. 0 1 1 0     4 1  Question 30 How many of the following statements are true? All square matrices have an inverse. The inverse of a matrix could be the same as the transpose of that matrix. If the determinant of a matrix is equal to zero, then the inverse does not exist. It is possible to take the inverse of an identity matrix. A. 0 B. 1 C. 2 D. 3 E. 4 Question 31 A species of bird has a life span of three years. The females in this species do not reproduce in their first year but produce an average of four female offspring in their second year, and three in their third year. The Leslie matrix, L, below is used to model the female population distribution of this species of bird.  0 4 3  L  0.2 0 0   0 0.4 0  The element in the second row, first column states that on average 20% of this population will A. be female. B. never reproduce. C. survive into their second year. D. produce offspring in their first year. E. live for the entire lifespan of three years.  19 2023 GENMATH 1 Question 32 For one particular week in a school year, students at Phyllis Island Primary School can spend their lunch break at the playground (P), basketball courts (B), oval (O) or the library (L). Students stay at the same location for the entire lunch break. The transition diagram below shows the proportion of students who change location from one day to the next. 0.40 0.40 P B 0.30 0.20 0.40 0.10 0.30 0.20 0.10 0.10 L O 0.10 0.30 The transition diagram is incomplete. On the Monday, 150 students spent their lunch break at the playground, 50 students spent it at the basketball courts, 220 students spent it at the oval, and 40 students spent it in the library. Of the students expected to spend their lunch break on the oval on the Wednesday, the percentage of these students who also spent their lunch break on the oval on Tuesday is closest to A. 27% B. 30% C. 33% D. 47% E. 52% TURN OVER 2023 GENMATH 1 20 Networks and decision mathematics Question 33 Consider the following graph. How many of the following five statements are true? The graph is a tree. The graph is connected. The graph contains a path. The graph contains a cycle. The sum of the degrees of the vertices is eight. A. 1 B. 2 C. 3 D. 4 E. 5 Question 34 A bipartite graph is typically used to display which one of the following? A. the allocation of tasks on a construction site B. the path used to visit five different construction sites C. the total distance travelled between two construction sites D. the critical path of activities to be completed in a construction project E. the minimum length of cable required to connect six construction sites  21 2023 GENMATH 1 Question 35 Consider the weighted graph shown below. 15 10 8 12 9 7 6 The weight of the minimum spanning tree is A. 30 B. 32 C. 40 D. 42 E. 52 Question 36 Four employees, Anthea, Bob, Cho and Dario, are each assigned a different duty by their manager. The time taken for each employee to complete duties 1, 2, 3 and 4, in minutes, is shown in the table below. Duty 1 Duty 2 Duty 3 Duty 4 Anthea 8 7 7 8 Bob 10 8 10 9 Cho 8 9 7 10 Dario 7 7 8 9 The manager allocates the duties so as to minimise the total time taken to complete the four duties. The minimum total time taken to complete the four duties, in minutes, is A. 29 B. 30 C. 31 D. 32 E. 33 TURN OVER 2023 GENMATH 1 22 Question 37 The adjacency matrix below represents a planar graph with five vertices. J K L M N 0 1 0 1 1 J 1 0 2 1 1  K  0 2 0 1 1 L   1 1 1 0 1 M 1 1 1 1 0  N The number of faces on the planar graph is A.   5 B.   7 C.  9 D. 15 E. 17  23 2023 GENMATH 1 Question 38 A particular building project has ten activities that must be completed. These activities and their immediate predecessor(s) are shown in the table below. Activity Immediate predecessor(s) A – B – C A D A E B F D, E G C, F H F I D, E J H, I A directed graph that could represent this project is A. C A G D F H start B finish E I J B. C dummy A G D F H start B finish E I J C. C A G D F dummy start H B finish E I J D. C A G D dummy start F H B finish E I J E. C dummy A G D F H start B finish E I J TURN OVER 2023 GENMATH 1 24 Use the following information to answer Questions 39 and 40. The network below shows the one-way paths between the entrance, A, and the exit, H, of a children’s maze. The vertices represent the intersections of the one-way paths. The number on each edge is the maximum number of children who are allowed to travel along that path per minute. B 15 G 12 H 10 exit 10 A 12 10 F entrance 10 6 C 8 4 4 8 5 E 7 D Question 39 Cuts on this network are used to consider the possible flow of children through the maze. The capacity of the minimum cut would be A. 20 B. 23 C. 24 D. 29 E. 30 Question 40 One path in the maze is to be changed. Which one of these five changes would lead to the largest increase in flow from entrance to exit? A. increasing the capacity of flow along the edge CE to 12 B. increasing the capacity of flow along the edge FH to 14 C. increasing the capacity of flow along the edge GH to 16 D. reversing the direction of flow along the edge CF E. reversing the direction of flow along the edge GF END OF MULTIPLE-CHOICE QUESTION BOOK Victorian Certificate of Education 2023 GENERAL MATHEMATICS Written examination 1 FORMULA SHEET Instructions This formula sheet is provided for your reference. A multiple-choice question book is provided with this formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. © VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2023 GENMATH EXAM 2 General Mathematics formulas Data analysis xx standardised score z sx lower and upper fence in a boxplot lower Q1 – 1.5 × IQR    upper Q3 + 1.5 × IQR sy least squares line of best fit y = a + bx,  where   b = r   and   a  y  bx sx residual value residual value = actual value – predicted value actual figure seasonal index seasonal index = deseasonalised figure Recursion and financial modelling first-order linear recurrence relation u0 = a,  un + 1 = Run + d  n  effective rate of interest for a r  reffective   1    1  100% compound interest loan or investment  100n    3 GENMATH EXAM Matrices a b  a b determinant of a 2 × 2 matrix A  ,   det A   ad  bc c d  c d 1  d b  inverse of a 2 × 2 matrix A1  ,  where  det A ≠ 0 det A  c a  recurrence relation S0 = initial state,   Sn + 1 = T Sn + B Leslie matrix recurrence relation S0 = initial state,   Sn + 1 = L Sn Networks and decision mathematics Euler’s formula v+f=e+2 END OF FORMULA SHEET