Question designs · not yet written

New separator ideas

Descriptions of questions that would plausibly separate a cohort: in scope for the current study design, aimed at the weaknesses the examiner reports document year after year. Designs only — the questions themselves are not written here.

A design brief of 64 question ideas for VCE General Mathematics Units 3 & 4 (2023–2027)

Compiled 13 September 2026. These are descriptions, not questions. Each entry is a specification for someone who will later write the actual item.

A separator is a question where 50% or fewer of the state earns full marks. Every idea below is engineered against a failure mode that VCAA's own examination reports document, and every one sits inside the 2023–2027 key knowledge and key skills (01-study-design.md).

0. How to read this document, and how the bands were calibrated

0.1 Band definitions

Band	Full-marks rate	Meaning
Brutal	< 10%	The corpus floor. Only ever achieved in Exam 2.
Severe	10–24%	The standard "last part of the question" difficulty.
Hard	25–39%	A genuine discriminator that most well-taught students still miss.
Tricky	40–50%	Sits just under the separator line; a coin-flip item.

0.2 The Exam 1 ceiling on difficulty — an honest constraint

Since the move to four options (A–D) at the November 2024 sitting, the guessing floor is 25%. The lowest facilities ever recorded in the four-option era are 2024 Exam 1 Q38 (18%) and Q39 (19%); in the five-option era the floor was 2008 Exam 1 Q7 (11%, planar faces), 2020 FM Exam 1 Q7 (11%) and 2006 Exam 1 Q9 (17%, crashing). No multiple-choice item in the entire corpus has ever fallen below 10%.

Therefore: no Exam 1 idea in this document is predicted brutal. All 11 brutal ideas are Exam 2. That is not timidity — it is what the data allows. Making an MC item brutal requires a distractor that attracts more than three-quarters of the cohort, which in practice means an ambiguous item, and ambiguous items are exactly what §6 rules out.

0.3 The five mechanisms that actually produce separators

Counted across separators.csv (905 rows) and the trap inventories in 02–05 and 07:

#	Mechanism	Signature evidence
1	Two distinct elements required for one mark	2022 E2 Q3d Networks, 1 mark, 2% — "Some students listed the vertices correctly, but not the new capacity."
2	Re-scan / re-check after a change (crashing, rate change, culling)	2019 E2 Q3e Networks, 2 marks, 3%; 2021 E2 Q9b Recursion, 1 mark, 8%
3	Interpretation in context, brevity enforced, further engagement punished	2019 E2 Q3c Networks — "if they … then stated an incorrect float time, they could not be awarded marks"; 2025 E2 Q18c, 47%
4	Read from a graph / table rather than compute	Every Exam 1 report 2016–2022: "Questions that required the use or analysis of graphical or tabular information were answered less well."
5	Rounding discipline and sign discipline	2025 E2 Q13b.i Matrices, 1 mark, 9% — "Rounding should only be done at the final step"; 2025 E2 Q4b Data analysis, 1 mark, 21% — sign of r

1. DATA ANALYSIS

Exam 1: Q1–16 (16 marks). Exam 2: 24 marks.

1A. Exam 1 — multiple choice (7 ideas)

D1. The whisker that stops short

Working title: The whisker that stops short
Target skill: "the five-number summary and boxplots (including the designation and display of possible outliers)"; "construct stem and dot plots, boxplots … and use them to describe and interpret the distributions of numerical variables".
What the question does: Present a dot plot of about 22 values whose upper tail contains two values beyond Q3 + 1.5 × IQR and one value just inside it. Give the five-number summary in a small table. Ask: at which value does the upper whisker of the boxplot end? The four options are the upper fence, the largest data value, the largest non-outlier data value (key), and Q3.
Why it separates: 2020 Exam 2 Q2c (2 marks, 17%) — "Those who found the two outliers often extended the whiskers to the lower and upper fence values rather than to the second-lowest and second-highest values. Lower and upper fences are not part of the data set and therefore should not be plotted on a box plot." The reciprocal error is documented at 2015 Exam 1 Q6 and 2017 Exam 1 Q2 — outliers "are still valid data points … and must be used as maximum and minimum values if appropriate". Offering both errors as options forces the student to hold both rules simultaneously.
Marks: 1.
Predicted band: Hard (25–39%). Two distinct rules must both fire; each error has its own dedicated distractor with documented pulling power.
Fairness check: Fences are on the formula sheet and boxplot construction is core Unit 3 content; the dot plot makes every value visible, so nothing is hidden.

D2. The sign that survives a transformation

Working title: The sign that survives a transformation
Target skill: "calculate the coefficient of determination, r², and interpret in the context of the association being modelled"; "interpretation of the slope … of the least squares line"; data transformation applied to one axis only.
What the question does: Give the equation of a least squares line fitted to log₁₀-transformed data, with a negative slope, plus the value of r² to three decimal places. Ask for the value of r for the transformed association, correct to three decimal places. Options: +√r², −√r² (key), r² itself, and −r².
Why it separates: 2025 Exam 2 Q4b (1 mark, 21%) — "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." Also 2020 Exam 2 Q5f (1 mark, 25%). At 2019 Exam 1 Q10 the +√r² distractor beat the key 46% to 41%, with wording copy-pasted from the 2018 Exam 1 Q9 report. Putting the slope on a transformed scale adds a second thing to ignore correctly — the transformation is irrelevant to the sign rule, but students distrust that.
Marks: 1.
Predicted band: Hard (25–39%). The bare version already runs at 21–25%; four options lift it, the transformation noise lowers it.
Fairness check: The relationship between the sign of r and the sign of the slope is standard key knowledge; the transformation is a red herring, not a trick, because the equation is supplied in full.

D3. Deseasonalising upwards

Working title: Deseasonalising upwards
Target skill: "calculate, interpret and apply seasonal indices"; the formula-sheet relation seasonal index = actual figure ÷ deseasonalised figure.
What the question does: State a seasonal index below 1 (e.g. 0.80) for a named quarter, and ask: to correct for seasonality, the actual figure for that quarter should be increased, to the nearest whole percentage, by … Options 20%, 25% (key), 80%, 125%.
Why it separates: 2025 Exam 1 Q16 (50%) ran the mirror image with an index above 1 — "Multiplying by 0.57 is equivalent to a 43% reduction (100 − 57)%". 2017 Exam 1 Q16 and 2014 Exam 1 Q12 are the same design; the 2014 report records "The majority of students chose one of the two options with a 25% in the answer." Flipping to an index below 1 breaks the memorised "100 − index" shortcut, because 1 ÷ 0.80 = 1.25 and the answer is 25%, not 20%. The 20% distractor is the memorised rule misapplied.
Marks: 1.
Predicted band: Hard (25–39%). The upward-facing version has never been set; the rote rule fails and produces a beautifully plausible wrong option.
Fairness check: The seasonal index definition is on the formula sheet and the arithmetic is a single division. Nothing outside "apply seasonal indices".

D4. Counting the crosses

Working title: Counting the crosses
Target skill: "numerical smoothing of time series data using moving means with consideration of the number of terms required (using centring when appropriate)".
What the question does: State that a time series of 40 monthly observations is smoothed using six-mean smoothing with centring, and ask how many smoothed points will appear on the smoothed plot. Options: 34, 35 (key), 36, 40.
Why it separates: 2022 Exam 2 Q5c.ii (1 mark, 34%) — answer 28 for nine-mean over 36 months; "An incorrect response often seen was 4." 2019 Exam 1 Q16 is the odd-terms version. Even-order smoothing with centring loses n points, not n − 1 — an asymmetry that no formula sheet supplies and that most students derive wrongly under time pressure. The off-by-one distractors are the two most common derivations.
Marks: 1.
Predicted band: Hard (25–39%). "Off-by-one on a count" is documented distractor pattern #6 in 07-exam-craft.md §5.1.5, and this variant has the extra centring step.
Fairness check: Centred moving means are explicit key knowledge; the count is derivable by drawing the first and last window.

D5. Choosing the transformation from the residuals

Working title: Choosing the transformation from the residuals
Target skill: "construct a residual analysis to test the assumption of linearity and, in the case of clear non-linearity, transform the data to achieve linearity"; "data transformation and its use in transforming some forms of non-linear data to linearity using a square, logarithmic (base 10) or reciprocal transformation (applied to one axis only)".
What the question does: Show only a residual plot (no scatterplot) with a clear single-curve pattern, plus the direction of the fitted slope. Ask which transformation applied to the explanatory variable would be most likely to linearise the data. Options: (x)², log₁₀(x), 1/x, (y)².
Why it separates: 2024 Exam 1 Q12 (48%) required identifying the point furthest from the regression line after a reciprocal transformation. 2020 Exam 2 Q6d (2 marks, 21%) — "A small number tried to work with a reciprocal transformation or a logarithm transformation." Mechanism 4: the student must reconstruct the shape of the raw scatterplot from the residual pattern instead of being shown it, which the reports say is where the cohort collapses ("Questions that required the use or analysis of graphical or tabular information were answered less well" — every Exam 1 report 2016–2022).
Marks: 1.
Predicted band: Hard (25–39%). Reading a residual plot backwards is a step VCAA has not yet demanded in Exam 1, and the fourth option tests that a y-transformation is a legitimate but different choice.
Fairness check: Both residual analysis and the three named transformations are key skills; the residual plot is unambiguous, and one transformation is clearly better than the others.

D6. The conditional the other way round

Working title: The conditional the other way round
Target skill: "construct two-way tables and use them to identify and describe associations between two categorical variables"; percentage segmented bar charts.
What the question does: Give a 3 × 2 two-way frequency table of counts. Ask for the percentage of respondents in the response category who came from one explanatory level — i.e. the reversed conditional, requiring the row total when the table is naturally read by column. Options: the correct reversed percentage (key), the forward conditional, the grand-total percentage, and the raw count expressed as a percentage of the column.
Why it separates: Distractor pattern #1 in 07-exam-craft.md §5.1.5 — "wrong base / wrong denominator in a percentage" — is the single most frequently named distractor family. 2013 Core Q4 report: "the wrong base was used in calculating the percentage … 3/82 × 100% = 3.65% … This answer rounds to 4% (option B)". 2023 NHT: "A number of students incorrectly chose D, which gives the percentage of workers with symptoms in each exposure level." 2022 Exam 2 Q3c (1 mark, 48%) — "0.6 was a common incorrect response."
Marks: 1.
Predicted band: Tricky (40–50%). A well-drilled student gets it; the reversal costs perhaps half the cohort, consistent with 48% at 2022 E2 Q3c.
Fairness check: Pure two-way-table arithmetic; the table is complete and the question names the sub-group explicitly.

D7. Standardising a deseasonalised value

Working title: Standardising a deseasonalised value
Target skill: Recombination. "solve problems using z-scores and the 68–95–99.7% rule" × "seasonal adjustment including the use and interpretation of seasonal indices".
What the question does: Give a quarterly sales figure, the seasonal index for that quarter, and the mean and standard deviation of the deseasonalised series. Ask for the standardised value (z-score) of that quarter's deseasonalised figure. Options include the z-score of the raw figure, the z-score of the deseasonalised figure (key), the z-score computed with the index multiplied rather than divided, and a sign-flipped value.
Why it separates: Two chained steps, each with its own documented failure. Deseasonalising direction: 2023 Exam 2 Q4c (2 marks, 21%) required both a mean and an index calculation — "Both calculations were required to attract the 2 marks". z-score set-up: 2013 Exam 2 Q3a report — "Many students failed to use brackets: (91 − 85.6) ÷ 2.99. Without brackets, this calculation becomes 91 − 85.6/2.99 = 62.371…". VCAA has never combined z-scores with seasonal adjustment, so no rehearsed template exists.
Marks: 1.
Predicted band: Severe (10–24%). Two independent chances to fail, and the "divide vs multiply" distractor is documented as the dominant seasonal-index error.
Fairness check: Both components are explicit key skills examined every year; the formula sheet supplies both the z-score and the seasonal-index relations. The only new thing is that they appear in the same stem.

1B. Exam 2 — extended response (11 ideas)

E1. Two roads to the same seasonal index

Working title: Two roads to the same seasonal index
Target skill: "seasonal adjustment including the use and interpretation of seasonal indices and their calculation using seasonal and yearly means".
What the question does: Give a table of monthly figures for four complete years. Ask the student to calculate the seasonal index for one named month, rounded to three decimal places, showing working. Two legitimate routes exist: (a) compute the index for that month in each of the four years and average them; (b) divide the mean of that month's four values by the overall monthly mean across all 48 values. Both must be accepted.
Why it separates: 2025 Exam 2 Q6b (2 marks, 12%; 74% scored zero) is exactly this structure and VCAA published both routes: "Seasonal Index = (0.4986 + 0.6780 + 0.6704 + 0.5042) / 4 = 0.588 OR … 17.5 / 29.79 = 0.587". 2023 Exam 2 Q4c (2 marks, 21%) — "Both calculations were required to attract the 2 marks." The item punishes students who know the formula but cannot organise a 48-cell table, and rounding to three decimals means the two routes give different last digits, which is itself a documented anxiety point.
Marks: 2 — one for a correct set of monthly means or yearly ratios, one for the correct index at the stated precision.
Predicted band: Severe (10–24%), at the bottom of the band. The 2025 original ran at 12% with a three-decimal instruction; this version is its twin.
Fairness check: This is the single most explicitly worded calculation dot point in the time series sub-topic, and both accepted methods are the ones VCAA itself publishes.

E2. The residual that lives on the transformed scale

Working title: The residual that lives on the transformed scale
Target skill: "interpretation and use of the equation of the least squares line fitted to the transformed data to make predictions"; "residual value = actual value − predicted value"; "construct a residual analysis to test the assumption of linearity".
What the question does: Give a least squares line fitted to reciprocal-transformed data and a table of raw values. Part (i): calculate the residual for one named data point on the transformed scale, showing all working. Part (ii): mark the corresponding point on a supplied residual plot whose vertical scale is in transformed units.
Why it separates: 2021 Exam 2 Q5b (1 mark, 14%) — "A common response was 0.2 gained by using the correct coefficients but using difference rather than the reciprocal of difference." 2024 Exam 2 Q3g.i (1 mark, 47%) — "Some students did not understand the connection between a scatterplot and a residual plot." 2019 Exam 2 Q5d — "Many found the predicted value correctly but then subtracted from 1013 rather than from the pressure 3 pm value". Three documented errors stack here: transform direction, residual direction, and plotting precision (2025 Exam 2 Q4f.ii, 45% — "Students need to be precise when marking a point on a grid").
Marks: 2 — one for the residual with visible substitution, one for the plotted point.
Predicted band: Severe (10–24%). Each sub-part independently runs at 14–47%; requiring both for full marks compounds them.
Fairness check: Transformation-then-predict and residual computation are both listed key skills, and 2026 NHT Exam 1 Q11–12 has already examined a residual on a transformed scale.

E3. Whiskers to the data, not to the fences

Working title: Whiskers to the data, not to the fences
Target skill: "the five-number summary and boxplots (including the use of the lower fence … and upper fence … to identify and display possible outliers)".
What the question does: Supply an ordered stem plot of about 25 values and a blank scale. Ask the student to construct a boxplot showing any outliers as separate points, and to write down how many values lie beyond the upper fence. The data is constructed so that exactly one value sits between Q3 + 1.5 × IQR and the next value up, and the second-highest value is close to the fence.
Why it separates: 2020 Exam 2 Q2c (2 marks, 17%) — "Only a small number of students obtained full marks. Many drew a boxplot from the five-number summary without consideration of outliers. Those who found the two outliers often extended the whiskers to the lower and upper fence values". The counting sub-part attacks 2024 Exam 2 Q2b (1 mark, 3%) — "The question asked for the number of heights, not an actual height."
Marks: 2 — one for the box and correctly placed quartiles/median, one for the whiskers terminating at actual data values with outliers plotted separately.
Predicted band: Severe (10–24%). The 2020 original ran at 17%; adding the explicit count keeps it there rather than lower, because the count is now signposted.
Fairness check: Entirely routine construction with the fence formula supplied; the only demand is that the student apply the definition rather than a remembered picture.

E4. Association by the statistic you were told to use

Working title: Association by the statistic you were told to use
Target skill: "construct parallel boxplots and use them to identify and describe associations between a numerical variable and a categorical variable"; "answer statistical questions that require a knowledge of the associations between pairs of variables".
What the question does: Supply three parallel boxplots for an ordered three-level categorical variable. Ask whether the numerical variable is associated with the categorical variable, instructing the student to refer to the medians in the response. The medians move monotonically but the IQRs do not, so a student who volunteers spread as well says something false.
Why it separates: 2020 Exam 2 Q3d (2 marks, 36%) — "A statement that an increase or change in median (or IQR) signals an association was required for the first mark." 2022 Exam 2 Q2b (2 marks, 41%) — "Many students gave the correct values for each median but did not specifically mention a change or difference." 2025 Exam 2 Q1c.ii (1 mark, 47%) — "The question clearly indicated using the information in Table 2, so using other statistics such as range and IQR was not appropriate." The further-engagement penalty (10 reports, 07-exam-craft.md §7 item 10) does the rest: the non-monotone IQRs make volunteering spread fatal.
Marks: 2 — one for a direct yes/no plus an explicit statement that the medians change across levels, one for quoting the three median values in context.
Predicted band: Hard (25–39%). Two distinct elements are required for the first mark and the over-answerer is punished; historical analogues sit at 36–47%.
Fairness check: The stem names the statistic, so there is no guesswork — a well-prepared student writes two sentences and stops.

E5. Outside the data, inside the grid, on a transformed axis

Working title: Outside the data, inside the grid, on a transformed axis
Target skill: "use of the rule of the fitted line to make predictions being aware of the limitations of extrapolation"; "interpretation and use of the equation of the least squares line fitted to the transformed data".
What the question does: Give a least squares line fitted to log₁₀-transformed explanatory data, plotted on a grid whose axis extends well beyond the data. Ask the student to make a prediction for a value that is inside the printed grid but outside the range of the explanatory data, then state whether the prediction is interpolation or extrapolation and justify the answer.
Why it separates: 2025 Exam 2 Q4d (1 mark, 27%) — "Students needed to be careful that they recognised … 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." 2023 Exam 2 Q3f (1 mark, 41%) — "Even though the value of average temperature (−6 °C) fits on the axes given, it is outside the range of data values". 2020 Exam 2 Q5c (1 mark, 38%) — "students … may have been confused because the point 65 was within the grid of the graph". Three separate years, same trap; putting it on a transformed axis means the student must also know which variable's range matters after transformation.
Marks: 1 — the naming and the justification are both required for the single mark.
Predicted band: Hard (25–39%). The untransformed version has run at 27%, 38% and 41%; the transformation adds confusion but the question is heavily signposted.
Fairness check: The data range is visible in the supplied table; the distinction is core key knowledge stated in the study design as "being aware of the limitations of extrapolation".

E6. Deseasonalise, fit, forecast, re-seasonalise

Working title: Deseasonalise, fit, forecast, re-seasonalise
Target skill: "modelling trend by fitting a least squares line to a time series with time as the explanatory variable (data de-seasonalised where necessary), and the use of the model to make forecasts (with re-seasonalisation where necessary)".
What the question does: Give twelve quarters of actual data and a complete set of four seasonal indices. Part (a): deseasonalise a named quarter, rounded as instructed. Part (b): fit a least squares line to the deseasonalised series with quarter number as the explanatory variable, giving coefficients to a stated number of significant figures. Part (c): forecast the actual (re-seasonalised) figure for a future quarter. Part (c) is fully dependent on (b).
Why it separates: Significant figures are named as a separate recurring complaint in 11 reports (07-exam-craft.md §7 item 8): 2025 Exam 2 Q5a (2 marks, 28%) — "Significant figures remain problematic and many students did not interpret values from their calculator that were written using exponent notation (such as 1.05E6)"; 2024 Exam 2 Q1d (2 marks, 39%) — same complaint with 5.16E-3. 2019 Exam 2 Q6a (2 marks, 32%) — "the calculated value for autumn of 0.9975 was often rounded to the nearest whole number rather than to two decimal places". The re-seasonalisation step reverses the division and is the mirror of the error VCAA names every year.
Marks: 3 — one for the deseasonalised value, one for the trend line at the stated precision, one for a correct re-seasonalised forecast (consequential on (b) if the substitution is shown).
Predicted band: Severe (10–24%) for all three marks. Each part alone sits at 28–39%; the product of three dependent steps plus a significant-figure gate lands well below 25%.
Fairness check: The dot point quoted above names this exact chain. All computation is CAS-native and the rounding instruction is explicit.

E7. How many, not which one

Working title: How many, not which one
Target skill: "answer statistical questions that require a knowledge of the distribution(s) of one or more numerical variables"; boxplots with outliers displayed.
What the question does: Supply three parallel boxplots (three groups, stated group sizes) with outliers shown as separate points on two of them. Ask for the smallest possible number of individuals across all three groups whose measurement exceeds a stated threshold that falls between Q3 and the upper whisker of one group. The answer requires reasoning from quartile proportions plus explicit counting of the plotted outliers, with no double counting.
Why it separates: 2024 Exam 2 Q2b (1 mark, 3% — the hardest single mark in the current era) — "Students did not recognise that only 1 value is needed to extend the whisker beyond the IQR. The emphasis on 'smallest' in the question was overlooked … The question asked for the number of heights, not an actual height." 2020 Exam 2 Q3c (1 mark, 22%) — "Many students found 25 as their answer by adding all the outliers to the top quarter of 'above average', thus counting the two outliers in above average twice." This idea fuses both failures into one stem.
Marks: 1.
Predicted band: Brutal (< 10%). Its two parent items ran at 3% and 22%; requiring both the "smallest" reasoning and the no-double-count rule puts it at the corpus floor.
Fairness check: Every number needed is on the plots and in the stem; the word "smallest" is doing honest work, and the reasoning is the standard quartile argument. It is hard because it is precise, not because it is obscure.

E8. Least squares from summary statistics, sign supplied by the picture

Working title: Least squares from summary statistics, sign supplied by the picture
Target skill: "the determination of the coefficients a and b using technology, and the formulas b = r·(s_y/s_x) and a = ȳ − b·x̄".
What the question does: Give x̄, s_x, ȳ, s_y and r² (not r), together with a small scatterplot showing a clearly negative association but no raw data. Ask for the equation of the least squares line with coefficients to a stated number of significant figures, using the variable names.
Why it separates: 2021 Exam 2 Q2b (2 marks, 29%) — "Many students struggled with the calculation of Pearson's correlation coefficient from summary statistics. Some students did not attempt this question." 2019 Exam 1 Q11 offers r·s_y/s_x, r alone, and the swapped r·s_x/s_y as options. Three failure points combine: recovering r from r² with the sign read off the picture (2025 E2 Q4b, 21%); the s_y/s_x order (documented distractor pattern #5); and naming the variables rather than x and y (2008 Q4b — "The variables arm span and height were required in the equation rather than y and x").
Marks: 2 — one for a correct slope, one for a correct intercept with correct variable names at the stated precision.
Predicted band: Hard (25–39%). The parent item ran at 29%; the sign-from-picture step is new but the formulas are supplied.
Fairness check: Both formulas are printed on the formula sheet and the dot point names them explicitly. No raw data is needed, and the scatterplot removes any ambiguity about direction.

E9. What you are assuming when you forecast

Working title: What you are assuming when you forecast
Target skill: "use the model to make forecasts with consideration of the limitations of extending forecasts too far into the future"; Outcome 2 — "discussing the assumptions in application of these models".
What the question does: After a "show that" part has supplied a forecast value for a year well beyond the data, ask: what assumption is being made when this equation is used to make that forecast? One mark, one sentence, no calculation.
Why it separates: 2021 Exam 2 Q3e.ii (1 mark, 18%; 82% scored zero) — "Same decreasing trend continues in the future. 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 Exam 2 Q4b.iii is the same item with the same answer. This is the purest instance of 07-exam-craft.md §7 item 14 — students reproduce a bound-reference sentence about extrapolation instead of answering.
Marks: 1.
Predicted band: Severe (10–24%). Directly precedented at 18% twice, and nothing about the current cohort suggests improvement — the 2024 and 2025 reports still name terminology precision as a top complaint.
Fairness check: One sentence, no computation, and the required idea ("the trend continues") is a standard interpretation taught in every course. The mark is withheld only for generic non-answers.

E10. Draw the line on the truncated axis

Working title: Draw the line on the truncated axis
Target skill: "use of the least squares line of best fit to model and analyse the linear association between two numerical variables"; Outcome 3 — "use appropriate domain and range specifications to illustrate key features of graphs".
What the question does: Supply a scatterplot whose horizontal axis begins at a non-zero value (e.g. 10, not 0) and the equation of the least squares line. Ask the student to draw the line on the plot. A follow-up part asks them to read a predicted value off the drawn line to the nearest unit.
Why it separates: 2017 Exam 2 Q3b.i (1 mark, 26%) — "Many students did not realise that the scale on the horizontal axis started at 10 and not 0. It was clear that some students did not take a ruler into the examination." 2020 Exam 2 Q5a (1 mark, 41%) — "It is necessary to calculate appropriate points and plot them in order to get an accurate line. Students should ensure that their line extends across the whole set of axes." 2021 Exam 2 Q4a (1 mark, 46%) — "Students who wrote the endpoints of (1908, 75.256) and (2020, 48.04) on the graph tended to be successful. Many students left the question out entirely." The report also names this as a chronic non-attempt item.
Marks: 2 — one for the line drawn accurately across the full grid, one for the read-off value (consequential on a reasonable line).
Predicted band: Hard (25–39%). Parent items span 26–46%; the truncated axis is what pushes it toward the lower end.
Fairness check: The equation is given, a ruler is standard equipment named in the reports, and accuracy tolerance is generous in VCAA's own guides.

E11. Two lines, two subgroups, one comparison

Working title: Two lines, two subgroups, one comparison
Target skill: "use of the coefficient of determination, r², to assess the strength of the association in terms of explained variation"; "interpretation of the slope and intercepts of the least squares line in the context of the situation being modelled"; "answer statistical questions that require a knowledge of the associations between pairs of variables".
What the question does: Give two least squares equations fitted to the same pair of variables for two subgroups (e.g. two cities), each with its own r², plus the data ranges. Ask the student to use both the slopes and the coefficients of determination to compare the two associations, in context, in two sentences.
Why it separates: 2018 Exam 2 Q3e (2 marks, 2% — the lowest Data analysis figure in the corpus) is this design. It demands two distinct elements for the marks (rate of change and explained variation) while the further-engagement rule punishes any extra statistic. It also lands on the two interpretation templates the reports police most tightly: slope — 2020 Exam 2 Q5d (44%), "Many students gave a response that was close to being correct but failed to reference the 1 cm increase"; r² — 2025 Exam 2 Q5b (44%), "75% of the variation in sale price can be explained by the variation in days".
Marks: 2 — one for a correct comparative slope statement with units and context, one for a correct comparative r² statement in explained-variation language.
Predicted band: Brutal (< 10%). Its direct ancestor ran at 2%; both interpretation templates independently sit at 44%, so requiring both, comparatively, collapses the joint rate.
Fairness check: Both interpretations are named dot points that appear in some form every year. The question supplies both equations and both r² values — there is nothing to compute, only to say correctly.

2. RECURSION AND FINANCIAL MODELLING

Exam 1: Q17–24 (8 marks). Exam 2: 12 marks.

Structural note: this is the hardest Exam 1 block in the current era — mean facility 49.8% (2023),
57.0% (2024), 54.9% (2025), lowest or second-lowest every year. VCAA's own diagnosis:
"Students did not score as highly on questions involving the use of recurrence relations or the
finance solver, or where questions required two or more steps."

2A. Exam 1 — multiple choice (6 ideas)

R1. The line where the rate changed

Working title: The line where the rate changed
Target skill: "use a table to investigate and analyse on a step-by-step basis the amortisation of a reducing balance loan or an annuity, and interpret amortisation tables".
What the question does: Present six consecutive lines of an amortisation table in which the interest rate changed immediately before one of the middle payments. Ask by how many percentage points per annum the rate changed. The student must compute the per-period rate from interest ÷ previous balance on two different lines and annualise both.
Why it separates: 2018 Exam 1 Q23 (29%) is the closest ancestor — "The interest rate for this loan changed immediately before repayment number 28. This change in interest rate is best described as". 2023 Exam 1 Q24 (27%) and 2023 Exam 1 Q23 (37%) are the other two-step table items. The reports name the exact failure: "some students give the per-period rate where the annual is wanted, others give the annual where the per-period is wanted" — named in 2017, 2018, 2019, 2020 (twice), 2021, 2022, 2023, 2024 and 2025.
Marks: 1.
Predicted band: Hard (25–39%). Three chained steps (two rate extractions plus an annualisation) in an item worth one mark, with a natural distractor at each step.
Fairness check: Amortisation-table interpretation is an explicit key skill; every number needed is printed in the table.

R2. Lower rate, worse deal

Working title: Lower rate, worse deal
Target skill: "the difference between nominal and effective interest rates and the use of effective interest rates to compare investment returns and the cost of loans when interest is paid or charged, for example, daily, monthly, quarterly".
What the question does: Offer four investment accounts with different nominal rates and different compounding frequencies, arranged so that the lowest nominal rate is not the best return and the highest compounding frequency is not the best either. Ask which account gives the greatest return over one year.
Why it separates: 2018 Exam 1 Q19 report — "This question required an understanding that both the interest rate and compounding period in combination determine the overall cost of the loan. Students who selected option A seemed to simply select the lowest interest rate without considering the compounding period." 2022 Exam 1 Q21 (33%) is the four-statement version of the same idea. Four effective-rate computations in a one-mark item is exactly the "multiple options being checked" pattern the 2024 report names as the hardest question type.
Marks: 1.
Predicted band: Hard (25–39%). Precedented at 33%; four CAS operations under time pressure with a tempting shortcut that fails.
Fairness check: The effective-rate formula is on the formula sheet and the comparison is the dot point verbatim.

R3. Effective given, nominal wanted

Working title: Effective given, nominal wanted
Target skill: "the difference between nominal and effective interest rates and the use of effective interest rates to compare"; "the future value of a compound interest investment or loan after n compounding periods and its use to solve practical problems".
What the question does: State the effective annual rate for an account compounding weekly, plus a principal and a term in years, and ask for the interest earned. The student must invert the effective-rate formula to recover the nominal rate before using the finance solver.
Why it separates: 2025 Exam 1 Q23 (41%) is precisely this, and the report spells out the three-step solution: "Step 1. The nominal interest rate = nom(4.51,26) = 4.415% p.a. Step 2. Use Finance Solver … Step 3. The interest earned = $6233.89 − $5000". 2019 Exam 1 Q24 is the two-stage sibling. Students who substitute the effective rate directly into the solver get a plausible, slightly-too-large answer — distractor pattern #3, "the value one step before the final answer".
Marks: 1.
Predicted band: Hard (25–39%). Precedented at 41% with five options; four options and a weekly compounding period pull it lower.
Fairness check: The formula sheet gives the effective-rate relation in the direction needed for inversion; solving it is Units 1–2 algebra plus CAS.

R4. Where flat rate overtakes reducing balance

Working title: Where flat rate overtakes reducing balance
Target skill: "use of a recurrence relation to model and compare (numerically and graphically) flat rate, unit cost and reducing balance depreciation of the value of an asset with time".
What the question does: Give an asset depreciated two ways — a stated dollar amount per unit of use with a stated annual usage, and a stated reducing-balance percentage — and ask after how many whole years the unit-cost value will first be lower than the reducing-balance value.
Why it separates: 2025 Exam 1 Q19 (49%) is the flat-rate version; the report gives both routes and notes "n = 5.582. The value using flat rate depreciation will first be lower after 6 years" — an explicit ceiling step, distractor pattern #6. Replacing flat rate with unit cost adds the failure VCAA names repeatedly: "$4.40 was found by ignoring the eight years" (2016 Q6c); "$14 760 was a common incorrect answer, with some students not accounting for 52 weeks per year" (2024 Q5b).
Marks: 1.
Predicted band: Tricky (40–50%). The flat-rate ancestor sits at 49%; the usage-rate conversion costs a few more points without changing the shape.
Fairness check: All three depreciation methods are named key knowledge; the comparison can be done by a short CAS table without any algebra.

R5. Interest-only, then the real repayments start

Working title: Interest-only, then the real repayments start
Target skill: "use of technology with financial modelling functionality to solve problems involving reducing balance loans … including the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan".
What the question does: A loan runs interest-only for the first two years, after which the interest rate changes and the borrower must clear the loan within the original total term. Ask for the new monthly repayment.
Why it separates: 2021 Exam 1 Q24 is the direct ancestor ("Bob decided … to make interest-only repayments for the first two years. After these two years the interest rate changed …"). The 2021 report's verdict on the whole block: "Students did not answer well the questions involving the use of the finance solver or questions involving a change in condition part way through the problem (Questions 19, 20, 21, 23 and 24)" — Q19 44%, Q20 44%, Q21 40%, Q23 31%, Q24 33%. Interest-only adds a second trap: the balance after two years is unchanged, which students who run the solver blindly will not reproduce.
Marks: 1.
Predicted band: Severe (10–24%). Change-of-condition MC items cluster at 31–44%; adding the interest-only constant-balance insight, which the 2017 Exam 1 Q19/Q20 report treats as a separate idea, takes it below 25%.
Fairness check: Interest-only repayments have been examined in Exam 1 (2017 Q19/Q20), Exam 2 (2023 Q7b.i) and NHT (2026 Q7). Two solver operations, both routine.

R6. Which of these is not the same loan

Working title: Which of these is not the same loan
Target skill: "model and analyse growth and decay in financial contexts using a first-order linear recurrence relation of the form u₀ = a, u_{n+1} = R·u_n + d"; "use a rule for the future value of a compound interest investment or loan, or a depreciating asset".
What the question does: Describe one reducing-balance loan in words, then offer four representations — a recurrence relation, a closed-form rule, a finance-solver parameter set, and a table of the first three balances. Ask which one does not describe the loan. The false option is one that would be right for the corresponding flat-rate or annuity-investment model.
Why it separates: Recurrence-versus-rule confusion is named in seven consecutive years: 2016 Q6b, 2017 Q6b, 2019 Q7d ("A few wrote both a recurrence relation and a rule … which could not be accepted"), 2020 Q7d (44%), 2022 Q6c, 2024 Q5c, 2025 Q8a.ii (50%) — "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." 2024 Exam 1 Q17 (36%) is the abstract version. The "not true" stem is a documented Exam 1 device that students routinely answer with the true statement.
Marks: 1.
Predicted band: Hard (25–39%). Four representations to check against one description; the "not" stem is worth several points of facility on its own.
Fairness check: All four representations are explicitly examinable, and the sign convention for the solver is standard. There is exactly one wrong option and it is wrong unambiguously.

2B. Exam 2 — extended response (9 ideas)

R7. Show the recursion across the change

Working title: Show the recursion across the change
Target skill: "demonstrate the use of a recurrence relation to determine the depreciating value of an asset or the future value of an investment or a loan after n time periods for the initial sequence".
What the question does: Give a loan's opening balance, rate and repayment. Using recursion, and showing each step, calculate the balance after the first three repayments, correct to the nearest cent — where the repayment amount increases at the third payment. The student must write out three lines of the form V₁ = 1.00xx × V₀ − d.
Why it separates: "Use recursion to show that" is the second-most complained-about question type in the corpus (26 reports). 2021 Exam 2 Q9a (2 marks, 21%) — "Many students correctly calculated the factor of 1.00425 but either added the $900 or did not show the recursive calculations in full." 2022 Exam 2 Q8a (1 mark, 44%) — "Transcription and rounding errors were common. Rounding to the nearest cent was required." 2024 Exam 2 Q7a (1 mark, 49%) — "Note that an answer of $297 477.4 is not correct to the nearest cent." Changing the repayment at line three defeats copy-the-pattern.
Marks: 2 — one for correct recursive working shown line by line, one for the final value at the nearest cent.
Predicted band: Severe (10–24%). The two-mark ancestor ran at 21%; the mid-sequence change adds a fourth way to fail.
Fairness check: "Use recursion" is a standing VCAA instruction with a published meaning; the arithmetic is three multiplications and three subtractions.

R8. The top-up that clears the loan on time

Working title: The top-up that clears the loan on time
Target skill: "use of technology with financial modelling functionality to solve problems involving reducing balance loans, such as repaying a personal loan or a mortgage, including the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan".
What the question does: A mortgage runs for two years at one rate; the rate then changes and the borrower wants to keep both the repayment amount and the original finishing date. Ask for the one-off extra amount that must be paid into the loan at the moment of the rate change. Requires two finance-solver runs — the actual balance after two years, and the balance that would be needed to amortise at the new rate over the remaining periods — and then their difference.
Why it separates: 2021 Exam 2 Q9b (1 mark, 8%) is exactly this, and VCAA published both solver set-ups: "FV = 146 073.7405 … PV = 153 112.9399 … Extra amount to be added = 153 112.94 − 146 073.74 = 7039.20". 2020 Exam 2 Q11 (2 marks, 19%) is the sibling — "Some students gave the final answer as 17 repayments. This resulted from entering both PV and FV as positive". The item combines two solver runs, a sign convention and a subtraction, with the 2021 report also noting the answer "$7039.2" (one decimal place) as a separate loss.
Marks: 2 — one method mark for either solver run shown with its parameters, one for the correct difference to the nearest cent.
Predicted band: Brutal (< 10%). Its direct ancestor scored 8% for a single mark.
Fairness check: Two routine solver operations with a clear real-world meaning. Sign conventions are examinable and the reports have published this exact set-up, so it is teachable.

R9. The last repayment is slightly different

Working title: The last repayment is slightly different
Target skill: "use of technology with financial modelling functionality to solve problems involving reducing balance loans"; "use a table to investigate and analyse … the payment made, the amount of interest paid, the reduction in the principal and the balance".
What the question does: A loan amortises to a residual balance a few cents either side of zero at the scheduled number of payments. Ask for the value of the final repayment, which will be slightly different from all the others, correct to the nearest cent.
Why it separates: Four separate years, four separate errors. 2021 Exam 2 Q8c (1 mark, 15%) — "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." 2022 Exam 2 Q8c (1 mark, 23%) — "some students did not clearly indicate the small amount less than one dollar as asked." 2023 Exam 2 Q5c (1 mark, 25%) — "Some students added the $0.14 future value instead of subtracting." 2024 Exam 2 Q8 (2 marks, 13%) — "Some students incorrectly used 289 payments to find a final payment of $4.20 (not slightly different)."
Marks: 2 — one for the number of scheduled repayments, one for the adjusted final repayment to the nearest cent.
Predicted band: Severe (10–24%). Four precedents spanning 13–25%, with the add-versus-subtract direction as the dominant error.
Fairness check: Every part is a finance-solver operation; the phrase "slightly different" is a genuine sanity check, not a riddle.

R10. Complete the table, not the solver

Working title: Complete the table, not the solver
Target skill: "use a table to investigate and analyse on a step-by-step basis the amortisation of a reducing balance loan or an annuity, and interpret amortisation tables".
What the question does: Give three lines of an amortisation table and the annual rate. Instruct: using the values in the table, complete the next line — payment, interest, principal reduction and balance — rounding all values to the nearest cent. The next line is the first interest-only repayment, so the payment differs from the pattern above it.
Why it separates: 2025 Exam 2 Q7c (1 mark, 40%) — "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." The assessment guide marks it "Designated rounding question. Accept only this answer." 2023 Exam 2 Q6b (1 mark, 40%) — "Incorrect rounding of values in the calculation sequence was frequently observed. Some students wrote only the repayment amount." 2026 NHT Exam 2 Q7 is the interest-only version. The item punishes students who reach for the solver because it is faster — the opposite of the usual advice.
Marks: 2 — one for the correct interest and payment, one for all four values correct at the nearest cent.
Predicted band: Hard (25–39%). Precedents at 40% each; adding the interest-only switch and a second mark pushes it below.
Fairness check: The relationships (interest = balance × rate; principal reduction = payment − interest) are standard, and the instruction "using the values in the table" is VCAA's own wording.

R11. Show that rate, without the solver

Working title: Show that rate, without the solver
Target skill: "use a rule for the future value of a compound interest investment or loan"; Outcome 3 — "specify the process used to develop a solution to a problem using technology and communicate the key stages of mathematical reasoning".
What the question does: Give a recurrence relation whose multiplier is stated to four decimal places and the compounding frequency. Ask the student to show that the annual interest rate is a stated percentage. The response must display the arithmetic (multiplier − 1) × periods per year × 100 with no CAS syntax anywhere.
Why it separates: 2021 Exam 2 Q8b (1 mark, 27%) — "Students must be aware that any response including CAS syntax/notation cannot attract full marks. In the instance of 1-mark questions (particularly 'show that' questions such as this one), the student may not be awarded the mark." 2023 Exam 2 Q7a (1 mark, 32%) — "(1.0015 − 1) × 52 × 100 = 7.8%. Students needed to show all working steps that led to the given result. A response with 'solve' CAS syntax was not considered appropriate." The distilled show-that rules (07-exam-craft.md §2.3) forbid working backwards from the given value — which is exactly what a solver-first student does.
Marks: 1.
Predicted band: Severe (10–24%). Two precedents at 27% and 32%; a four-decimal multiplier with a non-standard compounding frequency (e.g. fortnightly, 26 periods) drops it further.
Fairness check: Two lines of arithmetic. The show-that conventions are published in every examination report and in the assessment guides.

R12. How much do you need so it never runs out

Working title: How much do you need so it never runs out
Target skill: "simple perpetuity as a special case of an annuity that lasts indefinitely"; "use of technology to solve problems involving annuities including determining the amount to be invested … to provide a regular income".
What the question does: Part (a): given a required monthly payment and an annual rate compounding monthly, determine the principal needed for a perpetuity. Part (b): state the balance of that perpetuity after five years of payments. Part (c): explain, in one sentence, why the balance does not change.
Why it separates: Perpetuities are named as a persistent weakness in 2015, 2018, 2021, 2022 and 2024. 2021 Exam 2 Q6b (1 mark, 47%) — "The definition of a perpetuity continues to be misunderstood by many students"; the report adds "sometimes a mark is available for students who simply know definitions". 2022 Exam 2 Q7d (1 mark, 49%) — "Many students were unable to provide the definition of a perpetuity. The annuity payment of $5214.28 was the most common mistake." 2023 Exam 1 Q24 (27%) is the Exam 1 version. The three-part structure means a student who does not know the definition loses all three marks.
Marks: 3 — one for the principal, one for the unchanged balance, one for the explanation (payment equals the interest earned each period).
Predicted band: Hard (25–39%) for all three marks. Each part individually sits near 47–49%; requiring all three, where they share one misconception, lands in the low 30s.
Fairness check: Perpetuity is a named dot point; the condition "payment = interest per period" is standard content that the study design defines explicitly as a special case of an annuity.

R13. Why the advertised rate is lower

Working title: Why the advertised rate is lower
Target skill: "the concepts of financial mathematics including simple and compound interest, nominal and effective interest rates"; Outcome 2 — "interpret and report the results … in terms of the context under consideration".
What the question does: Give an advertised (nominal) annual rate and the computed effective rate for an account compounding fortnightly, and ask the student to explain in simple terms why the effective rate is higher, referring to the context.
Why it separates: 2024 Exam 2 Q6b (1 mark, 26%) — "It does not take into account the fortnightly compounding. This question was not answered well. Many students had some understanding of the effective interest rate and could even define it, but could not explain in simple terms why the nominal rate was lower." The assessor professional-learning note is explicit: the mark went to responses "including that the effective is based on compounding periods"; a response that merely defined effective interest did not get it. This is the pure "definition-instead-of-explanation" failure, item 14 in the ranked complaints.
Marks: 1.
Predicted band: Hard (25–39%). Precedented at 26%; specifying "in the context of this account" gives a slight lift.
Fairness check: One sentence, no computation, and the required idea is the defining distinction between nominal and effective rates.

R14. Per year, per unit, per what

Working title: Per year, per unit, per what
Target skill: "use of a recurrence relation to model and compare … unit cost … depreciation of the value of an asset with time"; "use of the rules for the future value of an asset after n depreciation periods".
What the question does: An asset depreciates by a fixed amount per unit of use. The usage rate changes after year three. Part (a): complete a recurrence relation in the boxes, in terms of the number of units of use, not years. Part (b): determine the value after five years.
Why it separates: 2021 Exam 2 Q7c (1 mark, 21%) — "The correct numbers were 12 000 and 0.05. Many students did not recognise that the value was of the machine in terms of the number of cups of coffee made." Rate-conversion failures are named in nine consecutive years, with the specific unit-cost variants "$4.40 was found by ignoring the eight years" (2016 Q6c) and "some students not accounting for 52 weeks per year" (2024 Q5b). Changing the usage rate mid-model kills the "one multiplier, five times" shortcut.
Marks: 2 — one for a recurrence relation indexed correctly on units of use, one for the value after five years.
Predicted band: Severe (10–24%). The one-mark ancestor sat at 21%; the change of usage rate adds the "change in condition part way through" mechanism VCAA names as its hardest type.
Fairness check: Unit-cost depreciation and its recurrence form are both explicit dot points; the box-completion format is VCAA's own.

R15. What the loan actually cost

Working title: What the loan actually cost
Target skill: "the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan".
What the question does: A loan is repaid at one rate for a stated number of years and at a different rate thereafter, with the repayment adjusted at the change so the original term is preserved. Ask for the total cost of the loan and the total interest paid, both to the nearest cent.
Why it separates: 2024 Exam 2 Q8 (2 marks, 13%) — "Total cost = $884 633.62. Number of payments = 288. Many students found the 288 but left the total cost blank. Some students misunderstood the meaning of the total cost of the loan." 2022 Exam 2 Q8d (1 mark, 19%) — "This was a challenging final question for the majority of students." The item is mechanism 1 (two distinct elements) layered on mechanism 2 (re-scan after a change): the two repayment amounts must be counted over the correct number of periods each, and total interest is total cost minus principal — a relationship not on the formula sheet.
Marks: 2 — one for a correct total of all repayments across both phases, one for the total interest.
Predicted band: Brutal (< 10%). The single-phase ancestor scored 13% for two marks; splitting the term into two phases with different repayments adds a counting error to a definitional one.
Fairness check: "Total interest paid and the total cost of the loan" is quoted verbatim from the study design. Every value is obtainable from two solver runs and a multiplication.

3. MATRICES

Exam 1: Q25–32 (8 marks). Exam 2: 12 marks.

3A. Exam 1 — multiple choice (6 ideas)

M1. When the roster comes back around

Working title: When the roster comes back around
Target skill: "binary and permutation matrices, and their properties and applications"; "elementary matrix operations (… product and power)".
What the question does: Give a permutation matrix applied to a weekly roster of six tasks, described in words or as a diagram. Ask for the number of weeks after which every task returns to its original position — i.e. the least n with Pⁿ = I. Construct the permutation from two disjoint cycles of different lengths (e.g. 2 and 3) so the answer is their lowest common multiple, not either cycle length.
Why it separates: 2025 Exam 2 Q14c (1 mark, 11%) — "Order of the activities rotates on a four-day cycle. There are 10 cycles completed in the 40-day program. Only a small proportion of students showed an understanding of the effect of the identity matrix." 2024 Exam 1 Q29 (49%) required four successive permutation applications. Disjoint cycles of unequal length mean the two obvious cycle lengths are both available as distractors.
Marks: 1.
Predicted band: Hard (25–39%). A student who computes P², P³, P⁴… on CAS gets there; one who reasons from a single cycle does not.
Fairness check: Permutation matrices and their properties are named key knowledge, and the whole thing is four CAS multiplications.

M2. Which Leslie population survives

Working title: Which Leslie population survives
Target skill: "use of the matrix recurrence relation S₀ = initial state matrix, S_{n+1} = L·S_n where L is a Leslie matrix … to generate a sequence of state matrices"; Outcome 3 — "produce results, using a technology, which identify examples or counter-examples for propositions".
What the question does: Give a 3 × 3 Leslie matrix with one birth rate left as a parameter k, plus an initial state matrix. Offer four statements about the long-term total population for different values of k (increases, decreases, stabilises) and ask how many are true.
Why it separates: The GM sample Exam 1 Q29 uses exactly this design ("When k = 2, the population will decrease in the long term. / When k = 3, the population will increase in the long term."). 2025 NHT Exam 1 Q27's published solution shows the required method: "Set up a sample calculation L × Sₙ … By cycle number 25 the population will be 359, and by cycle 50 it will be 334. Total population is expected to decrease in the long term." The "how many statements are true" format is "an entirely off-by-one distractor set by construction"; at 2022 Exam 1 Q21 the key scored 33% against 45% for one-fewer, and at 2023 Q30 it was 39% against 42%.
Marks: 1.
Predicted band: Severe (10–24%). Four separate long-run CAS investigations in a one-mark item, in the highest-inversion-rate MC format in the corpus.
Fairness check: Long-run behaviour of a Leslie model is precedented (2025 NHT Exam 1 Q27) and the formula sheet supplies the recurrence. The question asks about total population direction, never a "steady state" — a Leslie matrix is not regular, so steady-state language would be out of scope (see §6).

M3. Where did next week's group come from

Working title: Where did next week's group come from
Target skill: "use of transition diagrams, their associated transition matrices and state matrices to model the transitions between states in discrete dynamical situations".
What the question does: Give a three-state transition diagram (not matrix) and an initial state matrix. Ask: of those in state B next week, what percentage were in state A this week? The student must compute the inflows into B from all three states, then express one as a percentage of that total.
Why it separates: This is the hardest routine mark in the topic. 2023 Exam 1 Q32 (18%, with a distractor at 54%) — "A common error in this question was to use the 0.3 (t₃₃) from the transition diagram." 2021 Exam 2 Q3c (1 mark, 17%) — "85% taken directly from the transition matrix was a very common incorrect answer." 2016 Exam 2 Q3d (1 mark, 17%) — "Common incorrect answers were 65% and 94%." 2018 Exam 2 Q3d (1 mark, 17%). Four independent instances, all at 17–18%.
Marks: 1.
Predicted band: Severe (10–24%). Four precedents, all between 17% and 18%, and the diagram form forces an extra translation step.
Fairness check: Backward percentages are an application of the transition model, not a new technique; every proportion needed is on the diagram.

M4. The state matrix you were given is S₁

Working title: The state matrix you were given is S₁
Target skill: "use of the matrix recurrence relation S₀ = initial state matrix, S_{n+1} = T·S_n + B to extend modelling to populations that include culling and restocking".
What the question does: Give T, B, and a state matrix explicitly labelled S₁ (not S₀), then ask for a named element of S₄. The student must apply the recurrence three times, not four, and must apply B after each transition rather than once at the end.
Why it separates: Two documented errors, both catastrophic. Off-by-one: 2014 Exam 2 Q2d — "The power by which the transition matrix must be raised is one less than the number of the required state matrix"; 2010 Exam 2 Q2b — "the initial state matrix was given as S₁ rather than S₀ and so S₂ = T · S₁"; 2008 Exam 2 Q2a.i — "Many students renamed the state matrix S₁ as S₀ and then squared the transition matrix." B-placement: 2008 Exam 2 Q3b — "The added column matrix at each year is applied after each transition matrix is applied. Consequently, simply squaring the transition matrix is not appropriate." 2024 Exam 1 Q32 (33%) is the reverse-iteration version of the same trap.
Marks: 1.
Predicted band: Hard (25–39%). Precedented at 33%; the two errors have dedicated distractors that are both easy to generate.
Fairness check: The recurrence is on the formula sheet and the labelling of the given matrix is unambiguous. Three CAS steps.

M5. Four products, four different questions

Working title: Four products, four different questions
Target skill: "elementary matrix operations (sum, difference, multiplication of a scalar, product and power)"; "use of matrices to represent numerical information presented in tabular form".
What the question does: Give a table of quantities and a table of unit costs as two matrices, plus a summing matrix of ones. Ask which matrix expression gives one named total (e.g. total cost for one particular department). Every distractor is a valid, defined product that answers a different legitimate question — the total across departments, the per-item total, the grand total.
Why it separates: This is VCAA's own design, published in full. 2022 Exam 1 Q8 (31%) report: "Option A is correct. Option B finds the total time for the four different departments. Option C finds the time for each of Henry, Irvine or Jean to service the laptops separately from the desktops. Option D finds the time for each … at the four different departments. Option E finds the total time altogether." 2018 Exam 1 Q22 (29%) and Q23 (29%) are siblings. Order-of-multiplication errors are the single most-repeated Matrices complaint (§3.1 of 04-matrices.md).
Marks: 1.
Predicted band: Hard (25–39%). Precedented at 29–31%; four options rather than five raises it slightly.
Fairness check: No distractor is undefined or ill-formed — each is honest mathematics answering a different question, which is precisely the discrimination the dot point asks for.

M6. The value that kills the inverse

Working title: The value that kills the inverse
Target skill: "the inverse of a matrix and the condition for a matrix to have an inverse, including determinant".
What the question does: Give a 2 × 2 matrix containing a parameter g in two positions, so that det = 0 yields a quadratic with two roots. Ask for the value(s) of g for which the matrix has no inverse. Options: one root, the other root, both roots (key), and "no value of g".
Why it separates: 2024 Exam 1 Q27 (50%) is the linear version — "The inverse does not exist if the determinant = 0. Equate the determinant to zero and solve for g." 2022 Exam 2 Q2b.ii (1 mark, 21%) — "A unique solution exists, as the determinant is not zero. This question was not answered well, with few students able to demonstrate how to interpret the determinant value. The response also needed both parts." 2020 Exam 2 Q2a (1 mark, 37%) — "Some stated that the determinant had to be greater than 0." The quadratic form produces two roots, and the single-root distractors are exactly what a hurried solve produces.
Marks: 1.
Predicted band: Tricky (40–50%). The linear ancestor sits at 50%; the second root costs perhaps 8–10 points.
Fairness check: Both the determinant and the inverse of a 2 × 2 are on the formula sheet; solving a quadratic is Units 1–2 assumed knowledge and is CAS-native. Critically, the question must not frame this as solving simultaneous equations — the matrix-inverse method for systems of linear equations was removed from the 2023+ design.

3B. Exam 2 — extended response (9 ideas)

M7. A Leslie population that is being culled

Working title: A Leslie population that is being culled
Target skill: Recombination. "use of the matrix recurrence relation … S_{n+1} = L·S_n where L is a Leslie matrix" × "use matrix recurrence relations to model populations with culling and restocking".
What the question does: Give a 3 × 3 Leslie matrix for a pest species plus a culling matrix B with negative entries applied to the two older age groups only. Part (a): explain what the zero in a named position of B means in context. Part (b): showing relevant calculations, determine the total population after three cycles, rounded as instructed. Part (c): state whether the cull is sufficient to reduce the total population below a stated threshold, justified by the calculation.
Why it separates: VCAA has combined T with B many times but has never published an L + B item — so there is no rehearsed template. All three parent failure modes are live: B applied before T rather than after (2008 Exam 2 Q3b, 2009 Exam 2 Q4a — "Students should note that L₃ ≠ T² · L₁ − [5;7]"); the total-population interpretation of a Leslie model (2026 NHT Exam 2 Q11c, VCAA's own two-mark answer is a total-then-percentage chain); and rounding at intermediate steps (2025 Exam 2 Q13b.i, 1 mark, 9% — "Rounding should only be done at the final step").
Marks: 3 — one for the contextual interpretation of the zero, one for correct iteration with working shown, one for the threshold judgement with its supporting number.
Predicted band: Brutal (< 10%) for all three marks. Its nearest ancestors (2021 Exam 2 Q4, 2 marks, 10%; 2025 Exam 2 Q13b.ii, 2 marks, 17%) are already there, and this has one more dependent part.
Fairness check: The key skill "use matrix recurrence relations to model populations with culling and restocking" is not restricted to transition matrices, and both S_{n+1} = T·S_n + B and S_{n+1} = L·S_n are printed on the formula sheet. Nothing here asks for equilibrium or steady state, which a Leslie matrix does not have.

M8. Read the first row, then follow the survivors

Working title: Read the first row, then follow the survivors
Target skill: "use of transition diagrams, their associated transition matrices and state matrices to model … practical situations such as the modelling and analysis of an insect population comprising eggs, juveniles and adults".
What the question does: Part (a): interpret each of the three elements of the first row of a Leslie matrix in context, for one mark — so all three must be right. Part (b): determine what percentage of individuals in their first age group survive to their third age group, which requires multiplying two sub-diagonal survival rates.
Why it separates: 2026 NHT Exam 2 Q11a/b is this design; VCAA's guide answers are "Female(s) are expected to have 0 offspring in their first year of life, 2 offspring in their second year of life, 1 offspring in their third year of life" and "0.35 × 0.22 = 0.077 → 7.7%". Part (a) is mechanism 1 (three distinct elements for one mark) applied to a structure that is not on the formula sheet — the sheet gives S_{n+1} = L·S_n and nothing about what the rows mean. 2024 Exam 2 Q11a.i (1 mark, 24%) shows the cohort's grip on Leslie structure is weak. Part (b) is a chained product that students consistently read as a single element (2021 Exam 2 Q2c, 1 mark, 21% — "Most students appeared not to interpret this question correctly").
Marks: 2 — one for all three first-row interpretations, one for the chained survival percentage.
Predicted band: Severe (10–24%). The all-or-nothing interpretation mark plus a two-step product; the 2024 Leslie-structure item ran at 24%.
Fairness check: The insect-population example is quoted in the study design itself, and interpretation in context is Outcome 2 key skill. The percentages are supplied.

M9. The ranking that needs the second step

Working title: The ranking that needs the second step
Target skill: "communication and dominance matrices and their use in analysing communication systems and ranking players in round-robin tournaments".
What the question does: Give the results of a five-player round robin as a directed graph. Part (a): construct the one-step dominance matrix. Part (b): the one-step row sums produce a tie for first place; determine the overall ranking using two-step dominance, stating which matrix sum was used.
Why it separates: 2022 Exam 2 Q4b (2 marks, 39%) — "Ali with 280 votes. Many students showed no working for this question. Whole number rounding applied." The dominance construction (A + A²) is on the "not on the formula sheet" list, and the 2006 report names "explaining two-step (or more) dominance" as an area of weakness. Presentation failures compound it: arrows on every edge of a directed graph are required (2024 Exam 2 Q11a.i, 24% — "All edges required arrows otherwise it is no longer a directed graph") and matrix brackets are mandatory (2007 report).
Marks: 2 — one for the correct one-step dominance matrix, one for the ranking with the two-step method identified.
Predicted band: Hard (25–39%). A tie at the top is the structural feature that stops a one-step answer scoring; 39% is the observed rate for the comparable working-required item.
Fairness check: Dominance matrices and their ranking application are named key knowledge; the graph is unambiguous and the arithmetic is two CAS operations.

M10. Where the restocked animals came from

Working title: Where the restocked animals came from
Target skill: Recombination. Backward-percentage reasoning × "use matrix recurrence relations to model populations with culling and restocking".
What the question does: A three-region wildlife model runs S_{n+1} = T·S_n + B, where B adds a fixed number of animals to one region each year. Ask: of the animals in that region at the start of next year, what percentage were in a different named region this year? The restocked animals are in the denominator but in none of the transition inflows, so the naive proportion is wrong.
Why it separates: Backward percentage alone runs at 17–18% four separate times (2023 Exam 1 Q32, 18%; 2021 Exam 2 Q3c, 17%; 2016 Exam 2 Q3d, 17%; 2018 Exam 2 Q3d, 17%). Adding B attacks the second-most-cited counting failure: 2025 Exam 2 Q12b (1 mark, 13%) — "Many students answered 115 by including the children who had left the centre"; 2024 Exam 2 Q12c (1 mark, 12%) — "Some students did not subtract the departed workers and answered 190." VCAA has not yet combined a backward percentage with a + B model.
Marks: 1.
Predicted band: Brutal (< 10%). The unaugmented version has never exceeded 18%; the extra denominator term removes the last shortcut.
Fairness check: Both components are examined most years. The B matrix is given explicitly and its meaning is stated in the stem, so the denominator is fully determined.

M11. The peak is not the long run

Working title: The peak is not the long run
Target skill: "use of the matrix recurrence relation … S_{n+1} = T·S_n + B"; "informal identification of the equilibrium state matrix in the case of regular transition matrices".
What the question does: Give a T + B model whose sequence for one state rises for three or four periods and then falls back toward its long-run value. Ask for the maximum number in that state at any time during the modelled period, and the period in which it occurs. The long-run value is deliberately lower than the peak.
Why it separates: 2022 Exam 2 Q3c (1 mark, 19%) — "219. The maximum occurred three days after the original date. Many students incorrectly assumed that the long run number was required." 2017 Exam 2 Q3d — "239 was the most common incorrect response from the steady state matrix." 2013 Exam 2 Q2c.ii — "It was not sufficient to assume that the number of adult trout was decreasing without checking the values given by S₂ and S₃." Requiring the period as well as the value makes it a two-element mark (mechanism 1).
Marks: 2 — one for the maximum value at the stated rounding, one for the period in which it occurs.
Predicted band: Severe (10–24%). The one-element ancestor ran at 19%; adding the period keeps it there.
Fairness check: Entirely a matter of generating the sequence on CAS and reading it — which is Outcome 3 key skill "produce tables of values … using technology". The word "maximum" is unambiguous.

M12. The ones who left do not count

Working title: The ones who left do not count
Target skill: "use of the matrix recurrence relation S₀ = initial state matrix, S_{n+1} = T·S_n"; interpretation in context.
What the question does: A four-state transition matrix has a 1 on the leading diagonal for a "departed" state. Part (a): state the long-run number in one of the active states, and explain the value. Part (b): state the total number still active after ten periods — which requires excluding the absorbing state from the sum.
Why it separates: 2024 Exam 2 Q12b (1 mark, 24%) — "Students who recognised that the 1 in the transition matrix would result in an absorbing state performed well on this question"; only 24% did. 2024 Exam 2 Q12c (1 mark, 12%) — "Some students did not subtract the departed workers and answered 190." 2025 Exam 2 Q12b (1 mark, 13%) — "Many students answered 115 by including the children who had left the centre." Three separate items across two years, all at 12–24%, all the same error.
Marks: 2 — one for the long-run value with the absorbing-state explanation, one for the corrected active total.
Predicted band: Severe (10–24%). Three precedents at 12%, 13% and 24%; requiring the explanation as well pins it low.
Fairness check: Absorbing states arise naturally from a transition matrix with a 1 on the diagonal; nothing beyond generating Sₙ and reading it correctly is required.

M13. Do not round the people

Working title: Do not round the people
Target skill: Outcome 3 — "the difference between exact numerical and approximate numerical answers when using technology to perform computation, and rounding to a given number of decimal places or significant figures"; matrix recurrence relations.
What the question does: A three-stage model runs for several periods; an intermediate state value is a non-integer number of individuals. Part (a): find the expected number in one stage after four periods, rounded to the nearest whole number. Part (b): use the unrounded value from the same calculation to find a percentage, correct to one decimal place — where rounding at part (a) shifts the part (b) answer out of tolerance.
Why it separates: 2025 Exam 2 Q13b.i (1 mark, 9%) — "Rounding should only be done at the final step. The expected number of children should not have been rounded to 10." Q13b.ii (2 marks, 17%) — "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." The same complaint appears in the 2006, 2007 and 2009 general reports in identical words: "Rounding off should only occur at the final answer stage in a question, not halfway through the calculation." 2017 Exam 2 Q2c — "students … often ended up with a total of 749 due to rounding."
Marks: 2 — one for the rounded whole number, one for the percentage computed from the unrounded value with working shown.
Predicted band: Brutal (< 10%). The parent one-mark item ran at 9%; this version makes the trap structural rather than incidental.
Fairness check: The examination's own standing instruction ("you should only round your answer when instructed to do so") governs this exactly, and part (a) is a genuine rounding instruction, not a trap — part (b) simply asks for a fresh calculation. The reports have warned about it for twenty years.

M14. Three different percentage changes, one matrix

Working title: Three different percentage changes, one matrix
Target skill: "matrix arithmetic: … types of matrices (row, column, square, diagonal …) and elementary matrix operations (… multiplication of a scalar, product …)"; "use of matrices to represent numerical information presented in tabular form".
What the question does: A 3 × 4 matrix of stock levels must be adjusted by a different percentage change for each of the three rows — one increase, one decrease, one unchanged. Ask the student to write down the matrix that must pre-multiply the stock matrix to produce the new values, and to state its order and its type.
Why it separates: 2020 Exam 2 Q1e (1 mark, 24%) — "Most students did not recognise the need to form a 3 × 3 matrix and give a row or column matrix. Many students also were not able to obtain the correct multiplying factors from the percentage changes." 2021 Exam 2 Q1b (1 mark, 48%) — "The correct answer was 1.05. … Students often responded with 0.05." 2018 Exam 2 Q2b (1 mark, 40%) — "Numbers such as 0.04, −0.01 and −0.02 were often included in students' answers." 2025 Exam 2 Q11c (1 mark, 44%) — "Many responses indicated a poor understanding of what a diagonal matrix is." Four failure modes: the shape, the pre- vs post-multiplication side, the 1 + r factor, and the name.
Marks: 2 — one for the correct 3 × 3 diagonal matrix with 1 + r entries, one for its order and type correctly stated.
Predicted band: Hard (25–39%). The shape mark alone ran at 24%; adding the order/type mark gives a partial-credit route that lifts the full-marks rate into the 30s.
Fairness check: Diagonal matrices are a named type in the key knowledge, and the percentage-to-multiplier conversion is Units 1–2 assumed knowledge.

M15. Name the two-step routes

Working title: Name the two-step routes
Target skill: "communication and dominance matrices and their use in analysing communication systems".
What the question does: Give a five-person communication matrix. Part (a): compute a named element of M + M² and explain what it counts. Part (b): list the actual two-step routes by which one named person can pass a message to another — not how many, but which.
Why it separates: 2021 Exam 2 Q2c (1 mark, 21%) — "The correct answer was Alex – Brie – Dex and Alex – Elena – Dex. This question was not answered well. Most students appeared not to interpret this question correctly." 2023 Exam 2 Q10b.ii (1 mark, 25%) — "Managers can directly communicate with directors but not indirectly via one other person. Students who dealt with each element separately tended to write more accurate responses." The interpretation of M² as a count of two-step links is not on the formula sheet, and the "list them" form defeats the memorised sentence.
Marks: 2 — one for the element value with its meaning in context, one for both routes listed correctly.
Predicted band: Severe (10–24%). Both ancestors sit at 21% and 25%; requiring both parts together lands below.
Fairness check: Communication matrices and their powers are explicit key knowledge; part (b) can be answered straight off the given matrix by inspection.

4. NETWORKS AND DECISION MATHEMATICS

Exam 1: Q33–40 (8 marks). Exam 2: 12 marks.

Structural note: this topic owns the corpus floor. Of the ten lowest Exam 2 percentages in the four
current content areas, six are Networks: 2% (2022 Q3d), 3% (2019 Q3e), 6% (2021 Q3b.ii),
7% (2024 Q15e), 9% (2021 Q4c), 9% (2023 Q14e). Five of those six are crashing or
improve-the-flow items.

4A. Exam 1 — multiple choice (6 ideas)

N1. The activity worth crashing

Working title: The activity worth crashing
Target skill: "use of crashing to reduce the completion time of the project or task being modelled".
What the question does: Give an activity network with two paths of nearly equal length. State that exactly one activity may be shortened, by up to three time units, at a fixed cost per unit. Ask by how much the project completion time can be reduced. The critical activity chosen can only be usefully shortened until the second path becomes critical, so the answer is smaller than the permitted reduction.
Why it separates: 2006 Exam 1 Q9 (17%) — "When choosing an activity to be crashed on the critical path, care needed to be taken to ensure that crashing this activity did not create a new critical path. This restricted the amounts by which critical path activities C, E, H or J could be crashed to one hour. … Note that D is a dummy activity, so it could not be crashed." 2020 Exam 1 Q10 (24%) is the resource-conflict version; 2013 Exam 1 Q8 (23%) the cost version. The 2007 report names the required insight as an area of weakness in its own right: "recognising that crashing activities on a critical path may create a new critical path".
Marks: 1.
Predicted band: Severe (10–24%). Three precedents at 17%, 23% and 24%; the full permitted reduction is always available as a distractor and always attracts the majority.
Fairness check: Crashing is an explicit dot point; the network is small enough to re-scan by hand in under a minute.

N2. Redraw it before you count

Working title: Redraw it before you count
Target skill: "the conventions, terminology, properties and types of graphs … and Euler's formula for planar graphs and its application".
What the question does: Present a graph drawn with three crossing edges that is planar. Ask for the number of faces. Options: the number of enclosed regions in the drawing as printed, the correct planar face count (key), that count minus the infinite face, and "the graph is not planar".
Why it separates: The single most reliable Exam 1 collapse in the topic. 2008 Exam 1 Q7 (11%) — "By not redrawing the graph in planar form before trying to determine the number of faces, 72 per cent of students erroneously obtained the answer 9." 2021 Exam 1 Q3 — key 34%, distractor 61%: "It would appear that many students did not redraw the graph in its planar form before counting the faces." 2022 Exam 1 Q4 (21%) — "The given graph is planar. It can be redrawn with no crossing edges." Three years, three collapses, same cause.
Marks: 1.
Predicted band: Severe (10–24%). Precedented at 11%, 21% and 34%; the as-printed count is the strongest distractor in the corpus.
Fairness check: Euler's formula is on the formula sheet and provides an independent check (v + f = e + 2). The graph genuinely is planar and can be redrawn in well under a minute.

N3. What must be true of this graph

Working title: What must be true of this graph
Target skill: "the conventions, terminology, properties and types of graphs; edge, face, loop, vertex, the degree of a vertex, isomorphic and connected graphs"; "Eulerian trails and circuits, and Hamiltonian paths and cycles".
What the question does: Describe a simple connected graph only by its vertex count and degree sum (no diagram). Offer four statements — about the number of odd-degree vertices, the existence of an Eulerian trail, the existence of a Hamiltonian path, and the number of edges — of which exactly one must be true for every such graph. The other three are true of some drawings only.
Why it separates: 2024 Exam 1 Q38 (18%, strongest distractor 33%) is exactly this. The report's method: "The most efficient way to solve this question to draw some examples of possible graphs … Check this diagram against each statement." This is documented distractor pattern #13 — "necessary but not sufficient / possible but not always true" — and the key finished third in the option ranking. The handshake relation (sum of degrees = 2e) and the odd-vertex conditions are both on the "not on the formula sheet" list.
Marks: 1.
Predicted band: Severe (10–24%). Precedented at 18%; students who test one drawing rather than several get the wrong answer with confidence.
Fairness check: Every statement can be settled by sketching two or three graphs, which is what the report itself recommends. All terminology is study-design vocabulary.

N4. One more worker changes the allocation

Working title: One more worker changes the allocation
Target skill: "the matching problem and the Hungarian algorithm"; "determination of the optimum assignment(s) of people or machines to tasks by inspection or by use of the Hungarian algorithm".
What the question does: Give a 4 × 4 cost table with a stated optimal total. A fifth worker is then added with a stated cost row, and one task may now be left unassigned. Ask for the new minimum total.
Why it separates: 2024 Exam 1 Q39 (19%, strongest distractor 45%) — the report works the whole two-stage solution: "Step 1. Determine the initial allocation. … Total time = 9 + 7 + 11 + 16 = 43 hours. Step 2. Include Edgar and develop new allocation." 2018 Exam 1 Q8 is the perturbation ancestor ("This allocation will achieve the minimum total completion time if the value of k is at least"). The failure mode is documented: students re-use the old allocation and swap only the obviously cheaper cell (2014 Exam 2 Q2c.ii — "A common incorrect answer was 21, the total of all the numbers on Table 3").
Marks: 1.
Predicted band: Severe (10–24%). Precedented at 19%; the arithmetic is easy but the re-optimisation is the step almost nobody takes.
Fairness check: The Hungarian algorithm is named key knowledge and a 5 × 4 table is solvable by inspection within the time available.

N5. The cut that does not separate

Working title: The cut that does not separate
Target skill: "solution of small-scale network flow problems by inspection and the use of the 'maximum-flow minimum-cut' theorem to aid the solution of larger scale problems"; "use of networks to model flow problems: capacity, sinks and sources".
What the question does: Show a flow network with two sources and one sink, with four labelled cuts drawn on it. One cut has the lowest capacity but fails to separate both sources from the sink; another is the true minimum cut. Ask which cut determines the maximum flow.
Why it separates: 2014 Exam 1 Q9 report — "A flow network with two sources (two car parks) and a single sink (the exit) was provided. While Cut C had the minimum capacity, neither it nor Cut A separated both sources (car parks) from the sink. Thus, neither of these cuts could be used to determine the minimum flow." 2012 Exam 1 Q7 (23%) — "Line 1 does not separate the source from the sink." 2008 Exam 1 Q6 (35%) — "It was surprising that 39 per cent of students apparently did not understand this key idea." Add the wrong-way edge rule (2007 Exam 2 Q3a — "The edge with the 10 should not have been counted as its flow was in the reverse direction") and there are two independent screens.
Marks: 1.
Predicted band: Hard (25–39%). Precedents at 23% and 35%; four options rather than five lifts it into the low 30s.
Fairness check: The cuts are drawn for the student; the separation requirement is the definition of a cut, and multi-source networks are precedented in both exams.

N6. Which network is that table

Working title: Which network is that table
Target skill: "construction of an activity network from a precedence table (or equivalent) including the use of dummy activities where necessary".
What the question does: Give a precedence table containing two activities that each require the same three immediate predecessors plus one that differs, so a dummy is unavoidable. Offer four activity networks and ask which one correctly represents the table. Distractors omit the dummy, reverse its direction, or place it between the wrong pair.
Why it separates: 2024 Exam 1 Q40 (32%) — the report checks every option: "Option A has one of these activities as G and the other as N. The table does not allow for the three-step path to G. Incorrect." 2025 Exam 1 Q40 (39%) — "It requires a dummy activity between the end of B and start of G." Dummy-activity errors are documented across nine separate reports; 2024 Exam 2 Q15c (1 mark, 10%) — "The dummy was often drawn without an arrow or label." Direction matters and half the cohort does not check it.
Marks: 1.
Predicted band: Hard (25–39%). Two precedents at 32% and 39%; forcing the dummy's direction to discriminate holds it near the lower figure.
Fairness check: Dummy activities are named in the study design, and each option can be verified against the table row by row.

4B. Exam 2 — extended response (10 ideas)

N7. Crash it by four weeks, waste nothing

Working title: Crash it by four weeks, waste nothing
Target skill: "use of crashing to reduce the completion time of the project or task being modelled"; Outcome 2 — "apply mathematical processes in non-routine contexts".
What the question does: Five named activities can each be reduced by up to two periods at a uniform cost. Require the project's overall completion time to be reduced by a stated amount at minimum cost, and ask the student to complete a table listing the reduction (0, 1 or 2) for each of the five activities. Two or three distinct optimal schedules exist and all must be accepted.
Why it separates: 2019 Exam 2 Q3e (2 marks, 3%) is this exact item; VCAA published a whole matrix of alternative acceptable schedules that "also reduced the overall time by four weeks and did not involve unnecessary wastage". 2023 Exam 2 Q14e (1 mark, 9%) is the budget-constrained version. 2022 Exam 2 Q2c (1 mark, 27%) the two-activity version. The mechanism is pure re-scan: every reduction changes which paths are critical, and a student who does not redraw or re-scan wastes reductions on paths that are no longer binding.
Marks: 2 — one method mark for any schedule that achieves the target reduction, one for a schedule that achieves it with no wasted reduction.
Predicted band: Brutal (< 10%). Directly precedented at 3%, and the design has not become easier.
Fairness check: The network, the activity durations and the permitted reductions are all supplied; every optimal schedule is accepted. Difficulty comes from the number of re-scans, not from ambiguity.

N8. The cheapest way to save three days

Working title: The cheapest way to save three days
Target skill: "use of crashing to reduce the completion time of the project"; "use of earliest starting times and latest starting times to identify the critical path … and determine the float times".
What the question does: Give a project network plus a table of different per-period crashing costs for six activities. Ask for the minimum additional cost of completing the project three periods earlier than currently possible. The cheapest activities are not all on the critical path, and after the first two reductions a second path becomes critical.
Why it separates: 2024 Exam 2 Q15e (1 mark, 7%) — "Only a small percentage of students were successful on this final question. Those who were successful tended to use a very systematic trial approach … This response shows all the possible options considered before finding the minimum cost." 2021 Exam 2 Q4c (1 mark, 9%) — "Few students recognised the need to reduce A by one week and L by two weeks." 2025 Exam 2 Q18d (1 mark, 21%) — "Reduce: A − 2, H − 1, K − 1. Minimum additional cost = $1400." 2017 Exam 2 Q4c.ii (1 mark, 15%) — "did not realise that activity G also had to be reduced as it was part of a new critical path formed."
Marks: 2 — one for a correct set of reductions achieving the target time, one for the minimum cost.
Predicted band: Brutal (< 10%). Four precedents at 7%, 9%, 15% and 21%; unequal costs, which VCAA has used only twice, push it to the bottom of that range.
Fairness check: All costs and durations are tabulated. The systematic trial approach VCAA itself endorses will find the answer; it simply takes discipline.

N9. Which pipe, and how big

Working title: Which pipe, and how big
Target skill: "use of networks to model flow problems: capacity, sinks and sources"; "the flow problem, and the minimum cut/maximum flow theorem".
What the question does: After establishing the maximum flow, state that it can be increased by enlarging one edge. Ask the student to complete a sentence with boxes: the pipe that should be enlarged runs from vertex ☐ to vertex ☐, and its new capacity should be at least ☐. The correct edge lies on the minimum cut, and the new capacity is set by the second-smallest cut, not by any single edge.
Why it separates: 2022 Exam 2 Q3d (1 mark, 2% — the lowest Networks figure in the corpus) — "Very few students completed the question correctly. Some students listed the vertices correctly, but not the new capacity." 2021 Exam 2 Q3b.ii (1 mark, 6%) — "The correct answer was 780. This question was not answered well." This is mechanism 1 in its purest form: three boxes, one mark, and the third box is the one nobody fills. The second-cut insight requires re-computing all cuts after the hypothetical change.
Marks: 1.
Predicted band: Brutal (< 10%). Two precedents at 2% and 6%.
Fairness check: The network is small, all capacities are shown, and the max-flow min-cut theorem is the named tool. The box format makes the required answer unambiguous.

N10. Which activity, and nothing else

Working title: Which activity, and nothing else
Target skill: "use of earliest starting times and latest starting times to identify the critical path in the network and determine the float times for non-critical activities".
What the question does: Give a network with two critical paths and ask which single activity has the greatest float time. One mark. The instruction asks for the activity only. Two non-critical activities have floats differing by one period, so a student who miscalculates either float names the wrong one — and a student who names the right one but volunteers a wrong float value loses the mark.
Why it separates: 2019 Exam 2 Q3c — "The answer is J. Students should be aware that if they did not leave their answer as J but further engaged with the question and then stated an incorrect float time, they could not be awarded marks." 2025 Exam 2 Q18c (1 mark, 47%) — the assessment guide instruction is "Reject, as further engagement, if incorrect float time given for F." 2021 Exam 2 Q4b (1 mark, 22%) — "Many students did not seem to realise there were two critical paths." 2016 Exam 2 Q3c (37%) — "Activity H. A common incorrect answer was activity M."
Marks: 1.
Predicted band: Hard (25–39%). Precedents span 22–47%; the two-critical-path structure is what moves it toward the lower end.
Fairness check: Forward and backward scanning are explicit dot points and the network is fully labelled. The further-engagement rule is published in every report and in the assessment guides — a student who answers exactly what was asked is safe.

N11. Two roads twice, ending somewhere else

Working title: Two roads twice, ending somewhere else
Target skill: "Eulerian trails and Eulerian circuits: the conditions for a graph to have a Eulerian trail or a Eulerian circuit, properties and applications"; "determination of the shortest path between two specified vertices in a graph, digraph or network by inspection".
What the question does: A graph has four odd-degree vertices. An inspector must travel every edge at least once, starting at a named vertex and finishing anywhere. Part (a): explain why some edges must be repeated. Part (b): complete two box-pairs naming the two edges that will be travelled twice for the minimum total distance. Part (c): state that minimum total distance.
Why it separates: 2023 Exam 2 Q13c (1 mark, 12%) — "vertex L and vertex N / vertex J and vertex M. This question was not answered well. Very few students had both pairs correct." 2018 Exam 2 Q4b (1 mark, 14%) — "The extra two edges (PQ and ST) needed to be added to the sum of all distances (86) on the graph." 2025 Exam 2 Q17b (1 mark, 27%) — 196 m; Q17a (40%) is the odd-vertex explanation. 2012 Exam 2 Q1a.iii — "This question was very poorly answered, with a common incorrect answer of 1180" (the un-augmented total). The pairing choice is a small combinatorial optimisation that students substitute with "repeat the two shortest edges".
Marks: 3 — one for the odd-vertex explanation, one for both repeated edges, one for the total distance.
Predicted band: Severe (10–24%) for all three marks. The middle part alone runs at 12–14%.
Fairness check: The odd-degree condition is core key knowledge and VCAA has set this shape in 2018, 2020, 2023, 2025 and three NHT papers. All edge weights are shown.

N12. One group, one route

Working title: One group, one route
Target skill: "use of networks to model flow problems: capacity, sinks and sources"; "solution of small-scale network flow problems by inspection".
What the question does: After a maximum-flow part, state that a single group must travel together along one route from source to sink, and ask for the largest possible group. The answer is the largest bottleneck over all single paths — necessarily smaller than the maximum flow, and usually smaller than the largest first edge.
Why it separates: 2007 Exam 2 Q3c — answer 7; "A common incorrect answer was 8." 2017 NHT Exam 2 Q3c and 2024 NHT Exam 1 Q39 are the same design. The failure is a category error: students reuse the max-flow answer they have just computed, exactly as at 2018 Exam 2 Q1b (1 mark, 32%) — "The answer of 9 from part a. was often repeated." The 2007 report also records the reverse sanity failure: "Minimum cut ⇒ maximum flow. Despite this, some answers given here were greater than the answer for Q3a."
Marks: 1.
Predicted band: Severe (10–24%). The dependency on an earlier part makes the wrong answer maximally available, and no precedent for this sub-type has exceeded 32%.
Fairness check: "By inspection" is the study design's own phrase for small flow networks; the graph is small enough to enumerate paths. The stem makes the single-route constraint explicit.

N13. Draw a cut of exactly this size

Working title: Draw a cut of exactly this size
Target skill: "use of networks to model flow problems: capacity, sinks and sources"; "the minimum cut/maximum flow theorem".
What the question does: Ask the student to draw, on the supplied directed network, a cut passing through a stated number of edges whose capacity equals a stated value, and to label it Cut 2. One of the edges it must cross runs backwards, contributing zero.
Why it separates: 2026 NHT Exam 2 Q14b is this item and the guide's marking note is blunt: "MUST be labelled Cut 2". 2018 NHT Exam 2 Q3b is the ancestor. Two distinct failures: the wrong-way edge, named in eight separate reports (2007 Exam 2 Q3a — "The edge with the 10 should not have been counted as its flow was in the reverse direction"; 2018 Exam 2 Q1a.ii — "Some gave an answer of 14 by not allowing for the 1 against the flow"), and the labelling instruction, which the 2024 report elevates to a general rule: "In two questions students were required to provide a label on a diagram. This direction meant that the label needed to be drawn on the diagram as indicated."
Marks: 1.
Predicted band: Tricky (40–50%). The arithmetic is easy once the backward edge is spotted; the label requirement costs a slice of the cohort that would otherwise score.
Fairness check: The target capacity is given, so the student can verify their own cut. Drawing and labelling on a supplied diagram is standard practice in every recent paper.

N14. The spanning tree and the vertex you can drop

Working title: The spanning tree and the vertex you can drop
Target skill: "minimum spanning trees in a weighed connected graph and their determination by inspection or by Prim's algorithm"; "use of minimal spanning trees to solve minimal connector problems".
What the question does: Part (a): draw the minimum spanning tree on a supplied weighted graph of eight vertices, where two edges share the same weight so two minimum spanning trees exist (both accepted). Part (b): state how much would be saved if one named leaf vertex were removed from the network entirely — which requires recomputing the tree on the remaining seven vertices, not just deleting one edge.
Why it separates: 2015 Exam 2 Q1d.i — "Many students were unable to find this minimum spanning tree. Common incorrect trees excluded PO or KL instead of MN. Some students drew complete circuits." 2012 Exam 2 Q1b.i has the clearest diagnosis in the corpus: "This answer may have been found by starting at the pump and then selecting the shortest edge from only the very last vertex connected, rather than any of the already-connected vertices. … The apparent inability of many students to apply the algorithm for determining a minimal spanning tree is a weakness that needs to be addressed." 2025 Exam 2 Q15d — "Some students included an edge that was not part of the original graph."
Marks: 2 — one for a valid minimum spanning tree using only original edges with no circuits or isolated vertices, one for the saving.
Predicted band: Hard (25–39%). Part (a) alone is usually well done (2025 Q15d ran at 81% for an unweighted spanning tree), but the removal part requires a second, independent tree computation that few will actually redo.
Fairness check: Prim's algorithm is named key knowledge; both optimal trees are accepted; every weight is printed.

N15. The dummy, and what it stops

Working title: The dummy, and what it stops
Target skill: "construction of an activity network from a precedence table (or equivalent) including the use of dummy activities where necessary".
What the question does: Give a partially drawn activity network and a precedence table. Part (a): add the missing dummy activity to the diagram, with its arrow and a label. Part (b): explain, in the context of this project, what would go wrong if the dummy were removed.
Why it separates: 2024 Exam 2 Q15c (1 mark, 10%) — "The dummy was often drawn without an arrow or label. A dashed or solid line was acceptable." The assessment guide: "MUST have correct arrow. Label as 'dummy' or 'd' or 'D'." Part (b) attacks 2012 Exam 2 Q2b — "Many unacceptable answers seemed to be direct excerpts from notes about dummy activities in general, rather than clear explanations of the purpose of the specific dummy activity in the given context. Many students said '… we cannot have parallel activities …' without an explanation of what this meant in the context of the question." That is the bound-reference-copying failure, named in seven reports.
Marks: 2 — one for the dummy drawn with a correct arrow and label, one for a context-specific explanation naming the affected activities.
Predicted band: Severe (10–24%). The drawing mark alone ran at 10%; the explanation mark is where the copied-notes answer dies.
Fairness check: Dummy activities are a named dot point; VCAA accepts dashed or solid lines and several label forms, so only the arrow and the contextual reasoning are actually demanded.

N16. How many workers can get there

Working title: How many workers can get there
Target skill: Recombination. "use of networks to model flow problems: capacity, sinks and sources" × "use of crashing to reduce the completion time of the project or task being modelled".
What the question does: A project network is supplied along with a separate transport network that limits how many workers per day can reach the site. Crashing an activity by one day requires a stated number of extra workers on site that day. Part (a): determine the maximum number of workers that can reach the site each day (a max-flow question). Part (b): determine the greatest reduction in completion time achievable within that worker limit, and name the activities crashed.
Why it separates: VCAA has never combined flow with scheduling, but both halves are its own hardest material: flow max-flow determination at 2021 Exam 2 Q3a (1 mark, 24%) — "Many students did not demonstrate they had tried to find the minimum cut"; 2024 Exam 2 Q14b (1 mark, 29%) — "This was found to be a challenging question by most. Those that were successful usually found a minimum cut after repeated trials." Crashing under a resource constraint is precedented in Exam 1 at 2020 Q10 (24%) — "The project manager asks all the workers assigned to activity H to also work on activity F". Part (b) is fully dependent on part (a), which the reports identify as the structure that converts a hard item into a brutal one.
Marks: 3 — one for the maximum flow, one for a feasible crashing schedule within the worker limit, one for the resulting completion time.
Predicted band: Brutal (< 10%) for all three marks. Both halves sit at 24–29% independently and the second is conditional on the first; the joint rate falls below 10%.
Fairness check: Every technique used — minimum cut, forward/backward scanning, crashing — is a named key skill, and no new mathematics is introduced. The two networks are given separately and the worker requirement is stated as a plain number, so the modelling step is explicit rather than inferred.

5. COVERAGE TABLE

Idea count by topic × predicted band.

Topic	Brutal (<10%)	Severe (10–24%)	Hard (25–39%)	Tricky (40–50%)	Total
Data analysis	2	6	9	1	18
Recursion and financial modelling	2	5	7	1	15
Matrices	3	6	5	1	15
Networks and decision mathematics	4	7	4	1	16
Total	11	24	25	4	64

5.1 The same table, split by paper

	Exam 1 (MC)	Exam 2 (written)	Total
Data analysis	7	11	18
Recursion and financial modelling	6	9	15
Matrices	6	9	15
Networks and decision mathematics	6	10	16
Total	25	39	64

Band	Exam 1	Exam 2
Brutal	0	11
Severe	8	16
Hard	14	11
Tricky	3	1

5.2 Gaps this table makes visible

No brutal Exam 1 ideas, by design. No MC item in the 2006–2026 corpus has ever gone below 10%, and with four options the guessing floor is 25%. Anything predicted brutal in MC would have to rely on ambiguity. See §6.
Tricky is deliberately thin (4 of 64). A tricky item is a coin flip; it separates weakly and is expensive to place. One per topic is enough to sit just under the line.
Networks is brutal-heavy, Data analysis is hard-heavy. That mirrors reality: six of the ten lowest Exam 2 figures in the current content areas are Networks crashing/flow items, whereas Data analysis separators cluster in the 25–45% interpretation band. Data analysis carries twice the marks, so its separators need to be distributed rather than concentrated.
Matrices has the highest brutal density relative to its mark weight (3 of 15) — driven by rounding discipline, backward percentages and the untouched Leslie-plus-culling recombination.
Recursion has the fewest brutal ideas (2) despite being the hardest Exam 1 block. Its difficulty is broad rather than deep: many items at 25–45%, few at the floor. Two brutal ideas (R8, R15) are enough to represent the "multi-stage changed-condition" family the reports name every year.
Recombination ideas are 7 of 64 — D7 (z-score × seasonal index), E6 is a within-topic chain, M7 (Leslie × culling), M10 (backward percentage × restocking), N16 (flow × crashing), plus D2 and E2 which put familiar skills on a transformed scale. Deliberately a minority: recombination is the riskiest category for fairness.

6. WHAT WOULD BE UNFAIR — AND WHY THESE IDEAS STAY ON THE RIGHT SIDE OF IT

VCAA sets hard questions. It does not set these seven kinds of hard question, and neither does anything above.

6.1 Content removed from the 2023 study design

Geometry and measurement, Graphs and relations (including linear programming), Number patterns, Business-related mathematics, solving simultaneous equations by matrix inverse, population and sample / µ and σ, and non-causal explanations (common response, confounding, coincidence) are all out. Every one of them produced real separators in the Further Mathematics era and every one is off-limits now.

Guard applied: M6 stops at "for which value of g does the inverse not exist" and never mentions a
system of equations. D6 asks about association in a two-way table but never about confounding.

6.2 Formulas not on the sheet that a student could not reasonably hold

VCAA supplies 11 formulas. It routinely requires the depreciation rules, the compound-interest rule, the Eulerian conditions, Prim's, Dijkstra's and the Hungarian algorithms, and the dominance construction — all absent from the sheet but all standard bound-reference content. What it does not require is a formula that no textbook supplies: the closed-form annuity formula, the number of edges in a complete graph as a memorised expression (the 2007 report supplies n(n − 1)/2 in the report, not to students), or any expression for the Pearson correlation coefficient.

Guard applied: E8 supplies both least-squares formulas because they are on the sheet. R11 asks for
arithmetic on a supplied multiplier, not for an annuity formula. N3 can be settled by sketching, not
by recalling the handshake lemma.

6.3 Answers that depend on which of two equally reasonable methods you chose

Where two legitimate procedures give slightly different answers, VCAA accepts both — 2025 Exam 2 Q6b published two seasonal-index routes with answers 0.588 and 0.587, and both were correct.

Guard applied: E1 explicitly requires both routes to be accepted. N7 and N14 explicitly require all
optimal schedules and both minimum spanning trees to be accepted. N7's ancestor (2019 Q3e) had VCAA
publishing an entire matrix of acceptable alternatives.

6.4 Ambiguity dressed up as difficulty

The Exam 1 instruction is invariantly "Choose the response that is correct for the question" — VCAA asserts that exactly one option is correct, never merely best. When VCAA itself creates an ambiguity it pays for it: at 2007 Exam 1 Q6 it omitted a dummy's arrow and had to accept two options.

Guard applied: every MC idea above has a single defensible answer. N3's distractors are true of
some graphs and the key is true of all — which is a stated property, not an ambiguity. M5's
distractors are all defined products, each answering a different explicitly different question.

6.5 Arithmetic volume as a substitute for thinking

Exam 2 is 60 marks in 90 minutes — 90 seconds a mark. A question requiring eight independent CAS operations for one mark is a time tax, not a discriminator.

Guard applied: the heaviest computational ideas here are R8 (two finance-solver runs), N8
(systematic trial over six activities — which VCAA's own 2024 report endorses as the expected method),
and M2 (four long-run CAS checks, matching the published 2025 NHT Exam 1 Q27 method). None exceeds
what VCAA has already asked for the same marks.

6.6 Punishing a correct answer for the wrong reason

There is a real line between the published further-engagement rule — extra information must be correct — and penalising a student for saying something true but unrequested. VCAA's own position is precise: "in Question 15b only the activity E needed to be written to attract the mark. If the response also included the delayed time correctly as 3 hours, then the mark was still awarded."

Guard applied: N10 and E4 rely on the further-engagement rule, but in both the extra material a
student would volunteer is false in the given data (a miscalculated float; non-monotone IQRs). A
student who answers exactly the question asked scores. That is the rule as VCAA writes it, not a trap.

6.7 New notation, new terminology, or content the cohort cannot have met

The study design's vocabulary is the examinable vocabulary — "response/explanatory", not "dependent/independent"; "association", not "relationship"; R and d, not b and c. Introducing a term students have not been taught, or a matrix type not in the key knowledge list, would be out of bounds.

Guard applied: every entry above quotes or closely paraphrases a study-design dot point in its
Target skill field. M2 speaks of long-run total population, never "steady state", because a Leslie
matrix is not regular and the study design restricts equilibrium language to "regular transition
matrices". M7's culling vector is justified by the key skill "use matrix recurrence relations to model
populations with culling and restocking", which is not restricted to transition matrices, and both
recurrences it uses are printed on the formula sheet.

6.8 The one-line test

Every idea in this document passes the same test:

A student who has learned the study design, kept a good bound reference, reads the command word,
shows their working, and rounds only when told, can answer it in the time available.

They separate because most students do not do all five of those things at once — not because the mathematics is out of reach.

7. SOURCE INDEX

Evidence	File
Every November separator, hardest first	`corpus/export/separators.csv` (905 rows)
All graded questions, for contrast and band calibration	`corpus/export/nov_all.csv` (2150 rows)
Scope, dot points, formula sheet, what is not on it	`research/01-study-design.md`
Data analysis wording bank, traps §3, difficulty ranking §3.12	`research/02-data-analysis.md`
Recursion wording bank, traps §3, multi-stage discriminators §3.12	`research/03-recursion-financial.md`
Matrices wording bank, traps §3, Leslie §1.12, guaranteed items §4	`research/04-matrices.md`
Networks wording bank, traps §3, crashing §1.14, flow §1.10	`research/05-networks.md`
Command words, show-that rules §2.3, rounding §3, distractor design §5.1.5, ranked complaints §7	`research/07-exam-craft.md`