Networks and decision mathematics

Definitive VCAA question-type taxonomy and wording bank (2006–2026)

Corpus: corpus/text\*.txt — every VCAA Further Mathematics / General Mathematics exam paper and examination report 2006–2026 (main + NHT), plus sample papers, assessment guides and examination specifications. Every claim below is traceable to a named file (see §4 Source index).

Note on gaps: four corpus files are text-less (vector-only) PDFs with no content — Documents_exams_mathematics_2024_2024GeneralMaths1-w.txt, Documents_exams_mathematics_2024_2024GeneralMaths2-w.txt, 2025-05_2025-NHT-generalmaths1.txt, 2025-05_2025-NHT-generalmaths2.txt. For those four papers the question content is reconstructed from their reports / assessment guides (2025-03_2024generalmaths1-report.txt, 2025-03_2024generalmaths2-report.txt, 2025-04_2024generalmathematics2-assessment-guide.txt, 2025-04_General_Maths_Examination_2.txt (VCAA exam-review transcript), 2025-10_2025-NHT-general-maths1-report.txt, 2025-10_2025-NHT-general-maths2-report.txt, 2025-06_2025NHT-GeneralMath2-assessment-guide.txt). Documents_exams_mathematics_2011furmath1-w.txt has a corrupted font encoding; its Module 5 content is reconstructed from Documents_exams_mathematics_further1_assessrep_11.txt.

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0. STRUCTURAL MAP — where Networks sits, and how much it is worth

Specification, verbatim (2025-04_genmath-specs-w.txt):

"The examination will be divided into four content areas: data analysis, recursion and financial modelling, matrices, and networks and decision mathematics. The examination will consist of 40 multiple-choice questions worth 1 mark each. Of these 40 questions, 16 will be allocated to data analysis, 8 … recursion and financial modelling, 8 … matrices, and 8 will be allocated to networks and decision mathematics."
"[Examination 2] … 24 marks will be allocated to data analysis, 12 marks … recursion and financial modelling, 12 marks … matrices, and 12 marks will be allocated to networks and decision mathematics."

Formula sheet. The only networks formula supplied is Euler's formula, printed as:

Euler's formula: v + f = e + 2 (Documents_exams_mathematics_2006furmath1-w.txt p.1444; identical in 2007, 2019 and all intervening papers; retained in Documents_exams_mathematics_generalmaths1-formula-w.txt / generalmaths2-formula-w.txt.)

Module-election instruction (2006–2022 only). Every Exam 1 module opens with:

"Before answering these questions, you must shade the 'Networks and decision mathematics' box on the answer sheet for multiple-choice questions and write the name of the module in the box provided." (2008 onward; 2006–2007 omit the "and write the name of the module" clause.) This instruction disappears in 2023+ because the area is compulsory.

Marks distribution in Exam 2 (GM era), verbatim structure: - 2023: Q12 (4 marks), Q13 (3 marks), Q14 (5 marks) = 12 - 2024: Q13 (4), Q14 (3), Q15 (5) = 12 - 2025: Q15 (4), Q16 (2), Q17 (2), Q18 (4) = 12 - 2024 NHT: Q14 (3), Q15 (4), Q16 (5) = 12 - 2026 NHT: Q13 (4), Q14 (3), Q15 (5) = 12

Structural constant across all 21 years: the last question of the Networks Exam-2 block is always a project-network / critical-path / crashing question, and it is always the hardest (its final part typically has a 7–25% success rate).

Difficulty benchmark (mean % of module available marks, Exam 2): 2006 53% → 2007 49% → 2008 52% → 2009 49% → 2010 46% → 2011 44% → 2012 35% → 2013 41% → 2014 44% → 2015 49% → 2016 51% → 2017 51% → 2018 50% → 2019 27/45 marks-basis → 2020 19.4% selection share → 2021 28% selection share → 2022 31.1% selection share. (Figures from the "Networks and decision mathematics" row at the head of each examination report; note that from 2019 the reported figure changes meaning from mean score to module-selection percentage.)

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1. COMPLETE QUESTION-TYPE CATALOGUE

1.1 Graph terminology — degree, degree sum, edges, vertices, loops, multiple edges

What it looks like. Almost always the opening question of the module (Exam 1 Q1/Q2; Exam 2 Q1a). A picture of a small undirected graph, or a stated property. One mark. Never more than one mark.

Exam 1 vs Exam 2. Present in both, every year without exception. Exam 1 form: "…is" + 5 (later 4) numeric options. Exam 2 form: "Write down / What is / Calculate…", 1 mark.

Sub-forms observed:

Real examples.

• 2019 Exam 1 Module 2 Question 1 — "In the graph shown above, the sum of the degrees of the vertices is / A. 5 B. 6 C. 10 D. 11 E. 12". Answer E; 85% correct (Documents_exams_mathematics_2019_FM1_examrep19.txt).
• 2025 Exam 2 Question 15a (1 mark) — "Calculate the sum of the degrees of all the vertices in this graph." Answer: 2 + 4 + 2 + 3 + 3 = 14; 88% correct (2026-01_2025-GeneralMaths2-report.txt).
• 2009 Exam 2 Module 5 Question 3a (1 mark) — "Write down the degree of vertex U."
• 2026 NHT Exam 2 Question 13a (1 mark) — "What is the degree of vertex S?" Guide: "Connects to P, PO and F | 3".
• 2021 NHT Exam 1 Module 2 Question 1 — "For a connected graph with four vertices and four edges, the sum of the degrees of the vertices is". Answer D (8).
• 2025 Exam 1 Question 34 — "Consider the following graph. The number of bridges in this graph is / A. 1 B. 2 C. 3 D. 4". Answer C; report: "A bridge is an edge in a network whose removal will disconnect the network. Any of the top 3 edges, when removed, cause the graph to be disconnected."

Multi-statement "how many of these statements are true" variant (a recurring hard Exam 1 item): 2011 Exam 1 Q9, 2013 Exam 1 Q7, 2014 Exam 1 Q6/Q7, 2020 Exam 1 Q7, 2021 Exam 1 Q5, 2023 Exam 1 Q33, 2024 Exam 1 Q38. - 2013 Exam 1 Module 5 Question 7 — "A connected graph consists of five vertices and four edges. Consider the following five statements. • The graph is planar. • The graph has more than one face. • All vertices are of even degree. • The sum of the degrees of the vertices is eight. • The graph cannot have a loop. How many of these statements are always true for such a graph?" Answer C (3). Report: "only three are true for all graphs … the graph is planar / the sum of the degrees of the vertices is eight / the graph cannot have a loop." - 2023 Exam 1 Question 33 — "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." Answer D (4); 56% correct.

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1.2 Planar graphs, faces and Euler's formula (v + f = e + 2)

What it looks like. Exam 1: a "could have"/"the number of faces is" item, or "which graph is not planar", worth 1 mark. Exam 2: a fill-the-boxes item, 1–2 marks.

Frequency. Appears in every single Exam 1 from 2006 to 2026 except 2010, 2013, 2022 NHT. Appears in Exam 2 only in 2017, 2023, 2025 (the box-filling form).

Exam 1 forms, with years:

• "A connected planar graph has 12 edges. This graph could have / A. 5 vertices and 6 faces …" — 2007 Exam 1 Q2
• "A connected planar graph has 10 edges and 10 faces. The number of vertices for this graph is" — 2009 Exam 1 Q4
• "A planar graph has five vertices and six faces. The number of edges is" — 2015 Exam 1 Q2
• "A planar graph has five faces. This graph could have / A. eight vertices and eight edges …" — 2018 Exam 1 Q3, repeated verbatim as GM sample Exam 1 Q35 (Documents_exams_mathematics_genmath1-sample-w.txt)
• "A connected planar graph has seven vertices and nine edges. The number of faces that this graph will have is" — 2020 Exam 1 Q1 (89% correct)
• "Consider the graph below. The number of faces is" — 2021 Exam 1 Q3; "How many faces does this graph have?" — 2017 NHT Exam 1 Q3, 2021 NHT Exam 1 Q4
• "Euler's formula will be verified for this graph. What values of e, v and f will be used in this verification?" — 2019 NHT Exam 1 Q2
• "Euler's formula, relating vertices, faces and edges, does not apply to which one of the following graphs?" — 2006 Exam 1 Q8; "Euler's formula can be applied to which of the following graphs?" — 2024 NHT Exam 1 Q37
• "The adjacency matrix above represents a planar graph with four vertices. The number of faces (regions) on the planar graph is" — 2012 Exam 1 Q4; five-vertex version 2023 Exam 1 Q37
• "Which one of the following graphs is not a planar graph?" — 2018 Exam 1 Q6 (41%); "The number of edges that need to be removed for this graph to be planar is" — 2022 Exam 1 Q4 (answer A = 0; only 21% correct); "Which one of the following graphs can be redrawn as the planar graph above?" — 2016 Exam 1 Q5; "How many of these four graphs are planar?" — 2014 Exam 1 Q7, 2019 NHT Exam 1 Q6, GM sample Exam 1 Q39
• "A number of edges must be added to create a connected planar graph with 12 vertices and 3 faces. The number of edges that must be added is" — 2023 NHT Exam 1 Q4
• "A tree contains 11 vertices. One edge is added to the tree to make a new graph. How many faces does the resulting planar graph have?" — 2026 NHT Exam 1 Q36

Exam 2 form (the "boxes" form) — three near-identical occurrences:

• 2017 Exam 2 Module 2 Question 1c (1 mark):

  "Euler's formula, v + f = e + 2, holds for this graph.
  Complete the formula by writing the appropriate numbers in the boxes provided below.
  ☐ + ☐ = ☐ + 2
  v f e"

Answer 6 + 7 = 11 + 2; 83% correct. Report warning: "some students did not seem to check that the values were arithmetically correct, for example, 6 + 6 = 11 + 2." - 2023 Exam 2 Question 12b (2 marks) — identical stem, plus a second part:

"ii. Complete the sentence by writing the appropriate word in the box provided below.
Euler's formula holds for this graph because the graph is connected and ☐." (Answer: planar) - 2025 Exam 2 Question 15b (1 mark) — identical stem, "…in the boxes below". Answer 5 + 4 = 7 + 2; 91% correct. Guide: "All three boxes correct" for the single mark.

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1.3 Adjacency matrices ↔ graphs

What it looks like. Exam 1: matching a matrix to one of five/four drawn graphs (or vice versa), or counting entries. Exam 2: completing a matrix, completing a diagram from a matrix, or explaining a zero.

Exam 1 occurrences: 2007 Q3, 2010 Q3, 2012 Q4, 2017 Q3, 2017 NHT Q4, 2019 Q6, 2019 NHT Q7, 2020 Q8, 2021 Q7, 2023 Q37, 2023 NHT Q6, 2024 NHT Q34, 2025 NHT Q35/Q37, 2026 NHT Q39, GM sample Q38.

Exam 1 examples: - 2019 Exam 1 Module 2 Question 6 — "The map below shows all the road connections between five towns, P, Q, R, S and T. The road connections could be represented by the adjacency matrix …". Answer A. Report (Documents_exams_mathematics_2019_FM1_examrep19.txt): "It is possible to return to town P without passing through another town. This loop is identified as a 1 in the first row and first column of the matrix. Options C and D do not show this loop so can be discarded. There are three different ways of travelling directly between P and Q. Option A is the only remaining matrix that shows this." (Re-used verbatim as GM sample Exam 1 Q38.) - 2021 Exam 1 Module 2 Question 7 — "An adjacency matrix for this network is formed. The number of zeros in this matrix is". Answer C (10). Report: "A zero in an adjacency matrix designates no direct connection." - 2018 NHT Exam 1 Q5 — "In this matrix, a '0' indicates no friendship and a '1' indicates a friendship. How many zero elements will this adjacency matrix have?" - 2026 NHT Exam 1 Question 39 — matrix given; "For this graph, which one of the following statements is not true? A. The graph contains a Hamiltonian cycle. B. … an Eulerian circuit. C. … a spanning tree with four edges. D. The graph has a loop." (Matrix reading combined with terminology.)

Exam 2 occurrences: 2006 Q2, 2009 Q1, 2010 Q1, 2019 (none), 2021 NHT Q1d, 2022 Q2d, 2022 NHT Q2, 2023 NHT Q1d, 2024 Q13c, 2025 Q16.

Exam 2 examples: - 2010 Exam 2 Module 5 Question 1a (1 mark) — "Explain the meaning of a zero in the adjacency matrix." Model answer (Documents_exams_mathematics_further2_assessrep_10.txt): "Are not allowed to communicate with each other". Examiner note: "Responses had to be relevant to the context of the question. A number of students seemed to quote material directly from their notes and gave unacceptable answers such as 'there is no connection or edge' or 'cannot get to one point from another' with no link to the context at hand." - 2009 Exam 2 Module 5 Question 1a (1 mark) — "Explain why all values in the final row and final column are zero." Answer: "There is no land border between E and any other suburb." - 2006 Exam 2 Module 5 Question 2a (1 mark) — "Explain why the figures in bold in Matrix 1 are all zero." Answer: "No musician competes against him/her self." Examiner note: "Answers that referred to 'the absence of loops in the directed graph' were not accepted as they did not explain why these values are zero." - 2022 Exam 2 Module 2 Question 2d (2 marks) — "The adjacency matrix below shows road connections between the office and each cabin. … On the diagram below, add the new cabin, H, and any additional roads to the network." 80% got both marks. - 2025 Exam 2 Question 16 (2 marks) — "An adjacency matrix for this network is shown below. Some values in the matrix are given as x, y and z. … In this matrix, the '2' in row E, column C indicates that there are two ways of moving from the entry to the change room without passing through another area or backtracking. Write the values of x, y and z in the boxes below." Answers x = 2, y = 4, z = 2. Guide: A1 for "any two boxes correct", A1 for "all three boxes correct". Report: "The value for y was often given as 3." (20% got both marks.) - 2023 NHT Exam 2 Question 1d (2 marks) — "An adjacency matrix for this network is shown below. This matrix is incomplete. Complete the adjacency matrix above. (Answer on the adjacency matrix above.)"

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1.4 Isomorphic / equivalent graphs

Low frequency, Exam 1 only. 1 mark. - 2015 Exam 1 Module 5 Question 5 — friendship network K, L, M, N: "Which one of the following graphs does not contain the same information?" - 2023 NHT Exam 1 Module 2 Question 1 — same set-up ("The graph below represents a friendship network…"): "Which one of the following statements is true?" - 2019 Exam 1 Module 2 Question 4 — "Two graphs, labelled Graph 1 and Graph 2, are shown below. Which one of the following statements is not true? A. Graph 1 and Graph 2 are isomorphic. B. Graph 1 has five edges and Graph 2 has six edges. C. Both Graph 1 and Graph 2 are connected graphs. D. Both Graph 1 and Graph 2 have three faces each. E. Neither Graph 1 nor Graph 2 are complete graphs." - 2025 NHT Exam 1 Question 35 — "One approach is to consider the adjacency matrices … Graphs I and III are equivalent" (2025-10_2025-NHT-general-maths1-report.txt). VCAA's own recommended technique for isomorphism is build the adjacency matrices and compare. - 2013 Exam 1 Q6, 2015 Exam 1 Q6, 2022 NHT Exam 1 Q5, 2022 Exam 1 Q2, 2023 Exam 2 Q12c: "map → graph" matching, the practical cousin of isomorphism.

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1.5 Walks, trails, paths, circuits, cycles — and telling them apart

This is the single most heavily-penalised terminology area in the subject. See §3.1.

Exam 1 forms. - "Which one of the following is not a path for this graph?" — 2018 Exam 1 Q4 (88% correct) - "Which one of the following is not a Hamiltonian cycle for this graph?" — 2020 Exam 1 Q2 (93%) - "Which one of the following is not a Hamiltonian path?" — 2023 NHT Exam 1 Q2 - "A Hamiltonian circuit for the graph above is" — 2008 Exam 1 Q3 - "The number of Hamiltonian circuits involving all five vertices in the graph above is" — 2012 Exam 1 Q2 - "The number of Hamiltonian cycles in this graph, starting from E, is" — 2025 Exam 1 Q33 (answer B = 2; report: "Only 2 Hamiltonian cycles can be created when starting at E. EDCBAFE and EFABCDE") - "A bus starts at Kelly, travels through Nate and Lindon, then stops when it reaches Milton. The mathematical term for this route is A. a loop. B. an Eulerian path. C. an Eulerian circuit. D. a Hamiltonian path. E. a Hamiltonian cycle." — 2014 Exam 1 Q1 (87%)

Exam 2 form — the "what is the mathematical term" question. This is a fixture: it appears in 2010, 2012, 2013, 2017, 2017 NHT, 2019, 2019 NHT, 2020, 2021, 2022, 2023 NHT, 2024, 2024 NHT, 2025, 2025 NHT, 2026 NHT — i.e. 16 of the last 17 sittings. Always 1 mark.

Verbatim variants:

Set-up sentences that cue each term (learn these, they are the tell): - Hamiltonian path → "visit each … once, starting at X and ending at a different location" (2026 NHT Q13b), "visits each … once, ending at picnic area P1" (2013 Q1d), "visit each of the other locations once and end at her home" (2025 Q15c). - Hamiltonian cycle → "start and finish at the office, visiting each building only once" (2019 Q1b), "visit each exercise station and return to exercise station S" (2020 Q3), "visits each cabin once only before returning to the office" (2022 Q1a), "visits each attraction once, starting and finishing at the entrance" (2024 NHT Q15a). - Eulerian trail → "travel along each road once only" / "run along all tracks just once" (2020 Q3), "leave from … and travel along each road once only" (2022 Q1b). - Eulerian circuit → "travel along every road once and return to the start".

Counting variants: "In how many ways could he do this?" (2016 Exam 2 Q2c, Hamiltonian cycles — 31% correct, answer 4 with the note "Each Hamiltonian cycle can be reversed"); "Starting at vertex J, how many different Eulerian circuits are possible?" (2026 NHT Exam 1 Q35); "How many different ways are there to travel from Farnham to Carrie without passing through any town more than once?" (2011 Exam 2 Q1b, answer 6); "How many different ways may a vehicle travel from town A to town D without travelling along any road more than once?" (2008 Exam 2 Q2a, answer 7); "How many different routes from S to O are possible?" (2020 Exam 2 Q4a, answer 10 — only 30% correct, "9 was a common incorrect response").

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1.6 Eulerian trails and circuits — conditions on odd-degree vertices

The highest-value "explain why" territory in the topic. See §2.2 and §3.1.

Exam 1 forms: - "Which one of the following graphs has an Eulerian circuit?" — 2021 NHT Exam 1 Q2 - "How many of the graphs above have an Eulerian trail?" — 2022 NHT Exam 1 Q3; "How many of these four graphs have an Eulerian circuit?" — 2014 Exam 1 Q6 - "The minimum number of extra edges that are required so that an Eulerian circuit is possible in this graph is" — 2019 Exam 1 Q2 - "An Eulerian trail for the graph above will be possible if only one edge is removed. In how many different ways could this be done?" — 2017 Exam 1 Q6 (answer E = 5; only 29% correct). Report: "This question relied on the knowledge that a graph will have an Eulerian trail if exactly two of the vertices of that graph have an odd degree. The graph presented to the students had four odd-degree vertices and so an edge between any two of these odd-degree vertices could be removed. In this graph, there were five such edges." - "For the graph shown above, it is possible to find A. an Eulerian trail and an Eulerian circuit. B. an Eulerian trail and a Hamiltonian path. …" — 2018 NHT Exam 1 Q3 - "A simple connected graph has at least one vertex with an odd degree and at least one vertex with an even degree. Which one of the following could be the number of vertices with an odd degree in this graph?" — 2018 NHT Exam 1 Q7 (tests the handshaking corollary: the count of odd vertices is even) - "An undirected connected graph has five vertices. Three … even degree and two … odd degree. One extra edge is added. … In the resulting graph, it is not possible to have five vertices that are …" — 2009 Exam 1 Q8 (only 25% correct) - "An Euler path through a network commences at vertex P and ends at vertex Q. Consider the following five statements … How many of these statements are true?" — 2011 Exam 1 Q9. Report: "'The path could have included vertex Q more than once' is the only true statement."

Exam 2 forms (route + finish vertex): - 2009 Exam 2 Q3b — "Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark N. i. At which landmark must he finish his journey? ii. Regardless of which route Steven decides to take, how many of the landmarks (including those at the start and finish) will he see on exactly two occasions?" - 2010 Exam 2 Q3a — "If a cyclist started at intersection B and cycled along every road in this network once only, at which intersection would she finish?" - 2011 Exam 2 Q1c — "An engineer plans to inspect all of the roads in this network. He will start at Dunlop and inspect each road only once. At which town will the inspection finish?" - 2013 Exam 2 Q1ci — "A park ranger starts at the entrance and drives along every road in the park once. i. At which picnic area will the park ranger finish?" - 2015 Exam 2 Q1c — "An inspector checks the cables by walking along the length of each cable in one continuous path. To avoid walking along any of the cables more than once, at which vertex should the inspector start and where would the inspector finish?" - 2016 Exam 2 Q2a — "Nathan begins skating at ramp W and follows an Eulerian trail. At which ramp does Nathan finish?" - 2014 Exam 2 Q3a — "Charlie followed an Eulerian path through this network of train lines. i. Write down the names of the towns at the start and at the end of Charlie's path. ii. What distance did he travel?"

"How many extra edges" Exam 2 form: - 2018 Exam 2 Q2b — "For this graph, an Eulerian trail does not currently exist. For an Eulerian trail to exist, what is the minimum number of extra edges that the graph would require?" (Answer 2; 54%.) Examiner: "Some students gave a definition of an Eulerian trail rather than stating how many extra edges were required." - 2018 Exam 2 Q4a — "A road inspector will leave from town P to check all the roads and return to town P when the inspection is complete. He will travel the minimum distance possible. a. How many roads will the inspector have to travel on more than once?" (Answer 2; only 31%.) Examiner: "An Eulerian circuit was required; therefore, two extra edges were needed to make all vertices of even degree. Common incorrect answers were 3 and 4."

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1.7 Chinese-postman-style "repeat the minimum edges" questions

A distinctive and persistently poorly-answered sub-type: travel every edge, start and finish at the same vertex, minimum total distance.

• 2010 Exam 2 Module 5 Question 3b — "The next challenge involves cycling along every road in this network at least once. Teams have to start and finish at intersection A. The blue team does this and cycles the shortest possible total distance. i. Apart from intersection A, through which intersections does the blue team pass more than once? ii. How many kilometres does the blue team cycle?"
• 2012 Exam 2 Module 5 Question 1aiii — "The total length of all edges in the network is 1180 metres. A journey starts and finishes at the house and travels along every edge in the network. Determine the shortest distance travelled." Answer 1250 m. Report: "An Euler circuit would be an ideal solution but this is not possible due to the presence of two odd vertices … However, an 1180-metre long Euler path commencing at the house is possible, provided it ended at the other odd vertex. To return to the house, we must then add 70 metres for the length of the shortest path between these two odd vertices. This question was very poorly answered, with a common incorrect answer of 1180."
• 2018 Exam 2 Q4b — "Determine the minimum distance, in kilometres, that the inspector will travel." Answer 108 km; only 14% correct. Report: "The extra two edges (PQ and ST) needed to be added to the sum of all distances (86) on the graph."
• 2018 NHT Exam 2 Q2c — "The sum of all distances shown on the graph is 310 m. The dog starts and finishes at bush F and runs along every edge in the network. What is the shortest distance, in metres, that the dog could have run?" Answer 335 = "310 + 25 (repeated edge EF)".
• 2020 Exam 2 Q3c — "To complete training program 3 in the minimum distance, one track will need to be repeated. Complete the following sentence by filling in the boxes provided. This track is between exercise station ☐ and exercise station ☐." Answer S and T; only 25% correct.
• 2023 Exam 2 Question 13c — "Shyla would like to travel along all the roads. To complete this journey in the minimum distance, she will travel along two roads twice. Shyla will leave from landmark G but end at a different landmark. Complete the following by filling in the boxes provided. The two roads that will be travelled along twice are the roads between: • vertex ☐ and vertex ☐ • vertex ☐ and vertex ☐"
• 2025 Exam 2 Question 17 (2 marks) — see §2.2; 17a is the odd-vertex explanation, 17b "What is the minimum distance, in metres, that the gym owner will cover when completing the inspection?" Answer 196 m; only 27% correct.
• 2025 NHT Exam 2 Q14c — answer "G and H / B and L" (the two repeated roads).
• 2024 NHT Exam 2 Q15c — "…Rowan may need to visit one or more attractions twice. c. If Rowan visits each attraction, starting and finishing at the entrance, what is the length of the shortest path Rowan can take, in metres?" Answer 371 m.

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1.8 Trees, spanning trees, minimum spanning trees (Prim's / Kruskal's)

Exam 1 forms: - "Which one of the following graphs is a tree?" — 2013 Exam 1 Q1 (91%) - "Consider the graph with five isolated vertices shown below. To form a tree, the minimum number of edges that must be added to the graph is" — 2018 Exam 1 Q1; identical GM sample Exam 1 Q33 - "The following graph with five vertices is a complete graph. Edges are removed so that the graph will have the minimum number of edges to remain connected. The number of edges that are removed is" — 2016 Exam 1 Q3. Report: "The key … was to recognise that the answer would be a spanning tree, which, for a graph with five vertices, would have four edges. As the original graph had 10 edges, the number of edges to be removed was 10 − 4 = 6." - "Which one of the following is not a spanning tree for the network above?" — 2020 Exam 1 Q3 - "The plan shows the layout of a section of pipes … Which one of the following types of graph could be used to represent the layout … A. a bipartite graph B. a complete graph C. a loop D. a Hamiltonian path E. a tree" — 2015 Exam 1 Q3 - "Prim's algorithm can be used to find the A. critical path. B. shortest path. C. minimum cut. D. minimum allocation. E. minimum spanning tree." — 2022 Exam 1 Q1 (80%) - "For the network above, the length of the minimal spanning tree is" — 2010 Exam 1 Q5; "The weight of the minimum spanning tree is" — 2023 Exam 1 Q35 (74%) - "The minimal spanning tree for the network below includes two edges with weightings x and y. The length of the minimal spanning tree is 19. The values of x and y could be" — 2007 Exam 1 Q7 - "The minimum spanning tree for the network below includes the edge with weight labelled k. The total weight of all edges for the minimum spanning tree is 33. The value of k is" — 2016 Exam 1 Q4 - "The minimum spanning tree for this graph contains the edge with weight w. The length of this minimum spanning tree is A. 37 + w B. 44 + w …" — 2025 Exam 1 Q35 (65%) - "How many different minimal spanning trees are possible?" — 2017 NHT Exam 1 Q7; "How many edges with weight 2 will not be included in the minimal spanning tree?" — 2018 NHT Exam 1 Q6; "How many edges with weight 5 are not included in the minimum spanning tree?" — 2026 NHT Exam 1 Q38 - Applied wrappers: "The minimum length of cable, in metres, required to ensure that each of the seven points is connected to the main server directly or via another point is" (2019 Exam 1 Q5); "…required to ensure that all nine internet ports stay connected is" (2022 NHT Exam 1 Q4); "What is the minimum length of cable, in kilometres, that is necessary to make sure that each substation remains connected to the network?" (2019 NHT Exam 1 Q5); "The minimum total length of road, in kilometres, that needs to be cleared is" (2021 Exam 1 Q8); "The minimum length of cable required to connect all the camp sites is 53 m. The value of x, in metres, is at least" (2020 Exam 1 Q5); "The water pipes will cost $52 per metre. What is the minimum cost to link all the cabins to the water pump (P)?" (2024 NHT Exam 1 Q35). - "Based on the same set of vertices and edges, which one of the following graphs shows the cable layout (in bold) that would link all the computers with optical fibre cables for the minimum cost?" — 2013 Exam 1 Q3; "Which one of the following is the minimal spanning tree for the weighted graph shown above?" — 2014 Exam 1 Q5.

Exam 2 forms (draw it, then cost it): - 2008 Exam 2 Module 5 Question 1 (3 marks) — "The aim of the game is to connect the posts with ribbon using the shortest length of ribbon. This will be a minimal spanning tree. a. Draw in a minimal spanning tree for this network on the diagram below. (1 mark) b. Determine the length, in metres, of this minimal spanning tree. (1 mark) c. How many different minimal spanning trees can be drawn for this network? (1 mark)" — 1a answer: "Either of these two trees was accepted. A number of students drew circuits and these were not accepted." - 2011 Exam 2 Module 5 Question 2 (2 marks) — "All locations are to be connected using the smallest total length of water pipe possible. a. On the diagram, show where these water pipes will be placed. b. Calculate the total length, in metres, of water pipe that is required." Answer 510 m; "A consequential mark was available for the correct total length of any spanning tree drawn in Question 2a." - 2012 Exam 2 Module 5 Question 1b (2 marks) — "i. On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm. ii. What is the mathematical term that is used to describe this minimum length of pipe in part i.?" - 2015 Exam 2 Module 5 Question 1d (2 marks) — "i. The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree. Draw the minimum spanning tree on the plan below. ii. … How much would be saved in installation costs if the factory removed computer server P from its minimum spanning tree network?" - 2017 Exam 2 Module 2 Question 3a (2 marks) — "Electrical cables are required to power the rides. These cables will form a connected graph. The shortest total length of cable will be used. i. Give a mathematical term to describe a graph that represents these cables. ii. Draw in the graph that represents these cables on the diagram below." - 2023 NHT Exam 2 Question 2 (2 marks) — "The cheapest installation only requires pipes to be installed along some of the paths in the network. a. i. On the diagram below, draw the minimum length of pipe that is needed to supply water to all the flower gardens. ii. What is the mathematical term that is used to describe this minimum length of pipe drawn in part a.i.?" - 2025 Exam 2 Question 15d (1 mark) — "Using edges from the original graph, construct a spanning tree below." (Note: spanning tree, not minimum — unweighted graph.) 81% correct. Guide: "*Sample tree (multiple versions possible). NO new edges … 4 original edges with no isolated vertices". Report: "Some students included an edge that was not part of the original graph.*"

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1.9 Shortest path (Dijkstra or by inspection)

VCAA never uses the phrasing "Determine the shortest path from … to … . Write down the length of this path." In this entire 2006–2026 corpus the stem is always some form of "What is the shortest distance / minimum distance / shortest possible time / cheapest route…". Dijkstra is named only as a label in multiple-choice distractors (2018 Exam 1 Q2, 2021 NHT Exam 1 Q5), never as a required written procedure.

Exam 1: - "The shortest distance, in kilometres, to travel from town A to town B is" — 2007 Exam 1 Q4 - "The shortest distance, in kilometres, from Austin to Boyle is" — 2009 Exam 1 Q2 (93%) - "…the shortest time that it will take Stephanie to ride her bicycle from home to school is" — 2014 Exam 1 Q3 - "The shortest path for Niko from his home to the university could be found using A. a minimum cut. B. Prim's algorithm. C. Dijkstra's algorithm. D. critical path analysis. E. the Hungarian algorithm." — 2018 Exam 1 Q2 (68%) - "Using Dijkstra's algorithm, she finds that the A. critical path is 46 km … C. shortest path is 44 km. D. shortest path is 46 km. E. minimum spanning tree is 44 km." — 2021 NHT Exam 1 Q5 - "What is the shortest distance that Hunter can ride between home and school?" — 2017 NHT Exam 1 Q1 - "The shortest distance, in kilometres, for Margorie to travel from her home to the office is" — 2023 NHT Exam 1 Q5 - "Using this network of roads, the shortest distance that Sofia can take to travel from home to school, in metres, is" — 2025 Exam 1 Q38 (47% correct; 47% chose the distractor B — "The shortest path is 900 + 500 + 400 + 800 + 1100 = 3700") - "What is the minimum travel time, in minutes, from town C to town J?" — 2026 NHT Exam 1 Q37 - Three-person variant, 2025 NHT Exam 1 Q38 — "Alex will travel a shortest distance of 620 m … Bishnu … 620 m … Cassie … 630 m"

Exam 2: - "What would be the shortest possible time for a team to run from checkpoint X to checkpoint Y?" — 2010 Exam 2 Q2a - "In kilometres, what is the shortest distance between Farnham and Carrie?" — 2011 Exam 2 Q1a - "a. i. Determine the shortest distance between the house and the pump." — 2012 Exam 2 Q1ai (answer 160 m; "Incorrect answers included 70 + 60 + 80 = 210 and 50 + 40 + 60 + 80 = 230") - "What is the shortest distance, in metres, from the entrance to picnic area P3?" — 2013 Exam 2 Q1b - "What is the cheapest cost, in dollars, of installing cables between server K and server N?" — 2015 Exam 2 Q1b - "a. Bai considers travelling by bus along the route Northend (N) – Opera (O) – Seatown (S). How much would Bai have to pay? b. If Bai takes the cheapest route from Northend (N) to Seatown (S), which other town(s) will he pass through?" — 2017 Exam 2 Q1a/b (answer "Quigley and Rosebush"; "Some gave the path N-Q-R-S, which was accepted") - "What is the shortest distance, in metres, between the entrance and the seal exhibit (S)?" — 2019 NHT Exam 2 Q1a - "What is the shortest distance, in kilometres, covered in training program 1?" — 2020 Exam 2 Q3a (answer 3.2 km; "3.3 km was a common incorrect answer") - "What is the shortest distance, in kilometres, between Town G and Town M?" — 2021 Exam 2 Q2a (86 km, 85% correct). Examiner note: "A small number of students wrote the correct travel sequence but did not indicate the shortest distance as required by the question." - "What is the shortest distance, in kilometres, between Leah's home and the airport?" — 2021 NHT Exam 2 Q1b - "What is the minimum distance Eden could travel from G to M?" — 2023 Exam 2 Q13a - "Determine the shortest distance of the journey that starts and finishes at the dam and visits all the flower gardens, in metres." — 2023 NHT Exam 2 Q1c. Answer 855 m, route dam–T–R–G–L–P–T–dam. Note carefully: this is a shortest closed walk visiting every vertex, not a Hamiltonian cycle — the published answer revisits T. The stem deliberately avoids the word "once".

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1.10 Flow networks — capacity, cuts, max-flow / min-cut

Frequency. A flow question appears in every Exam 1 from 2006 to 2026 and in most Exam 2s. Exam 2 marks: 1–4.

Three canonical Exam 1 shapes: 1. Capacity of a named cut — "The capacity of the cut shown is" (2006 Q6, 2008 Q6, 2010 Q6, 2015 Q4, 2023 NHT Q3); "The capacity of Cut Q, in litres per minute, is" (2019 Q3 = GM sample Q34); "What is the capacity of Cut 1?" (2025 Q36). 2. Maximum flow — "The maximum flow from source to sink through the network shown above is" (2009 Q3); "The maximum flow between source and sink through the network is" (2010 Q7); "The maximum flow, in litres per minute, from the source to the sink is" (2016 Q2); "The maximum number of vehicles per hour that can travel through this network from the town onto the freeway is" (2012 Q7); "What is the maximum flow of water, in litres per minute, from source to sink?" (2025 Q37); "What is the maximum number of visitors able to walk from the entrance to the exit each hour?" (2024 NHT Q38); "The capacity of the minimum cut would be" (2023 Q39). 3. Which cut / which change — "Five cuts are drawn on the diagram. The maximum number of cars per minute that will reach the exit is given by the capacity of A. Cut A … E. Cut E" (2014 Q9); "…is given by the capacity of A. Cut A if x = 1 …" (2017 Q8); "The number of these cuts with a capacity equal to the maximum flow … is" (2020 Q9); "Which one of these five changes would lead to the largest increase in flow from entrance to exit?" (2023 Q40); "Which one of the following flow diagrams shows a cut that has a capacity of 19?" (2019 NHT Q4); "The maximum flow through this directed network is 35 litres per minute. This maximum flow can be achieved when A. w = 8, x = 6, y = 6 …" (2022 NHT Q7).

Exam 2 shapes: - "Determine the capacity of Cut 1." — 2007 Exam 2 Q3a, 2020 Exam 2 Q4b, 2022 Exam 2 Q3a, 2026 NHT Exam 2 Q14a. Preceded in 2020/2022 by the framing sentence "When considering the possible flow of [people/stormwater] through this network, many different cuts can be made." - "Cut A, shown on the graph, has a capacity of 10. Two other cuts are labelled as Cut B and Cut C. i. Write down the capacity of Cut B. ii. Write down the capacity of Cut C." — 2018 Exam 2 Q1a - "On the graph above, draw the cut (Cut 2) that has a capacity of 70 litres per minute. Label your answer clearly as Cut 2." — 2018 NHT Exam 2 Q3b; identical idea 2026 NHT Exam 2 Q14b: "A cut passing through four edges has a capacity of 109 megalitres per hour. Draw this cut on the directed network above. Label this cut as Cut 2." (Guide: "MUST be labelled Cut 2") - "Determine the maximum number of deliveries that can be made each day from the Central Mail Depot to the Zenith Post Office." — 2018 Exam 2 Q1b (answer 7; only 32% correct — "The answer of 9 from part a. was often repeated.") - "What is the maximum flow from S to O, in number of people per minute?" — 2020 Exam 2 Q4c - "Determine the maximum number of vehicles that can travel from the entrance to the exit per hour." — 2021 Exam 2 Q3a (answer 1330; only 24% — "Many students did not demonstrate they had tried to find the minimum cut.") - "What is the maximum flow of stormwater, in litres per minute, from the source to the sink?" — 2022 Exam 2 Q3b (answer 44; 37%) - "What is the maximum flow of water through the estate in megalitres per hour?" — 2026 NHT Exam 2 Q14c

"Improve the flow" sub-type (post-2021 favourite, almost always the lowest-scoring part on the paper): - 2021 Exam 2 Q3b — "The local council plans to increase the number of vehicles per hour … by increasing the capacity of only one road. i. Complete the following sentence by filling in the boxes provided. The road that should have its capacity increased is the road from vertex ☐ to vertex ☐. ii. What should be the minimum capacity of this road to maximise the flow of vehicles from the entrance to the exit?" (bii answer 780 — 6% correct.) - 2022 Exam 2 Q3c/d — "The maximum flow through this network may be increased either by reversing the direction of flow through one pipe or by increasing the capacity of one pipe. c. … The pipe that should have its flow reversed to cause the largest increase in flow from source to sink is the pipe that runs from vertex ☐ to vertex ☐. d. … The pipe that should have its capacity increased … is the pipe that runs from vertex ☐ to vertex ☐. Its new capacity, in litres per minute, should be at least ☐." (3d: 2% correct — the single lowest-scoring Networks part in the corpus. "Very few students completed the question correctly. Some students listed the vertices correctly, but not the new capacity.") - 2021 NHT Exam 2 Q2d — "Travelsafe Airlines can add eight extra seats to one of its flights in order to allow eight more passengers to travel from Melbourne (M) to London (L) on this day. Name two cities between which these extra seats could now be available." - 2011 Exam 2 Q4c — "The new pipe must enable the greatest possible rate of flow of stormwater into the ocean from Outlet 2. What minimum rate of flow through the pipe, in kilolitres per minute, will achieve this?"

"Single convoy / one group" sub-type — the path bottleneck, not the max flow: - 2007 Exam 2 Q3c — "On one particular train, 10 children set out from the West Terminal. No new passengers board … Determine the maximum number of children who can arrive at the East Terminal on this train." (Answer 7; "A common incorrect answer was 8.") - 2017 NHT Exam 2 Q3c — "A company would like to send one group of trucks … All trucks in this group must follow each other and travel along the same route … What is the maximum number of trucks that could be in this group?" (Answer 8.) - 2024 NHT Exam 1 Q39 — "A group of students set out from the entrance and walk to the exit. The students all walk together and travel along the same route … What is the maximum number of students that could be in the group?"

Multi-source / multi-sink (super-source) variants: 2011 Exam 2 Q4 (two sources, two outlets), 2014 Exam 1 Q9 (two car parks, one exit). Both are explicitly flagged in the reports as requiring the cut to separate all sources from the sink.

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1.11 Bipartite graphs, allocation and the Hungarian algorithm

Exam 1 shapes: - Bipartite reading — "The bipartite graph below shows the tasks that each of four people is able to undertake. All tasks must be allocated and each person can be allocated one task only. A valid task allocation is" (2012 Q3); "Each person completes a different task. Task 4 must be completed by" (2016 Q1, 97%); "Which one of the following statements is not true?" (2021 Q2); "Based on the bipartite graph, which one of the following allocations is not possible?" (2021 NHT Q3, 2026 NHT Q34); "A manufacturing business employs six different drivers to deliver their products to 10 different stores. This delivery structure could be represented graphically by … E. a bipartite graph" (2022 NHT Q2); "A bipartite graph is typically used to display which one of the following? A. the allocation of tasks on a construction site …" (2023 Q34, 77% correct). - Cost-table allocation (Hungarian or inspection) — always a 4×4 or 5×5 table. "The tasks are allocated so as to minimise the total time taken to complete the four tasks. This total time, in minutes, is" (2009 Q7); "If each person is allocated one task only, the minimum total time for this group of people to complete all four tasks is" (2010 Q9); "If each team member is allocated one task only, the minimum time in which this team would complete the four tasks is" (2013 Q4); "If each person is allocated a different task, the minimum total time for these four people to complete these four tasks is" (2015 Q7); "The minimum total time taken to complete the four duties, in minutes, is" (2023 Q36, 75%). - Maximising variants — "The athletics club will allocate each athlete to one event in order to maximise the total distance … Which allocation of athlete to event must occur in order to maximise the total distance?" (2022 Q3, 39% correct — the report works all five option totals: "Option A total distance = 59.6, Option B = 61.6, Option C = 60.4, Option D = 61.8, Option E = 60.8"); "Based on the estimates, which allocation of project parts will maximise the students' group score on the project?" (2024 NHT Q36). - Perturbation variants — "This allocation will achieve the minimum total completion time if the value of k is at least" (2018 Q8); "Before the duties are performed, it is found that Dinh will require 7 minutes for Duty 2 rather than 3 minutes. If the duties are allocated again, the minimum total time for all duties will A. remain the same. B. increase by 1 minute …" (2017 NHT Q8); "Diego and Eden swap their assigned tasks. This will result in an increase in the total completion time by" (2023 NHT Q7); "she finds that two allocations are possible. If each child starts their allocated job at the same time, then the first child to finish could be either" (2016 Q8 = GM sample Q40). - Algorithm-mechanics variant (new in 2025) — 2025 Exam 1 Question 39: "The Hungarian algorithm is used to determine the minimum cost to complete the works. Martha, the project manager, completes two steps of the Hungarian algorithm, as follows. • First, she subtracts the minimum entry in each row from each element in that row to obtain a new table of values. • Then, using this new table, she subtracts the minimum entry in each column from each element in that column. Which one of the following tables correctly displays the results after these two steps are completed?" (71% correct; the distractors are "row only", "column only", and "column then row".)

Exam 2 shapes (bipartite): - 2006 Exam 2 Q1 (3 marks) — "The following bipartite graph illustrates the positions that each is able to fill. a. Which musician must play the guitar? (1 mark) b. Complete the table showing the positions that the following musicians must fill in the band." (2 marks; "One mark was awarded if any two of these instruments were attributed to the correct person.") - 2014 Exam 2 Q1 (2 marks) — "a. How many of these four members have joined the steam trains interest group? b. Which interest group have both Brianna and Charlie joined?" - 2019 Exam 2 Q2a (1 mark) — "Each student will be allocated to only one sport. a. Complete the table below by allocating the appropriate sport to each student." - 2022 NHT Exam 2 Q1 (3 marks) — "a. Which student must be allocated the role of scribe? … b. Add Dar's potential roles to the bipartite graph below. c. … What role must Casey be given?" - 2024 NHT Exam 2 Q14 (3 marks) — "a. Which friend plans to visit the most attractions? b. How many attractions have only one friend planning to visit? c. … If the minimum number of different attractions are visited, list the names of the attractions that are visited." - 2019 NHT Exam 2 Q3c (1 mark) — "When all steps of the Hungarian algorithm are complete, a bipartite graph can show the allocation for minimum cost. Complete the bipartite graph below to show this allocation for minimum cost."

Exam 2 shapes (Hungarian mechanics) — 2008, 2012, 2014, 2017, 2019 NHT only: - 2008 Exam 2 Q3 (4 marks) — "The concerts will be allocated so as to minimise the total distance … The hungarian algorithm is to be used to find this minimum value. a. Step 1 of the hungarian algorithm is to subtract the minimum entry in each row from each element in the row. Complete step 1 for Tahlia … b. Explain why this table shows that Tahlia should attend Concert 2. c. Determine the concerts that could be attended by James, Dante and Chanel to minimise the total distance travelled … d. Determine the minimum total distance, in kilometres, travelled by the four cars." - 2012 Exam 2 Q3 (5 marks) — "…a. Complete step 1 for task X by writing down the number missing from the shaded cell in Table 2. … b. Following the Hungarian method, the smallest number of lines that can be drawn to cover the zeros is shown dashed in Table 3. These dashed lines indicate that an optimal allocation cannot be made yet. Give a reason why. c. Complete the steps of the Hungarian method to produce a table from which the optimal allocation of tasks can be made. d. Write the name of the task that each person should do for the optimal allocation of tasks." - 2014 Exam 2 Q2 (4 marks) — "a. Complete Table 2 by filling in the missing numbers for Andrew. … b. Explain why Andrew made this decision [that an allocation was not yet possible]. c. i. Which task should be allocated to Andrew? ii. How many hours in total are used to plan for the open day?" - 2017 Exam 2 Q2 (2 marks) — "Table 2 shows the final result of all her steps of the Hungarian algorithm. a. In Table 2 there is a zero in the column for Colin. When all values in the table are considered, what conclusion about minimum total planning time can be made from this zero? b. Determine the minimum total planning time, in minutes, for all four tours." - 2019 NHT Exam 2 Q3 (4 marks) — "The first step … involves row reduction … a. Write down the values of A, B, C and D. b. The next step … involves covering all the zero elements with horizontal or vertical lines. The minimum number of lines required to cover the zeros is three. Draw these three lines on Table 3 above. c. … Complete the bipartite graph below to show this allocation for minimum cost. d. Business 4 has changed its quote … How much is this reduction?"

Note on the required wording "The allocation that minimises the total time is …". That sentence does not appear in this corpus. VCAA's stems are always of the form "allocated so as to minimise the total time taken" (2008 Exam 2, 2009 Exam 1, 2016 Exam 1, 2023 Exam 1, GM sample Exam 1), "in order to minimise the total time taken" (2019 Exam 2), "so that the total time of completing the four tasks is a minimum" (2012 Exam 2), or "that will minimise the total production completion time" (2023 NHT Exam 1).

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1.12 Project networks: activity tables, precedence, drawing the network, dummy activities

Precedence-table → network (Exam 1 favourite): - "For a particular project there are ten activities that must be completed. These activities and their immediate predecessors are given in the following table. A directed graph that could represent this project is" — 2006 Exam 1 Q5; identical form 2023 Exam 1 Q38 (60% correct) and 2026 NHT Exam 1 Q40 (reversed: network given, "The activity table that could represent this project is"). - "A directed network for this project will require a dummy activity. The dummy activity will be drawn from the end of A. activity B to the start of activity C. …" — 2019 Exam 1 Q7 and 2022 Exam 1 Q7 (identical wording, different data). Report (2019): "Activity E has both B and C as immediate predecessors. A sketch of the directed network shows where the dummy activity is required. Students could have answered the question without sketching the entire network." - "When a directed network for this project is drawn, the number of dummy activities required is" — 2022 NHT Exam 1 Q8. Report: "A dummy is required because activity I comes after F but not B. A second dummy is required because activity H comes after D and E but not I or G. A third dummy is required because activity J comes after E, I and G but not D." - "Which one of the following directed graphs shows the sequence of these activities?" (from an EST/LST/duration table) — 2015 Exam 1 Q9. - "In this project, the number of activities that have exactly two immediate predecessors is" — 2023 NHT Exam 1 Q8; "The number of activities that have exactly two immediate predecessors is" — 2016 Exam 1 Q6. - "Which one of the following statements about this project is not true?" (about a precedence table, testing EST/LST/dummy reasoning) — 2024 NHT Exam 1 Q40, 2024 NHT (main GM1 Q40 analogue).

Exam 2 "complete the network / add the activity": - 2006 Exam 2 Q3a (2 marks) — "Use the information in the table above to complete the network below by including activities G, H and I." ("Arrows were required for full marks.") - 2009 Exam 2 Q4d (1 mark) — "A twelfth activity, L, with duration three weeks, is to be added without altering the critical path. Activity L has an earliest start time of four weeks and a latest start time of five weeks. d. Draw in activity L on the network diagram above." ("The answer required the correct edge with an arrow marked in the correct direction.") - 2013 Exam 2 Q2a (1 mark) — "Activity G is missing from the network diagram for this project … a. Complete the network diagram above by inserting activity G." - 2016 Exam 2 Q3ei (1 mark) — "The new activity, N, will take six days to complete and has a float time of one day. Activity N will finish at the same time as the project. i. Add activity N to the network below." (Only 21%. "This was a directed graph and an arrow needed to be included on the line for activity N.") - 2022 NHT Exam 2 Q4b (1 mark) — "Activity G is missing from this network. b. Draw activity G on the directed network above." - 2024 Exam 2 Q15c (1 mark) — add a dummy. Guide: "MUST have correct arrow. Line can be solid or dashed. Label as 'dummy' or 'd' or 'D'' or 'd''." Only 10% correct. VCAA review transcript: "Only 10% of students were able to position the dummy correctly and label it. Some students didn't attempt the question. Others drew a dummy but did not label as instructed … The student recognised that the latest start time of H being 17 meant that the dummy will be drawn from the end of D to the start of H."

Exam 2 "where does the dummy go / what does it mean": - 2012 Exam 2 Q2b (1 mark) — "A dummy activity starts at the end of activity B. Explain why this dummy activity is used on the network diagram." Full-mark answer: "Activity F has only activity B as a predecessor, while activities G and H have both B and C as predecessors. As there cannot be two activities called B, a dummy activity (with zero time) is drawn as a form of extension of B to the start of G and H to indicate that B is a predecessor for these two activities as well." - 2018 NHT Exam 2 Q4d (1 mark) — "A dummy activity is required, as shown on the revised directed network below. Explain what this dummy activity indicates on the revised directed network." Answer: "Both C and G are immediate predecessors of K." - 2018 Exam 2 Q3d (1 mark) — "This activity has a duration of one hour, an earliest starting time of five hours and a latest starting time of 12 hours. Complete the following sentence by filling in the boxes provided. The extra activity could be represented on the network above by a directed edge from the end of activity ☐ to the start of activity ☐." (Answer E → J; 25% correct.) - 2020 Exam 2 Q5a (1 mark) — "This network will require a dummy activity. a. Complete the following sentence by filling in the boxes provided. This dummy activity could be drawn as a directed edge from the end of activity ☐ to the start of activity ☐." (Answer B to C; 26% correct.) Identical in GM sample Exam 2 Q18a. - 2015 Exam 2 Q3fi (1 mark) — "This causes a slight change to activity G, which then cannot start until activity F has been completed. i. On the directed graph below, show this change without duplicating any activity." Report: "The dummy activity needed to indicate that F has become a prerequisite for G. This required an arrow on the line. Then, this connection needed to be identified as having a duration of zero or be labelled as dummy."

Exam 2 "immediate predecessors" questions: - "Write down all the activities that must be completed before activity G can commence." — 2010 Exam 2 Q4c - "What is the least number of activities that must be completed before activity F can commence?" — 2010 Exam 2 Q4a - "Write down the two activities that are immediate predecessors of activity G." — 2017 NHT Exam 2 Q1a - "Write down the two immediate predecessors of activity I." — 2017 Exam 2 Q4a (67% correct). Critical examiner note: "The dummy is not an activity and hence writing it as an additional predecessor could not be accepted." - "Which activities have more than one immediate predecessor?" — 2019 NHT Exam 2 Q2a - "Write down the immediate predecessor(s) for activity I." — 2023 Exam 2 Q14a - "List the activities that have exactly one immediate predecessor." — 2024 NHT Exam 2 Q16a - "How many of these activities have two immediate predecessors?" — 2022 Exam 2 Q2a (49% correct; answer 3 — "Activities H, I and J. The most common error was two.") - Exam 1 counting form: "The total number of activities that need to be completed before activity L may begin is" — 2009 Exam 1 Q5.

Complete-the-table form: - 2010 Exam 2 Q4 — Table 1 with columns Activity | EST (minutes) | LST (minutes) | Duration (minutes) | Immediate predecessor, one cell blank. - 2022 NHT Exam 2 Q4a (1 mark) — "Table 2 shows the earliest start time (EST) and the latest start time (LST) for activities A, B and C. The EST for activity C and the LST for activity B are missing. Use the information in Table 1 to complete Table 2." - 2015 Exam 2 Q3, 2020 Exam 2 Q5, 2023 Exam 2 Q14, 2025 NHT Exam 2 Q16a all present an Activity/EST/Duration/Predecessor table with gaps.

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1.13 Forward and backward scanning: EST, LST, float, critical path, minimum completion time

This is the largest single sub-topic — it accounts for ~40% of all Networks marks in Exam 2 and 2–4 of the 8 Exam 1 questions every year.

Earliest starting time. - "What is the earliest start time for activity E?" — 2009 Exam 2 Q4a - "What is the earliest start time for activity F?" — 2010 Exam 2 Q4b - "Determine the earliest starting time, in days, for activity E." / "…for activity H." — 2012 Exam 2 Q2a/2c - "Determine the earliest starting time of activity H." — 2013 Exam 2 Q2b (answer 7 hours: "Activity H cannot start until activities C and E have both been completed, taking a minimum of 7 hours in total.") - "Determine the earliest starting time of activity F." — 2014 Exam 2 Q4a (answer 7; "A common incorrect answer was 11 hours, which is the earliest finishing time of F.") - "Determine the earliest start time for activity M." — 2016 Exam 2 Q3a (answer 11 days: "Activity M cannot be started until path C – G – J is completed.") - "Determine the earliest starting time, in hours, for activity I." — 2018 Exam 2 Q3a (answer 10; "The most common incorrect answer given by students was 8, which indicated that they did not recognise that the longest path to I was required.") - "Determine the earliest starting time, in days, for activity I." — 2018 NHT Exam 2 Q4a - "What is the earliest start time, in weeks, of activity K?" — 2021 Exam 2 Q4a - "What is the earliest start time, in days, for activity J?" — 2023 Exam 2 Q14b - Exam 1: "The earliest start time for Activity K, in days, is" (2008 Q8); "The earliest start time for activity L, in hours, is" (2007 Q6); "The earliest starting time, in hours, for activity N is" (2017 Q4 = GM sample Q36); "The earliest start time, in hours, for activity G is" (2022 Q6).

Latest starting time. - "ii. find the latest starting time for activity D." — 2013 Exam 2 Q2cii (answer 14; "A common incorrect answer was 15 hours." Full model reasoning quoted in §3.3.) - "What is the latest starting time of activity L?" — 2014 Exam 2 Q4b ("Latest starting time of L = length of critical path − duration of L = 21 − 3 = 18") - "What is the latest starting time (LST) of activity D?" — 2015 Exam 2 Q3b - "Determine the latest starting time, in weeks, for activity C." — 2017 NHT Exam 2 Q1d - "Determine the latest start time of activity E." — 2019 Exam 2 Q3b (answer 12; only 45% — "Answers from 7 to 15 were observed.") - "Determine the latest start time of activity D." — 2023 NHT Exam 2 Q3b - "What is the latest starting time, in days, for activity I?" — 2024 NHT Exam 2 Q16c - "Determine the latest start time, in days, for activity E." — 2025 Exam 2 Q18b (answer 9; guide shows the required working "LST = 20 − 4 − 4 − 3 = 9") - "Determine the latest start time, in days, for activity H." — 2026 NHT Exam 2 Q15b (guide: "25 − 4 − 4 − 5 = 12") - "ii. What is the latest start time for activity N?" — 2016 Exam 2 Q3eii - Exam 1: "What is the latest starting time for Activity I, in days, so that the project is completed in the shortest time possible?" (2017 NHT Q6); "The activity that has the latest starting time of 12 hours is completed by …" (2018 NHT Q4).

Float / slack. - "Determine the slack time, in weeks, for activity D." — 2007 Exam 2 Q4b (the only use of "slack time" as a question stem; the 2007 report lists "determining the slack time for an activity" as an area of strength). "Slack" also appears in the 2006 report ("Every activity on a critical path has zero slack time (float)"). - "The project supervisor correctly writes down the float time for each activity that can be delayed and makes a list of these times. Determine the longest float time, in weeks, on the supervisor's list." — 2009 Exam 2 Q4c - "What is the float time, in minutes, for activity G?" — 2010 Exam 2 Q4d - "What is the float time of activity J?" — 2014 Exam 2 Q4c ("2 hours; LST − EST = 13 − 11") - "Which activity has a float time of two days?" — 2016 Exam 2 Q3c ("Activity H. A common incorrect answer was activity M.") - "Two of the activities have a float time of two hours. Write down these two activities." — 2018 Exam 2 Q3c (answer A and C; 45%) - "Which activity has the longest float time?" — 2019 Exam 2 Q3c; 2024 NHT Exam 2 Q16d; 2025 Exam 2 Q18c - "How many of these activities have zero float time?" — 2021 Exam 2 Q4b (answer 7; only 22% — "Many students did not seem to realise there were two critical paths.") - "Name the four activities that have a float time." — 2020 Exam 2 Q5c; sample version "Name the four activities that have a float time of at least one month." (GM sample Exam 2 Q18c) - "How many activities have a float time of zero?" — 2023 Exam 2 Q14c - "Which two activities have a float time of two days each?" — 2023 NHT Exam 2 Q3c - "State all the activities that have a float time of 2 minutes." — 2026 NHT Exam 2 Q15c - "Activity ☐ has the longest float time of ☐ weeks." — 2021 NHT Exam 2 Q3b (fill-boxes form) - "Which activity could be delayed for the longest time without affecting the minimum completion time of the project?" — 2017 NHT Exam 2 Q1e; 2024 Exam 2 Q15b - "Once the water has been turned off (Activity B), which of the activities C to I could be delayed without affecting the shortest time to complete all activities?" — 2011 Exam 2 Q3c - "How many activities could have their completion time increased by two weeks without altering the minimum completion time?" — 2021 NHT Exam 2 Q3c - Exam 1: "The number of activities that have a float time of 10 hours is" (2019 Q8; report shows the mandatory method "Activity G LST − EST = 16 − 6 = 10; Activity I LST − EST = 22 − 12 = 10; Activity N LST − EST = 29 − 19 = 10"); "How many of these activities could be delayed without affecting the minimum completion time of the project?" (2018 Q5); "To complete the project in minimum time, some activities cannot be delayed. The number of activities that cannot be delayed is" (2017 Q5 = GM sample Q37; 2021 NHT Q7); "When this project is completed in the minimum time, the sum of all the float times, in days, will be" (2022 Q8); "The project can still be completed in minimum time if activity C is delayed. The maximum length of the delay for activity C is" (2018 NHT Q8); "The float time, in days, of Activity B is" (2025 Q40).

Critical path. - "Write down the critical path for this project." — 2006 Exam 2 Q3c, 2009 Exam 2 Q4b, 2016 Exam 2 Q3b, 2019 NHT Exam 2 Q2b - "Write down the critical path for this network." — 2010 Exam 2 Q4f - "Write down the critical path." — 2018 Exam 2 Q3b (76% correct; answer B-E-G-H-J) - "In order, list the activities on the critical path." — 2012 Exam 2 Q2d (answer ABHILM; "Common incorrect answers included ABFJ and ACGM") - "Given that activity G is not on the critical path i. write down the activities that are on the critical path in the order that they are completed" — 2013 Exam 2 Q2ci - "Write down, in order, the activities on the critical path." — 2015 Exam 2 Q3d - "There are two critical paths. One of the critical paths is A–E–J–L–N. Write down the other critical path." — 2017 Exam 2 Q4bi (44%; "Quite a few students gave A–D–I–L–N.") - "How many activities are on the critical path?" — 2019 Exam 2 Q3a (60%; "Some responses attempted to give the critical path rather than state the number of activities.") - "This network contains two critical paths. State the activities that are common to both critical paths." — 2025 Exam 2 Q18a (44%; answer A, D, L. Guide: "In any order") - "State the critical path for this project." — 2026 NHT Exam 2 Q15a - Exam 1: "The critical path for this project is A. ADGK …" (2010 Q8); "The critical path for this project includes activities A. B and I …" (2011 Q7); "Which one of the following statements about critical paths is true?" (2014 Q8 — a pure-theory item); "There is one critical path for this project. Three critical paths would exist if the duration of activity …" (2016 Q7); "This project currently has one critical path. A second critical path, in addition to the first, would be created by …" (2008 Q9). - Also: "Determine the value of p, in days, that would create more than one critical path." (2018 NHT Exam 2 Q4b); "This project involves nine activities … There is only one critical path for this project. b. How many non-critical activities are there?" (2006 Exam 2 Q3b).

Minimum completion time. - "Determine the minimum time, in weeks, to complete this project." — 2007 Exam 2 Q4a - "What is the shortest time, in minutes, in which this construction project can be completed?" — 2010 Exam 2 Q4e - "Determine the shortest time in which activities A to I can now be completed." — 2011 Exam 2 Q3d/e - "Determine the minimum completion time, in weeks, for this project." — 2017 NHT Exam 2 Q1c - "What is the minimum completion time, in weeks, for this project?" — 2022 Exam 2 Q2b - "What is the minimum completion time, in weeks, for the library upgrade?" — 2022 NHT Exam 2 Q4c - "Determine the minimum time to complete this project, in days." — 2023 NHT Exam 2 Q3a - "What is the minimum number of days the Tilt-A-Whirl will need to be closed to complete this project?" — 2024 NHT Exam 2 Q16b - Given-value form (extremely common — the minimum completion time is supplied so the rest of the question is unlocked): "The minimum time in which all 13 activities can be completed is 21 hours." (2014 Exam 2 Q4b) • "All nine of these activities can be completed in a minimum time of 26 minutes." (2015 Exam 2 Q3a) • "The minimum completion time for the skateboard park is 15 days." (2016 Exam 2 Q3b) • "The minimum completion time for the project is 19 days." (2017 Exam 2 Q4b) • "The minimum completion time for the project is 15 hours." (2018 Exam 2 Q3b) • "The minimum completion time for the project is 35 weeks." (2019 Exam 2 Q3) • "The minimum completion time for this project is 18 hours." (2021 Exam 1 Q6) • "The minimum completion time for this project is 20 weeks." (2021 NHT Exam 2 Q3) - Exam 1: "The minimum completion time for this project, in hours, is" (2018 Q7); "The minimum completion time for this project, in days, is" (2020 Q6).

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1.14 Crashing — reducing activity times, its effect on the critical path and completion time

Exam 1 forms: - "The project is to be crashed by reducing the completion time of one activity only. This will reduce the completion time of the project by a maximum of" — 2006 Exam 1 Q9 (only 17% correct) - "The duration of every activity is initially 5 hours. For an extra cost, the completion times of both activity F and activity K can be reduced to 3 hours each. If this is done, the completion time for the project will be A. decreased by 2 hours … E. unchanged." — 2009 Exam 1 Q6 - "The duration of each activity can be reduced by one hour. To complete this project in 16 hours, the minimum number of activities that must be reduced by one hour each is" — 2012 Exam 1 Q8 - "Each of the five activities can have its completion time reduced by a maximum of one hour at a cost of $100 per hour. The least cost to achieve the greatest reduction in the time taken to finish the project is" — 2013 Exam 1 Q8 - "The cost of reducing the completion time of any activity in this project is $1000 per day. The landscape gardener has a maximum of $3000 to spend … The total completion time of the project can be reduced by three days by reducing A. activity K by one day and activity Q by two days …" — 2019 NHT Exam 1 Q8 - "The project manager asks all the workers assigned to activity H to also work on activity F … Which one of the following is correct? A. The completion time will be reduced by one week if activity F is completed before activity H is started …" — 2020 Exam 1 Q10 (a resource-conflict crash; only 24% correct) - "Activity D is no longer required and is removed. A new activity, activity M, is added … The minimum completion time for the Sunny Life caravan will be A. the same as the Holiday Fun caravan …" — 2021 NHT Exam 1 Q8

Exam 2 forms — three shapes:

(a) "Which activities are pointless to crash?" - 2007 Exam 2 Q4c (1 mark) — "The activities that can be reduced in time are A, C, E, F and G. c. Which of these activities, if reduced in time individually, would not result in an earlier completion of the project?" Answer: A, E and G. "While these three activities are not on a critical path, crashing any of these will not affect the completion time of the project." - 2013 Exam 2 Q2d (1 mark) — "Consider the following statement. 'If just one of the activities in this project is crashed by one hour, then the minimum time to complete the entire project will be reduced by one hour.' Explain the circumstances under which this statement will be true for this project." Model answer: "This will happen only if the crashed activity is on the (single) critical path A-F-I-M in this project." ("This question was generally poorly answered.")

(b) "Crash to a target, find the new time or the minimum cost" - 2007 Exam 2 Q4d/e — "d. Determine the minimum time, in weeks, for the project to be completed now that certain activities can be reduced in time. e. Determine the minimum additional cost of completing the project in this reduced time." (15 weeks; $25 000.) - 2013 Exam 2 Q2e — "Assume activity F is crashed by two hours. What will be the minimum completion time for the project?" (36 hours.) - 2014 Exam 2 Q4e — "Activity A can be crashed by up to four hours at an additional cost of $90 per hour. This may reduce the minimum completion time for the project, including activity X. Determine the least cost of crashing activity A to give the greatest reduction in the minimum completion time of the project." ($270.) - 2016 Exam 2 Q3d — "The completion times for activities E, F, G, I and J can each be reduced by one day. The cost of reducing the completion time by one day for these activities is shown in the table below. What is the minimum cost to complete the project in the shortest time possible?" ($2000; only 21%.) - 2017 Exam 2 Q4c — "The project could finish earlier if some activities were crashed. Six activities, B, D, G, I, J and L, can all be reduced by one day. The cost of this crashing is $1000 per activity. i. What is the minimum number of days in which the project could now be completed? ii. What is the minimum cost of completing the project in this time?" (17 days; $4000 — only 15%.) - 2019 Exam 2 Q3d — "The completion time for each of these five activities can be reduced by a maximum of two weeks. What is the minimum time, in weeks, that the renovation project could take?" (29; 24% — "Many failed to take two weeks off each of the critical activities.") - 2020 Exam 2 Q5d — "The project is to be crashed by reducing the completion time of one activity only. What is the minimum time, in months, that the project can be completed in?" (17; only 12% — "Reduce on critical path (20 months). Reduce B by 3 (same as B-F-H-I). 19 was the most common incorrect response.") Identical wording in GM sample Exam 2 Q18d. - 2021 Exam 2 Q4c — "The overall completion time for the roadworks can be reduced to 16 weeks. What is the minimum cost, in dollars, of this change in completion time?" ($380 000; 9% — "Few students recognised the need to reduce A by one week and L by two weeks.") - 2023 NHT Exam 2 Q3d/e — "d. What is the maximum amount of time that can be saved by using the new pumps, in days? e. What is the minimum amount of money that the owner will need to spend to save the maximum amount of time?" (4 days; $1900 — "Reduce A by one day, B by two days and K by two days.") - 2024 Exam 2 Q15d/e — "…What will be the new minimum completion time?" (30 weeks; "New critical path is A – D – H – J") and the cost part ($50 000; 7% — "Activities reduced (weeks): A (−2), D (−1), H (−1), B (−1). Total cost = 5 × $10 000"). - 2025 Exam 2 Q18d — "Frances would like to construct the home gym in three days less than was previously possible. What is the minimum additional amount Frances will need to pay?" ($1400; 21% — "Current time = 25 days. New time = 22 days. Reduce: A – 2, H – 1, K – 1"). - 2026 NHT Exam 2 Q15d (2 marks) — "…Each minute by which an activity's completion time is reduced has an associated cost of $20. Determine the new minimum completion time and the minimum associated cost to achieve this time." (21 minutes; $100. Guide: "On A–C–E–J–M–N, reduce A and M each by 2 minutes (now 21 minutes). On A–C–E–H–L–N, reduce H by 1 minute (now 21 minutes). ALL other paths are, at longest, 21 minutes.")

(c) "Complete the reduction table" (2 marks; introduced 2019, now standard) - 2019 Exam 2 Q3e (2 marks) — "The completion time for each of these five activities can be reduced by a maximum of two weeks. Fencedale High School requires the overall completion time for the renovation project to be reduced by four weeks at minimum cost. Complete the table below, showing the reductions in individual activity completion times that would achieve this. | Activity | Reduction in completion time (0, 1 or 2 weeks) |" Answer C 0, D 1, G 2, H 1, K 1. Only 3% got both marks. "A method mark was available for one of the following answers that also reduced the overall time by four weeks and did not involve unnecessary wastage." - 2022 Exam 2 Q2c (1 mark) — "One activity can have its completion time decreased by two weeks and another activity can have its completion time decreased by one week. These two changes result in the minimum completion time being reduced by three weeks. Complete the table below, showing the two activities that could have their completion times reduced and the reduction in individual activity completion time that would achieve the three-week reduction. | Activity | Reduction in completion time (1 week or 2 weeks) |" (A −2, I −1; 27%.) - 2023 Exam 2 Q14e (1 mark) — "The managers of the project have a maximum budget of $15000 … Complete the table below, showing the reductions in individual activity completion times that would achieve the earliest completion time within the $15000 budget. | Activity | Reduction in completion time (0, 1 or 2 days) |" - 2024 NHT Exam 2 Q16e (1 mark) — "Complete the table below, showing the reductions in individual activity times that would achieve the maximum reduction in completion time for the minimum cost. | Activity | Reduction in completion time (0, 1 or 2 days) |" (A 2, B 2, D 1, E 0, G 2, H 0.) - 2025 NHT Exam 2 Q16e — new minimum time 25 weeks, $16 000 = "Reduce: M, N, P, Q, R, S, W, Y gives 8 × $2000. Do not reduce T, U, V, X. ALL paths now take 25 weeks."

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2. THE VCAA WORDING BANK

These are the exact recurring stems. The right-hand column gives the answer form the reports prescribe, with mandatory elements in bold.

2.1 Degrees, edges, faces

2.2 The odd-vertex justification — THE single most prescribed answer in the topic

Stems: - "ii. With reference to the network diagram, explain why a motorist at A could not drive each of these routes once only and arrive back at A." — 2008 Exam 2 Q2bii - "i. Explain why this is not possible. Refer to the graph in your answer." — 2017 NHT Exam 2 Q2ci - "a. Explain why the dog could not follow an Eulerian circuit through this network." — 2018 NHT Exam 2 Q2a - "b. With reference to the degrees of the vertices on the graph on page 22, explain why the inspector is not able to walk such a route." — 2024 NHT Exam 2 Q15b - "a. Explain, with reference to the degrees of the vertices, why the gym owner's intended route must involve some repeated edges." — 2025 Exam 2 Q17a

Prescribed full-mark answers, verbatim from reports/guides:

The 2008 report's explicit rejection list — still the governing statement of what a generic answer costs:

"It is not sufficient to say 'As you can see, there are odd vertices.' This could simply be a transcription from a definition in the book of notes without showing any reference to the data in the question. Responses such as this do not indicate that the student understands what vertices are, let alone what odd vertices are. A response that does indicate an understanding of vertices could be 'Vertices at Q and P are of odd degree while all the other vertices are of even degree.'"
"Generic definitions that did not specifically refer to the vertices in this network were not accepted. Examples of this were 'a eulerian circuit requires all vertices of even degree', 'does not contain a eulerian circuit' and 'because all towns must have an even degree leading to it.' **None of these statements indicate that the student understands which, if any, of these vertices has an odd degree."

Mandatory elements (synthesised): (1) name or count the odd-degree vertices in this specific graph; (2) state the condition being violated (Eulerian circuit ⇒ all vertices even; Eulerian trail ⇒ exactly two odd). 2025's guide makes this explicit: "MUST mention vertices and type." 2025 Exam 2 Q17a scored only 40%.

2.3 Adjacency-matrix explanations

Rule: every adjacency-matrix / graph-feature explanation must be re-expressed in the context's own nouns (suburbs, musicians, drivers, roads) — never in abstract graph vocabulary alone.

2.4 Path / trail / circuit / cycle naming

Mandatory element: the two-word (or three-word) name in full. The bare noun ("path", "trail", "cycle", "circuit", "tree") never scores.

2.5 Writing a route

2.6 Minimum spanning tree

2.7 Shortest path

Canonical VCAA stems (the literal "Determine the shortest path from … to … . Write down the length of this path." does not occur): - "What is the shortest distance, in kilometres, between Town G and Town M?" (2021 Exam 2 Q2a) - "What is the shortest distance, in metres, between the entrance and the seal exhibit (S)?" (2019 NHT Exam 2 Q1a) - "What is the shortest distance, in metres, from the entrance to picnic area P3?" (2013 Exam 2 Q1b) - "What is the minimum distance Eden could travel from G to M?" (2023 Exam 2 Q13a) - "Determine the shortest distance between the house and the pump." (2012 Exam 2 Q1ai) - "What would be the shortest possible time for a team to run from checkpoint X to checkpoint Y?" (2010 Exam 2 Q2a) - "If Bai takes the cheapest route from Northend (N) to Seatown (S), which other town(s) will he pass through?" (2017 Exam 2 Q1b) — the only stem that asks for the route rather than the length; the report notes "Some gave the path N-Q-R-S, which was accepted."

Answer form: a bare number with the stated unit. 2021's report gives the governing rule: "A small number of students wrote the correct travel sequence but did not indicate the shortest distance as required by the question."

2.8 Flow, cuts and max-flow / min-cut

The cut-capacity counting rule, stated verbatim by VCAA four times:

"For an individual flow to contribute to the capacity of the cut the direction of the flow must be from the source region to the sink region. For the cut shown, one of the edges indicates a flow of 4 units in the reverse direction. This precludes this particular flow from contributing to the capacity of the cut." (further_maths1_assessrep_08.txt, 2008 Exam 1 Q6)
"The edge marked 6 is counted as zero since its direction is from the exit side to the entrance side of cut A." (2017 NHT Exam 2 Q3a)
"The edge with the 10 should not have been counted as its flow was in the reverse direction." (2007 Exam 2 Q3a)
"Cut 1 cuts 6 edges, 4 of which have the direction of flow from source to sink. 7 + 3 + 5 + 12 = 27*" (2025 Exam 1 Q36)

2.9 Bipartite allocation

Hungarian "explain" answers (prescribed, verbatim): - 2008 Exam 2 Q3b — "Explain why this table shows that Tahlia should attend Concert 2." → "She is the only child with a zero in the column for Concert 2." Report: "The hungarian method attempts to reduce the table or matrix to give zeroes in columns. A single zero in a column with this method indicates that an allocation is appropriate. Many students did not refer to the zero but said that the table at this stage showed that Tahlia 'had to travel the least distance to get to Concert 2.' This answer is too general." - 2012 Exam 2 Q3b — "Give a reason why [an optimal allocation cannot be made yet]." → "Four lines are needed before an allocation of four tasks to four people may be attempted and there are only three at the moment." - 2014 Exam 2 Q2b — "Explain why Andrew made this decision." → "The minimum number of lines to cover all zeros is less than four." Report: "Students' explanations needed to refer to the stage in the process of the Hungarian algorithm. This required reference to the required minimum number of lines though zeroes. Some students simply stated that there was 'no clear allocation to Brianna' … Another common but unacceptable answer was 'there are not enough zeroes'." - 2017 Exam 2 Q2a — "When all values in the table are considered, what conclusion about minimum total planning time can be made from this zero?" → "Colin must plan Tour 2." Report: "Students needed to state that Colin must plan Tour 2 (or equivalent). Many gave additional information usually from values in Table 1; however, Table 1 did not need to be considered at all. An answer such as 'Colin will plan Tour 2 and take 8 minutes to do it' was acceptable as the extra information was correct and did not negate the first part. However, an answer such as 'Colin will plan Tour 2 because he is the fastest' was not correct as Diane could plan Tour 2 more quickly than Colin. The algorithm gives the best overall allocation taking all values into account."

2.10 Project-network stems

Critical path notation — how it must be written. Across every report the accepted form is an ordered sequence of activity letters, written with hyphens, en-dashes or no separator; VCAA prints all three (A-B-E-J-K-O-Q-S 2019 Exam 1 report; A–E–I–L–N 2017 Exam 2 report; ABHILM 2012 Exam 2 report; A – C – H – J 2024 assessment guide; B-D-E 2013 Exam 1 report). Never a list of vertices, and never unordered. When two critical paths exist, both must be given unless the question says otherwise (2017 Exam 2 Q4bi; 2025 Exam 2 Q18a; 2021 Exam 2 Q4b failure).

2.11 "Complete the following sentence by filling in the boxes provided"

This exact sentence is a VCAA fixture in Networks Exam 2. Occurrences: - 2011 Exam 2 Q4a — "Complete the following sentence for this network of pipes by writing either the number 1 or 2 in each box. Stormwater from Source ☐ cannot reach Outlet ☐." - 2014 Exam 2 Q3c — "In the boxes below, write down the pair of towns that this train line connects. between ☐ and ☐" - 2018 Exam 2 Q3d — "The extra activity could be represented on the network above by a directed edge from the end of activity ☐ to the start of activity ☐." - 2020 Exam 2 Q3c — "This track is between exercise station ☐ and exercise station ☐." (also GM sample Q16c) - 2020 Exam 2 Q5a — "This dummy activity could be drawn as a directed edge from the end of activity ☐ to the start of activity ☐." (also GM sample Q18a) - 2021 Exam 2 Q3bi — "The road that should have its capacity increased is the road from vertex ☐ to vertex ☐." - 2021 NHT Exam 2 Q3b — "Activity ☐ has the longest float time of ☐ weeks." - 2022 Exam 2 Q3c/d — "…the pipe that runs from vertex ☐ to vertex ☐. Its new capacity, in litres per minute, should be at least ☐." - 2023 Exam 2 Q13c — "The two roads that will be travelled along twice are the roads between: • vertex ☐ and vertex ☐ • vertex ☐ and vertex ☐" - 2017 Exam 2 Q1c / 2023 Exam 2 Q12b / 2025 Exam 2 Q15b — the Euler's-formula boxes.

Answer form: boxes only. Marks are all-or-nothing per box-set (2025 guide, Q15b: "All three boxes correct"; 2026 NHT Q14b: "MUST be labelled Cut 2").

2.12 "Explain why …" in Networks — complete inventory

Governing rule for every "Explain why" in this area of study (2008 report, restated in 2010, 2012, 2014, 2021):

"Students are expected to show understanding of explanations or assertions; this may be demonstrated by mathematics presented in a logical and clear manner or by directly relating a mathematical definition to data stated in the question."
"Where descriptive answers are required to a question, students are strongly advised to keep answers brief. An answer in point form is acceptable and additional information should not be provided." (2022 Exam 2 report)

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3. TRAPS AND EXAMINER COMPLAINTS

3.1 Confusing path / trail / circuit / cycle, and Eulerian vs Hamiltonian

3.2 Float-time errors and "further engagement"

The signature VCAA penalty in this topic. If the question asks which activity has the longest float and the student also volunteers a float value that is wrong, the mark is lost.

• 2019 Exam 2 Q3c — "The answer is J. Students should be aware that if they did not leave their answer as J but further engaged with the question and then stated an incorrect float time, they could not be awarded marks.**"
• 2025 Exam 2 Q18c — "Students need to be careful that if they provide additional information in their response, it must be correct. Only the activity F needed to be written to obtain the mark." Assessment guide: "Reject, as further engagement, if incorrect float time given for F.**"
• 2024 Exam 2 report (general) — "in Question 15b only the activity E needed to be written to attract the mark. If the response also included the delayed time correctly as 3 hours, then the mark was still awarded. However, where a student further engaged to provide an incorrect delay time in their response, the mark was not awarded.**"

Other float errors: - 2011 Exam 2 Q3c — "A common incorrect answer included activity G, either by itself or with other activities. This activity was on the critical path and could not be delayed." - 2016 Exam 2 Q3c — "Activity H. A common incorrect answer was activity M." (37% correct) - 2018 Exam 2 Q3c — "*This question was not answered well, with many combinations of incorrect answers given by students.*" (45%) - 2014 Exam 2 Q4a — "*A common incorrect answer was 11 hours, which is the earliest finishing time of F*" — EST vs EFT confusion. - 2018 Exam 2 Q3a — "*The most common incorrect answer given by students was 8, which indicated that they did not recognise that the longest path to I was required.*" - 2011 Exam 1 Q7 — "*students needed to realise that it was the latest starting time for activity M, 24 minutes (option D), and not the earliest starting time of 18 minutes (option C).*" - 2021 Exam 2 Q4b — "*Many students did not seem to realise there were two critical paths.*" (22% correct) - 2019 Exam 2 Q3b — "Answers from 7 to 15 were observed" for a single LST.

3.3 Listing the critical path incorrectly

• 2012 Exam 2 Q2d — answer ABHILM; "Common incorrect answers included ABFJ and ACGM." And Q2c: "Few students were able to answer this question correctly. Many ignored the dummy activity and obtained the incorrect answer of 13."
• 2012 Exam 2 Q2e — "A consequential mark was available for a correct calculation that showed the addition of the times … Instead, most students wrote a single number here without showing the calculation and were ineligible for the consequential mark."
• 2017 Exam 2 Q4bi — "Quite a few students gave A–D–I–L–N" instead of A–E–I–L–N.
• 2019 Exam 2 Q3a — "Some responses attempted to give the critical path rather than state the number of activities."
• 2020 Exam 2 Q5c — answer A E F H; "Many students were unable to determine the critical path." (23%)
• 2013 Exam 2 Q2cii — LST of D; the full prescribed reasoning chain is worth memorising:

  "The critical path AFIM takes 37 hours. The latest starting time of activity D depends upon the latest starting time of activities G and J, of which G is the most critical. Activity G connects to the critical path as a prerequisite for activity I and must finish no later than 20 hours. Activity G takes 4 hours to complete and its latest starting time = 20 − 4 = 16. Then, activity D must end no later than 16 hours. Therefore, its latest starting time = 16 − 2 = 14. A common incorrect answer was 15 hours.**"

• 2010 Exam 2 Q4f — the report supplies VCAA's own table-only method for finding a critical path without drawing the network ("Activities that are not predecessors for any other activity will not have any successor before completion of the project … The minimum time to complete the project is the maximum of earliest start time + activity time for all these activities. The critical path for the project will then end at the activity that has this maximum value.").
• 2006 Exam 2 Q3c — "Every activity on a critical path has zero slack time (float). From the given information, this means that A and C cannot be on a critical path, which leaves only one complete path as an option."

3.4 Cut capacity — counting edges flowing the wrong way, and cuts that don't separate

Wrong-way edges (the most-repeated arithmetic trap in the topic): - 2007 Exam 2 Q3a — "This question was poorly answered, with many students giving an answer of 53. The edge with the 10 should not have been counted as its flow was in the reverse direction." - 2008 Exam 1 Q6 — only 35% correct: "*It was surprising that 39 per cent of students apparently did not understand this key idea and incorrectly chose option E.*" - 2017 NHT Exam 2 Q3a — "*7 + 11 + 0 + 8 = 26. The edge marked 6 is counted as zero since its direction is from the exit side to the entrance side of cut A.*" - 2018 Exam 2 Q1aii — "Some gave an answer of 14 by not allowing for the 1 against the flow." - 2024 Exam 2 Q14a — "The majority of students recognised that the edge with weight seven was against the flow, and therefore not included when summing the weights." (78% correct — improving.) - 2024 NHT Exam 1 Q38 — "*Minimum cut = maximum flow = 13 + 16 + 9 + 17 + 21 = 76 (12 is flowing from exit to entrance).*" - 2025 NHT Exam 2 Q15a — "*12 + 10 + 12 (4 and 8 are directed backwards) = 34*" - 2026 NHT Exam 2 Q14c — "*Cut through 19 – 12 – 22 – 18 – 24 – 18, with 22, 18 and 24 not counted (wrong way)*. 19 + 12 + 18 = 49"

Cuts that fail to separate source from sink: - 2012 Exam 1 Q7 — "The key … was to recognise that to apply the 'minimum cut-maximum flow' theorem, the cut must separate the 'source' of traffic (the town) from the 'sink' (the freeway). Of the cuts shown on the network, only those represented by lines 2, 3 and 4 separated the source from the sink … Line 1 does not separate the source from the sink." - 2014 Exam 1 Q9 — "*A flow network with two sources (two car parks) and a single sink (the exit) was provided. While Cut C had the minimum capacity, neither it nor Cut A separated both sources (car parks) from the sink. Thus, neither of these cuts could be used to determine the minimum flow. Of the three cuts that did separate both sources from the sink, Cut D had the minimum capacity and hence determined the maximum flow.*" - 2011 Exam 2 Q4bii — VCAA introduces the super-source device: "*To address this, a single Supersource* can be considered … It is now clearer that the required minimum cut must separate the Supersource from Outlet 2 and, in this case, includes the 200 pipe coming down from Source 1."

Max-flow sanity failure: - 2007 Exam 2 Q3b — "Minimum cut ⇒ maximum flow. Despite this, some answers given here were greater than the answer for Q3a." (A max flow can never exceed a stated cut capacity.) - 2013 Exam 2 Q3bii — "Often, group sizes were unreasonable since they exceeded 24, which was the number of students permitted initially on the most vacant track C to D." - 2021 Exam 2 Q3a — "Many students did not demonstrate they had tried to find the minimum cut." (24%) - 2024 Exam 2 Q14b — "Some students used a minimum cut, others an exhaustion of paths." (29%) VCAA review: "*This was found to be a challenging question by most. Those that were successful usually found a minimum cut after repeated trials*."

3.5 Not redrawing / not re-scanning the network after crashing

This is the defining failure of the last question of every Exam 2.

• 2006 Exam 1 Q9 (17% correct) — "When choosing an activity to be crashed on the critical path, care needed to be taken to ensure that crashing this activity did not create a new critical path. This restricted the amounts by which critical path activities C, E, H or J could be crashed to one hour. However, inspection shows that the remaining activity, B, could be crashed by a maximum of four hours without creating a new critical path. Note that D is a dummy activity, so it could not be crashed.**"
• 2006 Exam 2, areas of weakness — "recognising that crashing is relevant only for activities on a critical path."
• 2007 Exam 2, areas of weakness — "recognising that crashing activities on a critical path may create a new critical path and that this new one may also be reduced by crashing an activity on it."
• 2007 Exam 2 Q4d — "Reducing C and F by two weeks each reduces the length of the path B–C–F–H–I from 19 to 15 weeks. However, this creates a new critical path B–E–H–I which is 16 weeks long. This means we should reduce activity E by one week so that both paths … are new critical paths and both are 15 weeks long."
• 2013 Exam 1 Q8 — "Since path B-D takes at least 8 hours to complete, there is no point in paying for the completion time for path A-C to be reduced to less than 8 hours."
• 2013 Exam 2 Q2e — "If activity F is crashed by 2 hours, the path C-E-H-G-I-M (36 hours long) will then become the new critical path. Crashing activity F by more than one hour will not reduce the completion time of the project below 36 hours since F is not on the new critical path."
• 2014 Exam 2 Q4e — "Reducing any path that includes A below 18 hours is pointless since B-E-H-K becomes a critical path at 18 hrs."
• 2016 Exam 2 Q3d (21%) — "This makes the duration of A–E–I–K = 14 days. This is same as the duration of A–D–K, which cannot be crashed. Therefore, there is no point in crashing activity I any further.**"
• 2017 Exam 2 Q4cii (15%) — "Some students recognised the need to reduce the critical activities (I, J and L) but did not realise that activity G also had to be reduced as it was part of a new critical path formed."
• 2019 Exam 2 Q3d (24%) — "Many failed to take two weeks off each of the critical activities."
• 2020 Exam 2 Q5d (12%) — "Reduce on critical path (20 months). Reduce B by 3 (same as B-F-H-I). 19 was the most common incorrect response.**"
• 2021 Exam 2 Q4c (9%) — "Few students recognised the need to reduce A by one week and L by two weeks."
• 2024 Exam 2 Q15e (7%) — VCAA review: "Only a small percentage of students were successful on this final question. Those who were successful tended to use a very systematic trial approach … This response shows all the possible options considered before finding the minimum cost."
• 2025 NHT Exam 2 Q16eii — "Reduce: M, N, P, Q, R, S, W, Y gives 8 × $2000. Do not reduce T, U, V, X. ALL paths now take 25 weeks."
• 2026 NHT Exam 2 Q15d — "ALL other paths are, at longest, 21 minutes" — the required check.
• 2019 Exam 2 Q3e (3% for 2 marks) — VCAA published an entire matrix of alternative acceptable reduction schedules that "also reduced the overall time by four weeks and did not involve unnecessary wastage". Wastage = crashing beyond the point where another path becomes critical.**

3.6 Planar graphs — counting faces on a non-planar drawing

• 2008 Exam 1 Q7 (11% correct) — "the regions delineated by the edges in a graph only represent faces if the graph is drawn in planar form. The graph given in the question was planar, but not drawn in planar form … By not redrawing the graph in planar form before trying to determine the number of faces, 72 per cent of students erroneously obtained the answer 9."
• 2014 Exam 1 Q7 — "The majority of students chose option A, 0. This suggests that most students were unaware that intersecting edges in a graph do not automatically preclude the graph from being planar."
• 2021 Exam 1 Q3 — "When the graph is redrawn as planar there are only four faces. It would appear that many students did not redraw the graph in its planar form before counting the faces.**"
• 2022 Exam 1 Q4 (21% correct) — "The given graph is planar. It can be redrawn with no crossing edges." (Answer: 0 edges need removing.)
• 2006 Exam 1 Q8 — "Interestingly, 37 per cent of students incorrectly chose option A, perhaps unaware of the fact that any graph with four or less vertices is planar."
• 2018 Exam 1 Q6 — "some would have recognised option D as a complete graph with five vertices. Any complete graph with five, or more, vertices is non-planar.**"
• 2024 NHT Exam 1 Q37 — "Graph 1 Contains a complete pentagon, hence cannot be planar, hence the rule does NOT apply."
• 2024 Exam 1 Q35 — "The graph can be redrawn as planar. The number of faces can be counted or calculated using Euler's formula 7 + f = 11 + 2, gives f = 6."

3.7 Minimum spanning tree — algorithm slips

• 2012 Exam 2 Q1bi — the most detailed diagnosis in the corpus:

  "This answer may have been found by starting at the pump and then selecting the shortest edge from only the very last vertex connected, rather than any of the already-connected vertices. After choosing the third edge (from the pump) marked 40, the edge marked 60 should have been selected next rather than just choosing the smallest edge that immediately followed the 40. The apparent inability of many students to apply the algorithm for determining a minimal spanning tree is a weakness that needs to be addressed.**"

• 2008 Exam 2 Q1a — "A number of students drew circuits and these were not accepted."
• 2011 Exam 2 Q2a — "A number of students included circuits in their graphs or missed one or more vertices." Q2b: "A common error was for students to omit the length of one of the edges from their diagram."
• 2015 Exam 2 Q1di — "Many students were unable to find this minimum spanning tree. Common incorrect trees excluded PO or KL instead of MN. Some students drew complete circuits.**"
• 2025 Exam 2 Q15d — "Some students included an edge that was not part of the original graph." Guide: "NO new edges … 4 original edges with no isolated vertices."

3.8 Allocation / Hungarian slips

• 2020 Exam 2 Q2 — "Some students did not read the question carefully enough and attempted to find the minimum using the Hungarian Algorithm" when the question said maximise.
• 2014 Exam 2 Q2cii — "A common incorrect answer was 21, the total of all the numbers on Table 3**" (the reduced table, not the original cost table).
• 2017 Exam 2 Q2a — "Many gave additional information usually from values in Table 1; however, Table 1 did not need to be considered at all." (37% correct)
• 2017 Exam 2 Q2b — "Some responses showed evidence of the correct allocation but did not give the total planning time."
• 2022 Exam 1 Q3 (21% correct) — "The requirement in this question was to find the allocation that maximised the total distance."
• 2016 Exam 1 Q8 — there can be two optimal allocations; VCAA sets questions that hinge on it.
• 2012 Exam 2 Q3b — "This question was answered quite poorly by many students."

3.9 Dummy activities

• 2012 Exam 2 Q2b — "Many unacceptable answers seemed to be direct excerpts from notes about dummy activities in general, rather than clear explanations of the purpose of the specific dummy activity in the given context. Many students said that '… we cannot have parallel activities …', without an explanation of what this meant in the context of the question. Similarly, many students stated that the dummy was used 'in order to satisfy the two conventions', without explaining what these conventions were."
• 2017 Exam 2 Q4a — "The dummy is not an activity and hence writing it as an additional predecessor could not be accepted."
• 2012 Exam 2 Q2c — "Many ignored the dummy activity and obtained the incorrect answer of 13."
• 2024 Exam 2 Q15c (10% correct) — "The dummy was often drawn without an arrow or label. A dashed or solid line was acceptable." Guide: "MUST have correct arrow. Label as 'dummy' or 'd' or 'D'' or 'd''."
• 2015 Exam 2 Q3fi — "This required an arrow on the line. Then, this connection needed to be identified as having a duration of zero or be labelled as dummy."
• 2016 Exam 2 Q3ei — "This was a directed graph and an arrow needed to be included on the line for activity N."
• 2009 Exam 2 Q4d — "The answer required the correct edge with an arrow marked in the correct direction."
• 2007 Exam 1 Q6 — VCAA itself omitted the dummy's arrow and had to accept two options: "An arrow was not shown on the dummy activity … as is the normal convention. Due to this, credit was given to students who chose either option D (using a down arrow) or option E (using an up arrow)."
• 2015 Exam 2 Q3fii — "A number of students incorrectly stated that 'there is no effect since a dummy takes zero time', while others stated that 'there is no effect since G is not on the critical path'."

3.10 Answer-form and presentation rules that cost Networks marks

• One-mark questions: no sentences. "Many questions on the examination were worth one mark only, and with these questions the mark was awarded for a correct answer. Generally there was no need to put the answer into a sentence. Many students copied a correct answer incorrectly from calculations into a concluding sentence." (2017 Exam 2 report; repeated 2018, 2020, 2024.)
• Brevity, and no unrequested extras. "When descriptive answers are required to a question, brevity is encouraged … Some students correctly answered a question and then added extra irrelevant information. However, this extra information cannot be awarded extra marks and, if it is incorrect, can lead to marks not being awarded." (2017 Exam 2 report.)
• Method marks require visible working on 2-mark questions. "An incorrect answer will not be awarded any marks in a two-mark question; however, often a method mark can be awarded for an intelligible attempt." (2017, 2018, 2020, 2021, 2022 reports.)
• Consequential marks require the substitution to be shown. "working out needed to show a correct substitution of the previous reasonable, but incorrect, answer into a relevant calculation. The resulting answer then needed to match that substitution and be a reasonable answer." (2016 Exam 2 report.) Networks-specific instance: 2012 Exam 2 Q2e, 2011 Exam 2 Q2b.
• Crossed-out work is not marked. "Deleted work cannot be assessed at all." (2006, 2007, 2011, 2014 reports.)
• Drawing answers must be on the supplied diagram, and labelled as instructed. "In two questions students were required to provide a label on a diagram. This direction meant that the label needed to be drawn on the diagram as indicated." (2024 Exam 2 report.)
• Terminology precision. "Overall, students need to be very clear on the key knowledge and key skills contained in the VCE Mathematics Study Design 2023-2027 for General Mathematics, and to use the formal terminology within the course." (2024 Exam 2 report.) Repeatedly recommended remedy: "a glossary of relevant terms in the student's bound reference" (2006, 2008, 2010, 2012, 2015, 2016 reports).

3.11 Miscellaneous recurring Networks traps

• Reading a directed graph (reachability). 2020 Exam 1 Q4: "It is possible to reach vertices T and V directly and vertex X indirectly (via vertex V). Many students may have missed vertex X as the third possibility." Also 2006 Exam 1 Q2, 2012 Exam 1 Q6, 2014 Exam 1 Q2, 2021 Exam 1 Q4, 2009 Exam 2 Q2.
• Complete-graph edge counting. 2007 Exam 1 Q9: VCAA supplies the formula in the report — "A complete graph with n vertices has n(n−1)/2 edges" — but it is not on the formula sheet.
• Adjacency matrix of a map: loops and multiple edges. 2019 Exam 1 Q6 / GM sample Q38 / 2015 Exam 1 Q6.
• Counting paths/routes. 2020 Exam 2 Q4a (answer 10, "9 was a common incorrect response"); 2008 Exam 2 Q2a ("Few students managed to find that there were seven paths"); 2017 NHT Exam 1 Q5.
• Dominance matrices (pre-2016 only). One-step and two-step dominance appeared in Module 5 every year 2006–2015 (2006 Exam 2 Q2, 2007 Exam 1 Q8, 2008 Exam 2 Q4, 2009 Exam 1 Q9, 2013 Exam 1 Q5/Q9, 2014 Exam 1 Q4, 2015 Exam 1 Q8). From 2016 dominance moved out of Networks into the Matrices module and does not appear in Networks again. 2006 report weakness: "explaining two-step (or more) dominance."
• Bipartite-graph + real allocation trap. 2026 NHT Exam 1 Q34 ("Which one of the following allocations is not possible?") requires checking each row against the bipartite edges, not optimising.
• The "not true" stem. Used heavily in Exam 1 (2007 Q5, 2014 Exam 1 Q8 inverse, 2017 NHT Q2, 2018 NHT Q1, 2019 Q4, 2019 NHT Q7, 2020 Exam 1 Q2/Q3, 2021 Q2, 2022 Q5, 2022 NHT Q6, 2023 NHT Q2, 2024 NHT Q40, 2026 NHT Q39). Students who answer the true statement lose the mark; the reports repeatedly advise checking each option separately with the aid of a diagram (2017 Exam 1 Q7 report).

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4. SOURCE INDEX

All paths relative to corpus/text\.

Exam papers — Further Mathematics (Module 5 / Module 2 era) Documents_exams_mathematics_2006furmath1-w.txt, …2006furmath2-w.txt, …2007furmath1.txt, …2007furmath2.txt, …2008furmath1-w.txt, …2008furmath2-w.txt, …2009furmath1-w.txt, …2009furmath2-w.txt, …2010furmath1-w.txt, …2010furmath2-w.txt, …2011furmath1-w.txt (corrupt encoding), …2011furmath2-w.txt, …2012_2012furmath1-w.txt, …2012_2012furmath2-w.txt, …2013_2013furmath1-w.txt, …2013_2013furmath2-w.txt, …2014_2014furmath1-w.txt, …2014_2014furmath2-w.txt, …2015_2015furmath1-w.txt, …2015_2015furmath2-w.txt, …2016_2016furmath1-w.txt, …2016_2016furmath2-w.txt, …2017_2017furmath1-w.txt, …2017_2017furmath2-w.txt, …2018_2018furmath1-w.txt, …2018_2018furmath2-w.txt, …2019_2019furmath1-w.txt, …2019_2019furmath2-w.txt, …2020_2020furmath1-w.txt, …2020_2020furmath2-w.txt, …2021_2021furmath1-w.txt, …2021_2021furmath2-w.txt, …2022_2022furmath1-w.txt, …2022_2022furmath2-w.txt

Exam papers — NHT (Further Maths) …2017_nht_2017FM1-nht-w.txt, …2017_nht_2017FM2-nht-w.txt, …2018_nht_2018FM1-nht-w.txt, …2018_nht_2018FM2-nht-w.txt, …2019_NHT_2019FM1-nht-w.txt, …2019_NHT_2019FM2-nht-w.txt, …2021_NHT_2021FM1-nht-w.txt, …2021_NHT_2021FM2-nht-w.txt, …2022_NHT_2022furmath1-NHT-w.txt, …2022_NHT_2022furmath2-nht-w.txt, …2023_NHT_2023FM1-nht-w.txt, …2023_NHT_2023FM2-nht-w.txt

Exam papers — General Mathematics (2023+) Documents_exams_mathematics_2023_2023genmath1-w.txt, …2023_2023genmath2-w.txt, …2024_NHT_2024GM1-nht-w.txt, …2024_NHT_2024GM2-nht-w.txt, 2025-11_2025-GeneralMaths1_0.txt, 2025-11_2025-GeneralMaths2.txt, 2026-05_2026-NHT-GeneralMaths1.txt, 2026-06_2026-NHT-GeneralMaths2.txt, Documents_exams_mathematics_genmath1-sample-w.txt, …genmath2-sample-w.txt

Examination reports Documents_exams_mathematics_furthermaths1_assessrep_06.txt, …furthermaths2_assessrep_06.txt, …further1_assessrep_07.txt, …further2_assessrep_07.txt, …further_maths1_assessrep_08.txt, …further_maths2_assessrep_08.txt, …further1_assessrep_09.txt, …further2_assessrep_09.txt, …further1_assessrep_10.txt, …further2_assessrep_10.txt, …further1_assessrep_11.txt, …furthermaths2_assessrep_11.txt, …2012_FM1_assessrep_12.txt, …2012_fm2_assessrep12.txt, …2013_FM1_examrep13.txt, …2013_FM2_examrep13.txt, …2014_FM1_examrep14.txt, …2014_FM2_examrep14.txt, …2015_FM1_examrep15.txt, …2015_FM2_examrep15.txt, …2016_FM1_examrep16.txt, …2016_FM2_examrep16.txt, …2017_fm1_examrep17.txt, …2017_fm2_examrep17.txt, …2018_fm1_examrep18.txt, …2018_FM2_examrep18.txt, …2019_FM1_examrep19.txt, …2019_FM2_examrep19.txt, …2020_2020furmaths1-exam-report.txt, …2020_2020furmaths2-exam-report.txt, …2021_2021furthermath1-report.txt, …2021_2021furthermath2-report.txt, …2022_2022furmaths1-report.txt, …2022_2022furmaths2-report.txt

NHT reports …2017_nht_fm1nht_examrep17.txt, …2017_nht_fm2nht_examrep17.txt, …2018_nht_furthermaths1nht_examrep18.txt, …2018_nht_furthermaths2nht_examrep18.txt, …2019_NHT_fm1nht_examrep19.txt, …2019_NHT_fm2nht_examrep19.txt, …2021_NHT_2021-NHT-Further-Maths-1-exam-report.txt, …2021_NHT_2021-NHT-Further-Maths-2-exam-report.txt, …2022_NHT_2022furthmaths1-NHT-report.txt, …2022_NHT_2022-furthmaths2-NHT-report.txt, …2023_NHT_2023furthermaths1NHT-report.txt, …2023_NHT_2023furthermaths2NHT-report.txt, …2024_NHT_2024NHTgeneralmaths2-report.txt, 2026-02_2024-NHTgeneralmaths1-report.txt, 2025-10_2025-NHT-general-maths1-report.txt, 2025-10_2025-NHT-general-maths2-report.txt

General Mathematics reports, assessment guides, specifications 2025-07_2023generalmaths1-report.txt, 2025-03_2024generalmaths1-report.txt, 2025-03_2024generalmaths2-report.txt, 2025-04_2024generalmathematics2-assessment-guide.txt, 2025-04_General_Maths_Examination_2.txt (VCAA exam-review transcript), 2026-02_2025-GeneralMaths1-report.txt, 2026-01_2025-GeneralMaths2-report.txt, 2025-11_2025-GeneralMaths1-assessment-guide.txt, 2025-11_2025-GeneralMaths2-assessment-guide.txt, 2025-06_2025NHT-GeneralMath1-assessment-guide.txt, 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, 2026-06_2026-NHT-GeneralMaths1-assessment-guide_0.txt, 2026-06_2026-NHT-GeneralMaths2-assessment-guide.txt, 2025-04_genmath-specs-w.txt, Documents_exams_mathematics_generalmaths1-formula-w.txt, …generalmaths2-formula-w.txt, 2025-10_MCAS_GeneralMaths1.txt, Documents_exams_mathematics_2024_2024genmaths1-markguideresponses.txt