Data analysis

Full data analysis reference

This is the heart of the topic. VCAA recycles a small number of stems almost verbatim. For each, the question template, the years it recurs, the model answer form, and the mandatory elements as prescribed by the reports/marking guides.

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2.1 INTERPRETING THE SLOPE of a least squares line

Question template (near-identical wording, 2008 → 2026)

"Interpret the slope of [this / the] least squares line in terms of [variable 1] and [variable 2]." — 1 mark

Instances (exact, with file):

The examiner's model answers (verbatim from reports)

2008 Q4c — "On average, arm span increases by 1.09 cm for each 1 cm increase in height" (Documents_exams_mathematics_further_maths2_assessrep_08.txt)
2012 Q2d — "On average, it is predicted that the maximum temperature increases by 0.67 °C for each 1 °C increase in the minimum temperature." (...2012_fm2_assessrep12.txt)
2014 Q2c — "On average, population increases by 2680 people for each additional square kilometre of area." (...2014_FM2_examrep14.txt)
2015 Q3a — "On average, male life expectancy increased by 0.88 years for each one-year increase in female life expectancy." (...2015_FM2_examrep15.txt)
2017 Q3b.ii — "On average, egg density increases by 31.3 eggs/m² for each additional female moth caught." (...2017_fm2_examrep17.txt)
2017 NHT Q2b.ii — "On average, relative humidity decreases by 4.38% for each 1 °C increase in temperature."
2019 Q5a — "On average, pressure at 3 pm increases by 0.8894 hPa for each one unit (hPa) increase in pressure at 9 am." (...2019_FM2_examrep19.txt)
2019 NHT Q4c — "On average, life span decreases by 1.9 years for each additional hour of sleep time."
2020 Q5d — "On average, body density decreases by 0.001512 kg/l for each 1 cm increase in waist measurement." (...2020furmaths2-exam-report.txt)
2022 Q4c — "On average, relative humidity decreases by 3.417% for each 1 degree increase in temperature."
2022 NHT Q2d.i — "On average, the number of finishers increases by 0.3648 for each increase of 1 in the number of starters."
2023 NHT Q4d — "On average, foot width increases by 0.2479 cm for each 1 cm increase in foot length."
2024 Exam 2 Q3e — "On average, for each 1 metre increase in Wgold, Wbest increases by 0.86 metres." (2025-03_2024generalmaths2-report.txt)
2024 NHT Q4d — "On average, when the LLT increases by 1 m, the HHT decreases by 1.08 m."

The prescribed answer form — mandatory elements

The 2020 report gives the definitive accept/reject list (Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, Q5d):

"Three examples of acceptable answers are:
• Body density decreases by 0.001512 for each 1 cm increase in waist measurement.
• Body density decreases by 0.001512 for each additional cm in waist measurement.
• Body density decreases by 0.001512 for each cm increase in waist measurement.

Three examples of responses that were not acceptable answers are:
• Body density decreases by –0.001512 for each 1 cm increase in waist measurement. ← double negative
• Body density decreases by 0.001512 for each 1 cm in waist measurement. ← no change in x
• Body density decreases by 0.001512 for each increase in waist measurement. ← no quantified change in x
The latter suggests that the same body density decrease applies to any increase in waist measurement, be it 1, 2, 5 or 10."

The 2017 report is equally explicit (...2017_fm2_examrep17.txt, Q3b.ii):

"Interpretation of the slope needed reference to a ratio of changes – a change in response variable compared to a change in explanatory variable. Both variables needed to show change. Some acceptable answers were:
• egg density increases by 31.3 as the number of female moths increases by 1
• egg density increases by 31.3 for an increase of 1 in the number of female moths caught
• egg density increases by 31.3 for each additional female moth caught.
The following examples missed the key element that both variables have a quantifiable change:
• egg density increases by 31.3 for each female moth
• egg density increases by 31.3 for each increase in the number of female moths caught."

Other prescriptive statements: - 2022 report: "The interpretation of the slope must refer to a change in the response variable for each unit change in the explanatory variable." - 2024 marking guide (2025-04_2024generalmathematics2-assessment-guide.txt): "3e | (On average), for each 1 m increase in Wgold, Wbest increases by 0.860 m | A1 | Must be clear reference to the change in both variables" — note "(On average)" is bracketed, i.e. treated as optional in the 2024 guide. - 2024 Chief Assessor PL (2025-04_General_Maths_Examination_2.txt): "It was essential that students reference the change in the response variable for each one unit change in the explanatory variable." - 2019 report: "Many gave a response that was almost correct but failed to reference the one-unit increase in pressure 9 am. Interpreting the slope … required identifying the correct constant and then describing it. Describing both constants was to ignore the first step; specific knowledge was required, not various statements provided in the hope of including something relevant." - 2013 report (three-median line, Q4c) on units: "The interpretation of the slope had to relate to the average pay rate and the year. This means that appropriate units must have been given, in this case in dollars per hour per year. An answer that simply indicated 'an increase of 0.8 each year' did not explain what the number 0.8 referred to and was not accepted."

Mandatory-element checklist for a full-mark slope interpretation 1. Direction word matching the sign of b: increases / decreases (never write the minus sign and "decreases"). 2. The numerical value of the slope, correctly rounded as given. 3. The units of the response variable (or the response variable named so the units are implied). 4. "for each 1 [unit] increase in" — the explanatory variable's change must be quantified as one unit ("each additional", "each 1 cm", "for an increase of 1"). 5. BOTH variable names, in the correct roles. 6. "On average" — present in every VCAA model answer 2008–2024; bracketed as optional only in the 2024 marking guide. Always include it.

Fill-in-the-box variant (2023 onwards — the new format)

2023 Exam 2 Q1c.ii: "Complete the following sentence by filling in the box provided. This equation predicts that, on average, each 10 g increase in the weight of an oyster is associated with a ▢ cm³ increase in its volume." (1 mark)
2026 NHT Exam 2 Q3d: "Complete the following sentence by filling in the boxes below. The equation of the least squares line predicts that, on average, each 10 kg increase in body weight is associated with a ▢ gram ▢ in brain weight." (2 marks) — guide answer "13.9 / increase; One correct OR both given in incorrect order (A1); Completely correct (A1)"
2006 Exam 2 Q2a (earliest ancestor): "Complete the following sentence. On average, the height of a boy increases by ▢ cm for each one-month increase in age."

Note the deliberate shift: these use "is associated with" rather than a causal verb, and scale the x-change to 10 units, so the student must multiply the slope.

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2.2 INTERPRETING THE INTERCEPT

Question template

"Interpret the intercept of the least squares line in terms of the variables [response] and [explanatory]."

Model answers

2016 Q3b.ii (Documents_exams_mathematics_2016_FM2_examrep16.txt): "On average, when the actual temperature is 0 °C, the apparent temperature is −1.7 °C.
Another accepted answer was: When the actual temperature is 0 °C, the predicted apparent temperature is −1.7 °C.
A common incorrect answer confused intercept and slope. Another common incorrect answer mixed up the response variable (apparent temperature) and the explanatory variable (actual temperature)."
2012 Q2b: "On average, it is predicted that, when the minimum temperature is 0 °C, the maximum temperature will be 13 °C."

Mandatory elements 1. State the explanatory variable = 0 (with its units). 2. State the predicted / average value of the response variable (with units). 3. Both variables named, correct roles. 4. Use "on average" or "predicted" — the intercept is a model value, not an observation.

Related 1-mark disguise — 2025 Exam 2 Q4c asks for the intercept without naming it: "Use the equation of the least squares line to predict the sale price for a three-bedroom home, located in the city centre of Melbourne" → distance = 0 → $1 765 353.

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2.3 INTERPRETING THE COEFFICIENT OF DETERMINATION r²

Question template

"Interpret the coefficient of determination in terms of [variable 1] and [variable 2]." — 1 mark
or the calculation-only version: "What percentage of the variation in [response] can be explained by the variation in [explanatory]?"

Model answers (verbatim)

2006 — "56.9% of the variation in height is explained by the variation in age."
2009 — "80.81% of the variation in the rainfall can be explained by the variation in the percentage of clear days."
2015 — "95% of the variation in male life expectancy can be explained by the variation in female life expectancy."
2016 — "97% of the variation in apparent temperature can be explained by the variation in actual temperature."
2019 NHT — "41.6% of the variation in life span can be explained by the variation in sleep time."
2024 — "85.7% of the variation in Mgold can be explained by the variation in year."
2025 — "75% of the variation in sale price can be explained by the variation in days."
2026 NHT — "68.0% of the variation in the brain weight can be explained by the variation in the body weight." (guide adds: Accept "…is explained by..")

Prescriptive examiner advice

2009 report (Documents_exams_mathematics_further2_assessrep_09.txt): "The coefficient of determination refers to the relationship in which the variation in the dependent variable can be explained by the variation in the independent variable. It does not imply any causation between the two variables. Unacceptable causation terms for this relationship include 'is determined by', 'is due to', 'is caused by' and 'is because of'. Other common errors included students reversing the variables … Others incorrectly referred to the absolute values of one or both of the variables rather than to the variation in them."
2016 report: "Some students mixed up the response variable and the explanatory variable in this statement and wrote, '97% of the variation in actual temperature can be explained by the variation in apparent temperature'. Other common unacceptable answers used terms that suggested variation in actual temperature caused the variation in apparent temperature. An example was, '97% of the variation in actual temperature was due to the variation in apparent temperature'."
2006 report: "Common incorrect interpretations referred to causation. A common incorrect answer was '56.9% of the variation in height is due to the variation in age.'"
2019 NHT report: "Answers that referred to the variance in each variable were not acceptable."
2024 report: "Some students did not reference the variation in both variables. Some rounded to 86% in a question where rounding did not apply."
2024 Chief Assessor PL: "This student was not awarded the mark as they did not reference the variation in each variable, and the word 'attributed' suggests a causal relationship."

Mandatory-element checklist 1. The percentage (r² × 100), correctly rounded — 2024 guide: "Accept only 85.7%*". 2. The words "of the variation in" before the response variable. 3. "can be explained by the variation in" before the explanatory variable. 4. Both variables named, correct order (response first). 5. No causal verbs: due to, caused by, determined by, because of, attributed to, results in. 6. Say variation, not variance, and not the raw variable ("97% of apparent temperature" is wrong).

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2.4 SIGN AND VALUE OF r

Not an interpretation question, but a wording-sensitive calculation that recurs every single year.

• 2020 Exam 2 Q5f: "Write down the value of the correlation coefficient r. Round your answer to three decimal places." Answer −0.824. Report: "Many students did not recognise that the negative square root value was required. Students should understand that the sign of the correlation coefficient matches the sign of the slope."
• 2021 Exam 2 Q3b: answer −0.938. "Many students found 0.938 but left off the negative sign."
• 2025 Exam 2 Q4b: answer −0.284 (only 21% correct). "A large proportion of the cohort left their final response as positive 0.284, not recognising that the slope was negative in the equation of the least squares line provided in the question stem."
• 2017 NHT Exam 2 Q2a.i: "Since the gradient of the least squares line is negative, the correlation coefficient, r, must also be negative."
• 2009 Exam 2 Q3c.ii: "Many students failed to include the negative sign as indicated by a generally negative slope of the data in the graph."
• Exam 1 mirror: 2018 Q9 and 2019 Q10 — both "Many students incorrectly took the positive square root value and chose option E."

Association-not-causation MC stems (the recurring distractor design):

2025 Exam 1 Q10: "The correlation coefficient between games won and goals against is r = −0.466. Based on the correlation coefficient, it can be concluded that A. fewer goals scored against is associated with a smaller number of wins. B. fewer goals scored against causes a smaller number of wins. C. more goals scored against causes a smaller number of wins. D. more goals scored against is associated with a smaller number of wins." Report: "Reject Options B and C as causation cannot be concluded."
2008 Exam 1 Q10 — the earliest instance of the identical design: "A large study of Year 12 students shows that there is a negative association between the time spent doing homework each week and the time spent watching television. The correlation coefficient is r = −0.6. From this information it can be concluded that … D. if a student spends less time watching television, they will do more homework. [causal distractor] … E. an increased time spent watching television is associated with a decreased time doing homework. [correct]"
2023 Exam 1 Q10: "A study of Year 10 students shows that there is a negative association between the scores of topic tests and the time spent on social media. The coefficient of determination is 0.72. From this information it can be concluded that A. a decreased time spent on social media is associated with an increased topic test score. B. less time spent on social media causes an increase in topic test performance. … D. too much time spent on social media causes a reduction in topic test performance."
2026 NHT Exam 1 Q11 uses the same four-option association/causation grid on a reciprocal-transformed equation.

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2.5 "DESCRIBE THE ASSOCIATION BETWEEN …"

Question template

"[Use the scatterplot to] describe the association between [variable 1] and [variable 2] in terms of strength, direction and form." (3-element form: 2007, 2014, 2015, 2016, 2023)
"Describe the [linear] association between [v1] and [v2] in terms of strength and direction." (2-element form: 2012, 2019 NHT, 2020, 2022, 2025)
"Describe the association between [v1] and [v2] in terms of form and direction." (2024 NHT)
"Write down the direction of the association between [v1] and [v2]." (2026 NHT Q2c — single element)

Model answers

Prescriptive examiner advice — DO NOT ADD EXTRA ELEMENTS

2022 report (...2022furmaths2-report.txt general comments): "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. For example, in Core Question 4b., students were only asked to describe the association in terms of strength and direction. It was not appropriate to further comment on form or possible outliers."
2020 report general comments: "in Core Question 6b. students were asked to describe the strength and direction … Many students recognised that the association was strong and negative but then gave unnecessary additional information that the association was of linear (or non-linear) form." and at Q6b: "Some students gave additional information that negated an otherwise correct answer."
2015 report Q4a: "Many students did not state the form as linear." (when form was asked for)
2014 report Q4a: "Another common, incorrect answer offered a discussion about the coefficient of determination … Others referred to skewness, which is not applicable to a bivariate data plot."
2014 report Q2c: "Some students inappropriately referred to the data being skewed. The term 'skewed' only applies to univariate data plots, whereas this question referred to a bivariate data plot."
2016 report Q3a: "A relatively common incorrect response was 'Strong, linear and positively skewed'."
2007 report Q3e: "A common error was to analyse the residual plot for strength, direction and form, incorrectly concluding that there was no relationship between the variables."
2024 report Q3d and 2024 PL: "Some students wrote the strength as moderate." (recurring — the same complaint appears in 2022 Q4b: "Many students chose moderate for the strength.")

Mandatory elements: exactly the elements asked for, in the order given by the question/table, no more. Vocabulary is fixed: strength ∈ {weak, moderate, strong}; direction ∈ {positive, negative}; form ∈ {linear, non-linear}.

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2.6 "COMMENT ON / IS THERE AN ASSOCIATION?" — TWO CATEGORICAL VARIABLES

Question templates (three interchangeable wordings)

(A) "Does the [percentage segmented bar chart / information in Table X] support the [opinion / contention] that [variable 1] is associated with [variable 2]? Justify your answer by quoting appropriate percentages." — often with the concession "It is sufficient to consider one [category] only when justifying your answer."
(B) "Does the information in Table X support [person]'s belief? Explain your conclusion by comparing the values of two appropriate percentages." (2023)
(C) "Explain why … there is no association … quoting appropriate percentages to support your explanation." (2014 — the negative case)

Recurrences: 2008 Q2, 2011 Q2b, 2014 Q1c, 2018 NHT Q4c, 2019 NHT Q5b.iii, 2021 NHT Q1d.ii, 2023 Q2b.ii, 2024 NHT Q2b, 2025 NHT Q3b. Exam-1 MC mirrors: 2017 Q6, 2021 Q2.

The prescribed full-mark structure (two marks)

The 2019 NHT and 2021 NHT reports state the two-mark split most explicitly:

2021 NHT report: "A statement that clearly indicates the contention is supported with a change or difference in one category of event was required for the first mark. For the second mark a statement similar to the following (choosing one category) using appropriate percentages was required:
'The percentage of females who participated in the 12 km walk (34%) was higher than the percentage of males who participated in that event (16%).'"
2019 NHT report Q5b.iii: "A statement that clearly indicated the contention is supported with a change or difference in one category of likelihood of attack considered and a statement similar to one of the following using column percentages was required. Approximate percentages were acceptable.
• The percentage of animals with low likelihood of attack decreases with increased exposure to attack during sleep – low exposure 91%, medium exposure 89%, high exposure 11%
• The percentage of animals with medium likelihood of attack changes with increased exposure … – low exposure 6%, medium exposure 0%, high exposure 11%
• The percentage of animals with high likelihood of attack increases with increased exposure … – low exposure 3%, medium exposure 11%, high exposure 79%"

Other model answers:

2008 Q2 (..._further_maths2_assessrep_08.txt): "The percentaged segmented bar chart does support the opinion that lunch time activity (walked, sat or stood, ran) is associated with year level. For example, the percent that ran changed from around 78% to 40% to 10% from Years 6–8 and 8–10.
There are several ways of observing support from the bar chart. In general terms, an association is indicated by the percentage of girls undertaking a particular activity changing with the year level. Note that this change does not have to be a consistent increase or decrease, but it can be. …
Some students referred to the changes in the percentages for an activity but then did not comment on whether the segmented bar chart supported the opinion that the association exists. Others forgot to quote relevant percentages as required to support their argument."
2011 Q2b: "The data for each of the four age groups in the table does support the opinion that age at first marriage is associated with the year of marriage. For example, of all first marriages, the percentage of women aged 25–29 years increased from 23.4% (1986) to 31.7% (1996) to 34.5% (2006)." Report: "Many students tried to work down the table despite the direction to work across a row."
2018 NHT Q4c: "Students needed to give a statement that clearly indicated the contention is supported with a change or difference in one beak size considered and then give a statement similar to one of the following. Approximate percentages were acceptable.
• 50% of males had large beaks, which was higher than females, with 7%. …"
2024 NHT Q2b: "Yes, as the percentages differ from 'before' to 'on' to 'after'. The LLT occurred after the New Moon on 75% of the days compared to 8.3% on the New moon and 16.7% before the New moon."
2025 NHT Q3b guide: "YES. The percentage of taller daughters with no brothers (62.5%) was greater than the percentage of taller daughters with one or more brothers (39.6%) — A1: Yes or statement of affirmation and mention difference/comparison; H1: Correct percentages of taller daughters for both situations with proportion* … Ignore minor rounding errors"
2014 Q1c (no-association case): "All three countries have approximately the same percentage: 67%, 64% and 64%. … some students contradicted the given statement* and claimed that 'there was an association because…'"

Mandatory-element checklist 1. An explicit Yes / No / "does support" verdict (or unmistakable affirmation). 2. A comparison word: greater than, higher than, lower than, differs, changes, increases with, decreases with. 3. At least two percentages, quoted with their category labels, for the same response category across different groups (i.e. compare across the bars/columns, never down a single column). 4. The percentages must be conditional (column or group percentages), not percentages of the grand total. 5. Naming the variables/categories, so it is clear which percentage belongs to which group.

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2.7 "IS X ASSOCIATED WITH Y?" — NUMERICAL vs CATEGORICAL (boxplots / stem plots)

Question template

"Do the boxplots support the contention that [numerical variable] is associated with [categorical variable]? Refer to the values of an appropriate statistic in your response." (2020, 2022)
"Explain why the information given above supports the contention that [numerical] is associated with [categorical]. Refer to the values of an appropriate statistic in your response." (2019)
"The [parallel boxplots / back-to-back stem plot] suggest that [numerical] is associated with [categorical]. Explain why, quoting the values of an appropriate statistic." (2016, 2017, 2017 NHT)
"Explain why [variable] for these countries is associated with the year. Refer to specific statistical values in your answer." (2015)

Model answers (verbatim)

2019 Exam 2 Q3 (...2019_FM2_examrep19.txt): "A statement that a decrease or change in median (or IQR) signals an association was required for the first mark to be awarded. Median (or IQR) values for all three months needed to be quoted correctly for the second mark to be awarded.
Successful responses focused on one statistic only (usually the median) and quoted the values from the table rather than estimating from the boxplots.
Incorrect answers included using the word 'averages' or 'means' rather than medians and quoting only two medians rather than all three.
Some students went on to comment on the minimums and maximums; this additional information compromised an otherwise correct answer. Comments about the shape of the boxplots were also not appropriate.
Questions of this type require reasoning rather than comments about which season each month falls in."
2020 Exam 2 Q3d: "Yes, as seen by the increase in median BMI values as neck size increases. For below average neck sizes, the median BMI is 21.6, which increases to 24.6 for average neck sizes and increases again to 28.1 for above average neck sizes.
A statement that an increase or change in median (or IQR) signals an association was required for the first mark. Median (or IQR) values for the neck sizes needed to be quoted correctly for the second mark. …Some students went on to also quote other statistics, such as the maximums or ranges, and this additional information devalued an otherwise correct answer."
2022 Exam 2 Q2b: "Yes, as the medians differ from 9 am to 3 pm. The median at 9 am of 75 is greater than the median at 3 pm of 63.
Many students gave the correct values for each median but did not specifically mention a change or difference."
2017 Exam 2 Q2e: "Forest 21 / Grassland 30 / Medians differ from forest to grassland.
Full marks were awarded to students who described a change in the median values and quoted these medians for both places of capture.
The mean is not part of a stemplot five-number summary and was not appropriate to use because of the outlier of 52 in the forest data.
Students who initially gave the required median comparisons and then went further by quoting comparisons of other irrelevant statistics were not awarded full marks."
2016 Exam 2 Q2b.iii: "The medians for the two months differ. In May, the median maximum temperature is about 14.5 °C, while in July, the median maximum temperature is about 9 °C.
The answer needed to refer to the difference between the two median temperatures. Simply quoting the two median values was not sufficient. …
Alternatively, comparing the two interquartile range (IQR) values could have been used as the difference in the IQRs also indicates the presence of an association. …
Some students referred to average or mean temperatures; however, this cannot be accurately determined from a boxplot unless the distributions are clearly symmetric."
2017 NHT Q1d: "The median amount of solar energy collected differs from month to month as indicated for April (11 MJ), May (7 MJ) and June (5.8 MJ)."
2023 NHT Q2c: "Yes, as the median foot width of the boys, 9.15 cm, is greater than the median foot width of the girls, 8.8 cm. OR No, as the Interquartile range for boys and girls are both equal to 0.8." ← note VCAA accepted either verdict provided the statistic supported it.
2015 Exam 2 Q2b: "Median life expectancy changes with year. For example, in 1953 the median was 52, in 1973 the median was 63 and in 1993 the median was 69. Observing the change in IQR values … was also acceptable. Some students referred to mean values, which are not discernible from a box plot unless it is perfectly symmetrical."
2008 Exam 2 Q3b: "• The median arm span increases with year level • The IQR of arm span decreases with year level or the range of arm span decreases with year level. … To compare the boxplots, one appropriate summary statistic for arm span (dependent variable) had to be compared across the year levels (independent variable)."
2006 Exam 2 Q1f: "The median height increases with age. … Reference to a boxplot statistic was required, therefore, consideration of the mean height was not appropriate."

Mandatory-element checklist 1. A verdict word: Yes / supports / "the medians differ". 2. One statistic only — the median (or the IQR). Never mean, never average. 3. An explicit change/difference/increase/decrease statement. 4. Values quoted for EVERY group in the display (all three months, all three neck sizes, both times of day). 5. Values taken from the supplied table where one exists, not estimated off the plot. 6. Nothing else — no minimum/maximum, no range, no shape commentary, no contextual speculation.

Spread-comparison sibling (2022 Q2ai, 2025 Q1c.ii) — same rules but with a spread statistic:

2022 Q2a.i model: "More variable at 9 am. IQR at 9 am of 35 is higher than IQR of 29 at 3 pm." Report: "Some students did not demonstrate which was more variable despite calculating each IQR value correctly. Some stated the values of each IQR but gave no conclusion."
2025 Q1c.ii model: "House sale prices have a lower spread than apartment sale prices as shown by the lower standard deviation." Guide: "Must have comparative word, spread or standard deviation and the two variables." Report: "The question clearly indicated using the information in Table 2, so using other statistics such as range and IQR was not appropriate."

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2.8 "EXPLAIN WHY A RESIDUAL PLOT …" / JUSTIFYING LINEARITY

Question templates

(A) "A residual plot can be used to test an assumption about the nature of the association between two numerical variables. What is this assumption?" — 1 mark (2016, 2019, 2022 NHT)
(B) "Does the residual plot above support this assumption? Explain your answer." — 1 mark (2016)
(C) "Does this residual plot support the assumption of linearity that was made when fitting this line to this data? Briefly explain your answer." — 1 mark (2017 NHT, 2020, 2026 NHT)
(D) "The residual plot above does not support this assumption. Explain why." — 1 mark (2019)
(E) "In part a., a least squares line was fitted to the scatterplot. Does this residual plot justify this? Briefly explain your answer." (2023 NHT)
(F) "Explain why this residual plot supports the assumption of linearity for this relationship." (2007)

Model answers

Prescriptive advice

2016 report Q3d.ii: "Reference to a lack of pattern or the randomness of the plot was required. An unacceptable answer was that the '… points are all scattered evenly above and below …' with no mention of randomness."
2024 report Q3g.ii: "It was acceptable to say the residuals were randomly scattered. The number of points above and below the line was irrelevant."
2024 marking guide: "3gii | Yes, the residual plot shows no clear pattern | A1 | MUST say YES, (or implied affirmation). Accept randomly scattered"
2026 NHT guide: "2d | YES Plot shows a random scatter/distribution of points; Plot has no pattern | A1 | Must have YES (stated or implied), with reason"
2019 report Q5f.i: "A one-word answer was all that was required. Some gave long answers that contradicted what was expected of them in Question 5f.ii."
2019 report Q5f.ii: "A common mistake was to ignore the given statement that the residuals did not support the assumption and to state that linearity was confirmed. Many students had difficulty with responses to Questions 5f.i. and 5f.ii., often contradicting themselves or writing definitions directly from a bound reference without any link to the question. Some confused residuals and the correlation coefficient."

Mandatory elements: (1) an explicit Yes/No; (2) the words "no clear pattern" or "randomly scattered"; (3) nothing about counts of points above/below the axis; (4) for the assumption question, the single word linearity (or "the association is linear").

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2.9 "DESCRIBE THE SHAPE OF THE DISTRIBUTION" AND OUTLIER STATEMENTS

Question template

"Describe the shape of the distribution [of ... ]." — 1 mark. Sometimes "Note the number of possible outliers, if any." (2018 NHT)
Exam 1 form: "The shape of this distribution is best described as A. positively skewed with one or more possible outliers. …"

Model answers and accepted forms

2018 report general comments: "in Question 1f. of the Core section when asked to describe the shape of the distribution, some students gave additional information concerning the centre and spread. Attention needed to be given to the type of statistical description required."
2021 Exam 1 Q1 report (the canonical shape-reasoning paragraph): "The centre of this distribution is in the group 56–57. Disregarding the possible outlier, we can see that the graph is more spread to the left of the centre than it is to the right. That is, the lower tail is longer than the upper tail. Therefore, we say that the shape of the distribution is negatively skewed (skewed to the left), with a possible outlier."

Vocabulary permitted: positively skewed / negatively skewed / approximately symmetric, each optionally "with (n) outlier(s)" or "with no outliers". Banned: symmetrically skewed, evenly distributed, bell shaped, normally distributed (for a sample display), and skewed applied to a scatterplot.

Outlier show-that / explain-why wording

2013 Exam 2 Q2b: "Write down an appropriate calculation and use it to explain why the country with a development index of 70 is an outlier for this group of countries."
Model: "Q1 − 1.5 × IQR = 75 − 1.5 × 3 = 70.5. Therefore, 70 is an outlier because it is less than 70.5."
Report: "Many students calculated a value of 70.5 and then wrote that 'it is therefore an outlier'. Further explanation, including a direct comparison between 70 and 70.5, was expected. Other common, incomplete or unacceptable answers included • 70 is outside the bulk of the data – this does not compare 70 with Q1 − 1.5 × IQR • it is outside 2SD from the mean – standard deviations do not define an outlier • there is no other country close to it – this does not compare 70 with Q1 − 1.5 × IQR."
2008 Exam 2 Q3c: "Many students correctly calculated the lower boundary (fence) … and then said that 'it was therefore still an outlier' without any numerical comparison shown or suggested. As this does not explain what 'it' is, nor why 'it' is still an outlier, the actual value of 140 cm should be stated as still being lower than the calculated value of the fence."
2011 Exam 2 Q1c: "Most calculated the 28.25 but then did not discuss how 26.0 related to this. Mathematical symbols were sometimes used incorrectly; for example, 28.25 < 26.0 < 32.65."
2024 Exam 2 Q2c.ii report: "Some students did not clearly explain the conditions for an outlier because they tried to combine their explanations for both fences into one, which resulted in a mathematically incorrect statement."
2024 PL: "In this second sample response, this student did not qualify for the mark as there was no reference to either the lower or upper fence."

Mandatory elements for an outlier justification: (1) the IQR; (2) the fence calculation shown in full (Q3 + 1.5 × IQR or Q1 − 1.5 × IQR); (3) an explicit numerical inequality comparing the data value with the fence (52 > 50, 240 < 243.55); (4) the conclusion word therefore it is / is not an outlier. Never combine two fences into one chained inequality.

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2.10 COMPARING DISTRIBUTIONS USING A NAMED STATISTIC

Covered in §2.7 (median/IQR association) and the spread siblings. Additional formulations:

• 2018 Exam 2 Q3e — comparing two fitted lines: "Since 2008, the equations of the least squares lines for Sydney and Melbourne have predicted that future traffic congestion levels in Sydney will always exceed those in Melbourne. Explain why, quoting the values of appropriate statistics." (2 marks) Model: "2008 Sydney value higher, 29.2 > 24.5, and Sydney line slope greater, 1.15 > 0.7667". Report: "Only a small proportion of students answered the question entirely correctly. Many students focused on the starting values … from the table rather than the least squares lines. Some gave gradient figures … but did not clearly say that Sydney's slope was greater."
• 2025 NHT Exam 2 Q4g — which predictor is better: "Father's height as Coefficient of determination is higher: 14% > 9.5% or Correlation coefficient is higher: 0.3744 > 0.3082".

Pattern: whenever VCAA says "quoting the values of appropriate statistics" (plural), two statistics are required; "an appropriate statistic" (singular) means pick ONE and quote it for every group.

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2.11 "USE THE … TO PREDICT …" AND THE INTERPOLATION / EXTRAPOLATION JUSTIFICATION

Question templates

"Use the equation of the least squares line to predict [response] when [explanatory] is [value]. Round your answer to …" — 1 mark
"Is the prediction made in part b. an example of extrapolation or interpolation?" (2019) / "When using the equation of this least squares line to make the prediction in part b., are you extrapolating or interpolating?" (2020) / "Identify whether the prediction in part b. is an interpolation or an extrapolation." (2023 NHT) / "Is the prediction made in part b. an interpolation or an extrapolation?" (2022 NHT)
"Would the predicted sale price be an example of interpolation or extrapolation? Briefly explain your answer." (2025)
"Give a reason why this prediction may have limited reliability." (2017)

Model answers

The critical 2024–2025 prescription (repeated verbatim two years running)

2024 report (2025-03_2024generalmaths2-report.txt) Q3h: "While predictions from a regression line focus on the response variable, their effectiveness as interpolation or extrapolation comes from the explanatory variable. It was not appropriate to refer to either 1964 or 1.934 as being outside the data range."
2025 report (2026-01_2025-GeneralMaths2-report.txt) Q4d: "Students needed to be careful that they recognised in their answer that while predictions from a regression line focus on the response variable, their effectiveness as interpolation or extrapolation comes from the explanatory variable. It was not appropriate to refer to the sale price as being outside the data range."
2025 marking guide: "4d | Extrapolation as 2 km lies outside explanatory variable data range | A1 | OR equivalent MUST reference explanatory variable range"
2015 report Q5b.ii: "This question required a response related to the given statistical data rather than any sociological possibilities. Many students wrote about the likely impact of future technological advances, wars, famines, viruses or advances in medicine, or simply stated 'You don't know what's going to happen in the future'. Such responses were not accepted."

Mandatory elements: (1) the single word interpolation / extrapolation; (2) where a reason is asked, "because [value] lies outside/inside the range of the EXPLANATORY variable data"; (3) never justify by reference to the response variable or to real-world speculation.

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2.12 TIME SERIES TREND / SEASONALITY DESCRIPTION

Question templates

"Describe the trend in the time series plot." (2017 NHT)
"The time series plot contains irregular fluctuations. Give two other descriptions of the pattern in this time series plot." (2022)
"Excluding any possible outliers, identify two qualitative features of the time series plot." (2025) — 2 marks
"Identify one other qualitative feature of this time series plot." (2026 NHT)
"Identify a feature of this plot that is consistent with this time series having a seasonal component." (2023)
Exam 1: "Which one of the following options best describes the qualitative features of the time series graph above?" (2025) / "The time series plot is best described as having …" (2016, 2017, 2018 NHT, 2021, 2023 NHT)

Model answers

Prescriptive advice

2025 report Q6a: "There was no evidence of any increasing or decreasing trend. It should be noted that irregular fluctuations are present in all time series plots. Overall, students need to be very clear on the key knowledge and key skills contained in the study design … and to use the formal terminology within the course. This question asked for qualitative features of a time series plot. The possible features are all clearly named within the study design."
2024 report Q4b: "There was no evidence of an outlier, of seasonality or of structural change. Some students erroneously referred to 'an irregular fluctuation'. It should be noted that irregular fluctuations are present in all time series plots."
2024 PL: "To qualify for the mark, the response needed to include ONLY correct features. This response listed a feature for which there was no evidence and so did not qualify for the mark."
2018 NHT Exam 1 report: "Irregular fluctuations are a feature of all time series plots."
2021 Exam 1 Q12 report: "The times series plot shows a decreasing trend with irregular fluctuations. To indicate the presence of seasonality we would expect to see regular fluctuations in the plot as well."
2017 Exam 1 Q13 report: "The existence of 'peaks' and 'troughs' is not enough information to determine seasonality; they must exist with regular intervals of time between them."
2011 report Q3a: "Many students noted a 'decreasing trend from 1915 to 1970', without considering the two sections of the plot. Some gave very specific answers rather than answering in general terms. Many students said the data was either 'positively skewed' or 'negatively skewed', neither of which is an applicable expression for a time-series plot."
2009 report Q2c: "In answering such questions, students must always refer to the given data rather than their own personal experiences. Some students referred to the seasons of the year but there was no indication in the data to suggest that these figures belonged to the southern hemisphere."

The permitted vocabulary (from the study design, as VCAA insists): trend (increasing/decreasing/upward/downward), seasonality, irregular (random) fluctuations, structural change, outliers. Nothing else. Never skewed. Adding a feature not evidenced loses the mark.

Forecast-assumption wording:

2021 Q3e.ii: "Same decreasing trend continues in the future." Report: "Students found it hard to convey the idea of the trend continuing. It was common to see reference to extrapolation or linearity but without linking it to the given equation."
2021 NHT Q4b.iii: "A similar decreasing trend continues beyond 2018."

Seasonal-index interpretation wording:

2009 Q4c: "The autumn rainfall is 5% above the average for the four seasons of the year." (not "above the monthly average")
2025 NHT Exam 1 Q14 report: "The average season index is 1. 0.75 is 0.25, or 25%, less than the seasonal average."

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2.13 "SHOW THAT …" QUESTIONS IN DATA ANALYSIS

Question template

"Show that …" / "Write down an appropriate calculation and use it to explain why …" / "Show, with calculations, that …" / "Use the equation … to show that …"

Data-analysis instances: 2013 Q3b (LS equation from summary statistics); 2016 Q4b (two-mean centred smoothing = 157.25 mm); 2017 Q2d (outlier); 2019 Q5e.i (r = 0.966); 2020 Q5e (residual = −0.02); 2021 Q3e.i (predicted 49.252 s); 2022 Q1b (two outliers); 2023 Q4c (seasonal index 1.10); 2023 NHT Q4b (predicted foot width 9.8 cm) and Q4e (r = 0.641); 2024 Q2c.i (both fences); 2024 NHT Q1c (0.22 not an outlier) and Q4e (residual 0.052); 2025 Q4f.i (residual 27 984); sample GM2 Q2a ("Show, with calculations, that a boxplot constructed from this five-number summary will not include outliers").

The prescribed standard (identical across reports, 2018–2025)

2018 report: "Some questions on the examination asked students to 'show that' a particular answer could be obtained. Students must work towards the given result with all relevant steps shown in their response. The answer the student is asked to show may be relevant in a subsequent part of the question."
2020 report: "Students must work towards the given result with all relevant steps shown. … In other words, if a student starts with the answer they have been asked to show, they will not be eligible for the mark. … In each case above the result that is required must be the conclusion to the process. Marks are awarded for showing the process of obtaining the required result."
2019 report (worked illustration): "This means students must show each iteration by writing down the relevant calculation … To state V₁ = 54 000 without showing how this was obtained does not address the instruction to 'show'."
2024 marking guide: "2ci | … | A1 | *SHOW THAT question — MUST see BOTH fence calculations*"; "3f | Predicted value = 0.300 + 0.860 × 2.02 = 2.0372; Residual = 2.07 – 2.0372 = 0.0328 | M1A1 | MUST see the predicted value calculation / MUST see residual calculation"
2025 report Q4f.i: "Students needed to show all working that led to the given residual value."
2024 PL: "Question 3f was a show that question. Two clear calculations were required."
2016 report Q4b: "The data needed for these calculations should have been taken from the table rather than from the graph."
2024 report general comments — the self-check list: "Students are strongly encouraged to read the question again after writing their final answer, and consider the following: Has the question been fully answered? Does the answer match the value provided in a 'show that' question? Is the answer reasonable in a practical context?**"

Mandatory elements: every intermediate line written out (predicted value line, then residual line; both fences, not one; each smoothing average, then the centring average), and the given value must be the last line, reached — not assumed.

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2.14 ROUNDING INSTRUCTIONS AND HOW THEY ARE ENFORCED

The instruction wordings used

The blanket exam instruction (current era)

2025-03_2024generalmaths2-report.txt and 2026-01_2025-GeneralMaths2-report.txt both quote it verbatim:

"the instructions section at the beginning of the examination clearly states that 'In all questions where a numerical answer is required, you should only round your answer when instructed to do so.' Students often rounded when an exact answer should have been given. For example, in Question 1e the percentage quoted needed to be 85.7% not 86%."
2025: "in this question the percentage given needed to be 83.85% not 84%."

How it is enforced

• Rounding where none was asked = absolute error, zero marks.

  2024 PL: "If a student rounds in a question where rounding does not apply, this is considered an absolute error, and no answer mark is awarded."
  2018 report: "In Question 2c. of the Core section the required answer was 63.8% and since rounding did not apply an answer of 64% could not be accepted." (echoed at Q2c: "Rounding did not apply, so an answer of 64% was not accepted.")
  2017 report Q3b.iii: "Rounding did not apply to this question, so −412.5 was the only acceptable answer."
  2020 report: "in Core Question 4a.ii. the required answer was 1.065, and since rounding did not apply, an answer such as 1.07 was not accepted."
  2014 report Q1c: "An answer rounded to one decimal place was expected. A number of students simply wrote 16% or an unrounded figure. This answer was not accepted. It could not be considered as a rounding error as there was no evidence that rounding had occurred."

• Decimal places ≠ significant figures — flagged in the general comments of 2016, 2017, 2018, 2019, 2020, 2021 and in the specific comments of 2021 Q2a ("Many students rounded to three decimal places rather than three significant figures") and 2016 Q3b.i ("The number 0.9 has only one significant figure, whereas 0.90 has two significant figures.").
• Designated rounding questions — the marking guides flag some items "Designated rounding question — Accept only this answer*" (2025 guide Q1c.i, Q7c) or "Rounding to 3 dp applies". 2024 PL: "For a rounding error to be considered, firstly, rounding must apply to the question, then the assessor must see either a correct calculation shown prior to the final rounded answer, or additional correct decimal places."
• Round only at the end. 2025 report (Matrices 13b.i, but stated generally): "Rounding should only be done at the final step." 2017 report: "students should … keep all decimal places in their calculator for intermediate steps and then only round the final answer to the required accuracy." 2021 report Q4b: "rounding too soon from calculations was common."
• Calculator exponent notation is not an answer. 2025 report Q5a: "many students did not interpret values from their calculator that were written using exponent notation (such as 1.05E6)." 2024 report Q1d: "Significant figures remain a challenge and many students were not able to interpret values from their calculator that were written using exponent notation (5.16E-3)." 2014 report Q1b: "Technology syntax is not to be used in providing answers; standard mathematical notation is to be used" (re "2.944E7").
• Chained rounding across parts. 2017 report Q4b: "Some students did not recognise that the rounded slope value from part a. was a required value in part b." 2018 report Q3d: "Some who correctly rounded in Question 3c. used −1514.7556 in Question 3d."

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2.15 OTHER RECURRING MICRO-STEMS (with fixed accepted answers)

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2.16 QUICK REFERENCE — MANDATORY WORDS BY QUESTION TYPE

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Recursion and financial modelling

Full recursion and financial modelling reference

2.1 "Write down a recurrence relation, in terms of …"

Canonical template (2016–2020):

"Let [variable]n be the [quantity] after n [periods].
Write down a recurrence relation, in terms of [V]0, [V]n+1 and [V]n, that [could be used to model / models / can be used to determine] the [quantity]."

Template drift by year (all corpus-verified):

THE REQUIRED TWO-PART ANSWER — verbatim examiner rules

"A recurrence relation has the initial value written first, followed by the recurrence rule.
Common errors included:
- failure to include the initial value, C0
- writing the initial value as Cn, not C0
- using different symbols for different parts of the recurrence relation, e.g. V0 = 38 000, Cn+1 = Cn − 2750
- using a rule for the nth term Cn = 38000 − 2750n instead of the recurrence rule."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q6b

"Many students did not write a recurrence relation in the required form.
A recurrence relation has the initial value written first.
The name of the variable needed to be consistent in the recurrence relation.
Some students did not recognise the difference between the recurrence relation above and the rule An = 428 × 1.015^n."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, Q6b

"Responses generally used the notation well, but many wrote a recurrence relation reflective of reducing balance depreciation rather than flat rate.
A few wrote both a recurrence relation and a rule with Vn in terms of n which could not be accepted."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q7d

"Notation was a problem for some students. A common error was to finish the recurrence relation with 12500n."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q6c

"This question was not answered well. The multiplying factor of 1.0025 was frequently incorrect and often the 5214.28 was added. Notation was a problem, with many students using a mixture of 'P notation' and 'V notation'."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q7c

"Recurrence relations must have two components to be complete. The initial value, V0, and the relation itself. … Whilst correct use of notation is considered to be important, the way this student used the n+1 subscript did not preclude them from being awarded the mark. … The mark was not awarded to this student. It was common to see the n added at the end of an otherwise correct answer."
— 2025-04_General_Maths_Examination_2.txt (2024 Chief Assessor PL transcript), Q5c

"Some students incorrectly included an '[n]' on the end of the recurrence relation."
— 2025-03_2024generalmaths2-report.txt, Q5c

"The multiplying factor was sometimes given as 0.003, 0.997 or 1.036. A few students mixed up Sn and Vn."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, Q8b

"The question was done reasonably well, but some responses gave B0 as 350 000, or R as 1.048, and added rather than subtracted the 3517.28."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q9c

Marking-guide statements of the two-part requirement:

"V0 = 40 000 Vn+1 = Vn – 8000 | A1 | BOTH parts must be correct"
— 2025-11_2025-GeneralMaths2-assessment-guide.txt, Q8a.i
"V0 = 300 000 Vn+1 = 1.0025 × Vn – 1570.58 | A1 | Both parts must be correct"
— 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, Q7c
"B0 = 42 623.31 Bn+1 = 1.0045 × Bn – 2470.48 | A1A1 | Any two values correct / Fully correct recurrence relation" (2-mark version)
— 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, Q8b

MANDATORY ELEMENTS checklist for full marks: 1. Initial value written first, subscript 0 (not n, not 1). 2. Recurrence step with subscript n+1 on the left and n on the right. 3. Same variable letter throughout and the letter the question names. 4. No n in the recurrence step (that belongs only in the rule form). 5. Correct sign on d (subtract for loans/annuities/depreciation; add for savings plans). 6. Correct multiplying factor for the stated compounding period (per-period, not per-annum). 7. Do not also supply the closed-form rule — a competing rule invalidates the response.

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2.2 "Use recursion to show that …" — the reports are emphatic

Wording variants across years: - "Use recursion to write down calculations that show that the amount of money in Ken's savings account after two years, V2, will be $16224." (2016 Exam 2 Q5b) - "Recursion can be used to calculate the value of the van after two years. Complete the calculations below by writing the appropriate numbers in the boxes provided." (2017 Exam 2 Q5a) - "Write down a calculation to show that the balance in the account after one month, V1, is $12074.40" (2018 Exam 2 Q4b.i) - "Use recursion to show that the value of the tools after two years, V2, is $48600." (2019 Exam 2 Q7a; NHT 2017 Q5b; NHT 2021 Q6a) - "Showing recursive calculations, determine the value of the machine, in dollars, after two years." (2020 Q7b; 2021 Q9a; 2022 Q8a; 2023 Q5b; NHT 2022 Q5b; NHT 2023 Q6b; Sample Q6b) - "Show, using a recursive calculation, that the balance of the loan after one month is $598664.07" (NHT 2026 Q5b)

Note: the phrase "Recursively calculate …" does not occur anywhere in this corpus. The instruction is always one of the six forms above.

THE RULES — quoted verbatim

2016 (and NHT 2017) — the fullest statement:

"V0 = 15 000
V1 = 1.04 × 15 000 = 15 600
V2 = 1.04 × 15 600 = 16 224

'Using recursion' begins with writing the initial value (i.e. V0 = 15 000)

Then the first calculation using the recurrence rule must be shown and the answer labelled.
(V1 = 1.04 × 15 000 = 15 600 in this case)

Subsequent calculation(s) must then show the use of the prior answer to calculate the next value.
(V2 = 1.04 × 15 600 = 16 224 in this case)

Some students used only the explicit rule Vn = R^n V0 to write V2 = 1.04² × 15 600 = 16 224, but this was not using recursion as required."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q5b
(verbatim repeat in Documents_exams_mathematics_2017_nht_fm2nht_examrep17.txt, Q5b)

2019 — general comments section, the canonical statement:

"In Core question 7a. students were asked to use recursion to show that the value of the tools after two years, V2 = $48 600. This means students must show each iteration by writing down the relevant calculation:
V1 = 60 000 x 0.9 = 54 000
V2 = 54 000 x 0.9 = 48 600

To state V1 = 54 000 without showing how this was obtained does not address the instruction to 'show'.

Using V2 = 60 000 x 0.9² = 48 600 does not address the use recursion instruction."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

2020 — with the explicit non-example:

"In Core Question 7b. students were asked to show recursive calculations that would determine the value of the machine after two years. This meant students needed to show each iteration by writing down the relevant calculation:
V1 = 120 000 – 15 000 = 105 000
V2 = 105 000 – 15 000 = 90 000
The final answer on its own, or a statement such as V2 = 105 000 – (2 x 15 000) = 90 000, would not satisfy the requirements of this question."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

2022 — the shortest, bluntest version:

"Core Question 8a. required students to 'show recursive calculations'. The mark can only be obtained by showing the step-by-step approach. An answer on its own is not sufficient to attract the marks."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

2024 — marking guide instruction:

"E0 = 300 000
E1 = 1.003 × 300 000.00 – 2159.41 = 298 740.59
E2 = 1.003 × 298 740.59 – 2159.41 = 297 477.40
| A1 | SHOW THAT question. Show BOTH calculations and final answer is correct"
— 2025-04_2024generalmathematics2-assessment-guide.txt, Q7a

"Note that an answer of $297 477.4 is not correct to the nearest cent."
— 2025-03_2024generalmaths2-report.txt, Q7a

FULL-MARK RESPONSE STRUCTURE (mandatory elements): 1. Write the initial value with its subscript: V0 = 60 000. 2. Write the first iteration as a complete calculation with the operation visible and the result labelled: V1 = 0.9 × 60 000 = 54 000. 3. Write each subsequent iteration using the previous answer: V2 = 0.9 × 54 000 = 48 600. 4. Never use V0 × R^n, V0 − n × d, or a finance-solver screen. 5. The final value must equal the given number, correctly rounded (and to the nearest cent if a currency answer).

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2.3 The generic "show that" rule (applies to non-recursive show-that items)

"'Show that' questions give the answer to a basic and appropriate calculation, and students must write that calculation. The given number in a 'show that' question is sometimes needed in a following question. With this number, students can attempt the following question, even if they cannot complete the 'show that' question.
The given number in a 'show that' question is not to be used in a calculation; it must be the result of a calculation. The following calculations use the 2750 to show a different result than what is required and were not acceptable for this question:
- 38 000 − 8 × 2750 = 16 000 — shows how to find the depreciated value after eight years
- 16 000 + 8 × 2750 = 38 000 — shows how to find the initial value"
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q6a (near-identical text in Documents_exams_mathematics_2015_FM2_examrep15.txt)

"In the Recursion and financial modelling module Question 8b. students were asked to show that the annual interest rate is 4.2% from the given recurrence relation. Stating that 1 + r/1200 = 1.0035 and r = 4.2% amounts to recognising the well-known interest rate rule and then stating the conclusion. To show that the result is the case, some calculation must be provided:
r/1200 = 0.0035
r = 1200 x 0.0035 = 4.2%
… In each of these cases the result must be the conclusion of the reasoning process. Showing the process is the basis of achieving the mark."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

"Responses needed to show the calculation using basic arithmetic that resulted in an answer of 4.2%, not use the 4.2% and attempt to verify it."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, Q8b

"Questions that were not well answered often involved the instruction to 'show that' a particular answer could be obtained. Students must work towards the given result with all relevant steps shown. When a student started with the answer given, the mark could not be awarded. For example, in core Question 8b., students were asked to show that the compound interest rate for a loan was 2.6% per annum. A student who worked back from this value to find the factor of 1.001 in the recurrence relation could not be given the mark."
— Documents_exams_mathematics_2021_2021furthermath2-report.txt, general comments

"In other words, if a student starts with the answer they have been asked to show, they will not be eligible for the mark."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

"A question that asks to 'show that' a particular answer could be obtained was a problem for many students. Make sure when doing a 'show that' question, any equations or values used to 'show' the answer are copied correctly. Students should recall that they cannot simply substitute a value or use the fact they are attempting to 'show' as part of this process."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

"Does the answer match the value provided in a 'show that' question?" (in the list of self-checks students should run)
— 2025-03_2024generalmaths2-report.txt, general comments

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2.4 Rounding instructions and how examiners penalise

The instruction strings actually used (corpus counts across all files)

Also seen: "with V0 rounded to the nearest cent" (NHT 2026 Q8b), "Write your answer as a percentage" (2017 Q5b.ii), "Write your answer correct to two decimal places" (NHT 2017 Q7a; legacy 2015 Mod 4 Q2c).

Verbatim penalty rules

Nearest cent means exactly two decimal places — no further rounding:

"Currency answers rounded to the nearest cent needed to show two decimal places. Many students only wrote an answer of $22 203.7, which might arise from a calculator setting that restricts the number of displayed digits. Some students appeared to round all currency answers to the nearest five or ten cents. A common incorrect answer was $22 203.70, which had clearly been rounded from a written $22 203.664... This was assessed as a rounding error."
— Documents_exams_mathematics_2016_FM2_examrep16.txt, Q5d.ii

"In Recursion and financial modelling students are often asked to round their answer to the nearest cent. It is not appropriate to round to the nearest five cents or 10 cents in this instance."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, general comments

"In the Recursion and financial modelling module students sometimes continued working from a correct answer to an incorrect answer because they rounded to the nearest 5 cents or 10 cents. For example, in Question 5b. of the Core section, the required answer was $5484.23 and some students incorrectly rounded further to $5484.20 or $5484.25."
— Documents_exams_mathematics_2018_FM2_examrep18.txt, general comments

"…students sometimes further engaged from a correct answer to an incorrect answer because they rounded to the nearest 5 c or dollar. They needed to treat the answer to the nearest cent as if it were electronic money where no further rounding occurs. For example, in Core Question 9a. the required answer was $1704.03 and some students incorrectly rounded further to $1704.05 or $1704."
— Documents_exams_mathematics_2019_FM2_examrep19.txt, general comments

"In Recursion and financial modelling students are often instructed to answer questions to the nearest cent (which means two decimal places). Students sometimes unnecessarily changed from a correct answer to an incorrect answer because they rounded to the nearest 5c, 10c or dollar. In all questions where rounding applies, this is clearly stated in the question."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt, general comments

"In Recursion and financial modelling, students are often instructed to answer questions correct to the nearest cent (two decimal places). For example, in core Question 9b., the required answer to the nearest cent was $7039.20. A student who answered $7039.2 could not be awarded the mark. This given answer actually represents one decimal accuracy of any value between $7039.15 and $7039.24."
— Documents_exams_mathematics_2021_2021furthermath2-report.txt, general comments

"In Financial Mathematics, students should only round to the nearest dollar or nearest cent when specified."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, general comments

No instruction to round ⇒ exact answer required:

"In questions where no instruction to round is given, an exact answer is required as rounding does not apply."
— 2017, 2018, 2019, 2020, 2021, 2022 Exam 2 reports (identical sentence each year)
2018 example: "in Question 2c. of the Core section the required answer was 63.8% and since rounding did not apply an answer of 64% could not be accepted."
2024 example: "in Question 1e the percentage quoted needed to be 85.7% not 86%" (2025-03_2024generalmaths2-report.txt).
2024 PL: "This is an example where no instruction to round was given in the question. In all questions of this type, the exact answer must have been given. An exact answer may be a decimal, as long as it is a terminating decimal … 20.8% was the only acceptable answer, and not, for example, 21%." (2025-04_General_Maths_Examination_2.txt)

Premature rounding:

"Students should also take care to avoid premature rounding. They should use all the decimal places in their calculator for intermediate steps and then only round the final answer to the required accuracy level."
— Documents_exams_mathematics_2017_fm2_examrep17.txt, general comments
Worked example of the penalty: 2017 Q7a — "others knew the correct method but rounded 5.2 to two decimal places before multiplying by $360 000, giving $1 548."

The one-rounding-error concession (2020–2022 Core):

"In Core questions where a maximum of one rounding error was penalised, a second rounding error would not have resulted in the loss of marks if either:
- a correct calculation was shown prior to the final rounded answer
- additional correct decimal places could be seen earlier in the response or in the final answer."
— Documents_exams_mathematics_2020_2020furmaths2-exam-report.txt (repeated 2021, 2022)

The 2024+ "designated rounding question" regime — the current rule:

"For a rounding error to be considered, firstly, rounding must apply to the question, then the assessor must see either a correct calculation shown prior to the final rounded answer, or additional correct decimal places. If a student rounds in a question where rounding does not apply, this is considered an absolute error, and no answer mark is awarded. … The third rounding question type involved rounding to the nearest cent in relation to the recursion and financial modelling area of study. In addition to the calculations, the answer had to be $297,477 and 40 cents."
— 2025-04_General_Maths_Examination_2.txt, 2024 Chief Assessor PL

Marking-guide flags in the current era: - *** Rounding to nearest cent applies / *** Rounding to 2 dp applies / *** Rounding to 4 dp applies / *** Whole number rounding applies — 2025-11_2025-GeneralMaths2-assessment-guide.txt, 2025-06_2025NHT-GeneralMath2-assessment-guide.txt. - "Designated rounding question. Accept only this answer" — 2025 Q7c. - " Rounding to nearest cent applies, *providing Finance Solver working shown and additional decimal places shown" — 2024 Q8. - "H1 – Must see calculation using 9b.i answer *** Rounding to 4 dp applies" — 2025 Q9b.ii (a consequential/H mark).

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2.5 Interest rate wording and the conversion trap

Stem phrasings (frequency across corpus):

Other recurring stem sentences: - "Interest on this loan is charged at the rate of X% per annum, compounding [period]." (2016 Q7; 2023 Q5) - "Interest is calculated monthly and [name] makes monthly repayments." (2020 Q8; 2025 Q7; NHT 2021 Q5) - "Interest is calculated and paid monthly." (2017 Q7a perpetuity; NHT 2026 Q18) - "Interest is calculated immediately before each payment." (2019 Exam 1 Q20; NHT 2022 Exam 1 Q21) - "Immediately after the interest has been added/calculated each month, [name] deposits a further $X." (2016 Exam 1 Q23; 2017 Q7b; NHT 2017 Q7; 2021 Exam 1 Q22) - "If a customer does not pay the bill by the due date, interest is charged. Alex charges interest after the due date at the rate of 1.5% per month on the amount of an unpaid bill. The interest on this amount will compound monthly." (2017 Q6) - "Assume there are exactly 52 weeks in one year." / "Assume that there are exactly 26 fortnights in one year." (2025 Exam 1 Q22, Q23; 2025 Q9b) - "The annual interest rate for this loan is 3.6%." / "The interest rate for the annuity is 0.42% per month. Determine the interest rate per annum." (2023 Q6a)

THE CONVERSION TRAP — examiner evidence

(a) Per-period rate quoted as if annual (and vice versa).

"Many students did not read that the given interest rate was per month, so $200.25 was a common incorrect answer." — 2017 Q6a (Documents_exams_mathematics_2017_fm2_examrep17.txt)
"Many gave the annual interest rate percentage." — 2020 Q9b, where 0.3% (monthly) was wanted.
"0.24% per month was sometimes given." — 2020 Q10b, where 2.88% p.a. was wanted.
"0.2% was a common incorrect answer." — 2022 Q8b (2.4% p.a. wanted).
"Those who were incorrect often gave 0.3%, as this was the monthly rate." — 2024 Q7c.
"Some students gave the weekly interest rate rather than the annual rate, presumably through not reading the question carefully." — 2024 PL, Q5d.
"A weekly interest rate of 0.4% was a common incorrect response." — 2025-03_2024generalmaths2-report.txt, Q5d.

(b) Frequency of payment vs frequency of compounding.

"A common error was 0.18%, which did not account for the fortnightly payments." — 2018 Q6a.
"Some students gave the annual payment of $18 720." — 2017 Q7a.

(c) Rate and compounding period must be considered together.

"This question required an understanding that both the interest rate and compounding period in combination determine the overall cost of the loan." — 2018 Exam 1 Q19.

(d) The n in the rule must be the number of compounding periods.

"The growth factor per quarter was [1.0075]. As interest was compounding quarterly, the number of compounding periods in n years is equal to 4n, which is the required power." — 2020 Exam 1 Q26.

(e) Multiplying-factor derivation is expected to be shown.

"The multiplying factor is 1 + (3.6/12)/100 = 1.003. The repayment is (3.6/12)/100 × 225 000 = 675." — 2017 Exam 1 Q20 report.
2024 NHT Q7c: "Use this value in a calculation to show that the multiplication factor in the recurrence relation is 1.005."

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2.6 Amortisation table completion wording

"Using the values in the table, complete the next line of the amortisation table. Write your answers in the spaces provided in the table below. Round all values to the nearest cent." — 2023 Exam 2 Q6b

"Complete the next line in the amortisation table. Write your answers in the spaces provided in the table below." — NHT 2024 Exam 2 Q7d

"The interest rate on Declan's loan is 4.2% per annum, compounding monthly. Using the values in the table, complete the table below. Round all values to the nearest cent." — 2025 Exam 2 Q7c

"After one year, Timmy changes to making interest-only repayments. Complete the next line in the amortisation table below, rounding all values to the nearest cent." — NHT 2026 Exam 2 Q7

"Write down the values of P, Q and R, rounded to the nearest cent, in the boxes provided below." — NHT 2021 Exam 2 Q5c

MANDATORY: use the table's own numbers, not a solver.

"Students were instructed to use the values in the table. Using a finance solver resulted in a different balance to the nearest cent which was not appropriate."
— 2026-01_2025-GeneralMaths2-report.txt, Q7c

"Rounding errors occurred by those students who calculated their own interest rate rather than using the information given in the question. Rounding was required to the nearest cent."
— Documents_exams_mathematics_2022_2022furmaths2-report.txt, Q7b

Column relationships examiners expect students to state: - interest = (rate per period) × (previous balance) - principal reduction = payment − interest - new balance = previous balance − principal reduction - rate per period = interest ÷ previous balance (used to derive the rate — 2018 Exam 1 Q23; 2019 Exam 1 Q20; NHT 2023 Exam 1 Q19; NHT 2022 Exam 2 Q7a)

"Students must be familiar with the calculations that are used to create an amortisation table and be able to use these to complete questions such as this."
— Documents_exams_mathematics_2017_fm1_examrep17.txt, Q23

"The very small difference in these two values is due to the rounding of all values in the amortisation table. Either could be used to determine the interest at payment number 20."
— same, Q23

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2.7 Final / adjusted repayment wording

The method the reports prescribe (2019 Exam 1 Q23 model):

N     = 1
I%    = 7.5
PV    = 995.49        (balance outstanding after the second-last payment)
PMT   = solve   → −1001.71
FV    = 0
P/Y = C/Y = 12

"The negative indicates that this amount must still be paid back to the bank as the final payment."

Equivalent two-step alternative (2023 Exam 1 Q22 report): N = 299 to find the balance before the final payment ($2605.647…), then N = 1 to add one period's interest ($2614.1157…).

Or the lump adjustment (2018 Exam 1 Q22): run the full N, read the residual FV, and add it to the regular repayment.

Common error the reports name: giving the balance before the last payment instead of balance + interest. "A common incorrect response was $1197.39. This was the balance before the last payment but does not include the interest to be added to the outstanding balance for the last payment." (2021 Q8c)

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2.8 "In total, how much interest …" wording

The method VCAA prescribes (three interchangeable framings, all in reports): 1. Loans: total interest = (number of repayments × repayment) − (reduction in principal)

"Repayments − reduction in the principal = 12 × 800 − (70 000 − 65 076.22) = 9600 − 4923.78 = 4676.22" (2016 Q7a.ii)
"Paid = 78 × $1704.03 = $132 914.34; The principal reduced by: $350 000 − $262 332.33 = $87 667.67; Interest = $132 914.34 − $87 667.67" (2019 Q9b)
"Total repaid = 2228.40 × 5 × 12 = 133 704; Interest paid = Amount repaid − amount borrowed" (2024 Exam 1 Q21) 2. Annuities (payments received): interest = payments received − balance reduction "Balance start Year 2: 480 242.25; Balance end Year 2: 467 131.14; Balance reduction 13 111.11; Payments in Year 2 = 12 × 2800 = 33 600; Interest = 33 600.00 − 13 111.11 = 20 488.89" (2018 Q6b.ii) 3. Investments (no withdrawals): interest = final balance − initial balance "Interest earned = 2545.33 − 2500 = $45.33" (NHT 2019 Q7a)

Named errors: "A common incorrect answer was $4923.78, which is the reduction in the principal over the year. This failed to take into account the $9600 total of repayments made in the year." (2016). "Some students who answered parts a. and b. correctly gave $454.26 as the answer here, which was the total amount Lily was charged rather than the interest." (2017 Q6c).

Consequential marking: "An answer of $20 488.88, based on an unrounded value from Question 6b.i., was also awarded full marks." (2018 Q6b.ii). But: "Some students followed through with the incorrect response they obtained for Question 9a. but did not show the working required for the consequential mark." (2019 Q9b).

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2.9 "The value of the [asset] after n years can be determined using the rule Vn = …"

Template family: - "The amount of money in the account after n years, Vn, can also be determined using a rule. Complete the rule below by writing the appropriate numbers in the boxes provided. Vn = □ × □^n" (2016 Q5d.i) - "A rule of the form Vn = a × b^n can be used to determine the balance of Julie's account after n months. Complete this rule … balance = □ × □^n" (2018 Q4c.i) - "The value of the machine, in dollars, after n years, Vn, could also be determined using a rule of the form Vn = a + bn. Write down this rule for Vn." (2020 Q7d) - "Complete the rule below that gives the value of the coffee machine, Mn, in dollars, after n cups have been produced. Write your answers in the boxes provided. Mn = □ + □ × n" (2021 Q7c) - "Complete the rule below that gives the value of the car, Cn, in dollars, after n kilometres have been travelled. Cn = □ + □ × n" (NHT 2022 Q6b) - "The value of Marlon's guitar after n concerts, Gn, can be determined using a rule. Complete the rule below by writing the appropriate numbers in the boxes provided. Gn = □ − □ × n" (NHT 2019 Q6c) - "ii. Write a rule for Vn in terms of n." (2025 Q8a.ii) - "The value of the car, in dollars, after travelling n kilometres, Wn, can be modelled by a rule. Write the rule for Wn in terms of n." (NHT 2026 Q6c) - MCQ version: "A rule for the balance, Rn, in dollars, after n years is given by" (2020 Exam 1 Q26); "A rule for Gn, the value of the machine after n years is" (2023 Exam 1 Q18); "An equivalent rule to determine the value of the car after n years is" (NHT 2024 Exam 1 Q17).

Examiner statements on the rule/recurrence distinction:

"Some students were unsure of the distinction between a rule and a recurrence relation." — 2020 Q7d
"Many students did not demonstrate understanding of the distinction between a rule and recurrence relation and gave a similar answer again as in part a.i." — 2026-01_2025-GeneralMaths2-report.txt, Q8a.ii (only 50% correct)
"A common error was for students to write the two numbers in the incorrect boxes." — 2018 Q4c.i
"After completing a question, students should read the question again … For example in core Question 7c., students were asked to write a rule for the value of a coffee machine. Since an earlier question had related value in terms of time, many students failed to recognise the change to value in terms of number of cups produced." — 2021 general comments

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2.10 Other high-frequency stem furniture

• "The value of [asset], in dollars, after n years, Vn, can be modelled by the recurrence relation shown below." (2017, 2018, 2019, 2020, 2022, 2023 …)
• "The balance of the loan, in dollars, after n months, Ln, can be modelled by the recurrence relation" (2022 Q8; 2023 NHT Q6; 2026 NHT Q5)
• "Let Vn be the balance of the annuity after n monthly payments." (2021 Q9a)
• "can be determined using a recurrence relation of the form V0 = 60000, Vn+1 = 1.0015Vn − d" (2023 Q7) — invites solving for d.
• "Complete the calculations below by writing the appropriate numbers in the boxes provided." (2017 Q5a)
• "Write your answers in the boxes provided." (2021 Q6c, Q7c; NHT 2022 Q6b)
• "Write your answers in the spaces provided in the table below." (2023 Q6b; NHT 2024 Q7d)
• "For what value of k would this investment act as a simple perpetuity?" (2020 Q10c; Sample Q8c)
• "For what value of d does the recurrence relation generate a geometric sequence?" (2023 Q7d)
• "How much money did [name] initially deposit/invest/borrow?" (2016 Q5a; 2018 Q4a; 2018 NHT Q7a; 2024 NHT Q7a; 2025 Q7a) — the reliable opening 1-mark gift; 87–96% correct every year.
• "What monthly payment does Phil receive?" / "What amount does Timmy repay each month?" (2019 Q8a; 2026 NHT Q5a) — read d straight off the recurrence; still only 72–74% correct.

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Matrices

Full matrices reference

2.1 Order of a matrix

Stems (all 1 mark):

Full-mark answer form. rows × columns, in that order, written as m × n. - 2014 report: "Some students reversed the two numbers, writing 2 × 4, which was not accepted." - 2016 report: "A common incorrect answer was 1 × 4. An answer of (4, 1) was not accepted."

2.2 "The element in row i and column j of matrix M, mᵢⱼ, …"

Stem frames: - "The element in row i and column j of matrix M is mᵢⱼ." — 2014 Ex1 Q6, 2015 Ex1 Q1, 2015 Ex1 Q8, 2016 Ex1 Q5, 2017 Ex2 Q1b, 2017 Ex1 Q6, 2017 NHT Ex1 Q4, 2019 Ex1 Q3, 2020 Ex1 Q2, 2020 Ex1 Q6, 2020 Ex2 Q1cii, 2022 Ex1 Q1, 2023 Ex2 Q8a, 2024 NHT Ex1 Q28, 2025 Ex1 Q30, Sample Ex1 Q28. - Follow-ups: "What does the element m₁₂ indicate?" (2017 Ex2) · "What does the element q₂₃ represent?" (2020 Ex2) · "Which element shows the cost for one child ticket?" (2023 Ex2) · "Interpret the value of element c₃₁." (2022 NHT Ex2) · "Explain what the matrix element R₂₄ represents." (2008 Ex2) · "Explain what the matrix element r₃₂ represents." (2021 NHT Ex2) · "Explain what the element in row 3, column 2 of matrix Rᵀ represents." (2019 Ex2 Q1e) · "Element m₂₁ indicates that" (2023 Ex1 Q25) · "The elements are determined by the rule" (construct-from-rule form).

Full-mark answer structure (prescribed by reports). The answer must name all three of: 1. the quantity counted (students / shoppers / cars / rolls), 2. the row category, and 3. the column category — in context, not as coordinates.

Evidence: - 2017 report (Q1b): "The number of rolls sold in week 1, which was 24. … Some students simply gave the element 24." - 2020 report (Q1cii): "The number of shoppers (594) in the clothing area of Westmall (at 1.00 pm). Most students identified the key information of shoppers, clothing and Westmall. A small number of students misread the matrix to give merchandise shoppers in Grandmall – reading q₃₂ instead of q₂₃." - 2019 report (Q1e): "The answer is the number of cars parked in area C for two hours. Some students recognised area C and 2 hours but did not refer to the number of cars parked." - 2024 GM Exam 2 assessment-guide video: "Question 10a required an understanding of element notation. Many students were unable to demonstrate an understanding of the notation in the question." (54 % correct) - 2021 NHT report: "The number of Year 9 students awarded a credit in English." (all three components present)

2.3 "Explain what the element … represents in the context of this problem"

Applied to a transition matrix element, the full-mark answer must be a conditional rate sentence, not a statement about a fixed count:

"[p] % of those who [state A] this [period] will [state B] the next [period]."

Prescribed by the 2010 report (Q2a), which is the clearest statement VCAA ever made:

"20 % of those doing heavy training one week will change to moderate training the next week. … A common
incorrect answer was '20 % of players will change to moderate training the next week'. This would mean that
20 % of a fixed number of 300 will change, whereas it is only 20 % of a variable number (initially 90).
Others suggested that the change would (only) occur in the first week rather than a regular transitional
change."

For a 1 on the leading diagonal (absorbing state), the required form is a permanence sentence: - 2012 report (Q3b): "Once a worker leaves the site, they never return (or equivalent statement). … The most common unacceptable answer was '100 % of those who leave this year will leave the next year'." - 2015 report (Q2a): "All of the advanced-level students stay as advanced-level students." - 2025 report (Q13a), leading diagonal of zeros: "No child does the same activity in any two consecutive weeks." Guide: "No child does the same activity from one week to the next. | OR equivalent"

For a 0 element, the required form is a negation in context: - 2023 NHT report (Q2a): "No driver upgrades from a light-rigid licence one year to a heavy-rigid licence the next year." - 2025 NHT guide (Q13a): "No (0) resorts are classified as Budget one year and then Luxury the next. Or similar wording" - 2011 report (Q1aii): "No birds, lizards or insects eat birds." — with the three ways to lose the mark listed: "no lizards or insects eat birds (this omitted one of the zeros) / insects do not eat insects, birds or lizards (confused a column with a row) / none of these animals eat their own kind (confused a diagonal with a row)."

2.4 "Write down a matrix expression / calculation that can be used to determine …"

The classic product-with-ones / cost-matrix question. Five stable variants:

What exactly earns the mark:

1. A matrix expression, not a number. The answer must be a product of two matrices written in the correct order. 2015 report: "C × Q. Also accepted was the matrix calculation written as: [15 25 40] × [5 10 3 2; 15 30 9 6; 10 20 6 4]. Some students incorrectly gave Q × C."
2. The order of multiplication must be conformable. 2020 report: "Many students gave the correct matrix calculation. Some had the two matrices but in the incorrect order."
3. Row vs column matrix matters and the labelling may be required. 2008 report: "Despite a row matrix being required, many students wrote a column matrix instead. The question expected students to label the matrix as shown."
4. When asked to "write and evaluate", both are needed. 2006 report (Q2c): "The expression and the result were both required for full marks."
5. Result of a matrix product is a matrix — keep the brackets. 2016 report: "[6000]. The product of two matrices produces a matrix. The brackets needed to be included."; 2017 NHT report: "[21 200]. The product of two matrices produces a matrix. The brackets must be included."; 2011 report: "The correct answer shown is a 1 × 4 matrix with the four elements clearly separated by spaces … Some students wrote their four elements separated by commas or dots; however, this is not correct mathematical notation."
6. When the target number is given, a generic matrix product does not "show" it. 2016 report (Q3b): "0.65 × 520 + 0.25 × 320 + 0.25 × 80 + 0.5 × 80 = 478 customers. A matrix product such as TS₀ does not show an understanding of how 478 is calculated."; 2014 report (Q1e): "Some students copied the full given matrix multiplication and replaced the w with 1415. This was not acceptable."
7. Matrix brackets are required even for a hand-written data matrix. 2007 report (Q1a): "Matrix brackets were required."

2.5 "Write down the transition matrix T" (from a transition diagram)

Full-mark answer form. - Columns must be the "this period" state, rows the "next period" state, and the printed row/column labels must be respected. 2006 report: "Some simply copied the figures in order, as printed into the three columns." - Entries as proportions if the question says proportions. 2006 report: "Some students wrote percentages despite being asked for proportions. … The percentage, 8 %, was sometimes written as a proportion of 0.8." - Each column must sum to 1 (used as a check and as a question in its own right — 2017 Ex1 Q8 report: "Two of the unknown values in the matrix T could be found from the elementary concept of transition matrices – that is, the column values must add to one."; 2020 Ex1 Q10 report: "Since the columns of the transition matrix add to 1, then …")

And the reverse (completing a transition diagram): - All edges must carry arrowheads. 2024 GM Exam 2 report: "This question was not well answered. All edges required arrows otherwise it is no longer a directed graph." The assessment-guide video is blunter: "This response was not awarded the mark due to the absence of the arrows. The use of percentages rather than proportions was acceptable." - Loops count and must be labelled. 2013 report: "The 100 % cycle drawn at D … was a common omission."; 2015 report: "Another mark was available for the loop at L, including the 100 % label. A number of students missed the loop entirely, while others incorrectly labelled it as 1 %." - Do not add 0 % reverse edges. 2013 report: "Students should not add edges marked 0 % that are in the opposite direction to the given edges. Similarly, loops with 0 % at E and B should not be included."; 2015 report: "Others unnecessarily added a lot of 0 % transitions in the reverse directions." (Caveat: the 2024 assessment guide says "Accept additional edges with zero value" — so the 0 %-edge penalty is not universal; it is applied where the diagram was already complete without them.)

2.6 "Write down a matrix recurrence relation, in terms of S₀ and Sₙ₊₁, that can be used to determine …"

Exact frames in the corpus: - "Write down a recurrence relation, in terms of Sₙ₊₁ and Sₙ, that can be used to determine …" — 2017 NHT Exam 2 (Matrices module) - "Write down a recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ, that could be used to model …" — 2020 Exam 2, GM Sample Exam 2 - "Complete the recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ, that would model …" — 2021 Exam 2 - "Write down a matrix equation for Sₙ in terms of T, n and S₁." — 2008 Exam 2 Q2b - Presented-to-the-student forms: "The matrix recurrence relation shown below can be used to determine / to calculate / to predict …" — 2018 Ex2, 2019 Ex2, 2020 Ex2 (×2), 2021 NHT Ex2, 2022 NHT Ex2, 2023 Ex2, 2023 NHT Ex1, 2025 Ex2, 2026 NHT Ex2. - "Consider the matrix recurrence relation below." — 2017 Ex1 Q8, 2020 Ex1 Q10, Sample Ex1 Q32.

Full-mark answer structure. Both pieces, as on the formula sheet:

S₀ = [initial column matrix],   Sₙ₊₁ = T Sₙ        (or  Sₙ₊₁ = T Sₙ + B)

The 2008 report is explicit that the paper's own subscript convention must be honoured: "Sₙ = Tⁿ⁻¹ · S₁ — The initial state matrix defined as S₁ in the question was required here."

2.7 "In the long term / in the long run" and steady-state wording

Full-mark answer structure for a "show steady state" question. VCAA requires two consecutive state matrices that agree: - 2010 report: "In order to show a steady state had occurred 'after 7 weeks', it was necessary to demonstrate that a steady state had been achieved for two consecutive weeks. … such a pair of non-consecutive calculations did not eliminate the possibility that S₇ may not equal S₈." - 2006 report: "Any two products Tⁿ K₀ where n ≥ 38 … and this is the same as K₃₈." - 2008 report: "When two different calculations involving Sₙ = Tⁿ⁻¹ · S₁ produce the same result, the long-term state matrix has been produced. High powers for n, such as n = 50 and 51, could be used." - 2018 Exam 1 report gives the shortcut: "The long-term expectation involves raising the transition matrix to a high power, for example, 50. The steady state matrix is independent of the initial state, so one suggested approach is to use any initial state matrix where the values added to 500."

When the answer is a description rather than a number (2010 Ex2 Q4c/d), the report demands the limiting value be named:

"Attendance at Dinosaurs' games is expected to gradually increase to 3000, at which level it will remain
steady. A description of 'the way in which the number of people attending the Dinosaur's games would change'
was required. The correct answer had to acknowledge that a steady state of 3000 would occur after about 80
weeks. Many students simply calculated the predicted attendance after 80 weeks and gave this number as their
answer. Others merely said that 'the attendance was increasing.' However, neither of these responses was
accepted."

2.8 "Explain why matrix … does not have an inverse" / determinant wording

Note: VCAA has never used the literal string "does not have an inverse" as a question stem in this corpus. The actual family is:

Full-mark answer structure. - "The determinant is not equal to zero" — not "greater than zero". 2020 report: "The determinant is not equal to zero. Many students did not recognise that a non-zero determinant is required for an inverse matrix to exist. Some stated that the determinant had to be greater than 0." - For the interpret-the-determinant form, both halves are required: 2022 report: "A unique solution exists, as the determinant is not zero. … The response also needed both parts." - For not defined, name the matrices and the mismatching counts: 2026 NHT guide: "Must reference M & Q OR 1 and 3 for columns/rows."; 2006 report warns that "second number / first number" phrasing "did not indicate whether the student knew which numbers referred to rows and which to columns."

2.9 "Show that …" in matrix questions

Three distinct shapes, each with a different full-mark requirement:

1. Show that a determinant equals a value — 2022 Ex2 Q2bi: "Show that the determinant of the matrix [20 40; 30 50] is −200." Report: "20 × 50 − 40 × 30. This question was a 'show that' question, which required the calculation to be clearly demonstrated."
2. Show that a state-matrix element equals a value — the boxes format (§1.10c). The individual products must be visible; a matrix product is not enough (2016 report).
3. Show that a steady state exists — two consecutive state matrices (§2.7).
4. Show that an unknown parameter equals a value — 2021 NHT Ex2 Q3b "Show that the value of k is 0.62"; GM Sample Ex2 Q11bi "If R₁ = [500; 4000; 650; 7150], show that k = 500."
5. Showing relevant calculations — 2026 NHT Ex2 Q11c: "Showing relevant calculations, determine the level of threat faced by the numbat population 10 years later". Guide assigns A1 and H1 for calculation + interpretation separately.

The 2024 assessment-guide video adds a useful mark-scheme note about how much working "show" really needs:

"Sample response for Question 12d. The first one here, the student successfully showed each step towards the
solution. Underneath that, the second student simply gave the correct answer. Both of these samples were
awarded the mark." (i.e. for a 1-mark question a bare correct answer suffices; for a 2-mark question it
does not — see §2.10.)

2.10 Rounding and "write your answer to the nearest whole number"

Exact instruction wordings (Exam 2):

Assessment-guide marking annotations (2023+) now state the rounding rule as a marking condition: - 2025 GM Exam 2 guide: "| Whole number rounding applies" (Q12b, Q13bii) and "| One decimal place rounding applies" (Q13bi). - 2022 report uses the same language in prose: "Whole number rounding applied for this question." (Q3b, Q3c, Q4b).

The prescribed rounding discipline: - Round only at the final step. 2006 report: "Rounding off should only occur at the final answer stage in a question, not halfway through the calculation." 2009 report: "A number (for example, length) found during working out within a question should not be rounded off when used to determine the subsequent answer for that question." 2025 report (Q13bi): "Rounding should only be done at the final step. The expected number of children should not have been rounded to 10." (only 9 % scored this mark) - But a rounded answer carried from a previous part is not penalised. 2006 report: "in cases where the answer to a previous question must be used, the rounded version of the previous answer may be used without penalty. (Students could also use the unrounded version of the previous answer without penalty.)" Repeated verbatim 2007, 2009. - Rounding down instead of to nearest loses the mark. 2015 report (Q3c): "Some students rounded 42.55 to 42 despite the question instruction, 'Write your answer correct to the nearest whole number'." - Don't over-round a population count that is genuinely fractional. 2013 report (Q2bii): "On average, the state matrix predicts that there will be 331.25 adult trout after four years. Since this average is a predicted value, a decimal number of adult trout is valid. An answer of 331 adult trout was common and also accepted." - Calculator floating-point noise must be read as exact. 2007 report (Q2cv): "Others did not interpret a calculator answer such as 5.8460065E−30 as representing zero or 699.999828 as representing 700 in this practical context. Such unrounded answers were not accepted." Also listed in that year's Areas of weakness as "interpreting a calculator result of something such as 4.8E-34 as zero in context". - Convert to a percentage when asked. 2015 report (Q3b): "Many students did not convert 21 students into the required percentage."; 2018 report (Q3e): "Some students left the answer as 1532.1 and did not convert to the required percentage."

Working requirement for multi-mark questions (2025 report, verbatim):

"For questions worth more than one mark, students are advised to show working. An incorrect answer on its
own will not be awarded any marks. A method mark can, where appropriate, be awarded to students who have
shown the development of their answer."

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3. TRAPS AND EXAMINER COMPLAINTS

Ordered by how often and how expensively they recur.

Networks and decision mathematics

Full networks and decision mathematics reference

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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