New separator ideas

Descriptions of questions that would plausibly separate a cohort, aimed at the weaknesses the examination reports name year after year.

Compiled 15 September 2026. Companion to research/sm/01-study-design.md (scope and examination structure) and to the six area documents 02–08, whose catalogues, separator lists and traps are the raw material for everything below.

What this document is. Ninety-four question designs. Each entry says what would be given, what would be asked, how the parts would be ordered, precisely which step the middle of the cohort would fail, which archive questions and report sentences justify that prediction, the mark allocation and the paper and section it belongs in, and a predicted full-mark band anchored to a named archive comparison.

What this document is not. It contains no questions. There are no numbers to substitute, no functions with coefficients, no worked answers and no mark schemes. Every design is stated at the level of structure — a rational function whose numerator and denominator share a factor for one value of a parameter, not a rule with numbers in it. Anyone writing from these designs supplies the arithmetic themselves, and should expect the predicted band to shift by several points depending on how cleanly the numbers close.

Provenance of every percentage. All figures are VCAA's own published "percentage of students obtaining full marks", carried in corpus/sm/questions.json as pct, and every one quoted below has been read directly from that file rather than from the area documents. Where a mark distribution is quoted it is VCAA's published dist. NHT sittings publish no percentages and are cited for wording and scope only. The corpus defects catalogued in the area documents apply here unchanged: the 2011 papers are CID-shifted and unreadable, the 2024 November and 2025 NHT papers have no text layer, and a small number of topic tags are wrong in ways the source documents flag.

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Contents

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1. What the evidence says a separator needs

1.1 The definition, and the base rates

A separator is a question part with pct ≤ 50: fewer than half the state earned full marks. Across the whole graded archive there are 634 separators out of 1,428 parts carrying a published percentage — 44%. That is the background rate against which every prediction below has to be read. A question that separates is not unusual; a question that fails to separate is the unusual one on the technology-free paper.

Three structural variables predict separation far better than topic does. All figures recomputed directly from questions.json, November sittings only.

Mark value is the dominant signal (written parts only — Examination 1 plus Examination 2 Section B):

A three-mark written part is already a separator on the balance of probability. A four-mark part is one at five-to-one. Every five-mark written part set in twenty years of November papers has scored at or below 50%, and the median of those eight is 16%. The mechanism, as 07-exam-craft.md argues, is that mark value is a proxy for the number of independent instructions packed into one sentence: a four-mark question is not four times as hard as a one-mark question, it is four separate opportunities to be incomplete.

Position within a multi-part question is the second signal. Restricting to written questions with three or more graded parts (n = 152 questions):

Paper and section is the third.

Topic is a weak signal and should never carry a prediction on its own. Under the current design (November 2023–2025) the median full-mark rate by area of study runs from 51% to 67% — a sixteen-point spread, against the forty-six-point spread across mark values:

The practical consequence for design: the cheapest way to build a separator is to write a three- or four-mark written part and put it last. That is not a recommendation — it is a warning, because it is also the cheapest way to build a question that defeats the cohort without grading it. §1.3 is about the difference.

1.2 The constructions with the lowest full-mark rates

These are the ten deepest wells in the archive, taken across all six areas, with the mechanism each one exploits. They are the raw material for §2.

Four families account for all ten: symbolic generalisation (1%, 1%, 3%), change of variable in a definite integral (2%, 3%), enumerating cases over a parameter (2%, 3%, 4%), and a composite object whose parts must be identified before anything can be computed (2%, 2%, 4%).

1.3 The failure mechanisms the reports name most often

Eight mechanisms, with the archive evidence. Every design in §2 targets one of them explicitly, and the design notes name which.

M1 — Verification mistaken for proof. The most frequently repeated complaint in twenty years of reports. 2010 Exam 2 Section B Q3a at 5% asked students to verify a supplied solution of a differential equation and [RPT10 E2] records: "Many students attempted to solve the differential equation, despite this being asked for in part c." 2023 Exam 2 Section B Q2fii at 7%: "most students did not 'show that' the required result arose through a series of logical steps" [RPT23 E2]. 2022 Exam 2 Section B Q2ai at 34%: "A number of students apparently used a CAS to solve the given equation and then substituted their answers, again using CAS to verify the given result" [RPT22 E2]. 2006 Exam 2 Section B Q2c at 16%: "Most resorted instead to finding numerical approximations of the two angles" [RPT06 E2].

M2 — Necessary confused with sufficient. f''(a) = 0 does not establish a point of inflection. 2017 Exam 2 Section A Q10 at 6% — the lowest-scoring Calculus multiple-choice item ever set — turns entirely on a repeated factor giving no sign change. 2006 Exam 2 Section B Q4cii at 8%: "Most thought that it was sufficient to show that the second derivative needed to be zero" [RPT06 E2]. 2020 Exam 1 Q6b at 9%: "Few students attempted to justify that a point of inflection occurred at this point" [RPT20 E1]. 2025 Exam 2 Section A Q2 at 48%, nineteen years after the 2006 instance, with 40% choosing a counter-example to the converse.

M3 — Incomplete enumeration. One answer given where two or three exist. 2021 Exam 2 Section B Q1di at 6%: "Very few students gave all three values" [RPT21 E2]. 2013 Exam 2 Section B Q2b at 15%: "Most students found only two solutions for z" where there were four [RPT13 E2]. 2025 Exam 2 Section B Q5dii at 33.79%: "Many responses did not demonstrate that the modulus needed to be used and consequently only one of the solutions was found" [RPT25 E2]. 2021 Exam 1 Q6 at 26%: "Many students were able to find that p = ±2 for linear dependence but failed to conclude that p ∈ R \ {−2, 2} for independence" [RPT21 E1].

M4 — The answer is a set, and the parameter is not defined. 2006 Exam 2 Section B Q5e at 12%: "A large number of students who did manage to do something with this question found only those n values which were positive integers or zero" [RPT06 E2]. 2021 Exam 2 Section A Q6 at 23%. 2012 Exam 2 Section B Q2ei at 25%: "Others gave the correct general solution, but failed to define k" [RPT12 E2]. 2019 Exam 1 Q7d at 26%: "many students were unable to find a general solution" [RPT19 E1].

M5 — Change-of-variable bookkeeping. Four things change at a substitution — integrand, differential, and both terminals — and only three of them sit on the same line. 2019 Exam 2 Section B Q1e at 3%: "most simply replaced dx with dt" [RPT19 E2]. [RPT10 E1], reprinted verbatim in [RPT11 E1]: "Too many students changed the variable correctly but left the terminals unchanged; it should be emphasised that this is not logically correct, even if changing back to the original variable later enables them to obtain a correct answer."

M6 — The right number, the wrong object. 2021 Exam 2 Section B Q6d at 16%: "Some students calculated [the critical value] but did not proceed to answer the question correctly as a range of values" [RPT21 E2]. 2020 Exam 2 Section B Q2dii at 25%: "many did not fully describe the function as they did not include the domain" [RPT20 E2]. 2013 Exam 2 Section B Q4eiii at 10%, one mark: "The majority of students had scalar 0 as the result of the vector calculation, rather than the vector 0" [RPT13 E2]. 2020 Exam 2 Section B Q1bii at 35%: "Many students gave the (scalar) magnitude of the velocity rather than the required velocity" [RPT20 E2].

M7 — The sketch is a checklist and one item is missing. 2024 Exam 1 Q3c at 10%, 2022 Exam 1 Q10a at 13%, 2018 Exam 1 Q5 at 15%, 2025 Exam 1 Q9c at 16%, 2024 Exam 2 Section B Q1a at 17%, 2008 Exam 1 Q1 at 19%. [RPT08 E1] states the mechanism: "Many students were able to find the equations of the asymptotes, many were able to find the x-intercept and many were able to find the turning point, but few were able to complete all of these successfully."

M8 — The final part is not attempted. 2024 Exam 2 Section B Q3e at 4% and Q3d at 17%: [RPT24 E2] opens both comments with a variant of "Many students skipped this question". 2016 Exam 2 Section B Q3e at 23% and 2017 Exam 2 Section B Q3d at 26% — same opening sentence. 2021 Exam 2 Section B Q4e at 3% and 2021 Exam 2 Section B Q5d at 3% are both final parts of Section B questions. On the last two marks of a Section B question, a substantial fraction of the separation is pacing, not mathematics.

1.4 When a question discriminates rather than merely defeats

A separator is worth setting only if it grades. The archive contains both kinds and the published mark distributions tell them apart.

A discriminating separator has a spread distribution. 2018 Exam 1 Q4 (4 marks, 43%) has dist 5 / 8 / 8 / 36 / 43: every mark band is populated, 36% of the state sat at three of four, and the item separated a 43 from a 46 rather than a 30 from a 45. 2021 Exam 1 Q6 (4 marks, 26%) has dist 14 / 9 / 19 / 33 / 26: a third of the state sat at exactly three of four, having found the dependence condition and not written the complement. 2023 Exam 1 Q8 (4 marks, 21%) has dist 16 / 17 / 39 / 9 / 21: 39% scored exactly two — base case and assumption — which means the scheme was learnable and had been learned, and only the inductive step separated.

A defeating separator has a cliff. 2019 Exam 2 Section B Q6b (3 marks, 15%) has dist 67 / 15 / 2 / 15: two-thirds scored nothing, 15% scored full, and almost nobody scored two. Either you saw the structure or you did not, and the item carried no information about anyone in between. 2019 Exam 2 Section B Q4e (2 marks, 2%) has dist 96 / 2 / 2. 2021 Exam 2 Section B Q4e (3 marks, 3%) has dist 87 / 10 / 1 / 3.

Four conditions produce discrimination rather than defeat.

1. A reachable first step. The design must have an opening move that most of the cohort can make and that earns a mark. The mechanism is exactly the one VCAA states for "show that": [RPT06 E2] — "this instruction is intended to help keep them on track and enable access to subsequent marks for later parts of the same question, even if the student could not 'show' a given result."
2. A single named decision as the discriminator. The separation should turn on one identifiable choice — which acceleration form, which tail, which side of the asymptote, whether the cases are exhaustive — not on an accumulation of arithmetic. Compare 2012 Exam 1 Q8 at 28%, where [RPT12 E1] records that students who chose d/dx(½v²) over v dv/dx "were usually more successful, since squaring removed the square roots": one decision, two very different outcomes.
3. An answer whose form is checkable by the candidate. A design whose answer must be a set, an interval, a region, a pair of values or a vector gives a strong candidate a self-check and gives a weak one nowhere to hide. This is why M3, M4 and M6 produce such reliable separators.
4. Partial credit that is actually available. If the first two marks of a three-mark part depend on a step the last mark also depends on, the distribution will be bimodal. The fix is structural: split the part.

The corollary that governs every band prediction in §2. When a design is predicted below about 15%, it is predicted to defeat rather than to grade, and the entry says so. Those designs belong as the last one or two marks of a Section B question, where they cost nothing to the middle of the cohort and separate the top of it — which is exactly where VCAA puts them.

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2. The designs, by area of study

Format for every entry: the design (what is given, what is asked, how the parts are ordered) · the discriminator (which step the middle of the cohort fails, and why) · the evidence (archive refs with pct, and report quotations) · marks and placement · predicted band with its named archive anchor.

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2.1 Discrete mathematics — logic and proof

The area is four years old. questions.json holds twelve questions tagged Discrete mathematics, of which five are separators, and the pre-2023 archive contains zero induction, contradiction, contrapositive or counter-example questions. The designs below are therefore anchored partly on the nine live proof questions and partly on the ninety-odd "show that" parts whose marking commentary transfers.

Every Examination 1 since 2023 has carried exactly one proof question worth 3 or 4 marks, and induction in five of six sittings. Every Examination 2 since 2023 has opened Section A with a logic item.

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D1.1 — Induction on a recursively generated object

The design. Give an object defined by repeated application of an operation — the nth derivative of a product, the nth iterate of a function, the nth power of a supplied matrix — and a closed form for it. Ask for a proof by mathematical induction. The scheme is unscaffolded: no part asks for the base case separately.

The discriminator. Identifying what operation carries case k to case k+1. For a sum it is "add the next term"; for a derivative it is "differentiate once more". The middle of the cohort writes the base case and the assumption — both mechanical — and then applies the wrong operation, typically multiplying by the original object instead of operating on the assumed form.

The evidence. 2023 Exam 1 Q8 — 21%, dist 16 / 17 / 39 / 9 / 21. [RPT23 E1], verbatim: "A number of students either did not differentiate the function or differentiated incorrectly. Many students appeared to be thinking of index laws and assumed that f^(k+1)(x) was equal to f^(k)(x) × f(x)." The 39% who scored exactly two of four is the design working as intended: the ritual was learned, the step was not.

Marks and placement. 4 marks, Examination 1, mid-paper. Not Section B — [SPEC] puts Examination 1 on Outcome 1 and every live induction has been on Examination 1.

Predicted band. 18–26%, anchored directly on 2023 Exam 1 Q8 at 21%. A matrix-power variant would sit at the top of that band because the operation ("multiply by the matrix once more") is more visible than differentiation.

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D1.2 — Induction on an inequality with a threshold

The design. Two parts. Part (a) asks for the least value of n from which a stated inequality first holds, by finite checking — the one place in this area where testing examples is the intended method. Part (b) asks for a proof by induction that it holds for all n at or beyond that threshold. The inductive step requires a second, subsidiary inequality that is itself only true beyond the threshold.

The discriminator. Chaining inequalities in one direction with each link justified. A candidate who writes the chain with an unjustified middle link has a gap that the mark scheme will find, and the justification of that link is exactly where the threshold condition from the assumption gets used.

The evidence. Never set live. [SAMPLE] Examination 1 Q2 is VCAA's own demonstration of the type, split 1 + 3. The inequality context is named explicitly in the [SD] overview ("Proofs will involve concepts from topics such as: divisibility, inequalities, …"). The nearest live anchors are 2025 Exam 1 Q7 at 29% (a sum identity, mechanically simpler) and [RPT25 E1]'s named error "Assuming equality at the beginning of the inductive step" — an inequality proof makes that error impossible to commit invisibly, which is precisely why it discriminates.

Marks and placement. 1 + 3 marks, Examination 1.

Predicted band. 12–20% on part (b), below 2025 Exam 1 Q7's 29% because the step carries two justifications rather than one. Compare 2016 Exam 1 Q10 at 14% as the archive's benchmark for an Examination 1 item requiring several sequential correct decisions.

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D1.3 — Induction on divisibility by a composite divisor

The design. A statement that an expression in n — a difference of powers, or a polynomial — is divisible by a stated composite number for all natural n. Prove it by induction. The assumption must be stated with an explicit integer witness, and the step must produce the divisor times an integer.

The discriminator. Writing the assumption in a form that can be substituted. Without a witness variable there is nothing to substitute into the step, and the candidate stalls after two marks. The second discriminator is the closing sentence: naming the bracket as an integer is what makes the divisibility claim.

The evidence. [SAMPLE] Examination 1 Q3. 2026 NHT Exam 1 Q2 is the live parity variant at 3 marks, and its Assessment Guide shows VCAA's three-mark split: base case, assumption, then a single combined mark for step plus conclusion annotated "A1* given, must have working". The comparison that fixes the band is 2024 Exam 1 Q2 at 65% — the direct proof by integer parametrisation, and the only proof question ever listed among a paper's areas of strength [RPT24 E1].

Marks and placement. 4 marks, Examination 1.

Predicted band. 30–40%, between 2024 Exam 1 Q2 at 65% and 2025 Exam 1 Q7 at 29%: the parametrisation is as easy as the 2024 item, but the induction ritual adds two marks that the 2025 distribution shows the state only partially owns.

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D1.4 — Induction whose base case is not the first natural number

The design. A statement quantified over n at or beyond a stated value, or over a domain whose least element is not 1. Prove it by induction. Nothing else is unusual; the entire design is the quantifier.

The discriminator. Reading the domain and letting it determine three separate things: the base case, the range carried into the assumption, and the domain reproduced in the concluding sentence. The archive's five live and sample induction questions use four different domain statements, and the domain is the only thing that varies between them.

The evidence. 2023 Exam 1 Q8 quantifies over the positive integers; 2025 Exam 1 Q7 over the naturals; [SAMPLE] Examination 1 Q2 over n ≥ 5. [RPT25 E1] names "Misstating the assumption" as one of three common errors on the 29% item, and [RPT23 E1]'s published model answer makes the switch explicit — "true for some n = k" in the assumption, "true for all n" in the conclusion.

Marks and placement. 3 marks, Examination 1. This is the design to use when the paper needs an accessible proof question.

Predicted band. 35–45%, between 2024 Exam 1 Q2 (65%) and 2026 NHT Exam 1 Q2's three-mark shape. It is deliberately the mildest proof design in this set.

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D1.5 — Proof by cases

The design. A statement about integers whose natural proof partitions the domain exhaustively — by parity, or by remainder on division by a small modulus, or by sign. Prove it. Three or four cases, each short; the exhaustiveness sentence is a mark in its own right.

The discriminator. Stating the partition and justifying that it is exhaustive. A candidate who proves three cases and stops has done all the algebra and lost the mark that distinguishes a proof from a collection of verifications. The second discriminator is not proving the same case twice under two labels.

The evidence. Proof by cases is the only technique named in [SD] with no exemplar anywhere in the corpus — not in any live paper, not in [SAMPLE]. The content dot point is verbatim: "natural deduction and proof techniques: direct proofs using a sequence of direct implications, proof by cases, proof by contradiction, and proof by contrapositive." The transferable marking evidence is 2008 Exam 1 Q8c at 36% for one mark, where [RPT08 E1] records the state both over-proving and under-proving the same claim, and 2024 Exam 1 Q2 at 65%, whose Assessment Guide pays three marks for exactly three moves — parametrise, substitute, conclude.

Marks and placement. 3–4 marks, Examination 1.

Predicted band. 25–35%. The algebra per case is easier than an induction step, which pushes it above 2025 Exam 1 Q7 (29%); the exhaustiveness mark and the unfamiliarity of the scheme pull it well below 2024 Exam 1 Q2 (65%).

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D1.6 — Contradiction where the hypothesis must be retained

The design. An implication about integers or about real numbers — "if P then Q". The instruction names the technique: use proof by contradiction. The assumption that opens the proof must therefore be P and not-Q, not "not P".

The discriminator. Negating the conclusion while keeping the hypothesis. This is the single most common logical error in the area and it is invisible to a candidate who has only ever seen contradiction applied to a bare statement with no antecedent.

The evidence. [SAMPLE] Examination 1 Q4 is VCAA's own instance, and it is deliberately a statement that would be more natural to prove directly — the command word chooses the technique, the mathematics does not. The recognition item 2024 NHT Exam 2 Section A Q1 prints a model contradiction proof of a surd inequality; note that its statement has no hypothesis, so contrapositive is impossible, which is what makes it a one-mark item rather than a separator. The measured evidence for the underlying confusion is 2024 Exam 2 Section A Q1 at 72%, where 20% of the state chose the option that performed only one of the two contrapositive operations.

Marks and placement. 3 marks, Examination 1.

Predicted band. 25–35%, anchored between 2024 Exam 1 Q2 (65%, a direct proof of comparable algebraic weight) and 2025 Exam 1 Q7 (29%). The algebra is trivial; the opening line is the question.

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D1.7 — Written contrapositive, then a proof by it

The design. A quantified statement about integers. Part (a): state its contrapositive. Part (b): hence prove the original statement. Part (a) is one mark and is pure transformation; part (b) is two marks and requires the candidate to prove the contrapositive and then close with the equivalence.

The discriminator. Performing both operations — swap and negate — and preserving the leading quantifier. Then, in part (b), proving the right direction: a candidate who states the contrapositive correctly and then proves the original statement directly has answered a different question.

The evidence. Four live contrapositive items, all multiple choice, all at Section A Question 1: 2023 Exam 2 Section A Q1 at 85%, 2024 Exam 2 Section A Q1 at 72%, 2025 Exam 2 Section A Q1 at 93%, and 2026 NHT Exam 2 Section A Q1. Both reports that comment on them state the rule rather than the answer — [RPT24 E2]: "Asked for contrapositive — therefore, switch the hypothesis and the conclusion and negate both" — which is the tell that VCAA expects the operation to be unreliable. The 2024 item's 20% on the single-operation distractor is the measurement. [SAMPLE] Examination 2 Section A Q1 is the quantified variant, where two distractors are the negation of the whole statement offered as fake contrapositives.

Marks and placement. 1 + 2 marks, Examination 1.

Predicted band. 30–40% for the pair. The recognition versions run 72–93%; writing the sentence unprompted and then using it should cost roughly forty points, putting it near 2024 Exam 1 Q2's neighbourhood but below it.

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D1.8 — Written counter-example with both checks

The design. A plausible universal claim about functions — of the shape "if [a computable condition] holds at a point, then [a geometric property] holds there". Ask for a counter-example, and for a demonstration that the chosen object satisfies the hypothesis and fails the conclusion. Technology-active, because [SD]'s Outcome 3 key skill is "produce results, using a technology, which identify examples or counter-examples for propositions".

The discriminator. Verifying both halves. A candidate who exhibits an object and asserts that it works has done half the job; a candidate who verifies the conclusion fails but never checks that the hypothesis holds has produced an irrelevant object. The third failure, which the archive measures directly, is producing a counter-example to the converse.

The evidence. 2025 Exam 2 Section A Q2 at 48%, with 40% of the state choosing an option that is a counter-example to the converse, not to the stated claim. [RPT25 E2] sets out the standard: "the second derivative must equal zero at x = 0 and there must be a change of sign of the second derivative either side of x = 0." The same idea at 8% nineteen years earlier: 2006 Exam 2 Section B Q4cii, where [RPT06 E2] records "Most thought that it was sufficient to show that the second derivative needed to be zero." The over-supply failure is 2008 Exam 1 Q8c at 36%.

Marks and placement. 2–3 marks, Examination 2 Section B, as an opening or second part.

Predicted band. 30–40%. The multiple-choice version sits at 48%; requiring both verifications in writing should cost ten to fifteen points, which puts it beside 2017 Exam 1 Q2 (35%) as a comparably-marked written part.

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D1.9 — The vector proof that [SD] names and the archive has never set

The design. [SD] names three canonical vector proofs by title: the diagonals of a rhombus are perpendicular, the angle subtended by a diameter is a right angle, and the medians of a triangle are concurrent. The first two are in the archive. Set the third: position vectors for three vertices, define the midpoints, and ask for a vector proof that the three medians meet at a single point.

The discriminator. Parameterising two medians with different parameters, equating, solving for both, and then verifying that the resulting point lies on the third. A candidate who uses one parameter for both lines produces an unsolvable system — the same error [RPT25 E1] names on the 2025 lines-in-space question. The second discriminator is working in the general case rather than assuming convenient magnitudes.

The evidence. 2015 Exam 1 Q1b (rhombus) at 59%; 2022 Exam 1 Q6bii (semicircle) at 47%; 2010 Exam 2 Section B Q1dii at 49%; 2006 Exam 2 Section B Q2a at 70%. The special-case failure is documented three separate ways: [RPT10 E2] on Q1b (44%) — "simplifying assumptions such as |a| = |b| = 1/2, believing that a and b were orthogonal unit vectors" — and on Q1dii (49%) — "it was common to see |b| = |a| = 1 used for their scalar product to give zero." [RPT25 E1] on the parameter error: "It was common for students to use the same parameter for both lines. This did not result in viable equations to solve."

Marks and placement. 3–4 marks, Examination 1.

Predicted band. 15–25% — materially harder than either examined canonical result, because it requires two parameters and a third verification. Compare 2017 Exam 1 Q5 at 11%, the archive's hardest four-mark vector item.

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D1.10 — Identify the technique, with the hypothesis present

The design. Print a short, correct proof of an implication. Ask which technique it uses. Construct the proof so that the distinguishing line is the opening assumption, and include both contradiction and contrapositive among the options.

The discriminator. Distinguishing contradiction from contrapositive when the statement has a hypothesis — the only case in which the two look alike. A contradiction proof assumes the hypothesis and the negated conclusion and derives an absurdity; a contrapositive proof assumes the negated conclusion and derives the negated hypothesis. Every live instance so far has removed this difficulty by using a statement with no hypothesis.

The evidence. 2024 NHT Exam 2 Section A Q1 is the only instance, and its proof is of a bare surd inequality, so contrapositive is not even a candidate — the item tests recognition of "assume the negation" and nothing else. The comparison that sets the band is the pair of logic items on 2025 Exam 2: Section A Q1 (plain contrapositive) at 93% and Section A Q2 (counter-example, where the reasoning is genuinely required) at 48%.

Marks and placement. 1 mark, Examination 2 Section A, Question 1 or 2.

Predicted band. 40–55%, anchored on 2025 Exam 2 Section A Q2 at 48%. A logic item only separates when it requires a second thought; this design supplies one.

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D1.11 — Necessary and sufficient, stated as such

The design. A biconditional about a familiar object — a condition on a parameter that holds if and only if a geometric property holds. Part (a) asks which direction a supplied argument establishes. Part (b) asks for the other direction.

The discriminator. Recognising that two proofs are required and that they are different proofs. The forward direction is usually a computation; the converse usually requires starting from the property and reconstructing the condition. A candidate who proves one direction twice, in two notations, scores the marks for one.

The evidence. [SD]'s second content dot point names "implications, equivalences and if and only if statements (necessary and sufficient conditions)", and 02-discrete-proof.md records that a search of all 97 corpus files for "if and only if" and "necessary and sufficient" returns zero hits. The necessary-versus-sufficient distinction is nonetheless examined every year, inside calculus: 2017 Exam 2 Section A Q10 at 6%, 2006 Exam 2 Section B Q4cii at 8%, 2020 Exam 1 Q6b at 9%, 2025 Exam 2 Section A Q2 at 48%. The nearest thing to a live biconditional is 2008 Exam 1 Q8c at 36%, which asks for the minimal sufficient condition and nothing more.

Marks and placement. 1 + 3 marks, Examination 1.

Predicted band. 15–25% on the converse direction, anchored on the three sub-10% necessary-versus-sufficient items above. This design is close to the boundary of §1.4's discrimination test and should be scaffolded into two parts so that the forward direction is separately available.

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D1.12 — Prove exactly enough, for one mark

The design. A configuration already established to have one property; ask for a one-mark proof of a second property that follows from a single additional condition. The mark is for identifying and establishing that one condition.

The discriminator. Knowing the minimal sufficient condition. Over-proving does not lose the mark but loses the time, and in a 40-mark hour that is the same thing; under-proving — establishing a condition that is necessary but not sufficient — loses it outright.

The evidence. 2008 Exam 1 Q8c at 36% for one mark, with dist 64 / 36. [RPT08 E1], the best single paragraph in the archive on over-proving: "Many wasted time showing that the opposite pairs of sides had equal length or were parallel. Some students tried to show that all four angles were right angles… A few students correctly showed an adjacent pair of sides were at right angles, but then wasted time showing that adjacent sides were unequal in length, not realising that a square is a type of rectangle." [RPT10 E2] on 2010 Exam 2 Section B Q1b (44%): "Not all students understood clearly what they needed to show to prove that a given quadrilateral is a parallelogram."

Marks and placement. 1 mark, Examination 1, inside a multi-part vector question.

Predicted band. 30–40%, anchored on 2008 Exam 1 Q8c at 36%. This is a one-mark part that behaves like a three-mark part, which makes it unusually efficient.

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2.2 Functions, relations and graphs

This is the most hostile area on the technology-free paper in the whole archive: 42 of 63 Examination 1 parts are separators (67%), with a median full-mark rate of 39% (03-functions-graphs.md §1.4). The reason is structural — a sketch is marked against an instruction sentence, one mark per feature-set, and a perfect curve with one missing label scores one of three.

Examination 2 Section B Question 1 has been a functions-and-graphs question in thirteen of the last twenty years, and the Section A items cluster at Questions 1–5.

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D2.1 — Count the straight-line asymptotes of a parameter family

The design. A rational function with a parameter appearing in both numerator and denominator. Ask for every parameter value at which the graph has fewer than the generic number of straight-line asymptotes. Several values arise from cancellation, each for a different reason, and they must be listed separately.

The discriminator. Enumerating every cancellation case. The generic analysis is routine; what separates is recognising that a vertical asymptote disappears when the parameter makes a denominator root coincide with a numerator root, and that there is usually more than one way for that to happen. Targets M3.

The evidence. 2021 Exam 2 Section B Q1di at 6% — [RPT21 E2]: "Very few students gave all three values. Many responses included only one value." The companion Q1dii at 13%: "A common error was to include other incorrect values of k. Many students left this question blank." The multiple-choice version 2022 Exam 2 Section A Q3 at 38%, where the report's own explanation is "so only one vertical asymptote in this instance"; the easier 2024 version, 2024 Exam 2 Section A Q3, at 70%.

Marks and placement. 2 marks, Examination 2 Section B, late in the graphing question.

Predicted band. 5–15%, anchored directly on 2021 Exam 2 Section B Q1di at 6%. This is a defeating design by the §1.4 test and should be set as the second-last part of a ten-mark question, never as a stand-alone.

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D2.2 — Count the stationary points of the same family, including the degenerate case

The design. The same parameter family. Ask for the parameter values giving no stationary points, or exactly a stated number. The answer requires a discriminant condition on the derivative's numerator and the separate degenerate case in which cancellation reduces the function to a linear rule, which also has none.

The discriminator. The degenerate branch, and the boundary. [RPT24 E2]'s comment on the closest live relative is one sentence long and names the whole failure: "Many students did not include the equality sign."

The evidence. 2021 Exam 2 Section B Q1dii at 13%. 2024 Exam 2 Section B Q1di, Q1dii, Q1diii at 27%, 27% and 25% — three consecutive one-mark parts asking for the parameter conditions giving exactly one, three and five stationary points. 2020 Exam 2 Section B Q3eii at 2% is the extreme form: tabulate, over a parameter, how many points of inflection the family has.

Marks and placement. 1 + 1 + 1 marks, Examination 2 Section B.

Predicted band. 20–30% in the three-one-mark form, anchored on the 2024 triple at 25–27%; 10–20% if set as a single two-mark part, anchored on 2021 Exam 2 Section B Q1dii at 13%. The split is the difference between discriminating and defeating, and the 2024 paper demonstrates it: the same construction, roughly twice the full-mark rate.

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D2.3 — Examination 1 sketch with a hole and an oblique asymptote

The design. A rational function of degree three over degree two that factorises so one factor cancels. Part (a) asks for the quotient-plus-remainder form as a "show that". Part (b) asks for the sketch, with the full labelling instruction: asymptote equations, axial intercepts as coordinates, and an open circle at the removable discontinuity.

The discriminator. The checklist. Five separately marked features, of which the open circle is the one the state omits; and the oblique asymptote must be drawn as a ruled line with its equation, not as a gesture. Targets M7.

The evidence. 2025 Exam 1 Q9c at 16% — [RPT25 E1]: "An open circle to indicate the point of discontinuity at x = 1 needed to be shown… The point of discontinuity was often missing or was placed incorrectly. Students who were most successful used a ruler to draw the asymptotes." 2024 Exam 1 Q3c at 10%; 2018 Exam 1 Q5 at 15%; 2023 Exam 1 Q1b at 39% (oblique asymptote, no hole); 2008 Exam 1 Q1 at 19%, where [RPT08 E1] records "few were able to complete all of these successfully."

Marks and placement. 1–2 + 3 marks, Examination 1, Question 1 or 9.

Predicted band. 10–18%, anchored on 2025 Exam 1 Q9c at 16% and 2024 Exam 1 Q3c at 10%. Adding the hole to an oblique-asymptote sketch costs roughly twenty points against 2023 Exam 1 Q1b's 39%.

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D2.4 — Sketch the reciprocal of a printed graph

The design. Print the graph of a circular-polynomial function on a closed interval. Part (a): find its turning points in the open interval. Part (b): sketch the reciprocal on the same axes, labelling turning points and endpoints with coordinates.

The discriminator. Four transformation rules applied simultaneously — zeros become vertical asymptotes, maxima become minima at reciprocal height, points where the function equals one are fixed, and the endpoints map to reciprocal heights and stay closed. The archive shows the state losing the mark on the fixed points and on the endpoint convention.

The evidence. 2019 Exam 1 Q5b at 27% — [RPT19 E1]: "Common errors included neglecting to label the turning point at (π, 1), their graph not passing through the intersection points, and poor estimation of the location of the heights with respect to the given scale. Some students drew their graphs with an open circle at the endpoints." Part (a), 2019 Exam 1 Q5aii, at 42% — the open-interval restriction was itself a separator. 2006 Exam 1 Q3a at 39%.

Marks and placement. 2 + 3 marks, Examination 1.

Predicted band. 20–30%, anchored on 2019 Exam 1 Q5b at 27%.

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D2.5 — Write down the range of a product involving an inverse circular function

The design. One mark. A function that is the product of a power of the variable with an inverse circular function of a multiple of it. Write down the range.

The discriminator. Reasoning about the signs of the two factors — they are positive together, negative together and zero together, so the product is one-signed — rather than combining the two ranges arithmetically. There is no calculus in it and no calculation; it is a single observation.

The evidence. 2014 Exam 1 Q7a at 4% — the lowest-scoring one-mark question in the entire Functions area. [RPT14 E1]: "Few realised that x and the arctan function are both positive for the same values, negative for the same values and zero for the same values… Many students seemed to use the product of the range of each of the 'parts', some ignored one part and others found the product of the range of one part and the variable x."

Marks and placement. 1 mark, Examination 1, as the opening part of a longer question.

Predicted band. 5–15%, anchored on 2014 Exam 1 Q7a at 4%. A sum rather than a product would land nearer 20%; the multiplicative version is the one that defeats.

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D2.6 — A maximal domain that is an intersection

The design. Three one-mark parts. Parts (a) and (b) ask for the maximal domains of two components — one an inverse circular function with a linear argument, one a root or a reciprocal. Part (c) asks for the maximal domain of their sum or composite.

The discriminator. Recognising part (c) as an intersection of the two answers already obtained, not as a fresh computation. The secondary discriminator is endpoint inclusion after dividing a compound inequality by a negative number.

The evidence. 2012 Exam 1 Q10ai at 67%, Q10aii at 33%, Q10aiii at 26% — [RPT12 E1]: "This question involved finding the intersection of the domains found in the previous two parts. Many students seemed not to realise this." 2013 Exam 1 Q4a at 52% — [RPT13 E1]: "Some students reached −2 ≤ −2x ≤ 0 and then divided by −2 to write 1 ≤ x ≤ 0." 2009 Exam 1 Q10a at 34%, where the report lists submitted answers with the endpoints in the wrong order.

Marks and placement. 1 + 1 + 1 marks, Examination 1.

Predicted band. 20–30% on part (c), anchored on 2012 Exam 1 Q10aiii at 26%. The design is a good example of §1.4 condition 1: parts (a) and (b) are reachable, so the item grades rather than defeats.

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D2.7 — The derivative of an inverse circular composite, and where it is defined

The design. A function built as the reciprocal or a quotient of an inverse circular function. Ask for the derivative and for the largest set of values on which the derivative is defined.

The discriminator. The derivative's domain is strictly smaller than the function's, in two independent ways: the endpoints of the original domain go because the formula-sheet derivative has a root in the denominator, and any new denominator zero goes as well. The archive shows the state losing the second exclusion in particular.

The evidence. 2017 Exam 1 Q6 at 18% — [RPT17 E1]: "Some students confused the inverse function with the reciprocal function… Common errors for the domain included R, R \ {−1, 0, 1}, [−1, 1], (−1, 1) and [−1, 1] \ {0}. Many students did not exclude zero." 2018 Exam 2 Section B Q1ei at 21% — "The most common error was to include x = 0 in the domain. Another common error was to include the endpoints." 2010 Exam 1 Q5 at 35%, where [RPT10 E1] records the chain rule "often not used, despite the assistance of the formula sheet".

Marks and placement. 3 marks, Examination 1.

Predicted band. 15–25%, anchored on 2017 Exam 1 Q6 at 18% and 2018 Exam 2 Section B Q1ei at 21%.

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D2.8 — A hybrid derivative forced by a modulus, then sketched

The design. An inverse circular function of an even function of the variable. Part (a): find the derivative and specify it as a hybrid function over its maximal domain. Part (b): sketch that hybrid, showing asymptotes and excluded points.

The discriminator. Recognising that differentiating produces a square root of a square, hence a modulus, hence two branches; then splitting at the origin with correct strict and non-strict endpoints; then drawing both branches with the correct sign of gradient on each. The archive shows the state drawing a symmetric U-shape, which is the graph of the modulus rather than of the derivative.

The evidence. 2010 Exam 2 Section B Q4d at 30% and Q4e at 19%. [RPT10 E2] on Q4e: "The most popular response was a 'U shape' curve with a vertex at (0, 2)… These students ignored the graph in Question 4a., which clearly had negative gradients to the left of the y-axis. Some students had the correct curves and asymptotes, but did not exclude the y-intercept points." 2018 Exam 2 Section B Q1eiii at 39%; 2017 Exam 2 Section B Q3b at 49% — "Many students did not use a hybrid function."

Marks and placement. 2 + 2 marks, Examination 2 Section B.

Predicted band. 20–30% on the sketch, anchored on 2010 Exam 2 Section B Q4e at 19% and 2018 Exam 2 Section B Q1eiii at 39%.

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D2.9 — An inequality whose answer must be an interval

The design. A rational or reciprocal-circular inequality whose critical values include a zero of the denominator. The instruction specifies that the answer is to be given in interval notation.

The discriminator. Two independent failures. First, multiplying through by a quantity whose sign is not known — the algebraic route collapses without a sign table. Second, giving the critical values instead of the intervals between them, which the reports record as a separate and sufficient reason to lose marks.

The evidence. 2020 Exam 1 Q4 at 12% — [RPT20 E1]: "A quick sketch was helpful… Students who approached this problem algebraically were often unsure how to deal with the inequality signs. A number of students who found [the correct set] did not receive full marks as they did not write the final answer in interval notation." 2015 Exam 1 Q7b at 11% — "giving single value answers rather than intervals." 2017 Exam 2 Section A Q2 at 37%.

Marks and placement. 4 marks, Examination 1.

Predicted band. 10–20%, anchored on 2020 Exam 1 Q4 at 12% and 2015 Exam 1 Q7b at 11%. Dropping the interval-notation instruction would move the band up by roughly ten points.

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D2.10 — An exact trigonometric value needing a half-angle bridge and an explicit rejection

The design. Give the value of a circular function at a doubled or halved angle, together with an interval for the angle. Ask for the exact value of a different function at the base angle, in a prescribed surd form, and require the rejection of the other root to be explained.

The discriminator. Three sequential decisions: which identity gets from what is given to what is wanted in one step; what sign the companion ratio takes, fixed by the stated interval; and — the mark most often lost — writing down the reason the negative root is rejected. Substituting the answer back to check is explicitly not sufficient.

The evidence. 2006 Exam 1 Q5a at 22% — [RPT06 E1]: "Quite a few students whose working was correct failed to complete their solution, giving no proper explanation as to why the negative answer should be rejected… Substituting tan(π/8) = √2 − 1 into either the double angle formula or the compound angle formula was not sufficient to achieve full marks." 2007 Exam 1 Q10 at 20%; 2008 Exam 1 Q4 at 23%; 2012 Exam 1 Q10b at 13%; 2024 Exam 2 Section A Q4 at 27%, which turns on an interval forcing opposite signs for the angle and its half. [RPT16 E1] adds the plausibility check the state does not run: "Of great concern was the number of students who gave answers for sine or cosine that were either less than −1 or greater than 1."

Marks and placement. 4 marks, Examination 1.

Predicted band. 15–25%, anchored on 2006 Exam 1 Q5a at 22% and 2008 Exam 1 Q4 at 23%.

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D2.11 — The arc a parameter interval actually traces

The design. A path given parametrically on a restricted parameter interval, using a pair of functions that satisfy a Pythagorean identity. Part (a): show the Cartesian equation. Part (b): state the domain and range of the relation traced. Part (c): sketch only that arc, with endpoint coordinates and a direction arrow.

The discriminator. Part (b) asks for the image of the parameter interval, not the implied domain of the Cartesian rule — these are different sets and the archive shows the state giving the second. Part (c) then punishes the same misunderstanding graphically: a full conic where an arc was traced.

The evidence. 2019 Exam 2 Section B Q1b at 38% — [RPT19 E2]: "While most students stated the correct range, a significant number gave a domain which did not account for the restriction on t." 2013 Exam 1 Q7b at 22% — [RPT13 E1]: "A large number of students drew a complete hyperbola or the complete right-hand branch." 2007 Exam 1 Q6c at 22% — "Most students were unable to apply the domain restrictions correctly… Usually, a full ellipse was drawn." 2023 Exam 2 Section B Q1e at 22%; 2024 Exam 2 Section B Q4b at 41% — "Negative signs were often left off the coordinates of the end points. The direction of the path… was often left out or in the wrong direction."

Marks and placement. 2 + 2 + 3 marks, Examination 1 or Examination 2 Section B.

Predicted band. 18–28% on the sketch, anchored on the three independent 22% instances above.

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D2.12 — Hole or asymptote, as a condition on a parameter

The design. A hybrid function whose two branches meet at a point where the rational branch has a removable discontinuity, with a parameter in the numerator. Part (a): find the value of the constant that makes the hybrid continuous. Part (b): for which parameter values does the graph have a hole rather than a vertical asymptote?

The discriminator. Distinguishing a cancelling factor from a non-cancelling one, stated as a condition on a parameter rather than computed for a single case. Part (a) is mechanical and gives the item its reachable first step; part (b) is the separator.

The evidence. 2025 Exam 1 Q9b at 46% (the continuity value) and Q9c at 16% (the sketch with the hole). 2024 Exam 2 Section A Q2 at 48% — the multiple-choice version asking whether a hybrid rational function is continuous, has an asymptote, or has a point of discontinuity. 2026 NHT Exam 2 Section A Q2 asks for the parameter value making a piecewise inverse-circular/rational function continuous; 09-nht-specialist.md flags "continuous" as a term VCAA has so far trialled only in NHT.

Marks and placement. 1 + 2 marks, Examination 2 Section B.

Predicted band. 25–35% on part (b), anchored between 2025 Exam 1 Q9b (46%) and the parameter-family items at 25–27%.

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2.3 Algebra, number and structure — complex numbers

Complex numbers has appeared on every November Examination 1 from 2006 to 2025 — twenty out of twenty, for between 3 and 9 marks, median 4 — and Examination 2 Section B has carried a dedicated complex question every year, at Question 2 in thirteen of the last fourteen. The Examination 1 slice is the hardest: median full-mark rate 48.5% across 42 parts.

The area's separator profile is unusual. The densest cluster is not the algebra but the loci: rays, regions, areas and parameterised loci account for 33 of the 90 separators, and five of the six items in the whole area below 8% are loci questions.

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D3.1 — All integers for which a power is real, then purely imaginary

The design. Give a complex number whose argument is a simple rational multiple of π, in a form that requires conversion first. Part (a): find all integers n for which the nth power is real. Part (b): all n for which it is purely imaginary. Each answer is a set with a stated parameter domain.

The discriminator. Producing a general solution and defining the parameter as an integer. The archive is unambiguous that the state finds some values by inspection, states them as a list, and loses the marks. Targets M4.

The evidence. 2006 Exam 2 Section B Q5e at 12% — [RPT06 E2]: "A large number of students who did manage to do something with this question found only those n values which were positive integers or zero." 2012 Exam 2 Section B Q2ei at 25% — [RPT12 E2]: "The general solution for n eluded most students. Many gave some specific values for n… Others gave the correct general solution, but failed to define k." 2019 Exam 1 Q7d at 26% and Q7c at 39% — [RPT19 E1]: "Some students realised that if n was a positive or negative multiple of 6 then zⁿ was real, but were unable to express this mathematically." 2021 Exam 2 Section A Q6 at 23%.

Marks and placement. 2 + 2 marks, Examination 1.

Predicted band. 15–25%, anchored on 2012 Exam 2 Section B Q2ei at 25% and 2019 Exam 1 Q7d at 26%, with the "purely imaginary" half at the lower end because the index set is odd multiples rather than all multiples.

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D3.2 — The argument of a sum

The design. Two complex numbers described in terms of an unknown argument, with a "show that" in the preceding part establishing a relationship between them. Ask for the argument of their sum, in terms of that unknown.

The discriminator. Arguments add under multiplication and never under addition. The correct route is geometric — the sum of two equal-modulus complex numbers bisects the angle between them — or algebraic, by factoring out a common polar factor. The archive records students simply adding the arguments.

The evidence. 2010 Exam 2 Section B Q5d at 13% and the "show that" that sets it up, Q5c, at 18%. [RPT10 E2]: "Some students who did attempt the question used approaches such as Arg(u + w) = Arg(u) + Arg(w), which were incorrect… Quite a few students did a substantial amount of work in cartesian form to little avail. As this was a 'show that' question, it was important that all connecting steps were shown." The principal-value discipline that compounds it is documented across six reports; 2015 Exam 2 Section B Q2bii at 23% is the closest relative.

Marks and placement. 3 marks, Examination 2 Section B, after a "show that".

Predicted band. 10–20%, anchored on 2010 Exam 2 Section B Q5d at 13%. Below 15% this is a defeating design; supplying the geometric configuration as a printed diagram moves it towards 25%.

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D3.3 — A ray, its equation, and its Cartesian rule with a domain

The design. A ray drawn on a supplied Argand diagram, or described by its point of emanation and a point it passes through. Part (a): give its equation in the standard argument form. Part (b): give the rule of the corresponding function in Cartesian form, fully described.

The discriminator. Three separately marked things, of which the third is the separator: the principal-value angle, the open endpoint, and the domain restriction that distinguishes a ray from the line containing it. A candidate who writes the line's equation and stops has produced half an answer.

The evidence. 2020 Exam 2 Section B Q2dii at 25% — [RPT20 E2]: "While a high proportion of students gave the correct rule, many did not fully describe the function as they did not include the domain." 2023 Exam 2 Section B Q2dii at 18% — "While many students correctly identified z₀, finding the correct angle was a challenge for most." 2022 Exam 2 Section B Q2c at 25%; 2021 Exam 2 Section B Q2b at 31% — [RPT21 E2]: "The point of emanation is not part of the required ray and should be shown as an open circle." 2024 Exam 2 Section B Q2dii at 56%, where [RPT24 E2] records "The most common error was to quote the argument as −π/4" — a sign error and a quadrant error at once.

Marks and placement. 1 + 2 marks, Examination 2 Section B.

Predicted band. 15–25%, anchored on 2023 Exam 2 Section B Q2dii at 18% and 2020 Exam 2 Section B Q2dii at 25%.

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D3.4 — For which arguments does a ray meet a locus

The design. A line in the complex plane, given as a perpendicular bisector, and a ray emanating from the origin. Ask for the set of arguments for which the ray intersects the line.

The discriminator. The answer is an interval of arguments, with its endpoints determined by the direction of the line and by whether the limiting rays are parallel to it. There is no computation to be done; the whole item is the geometry of a half-line against a full line, and the archive shows almost nobody attempting it.

The evidence. 2016 Exam 2 Section B Q2f at 7% — the second-lowest complex-number item in the archive. Its structural twin 2017 Exam 2 Section B Q4f at 1% asks for one constant of a locus in terms of another. 2019 Exam 2 Section B Q2c at 7% asks for all real parameter values for which the roots of a quadratic satisfy a modulus inequality.

Marks and placement. 2 marks, Examination 2 Section B, final part.

Predicted band. 5–12%, anchored on 2016 Exam 2 Section B Q2f at 7%. This is explicitly a defeating design. It belongs as the last two marks of a twelve-mark question and nowhere else.

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D3.5 — A locus in terms of symbolic constants

The design. A quadratic over the complex field with symbolic real coefficients. Its non-real roots determine a circle or a line. Ask for the centre and radius, or for the perpendicular-bisector form, in terms of those coefficients.

The discriminator. Working symbolically to the end. There is never a number to check against, so every simplification must be exact and the candidate has no feedback. The design also exercises the conjugate-root theorem in a setting where the roots are never computed.

The evidence. 2019 Exam 2 Section B Q2d at 1%, 2 marks — the minimum-radius circle through the roots of a quadratic with symbolic coefficients. 2017 Exam 2 Section B Q4f at 1%, 1 mark. 2021 Exam 2 Section B Q2aii at 23% — [RPT21 E2]: "Many students used p(z) in the expanded form, which was less productive than using the factorised form directly."

Marks and placement. 2 marks, Examination 2 Section B, final part.

Predicted band. 1–8%. Both archive instances sat at 1%. A scaffolded version — part (i) the centre, part (ii) the radius — should reach 15–20% by the §1.4 splitting argument, and is the only version worth setting.

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D3.6 — An equation mixing a variable and its conjugate

The design. Part (a): solve a quadratic over the complex field with real coefficients — routine. Part (b): solve the equation obtained by replacing one occurrence of the variable with its conjugate.

The discriminator. Recognising that the conjugate blocks every polynomial technique, so the only route is to substitute the Cartesian form and solve two real simultaneous equations, one of them quadratic. The archive records a specific and instructive error: assuming part (a)'s answers carry over.

The evidence. 2021 Exam 1 Q8a at 70% and Q8b at 10% — a sixty-point drop across one symbol. [RPT21 E1]: "Students who were successful let z = x + iy, leading to two simultaneous real equations. Algebraic errors were often seen… A number of students assumed that the solutions to part a. were also solutions to part b., and some students confused the complex conjugate with the reciprocal."

Marks and placement. 1 + 3 marks, Examination 1.

Predicted band. 8–16%, anchored on 2021 Exam 1 Q8b at 10%. Part (a) is the reachable first step that makes the pair grade rather than defeat.

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D3.7 — A polynomial with a non-real coefficient

The design. A cubic or quartic over the complex field with at least one non-real coefficient, arranged so that grouping factorises it. Part (a): verify a supplied factor. Part (b): hence, or otherwise, solve, giving answers in Cartesian form.

The discriminator. Not applying the conjugate root theorem. Its hypothesis is that the coefficients are real, and VCAA tests the hypothesis at least as often as the conclusion — [SD] even chooses a cubic with non-real coefficients as one of its three worked factorisation examples. The secondary discriminator is giving solutions where solutions were asked for, not factors.

The evidence. 2007 Exam 1 Q2b at 42% — [RPT07 E1]: "Far too many students decided that the complex conjugate √5 + i was another solution despite the coefficients of the cubic polynomial not being real." 2024 Exam 1 Q1b at 48% — [RPT24 E1]: "With the known root, a small number of students tried inappropriately to apply the conjugate root theorem." 2015 Exam 1 Q4a at 40% — "Many students assumed that the Conjugate Root Theorem applied." 2006 Exam 1 Q9b at 16%, where the discriminant itself is non-real.

Marks and placement. 1 + 3 marks, Examination 1.

Predicted band. 30–42%, anchored on 2007 Exam 1 Q2b at 42% and 2024 Exam 1 Q1b at 48%. Making the discriminant non-real rather than the coefficients moves it to 2006 Exam 1 Q9b's 16%.

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D3.8 — Find all the coefficients, and show these are the only ones

The design. A monic cubic with non-zero real coefficients, constrained by a condition on the moduli or arguments of its roots. Find the coefficients, and show that the values found are the only possible ones.

The discriminator. The exhaustiveness argument. The constraint admits more root configurations than the candidate first sees, and the word "non-zero" in the stem is what excludes the extra configuration — which means the mark hangs on a single adjective being read.

The evidence. 2008 Exam 1 Q10c at 3%, 4 marks. [RPT08 E1]: "Some students then found the third root by solving z³ = 8 but the vast majority only considered the solution z = 2, ignoring the solution z = −2… Very few students realised that z + 2 was also a possibility, but when used, led to a cubic with some zero coefficients… which then had to be excluded to fully answer the question." 2021 Exam 2 Section B Q2aii at 23% is the technology-active relative.

Marks and placement. 4 marks, Examination 1, final question.

Predicted band. 3–10% unscaffolded, anchored on 2008 Exam 1 Q10c at 3%. Split into "find the values" and "explain why there are no others" and the first half should reach 30%, on the evidence of 2021 Exam 2 Section B Q2aii at 23% for a comparable symbolic determination.

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D3.9 — Roots in exact Cartesian form, then the translated root set

The design. Part (a): solve an equation of the form "the variable to a power equals a given complex number", answers in exact Cartesian form. Part (b): "hence" solve the equation obtained by translating the variable.

The discriminator. Part (a) discriminates on the conversion back to Cartesian with exact values and on giving all the roots. Part (b) discriminates on recognising that the new root set is the old one translated, rather than restarting — and on the direction of the translation.

The evidence. 2015 Exam 1 Q4a at 40% — [RPT15 E1] lists every failure: roots left in polar form, arithmetic errors in conversion, the conjugate-root theorem misapplied, "and a number of students gave only one solution for this cubic". Q4b at 44% — "Several students subtracted 2i from the answers in part a." 2013 Exam 1 Q8 at 23%, where the equation reduces to a quadratic in a square and the square roots of a non-real number are needed. 2020 Exam 1 Q3 at 37% — "Some students neglected to give the arguments for their final answers using principal values as required."

Marks and placement. 3 + 1 marks, Examination 1.

Predicted band. 30–42%, anchored on 2015 Exam 1 Q4a at 40% and Q4b at 44%. A reduction to a quadratic in a square first, as in 2013, lowers it to 20–28%.

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D3.10 — The area of a named region cut by a locus

The design. A circle and a ray, or a chord determined by two loci. Ask for the exact area of a named region — the minor segment, the major segment, or a portion of an annulus.

The discriminator. Identifying the central angle. It is not the ray's argument and it is not the inscribed angle, and the reports record all three being used. The second discriminator is the adjective: minor or major, which changes the formula entirely.

The evidence. 2017 Exam 2 Section B Q4g at 24% — [RPT17 E2]: "A significant number of students incorrectly used a sector angle of π/3." 2009 Exam 2 Section B Q2f at 25% — "Few students realised that the area of a portion of an annulus was to be found. Often elaborate approaches were set up to solve this simple problem." 2016 Exam 2 Section B Q2d at 39% (major segment); 2013 Exam 2 Section B Q2f at 41%; 2024 Exam 2 Section B Q2e at 52% (minor segment, a right-angle central angle). [RPT13 E2] lists "the major segment shaded" as a frequent error on a minor-segment question.

Marks and placement. 2 marks, Examination 2 Section B.

Predicted band. 25–38%, anchored on 2017 Exam 2 Section B Q4g at 24% and 2016 Exam 2 Section B Q2d at 39%. An annulus sector rather than a circular segment sits at the bottom of the band, on the evidence of the 2009 item at 25%.

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D3.11 — Two loci that meet in four points

The design. A circle centred at the origin, presented in the product-with-conjugate form rather than as a modulus. Intersect it with a relation that is secretly a pair of lines. Ask for all points of intersection, expressed in Cartesian complex form.

The discriminator. Recognising that a relation equating two moduli of conjugate combinations is a pair of lines, not one, and therefore producing four points. The secondary discriminator is the form of the answer — the reports record the state giving ordered pairs where complex numbers were asked for, and vice versa.

The evidence. 2013 Exam 2 Section B Q2b at 15% — [RPT13 E2]: "Most students found only two solutions for z." The sketch that precedes it, Q2a, at 20%. 2015 Exam 2 Section B Q2aiv at 44%; 2008 Exam 2 Section B Q5c at 33% — "A fairly common error was interchanging x and y coordinates in the second point of intersection." [RPT25 E2] on the labelling convention: "If they are using coordinates, they must not have i in the coordinate."

Marks and placement. 2 + 3 marks, Examination 2 Section B.

Predicted band. 12–22%, anchored on 2013 Exam 2 Section B Q2b at 15%.

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D3.12 — Roots of unity as machinery for a trigonometric identity

The design. Supply the factorisation of "the variable to the n minus one" into a linear factor and a cyclotomic one. Part (a): verify a stated root — one mark, reachable. Part (b): use De Moivre's theorem to show that a stated sum of cosines takes a stated value.

The discriminator. Two facts that are never on the formula sheet: the nth roots of unity sum to zero, and conjugate pairs of them contribute twice a cosine. The archive shows the state able to express the equation in powers of the root and unable to turn that into the identity by visible steps. Targets M1 as much as the complex algebra.

The evidence. 2023 Exam 2 Section B Q2fii at 7%, dist 85 / 8 / 7 — the lowest complex-number "show that" in the archive. [RPT23 E2]: "Many students were able to express the given equation in terms of powers of w but most students did not 'show that' the required result arose through a series of logical steps." The reachable parts of the same question: Q2e (verify the factorisation) at 54%, Q2fi at 47%, Q2a at 63%, Q2b at 61%, Q2c at 53%.

Marks and placement. 1 + 2 marks, Examination 2 Section B, late in the complex-numbers question.

Predicted band. 5–12% for the identity, anchored on 2023 Exam 2 Section B Q2fii at 7%. The preceding parts at 47–63% are what make the question as a whole grade; the identity itself separates the top few per cent and nothing else.

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2.4 Calculus

Calculus is the heaviest area under the current design — 27.8% of tagged marks 2023–2025, the first time it has led the subject. Its Examination 2 profile is unremarkable, but the median Calculus part on Examination 1 is itself a separator: 128 parts, median full-mark rate 43.5%. And the separator mass is concentrated in the multi-mark written questions: 158 separators carry 372 of the area's 623 published marks, which is 41% of the parts and 60% of the marks.

The ranking of Calculus separator families by depth is not about technique. The two deepest groups — verifying a supplied solution (median 23.5%) and modelling, interpretation and graphing (median 23.5%) — are hard instructions, not hard mathematics.

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D4.1 — Verify a supplied solution, and the initial condition

The design. Supply a closed-form solution of a differential equation, implicit or explicit. Instruct the candidate to verify it by differentiation and substitution into one named side, and — separately — to verify that it satisfies the stated initial condition. Both halves are marked.

The discriminator. Reading "verify" and not solving. The instruction prescribes the direction of work, and a candidate who separates the variables and integrates has done more mathematics and earned less. The second discriminator is the initial condition, which the reports record as routinely omitted. Targets M1.

The evidence. 2010 Exam 2 Section B Q3a at 5%, 3 marks — [RPT10 E2]: "Many students attempted to solve the differential equation, despite this being asked for in part c. Some students who solved instead of verifying the solution by substitution, verified the initial condition." 2016 Exam 2 Section B Q3d at 17% — "It was not always clear how expressions for the left side simplified to the right side. Verification that the given solution satisfied the initial conditions was often absent." 2013 Exam 2 Section B Q3a at 30%. The easy calibration point is 2022 Exam 2 Section B Q3d at 77%, which shows the design is not intrinsically hard — it is hard when the algebra is long enough that a candidate would rather solve.

Marks and placement. 3 marks, Examination 2 Section B, immediately after the part that asks for the solution.

Predicted band. 8–20%, anchored on 2010 Exam 2 Section B Q3a at 5% and 2016 Exam 2 Section B Q3d at 17%. Placing it before the "solve" part rather than after would raise it towards 40%.

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D4.2 — Write down an integral in a named variable, do not evaluate

The design. A solid of revolution generated by a parametrically described curve about one of the axes. Ask for the volume as a definite integral in terms of the parameter, with an explicit instruction not to evaluate it.

The discriminator. Four things change at a change of variable — integrand, differential, and both terminals — and only three of them appear on the same line. The terminals sit in a different visual position and are the ones the state leaves behind. Targets M5.

The evidence. 2019 Exam 2 Section B Q1e at 3%, 2 marks — 79% of the state scored zero. [RPT19 E2]: "Very few students answered this question correctly. The most common incorrect answer was an integral in terms of x. Of those that attempted to give an integral in terms of t, most simply replaced dx with dt." The Section A relatives: 2009 Exam 2 Section A Q10 at 23%, with 52% of the state on a single distractor. The scaffolded Examination 1 version, 2014 Exam 1 Q5b at 48%, asks for the equivalent integral "in terms of u only" and quarantines the evaluation into part (c).

Marks and placement. 2 marks, Examination 2 Section B; or 2 marks in the 2014 Exam 1 Q5b scaffolded form on Examination 1.

Predicted band. 3–12% in the unscaffolded form, anchored on 2019 Exam 2 Section B Q1e at 3%; 40–50% if the substitution is supplied and the evaluation is quarantined, anchored on 2014 Exam 1 Q5b at 48%. The gap between those two bands is the single clearest illustration in the archive of what scaffolding does to a separator.

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D4.3 — The second derivative of a modelled quantity, in terms of the quantity

The design. A differential equation giving the rate of change of a quantity as a function of the quantity itself. Part (a): express the second derivative in terms of that quantity. Part (b): hence find and classify the point of inflection of the solution curve, and interpret it in context.

The discriminator. The chain rule when differentiating with respect to the independent variable: every term picks up a factor which is the original right-hand side and must be substituted back, so that the answer contains no derivative. The archive records the state differentiating with respect to the wrong variable, or producing an expression in the independent variable rather than the dependent one.

The evidence. 2013 Exam 2 Section B Q3di at 9% — [RPT13 E2]: "A number of students found d²N/dt² in terms of t, while others found d²t/dN² and attempted to invert the result, which showed little understanding of the properties of second derivatives. A common error was to find d/dN(dN/dt)." Q3dii at 38%. 2023 Exam 2 Section B Q4ei at 21% — [RPT23 E2]: "A common error involved not recognising the need to use the chain rule when differentiating with respect to t." 2017 Exam 2 Section A Q6 at 46% is the one-mark version. The easy calibration point is 2020 Exam 2 Section B Q3d at 91%, where the right-hand side depends only on the independent variable and no substitution is needed.

Marks and placement. 2 + 2 marks, Examination 2 Section B.

Predicted band. 10–22% on part (a), anchored on 2013 Exam 2 Section B Q3di at 9% and 2023 Exam 2 Section B Q4ei at 21%.

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D4.4 — A point of inflection that is not one

The design. A function whose second derivative factorises with one repeated linear factor and one simple factor, so that only one of the two candidate values gives an inflection. Ask for the coordinates of any points of inflection, or — on Examination 1 — to determine them with justification.

The discriminator. f'' = 0 is necessary and not sufficient, and a squared factor produces no sign change. Every published instance of this construction has scored under 15%, across nineteen years and both papers. Targets M2.

The evidence. 2017 Exam 2 Section A Q10 at 6% — the lowest-scoring Calculus multiple-choice item ever set; [RPT17 E2]: "f''(x) does not change sign at −a", and the report's general comments name the whole-paper weakness as "understanding that f''(x) = 0 does not necessarily imply a point of inflection". Its written twin in the same paper, 2017 Exam 2 Section B Q1aiii, at 13% — "A common error was to erroneously include the point (0, 0), which is another point where f''(x) = 0, but it is not a point of inflection as there is no change of concavity." 2020 Exam 1 Q6b at 9% — "Few students attempted to justify that a point of inflection occurred at this point." 2006 Exam 2 Section B Q4cii at 8%. 2025 Exam 2 Section A Q2 at 48%, the counter-example form.

Marks and placement. 1 mark as a multiple-choice item, or 3 marks written on Examination 1, where the sign table must be shown.

Predicted band. 8–20%. The three-mark written version with an explicit "determine" should sit at the top of that band, on the evidence of 2026 NHT Exam 1 Q3's Assessment Guide, which marks a sign table plus the sentence "f'' does not change sign at …, so not a point of inflection".

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D4.5 — An arc length whose integrand is a perfect square

The design. A parametric curve engineered so that the sum of the squares of the component derivatives collapses to a perfect square, allowing the root to be removed by hand. On Examination 1, ask for the exact length; or, in the harder form, supply the integral with unknown integer constants and ask for them.

The discriminator. Spotting the perfect square before doing any algebra. The archive shows that candidates who instead combine the expression into a single algebraic fraction do not recover. The secondary discriminator is the sign of the root over the stated interval.

The evidence. 2018 Exam 1 Q10 at 2%, 5 marks, dist 35 / 22 / 24 / 17 / 0 / 2 — one of only two items in the whole archive at 2%. [RPT18 E1]: "Most students recognised that the arc length formula needed to be applied… Many students had difficulty simplifying (dx/dt)² + (dy/dt)²." 2020 Exam 1 Q9b at 14% — [RPT20 E1]: "Most students who successfully answered this question were able to identify the perfect square, which allowed the square root in the integrand to be removed. Few students who tried to write the term inside the square root as a single algebraic fraction were able to see the problem through to the conclusion." 2017 Exam 1 Q7 at 30%; 2017 Exam 2 Section B Q3e at 16%.

Marks and placement. 4 marks, Examination 1.

Predicted band. 10–20%, anchored on 2020 Exam 1 Q9b at 14% and 2017 Exam 1 Q7 at 30%. Note the 2018 distribution: zero per cent of the state scored four of five, which is the signature of an item where the last two marks are a single indivisible step. Splitting it into "simplify the integrand" and "evaluate" is the fix.

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D4.6 — Total surface area, curved surface plus ends

The design. Rotate an arc — parametric or Cartesian — about an axis to form a closed solid. Ask for the total surface area, which is the curved surface plus one or two circular discs whose radii come from the endpoint values.

The discriminator. Remembering the ends at all, and then getting their radii from the correct endpoint. The formula sheet supplies four surface-area formulas and no instruction about closure, so the word "total" — or the word "closed" — is the entire question.

The evidence. 2023 Exam 2 Section B Q3c at 38% — [RPT23 E2]: "Many students found the curved surface area only and did not include one or both ends. Of those who included two ends, errors with an incorrect radius were frequent." Q3d at 24%, where the same object is compared against a second solid with a variable upper terminal; [RPT23 E2]: "Some errors in the final value appeared to be due to a lack of brackets when entering expressions into a calculator." 2023 Exam 1 Q7 at 31% — "Some students did not use a substitution and instead tried to rely on inspection or recognition to find an antiderivative." 2023 Exam 2 Section A Q11 at 46%.

Marks and placement. 3 marks, Examination 2 Section B.

Predicted band. 25–38%, anchored on 2023 Exam 2 Section B Q3c at 38% and Q3d at 24%.

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D4.7 — Time as a definite integral, with the model starting part-way through

The design. A motion or filling model that only applies after an initial phase described separately. Part (a): write down an expression involving a definite integral that gives the time to reach a stated state. Part (b): evaluate it. Part (c): the same for the distance, remembering the first phase.

The discriminator. The lower terminal. It is the state at which the stated model begins, not zero — and the first phase's contribution must be added back at the end. The secondary discriminator is that the integrand is the reciprocal of the rate.

The evidence. 2017 Exam 2 Section B Q2di at 17% and Q2e at 16%. [RPT17 E2] is definitive: "This question was often misinterpreted by students, either by assuming that the model applied from the start of the skydiver's fall (integrating from 0 to 30) or by giving an answer that only gave the time after 2 seconds." 2015 Exam 2 Section B Q5div at 12%; 2018 Exam 2 Section B Q5ei at 41% — [RPT18 E2]: "An incorrect integrand, typically the reciprocal of the correct integrand, was common." 2014 Exam 2 Section B Q4c at 32% — "A number attempted to find t in terms of h, instead of using a definite integral."

Marks and placement. 2 + 1 + 3 marks, Examination 2 Section B.

Predicted band. 12–25%, anchored on 2017 Exam 2 Section B Q2di at 17% and Q2e at 16%. Removing the initial phase — making the model apply from rest — moves the band to 35–45%, on the evidence of 2018 Exam 2 Section B Q5ei at 41%.

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D4.8 — Euler's method with a negative step, shown

The design. A differential equation in two variables with a stated point. The target lies to the left of the start. Two or three steps. The instruction requires the working to be visible, and the answer to a stated accuracy.

The discriminator. The sign of the step, and then the visible intermediate value. The reports are explicit that a correct final answer with no tabulated working earns half the marks — which makes this a presentation separator as much as a computational one.

The evidence. 2017 Exam 2 Section A Q9 at 45%, where the step runs backwards. 2018 Exam 2 Section B Q3e at 25% — [RPT18 E2]: "Many students did not explicitly demonstrate their use of Euler's method." 2025 Exam 2 Section B Q3c at 38.93% — [RPT25 E2]: "Euler's method needed to be shown in some form to be awarded both marks. Several responses simply included the answer and did not show the development. A tabulated approach was acceptable as long as both [the intermediate value] and the final answer were shown." The technology-free version, 2007 Exam 1 Q7a at 36%, fails on fraction arithmetic: "Very often the lowest common denominator chosen for 10 and 11 was 121."

Marks and placement. 2 marks, Examination 2 Section B, or 2 marks on Examination 1 with exact fractions.

Predicted band. 25–40%, anchored on 2018 Exam 2 Section B Q3e at 25% and 2025 Exam 2 Section B Q3c at 38.93%.

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D4.9 — A solution curve on a printed slope field, read for a value

The design. Print a slope field. Ask for the solution curve through a stated point to be sketched on it, and then — "hence" — for a coordinate to be estimated to one decimal place from the sketch.

The discriminator. Following the ticks rather than crossing them, extending the curve across the whole printed domain, and reading the answer off the sketch as the word "hence" requires. A candidate who solves the equation algebraically and quotes a value inconsistent with their own graph loses the mark.

The evidence. 2017 Exam 1 Q8a at 17%. [RPT17 E1] lists six distinct failure modes in one sentence: "Several curves crossed the slope ticks rather than following them. Errors included: the final curve not being symmetrical; the curve not passing through (−1, 1); giving the value for x as around 1.2 (the value of the y-intercept); finding an approximate value from the solution in part b. even though this was inconsistent with the student's graph (part a. used the word 'hence'); Many graphs were almost flat between x = −0.5 and x = 0.5… Some drew the graph just to the x-intercepts rather than for the whole domain." 2007 Exam 1 Q8c at 13%; 2007 Exam 1 Q8a at 29% (sketching the field itself).

Marks and placement. 2 marks, Examination 1.

Predicted band. 12–22%, anchored on 2017 Exam 1 Q8a at 17%.

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D4.10 — A volume of revolution needing partial fractions with an irreducible quadratic

The design. A function presented under a square root, so that squaring it for the volume integral clears the root and leaves a rational function. The denominator has one linear factor and one irreducible quadratic, so the decomposition needs a linear numerator over the quadratic. The answer is required in a prescribed logarithm-plus-inverse-tangent form.

The discriminator. Three sequential decisions: the form of the decomposition, the modulus signs on the logarithmic term, and the prescribed final form. Each is separately marked and each has its own named failure in the reports.

The evidence. 2020 Exam 1 Q8 at 20%, 5 marks. 2017 Exam 1 Q10c at 14%, 4 marks. 2017 Exam 1 Q2 at 35% — [RPT17 E1] records both the wrong and the accidentally-right decomposition forms. 2007 Exam 1 Q4 at 35% — [RPT07 E1]: "The lack of modulus signs in this case leads to logarithms of negative numbers, so a correct answer cannot be properly obtained." [RPT13 E1] on the fake log law: "the following 'simplification' occurred often: 2logₑ3 − 5logₑ2 = logₑ(2·3/(5·2))."

Marks and placement. 5 marks, Examination 1.

Predicted band. 12–22%, anchored on 2020 Exam 1 Q8 at 20% and 2017 Exam 1 Q10c at 14%. Note the base rate: every five-mark written part in twenty years of November papers has been a separator, median 16%.

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D4.11 — Related rates with a unit conversion and a "hence"

The design. A geometric model whose linear dimension is given as a function of time, with the dimensions quoted in mixed units. Part (a) establishes one derivative. Part (b) asks, "hence", for the rate of change of an area or a volume at an instant.

The discriminator. Building the chain from variables that are actually related, and converting units before substituting. The "hence" is load-bearing: the archive records candidates who ignore it taking a far longer route and not finishing.

The evidence. 2009 Exam 2 Section B Q4e at 18% — [RPT09 E2]: "This related rates problem proved to be very difficult for most students. Often chain rule statements were used with variables which were unrelated to the problem." 2006 Exam 2 Section B Q1e at 22% — "Some did not use 'hence' and tried to use dA/dx to find dA/dt, which was a far more complicated approach." 2024 Exam 2 Section B Q3b at 37% — [RPT24 E2]: "Some students did not convert the depth measurement to metres." 2016 Exam 1 Q4 at 40% — [RPT16 E1]: "Quite a few students made errors with the formula for the surface area of a cube."

Marks and placement. 3 marks, Examination 2 Section B.

Predicted band. 18–32%, anchored on 2006 Exam 2 Section B Q1e at 22% and 2024 Exam 2 Section B Q3b at 37%.

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D4.12 — The modelling twist in the final part

The design. After a differential-equation model has been formulated, solved and interrogated across four or five parts, change exactly one assumption — turn off an inflow, impose a proportional harvest, introduce a delay before the model begins — and ask for one consequence.

The discriminator. Recognising that the earlier solution no longer applies and re-deriving from the model rather than reusing the answer. The second discriminator is attempting it at all: this design sits where §1.3's mechanism M8 operates, and the reports on every archive instance open with a variant of "Many students skipped this question."

The evidence. 2024 Exam 2 Section B Q3e at 4% — [RPT24 E2]: "Many students skipped this question without attempting to answer it. The most common error was not taking the 5-day delay into account." 2024 Exam 2 Section B Q3d at 17%, the part before it, with the same opening sentence. 2025 Exam 2 Section B Q3g at 18.05% — the tap is turned off and the model changes. 2021 Exam 2 Section B Q3c at 12%. 2023 Exam 2 Section B Q4g at 40% — a harvested logistic model, the mildest of the family because the twist is algebraic rather than temporal. 2011 Exam 2 Section B Q5e, the final part of a salt-tank staircase, at 11%.

Marks and placement. 2 marks, Examination 2 Section B, final part.

Predicted band. 5–20%, anchored on 2024 Exam 2 Section B Q3e at 4% at the bottom and 2023 Exam 2 Section B Q4g at 40% at the top; the difference is whether the twist changes the model or merely a parameter.

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2.5 Space and measurement

The largest area by parts (544 tagged, 878 marks), and the one the 2023 redesign changed most. 243 of 531 graded parts are separators (46%), and the median Examination 2 Section B part in this area is itself a separator at 49%.

The mechanics deletion removed 51 separators whose medians were 38% and 42% — the shallow end of the area. What survives is harder: the groups with the lowest medians are collisions and closest approach (27%), arc length and distance travelled (28%), velocity–time and multi-stage motion (30%), all of them current content.

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D5.1 — The distance between two skew lines

The design. Two lines in space in vector form, not parallel and not intersecting. Ask for the shortest distance between them. No formula for this is on the formula sheet.

The discriminator. Recognising the configuration and then constructing the formula — the common normal is the cross product of the two directions, and the distance is the scalar resolute of any connecting vector onto it. The archive shows the candidates who drew the configuration succeeding and the ones who worked from memory not.

The evidence. 2024 Exam 1 Q10 at 14%, 3 marks, dist 59 / 14 / 13 / 14. [RPT24 E1]: "A small number of students drew diagrams of skew lines and parallel planes to help motivate an appropriate formula for the distance between two skew lines. Other students tried to work from memory with varying results." The same report's general comments make the design legitimate: "Students may also have needed to recall or derive a formula (Question 10)."

Marks and placement. 3 marks, Examination 1, final question.

Predicted band. 10–20%, anchored on 2024 Exam 1 Q10 at 14%. Scaffolding it — part (i) a vector perpendicular to both, part (ii) the distance — should reach 30–35%, by comparison with 2019 Exam 2 Section B Q4d at 49% for the perpendicular-vector step alone.

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D5.2 — A family of parallel planes at a stated separation

The design. A plane, and a one-parameter family of planes parallel to it. Part (a): show that every member of the family is parallel to the given plane — one mark. Part (b): find all parameter values for which the separation equals a stated distance.

The discriminator. The distance formula carries a modulus, so the equation has two solutions. The archive measures this exactly: the state finds one. Targets M3.

The evidence. 2025 Exam 2 Section B Q5dii at 33.79%, 3 marks, dist 41.82 / 13.61 / 10.77 / 33.79. [RPT25 E2]: "Many responses did not demonstrate that the modulus needed to be used and consequently only one of the solutions was found." The reachable part (a), Q5di, at 67.92% — and even there, [RPT25 E2]: "Many responses did not identify that they were working with the normal vectors of the planes and many mixed up the multiple."

Marks and placement. 1 + 3 marks, Examination 2 Section B.

Predicted band. 25–38%, anchored on 2025 Exam 2 Section B Q5dii at 33.79%. This is the best-shaped separator in the current design's vector material: a reachable opener at 68%, a graded body, and a modulus that splits the top third.

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D5.3 — The angle between a line and a plane, asked as a sine

The design. A line and a plane in space. Ask for the sine of the angle between them, or for the acute angle correct to the nearest degree.

The discriminator. The angle between a line and a plane is the complement of the angle between the line's direction and the plane's normal. VCAA asks for the sine precisely because that is the quantity the complement makes natural, and the reports record the state computing the normal's angle and stopping.

The evidence. 2023 Exam 2 Section B Q5c at 50% — [RPT23 E2], one sentence: "A significant number of students did not proceed beyond finding the angle of [the normal]." [SAMPLE] Examination 1 Q15b asks for the sine explicitly, which is VCAA telling you the complement is the point. The contrast is 2023 Exam 2 Section A Q18 at 74%, which asks for the perpendicularity condition between two planes — normals only, no complement.

Marks and placement. 2 marks, Examination 1.

Predicted band. 30–45%, anchored on 2023 Exam 2 Section B Q5c at 50%, discounted for the absence of technology.

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D5.4 — Linear independence, where the answer is a complement

The design. Three vectors in space, one of which contains a parameter — ideally a parameter appearing squared, so the dependence condition has two roots. Ask for the values of the parameter for which the three vectors are linearly independent.

The discriminator. The negation. A candidate solves for dependence, finds the roots, and then has to write the complement of that finite set as the answer. A third of the state stops one line short. Targets M3 and M6 simultaneously.

The evidence. 2021 Exam 1 Q6 at 26%, 4 marks, dist 14 / 9 / 19 / 33 / 26 — 33% of the state scored exactly three of four. [RPT21 E1], verbatim: "Most students realised that they first needed to write down and solve a system of linear equations… Many students were able to find that p = ±2 for linear dependence but failed to conclude that p ∈ R \ {−2, 2} (or equivalent) for independence." The dependence version, 2008 Exam 1 Q3, at 16% — [RPT08 E1]: "A large number of students did not understand the concept of linear dependence." The multiple-choice versions run 57–76%.

Marks and placement. 4 marks, Examination 1.

Predicted band. 22–32%, anchored on 2021 Exam 1 Q6 at 26%. This is one of the best-shaped separators in the archive by the §1.4 test — every mark band populated, and a large, identifiable group one sentence from full marks.

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D5.5 — The volume of a pyramid whose height is a resolute

The design. Four points in space forming a pyramid on a parallelogram base. Part (a): the base area, by cross product. Part (b): a unit normal to the base. Part (c): the volume.

The discriminator. The height is the scalar resolute of an edge from the base onto the unit normal — not an edge length, and not the distance between two vertices. The archive's measurement of this is the most extreme in the whole area.

The evidence. 2019 Exam 2 Section B Q4e at 2%, 2 marks, dist 96 / 2 / 2. The preceding parts: Q4c (base area) at 22%, where [RPT19 E2] records "a significant proportion of students who multiplied the lengths of two adjacent sides of the parallelogram as if they were finding the area of a rectangle"; Q4d (unit normal) at 49%; Q4a (fourth vertex) at 35% — "A significant proportion of students did not correctly consider the order of the vertices."

Marks and placement. 2 marks, Examination 2 Section B, final part.

Predicted band. 2–12% as set in 2019. By the §1.4 test this defeats rather than grades; splitting part (c) into "find the height" and "hence the volume" should lift it to 20–25%, on the evidence of Q4d at 49% for a comparable single resolute step.

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D5.6 — An angle at a vertex with an unknown component

The design. Three points in space, one of them carrying a symbolic coordinate. The angle at one of the vertices is given exactly. Find the unknown.

The discriminator. Forming both displacement vectors out of the vertex rather than using position vectors — and then solving the resulting surd equation by hand. This is the archive's hardest pure dot-product item and the report names the error in one clause.

The evidence. 2017 Exam 1 Q5 at 11%, 4 marks, dist 25 / 5 / 45 / 13 / 11 — 45% of the state scored exactly two. [RPT17 E1]: "The most common errors involved finding the dot product of two position vectors rather than the vectors [out of the vertex]." 2006 Exam 2 Section B Q2b at 62% is the same construction with no unknown; 2015 Exam 2 Section A Q17 at 48% is the multiple-choice version; 2013 Exam 2 Section B Q4d at 44%.

Marks and placement. 4 marks, Examination 1.

Predicted band. 10–20%, anchored on 2017 Exam 1 Q5 at 11%. Giving the angle as a cosine rather than as an exact angle raises it by roughly ten points by removing one surd step.

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D5.7 — Express a vector as the sum of two resolutes, and name them

The design. Two vectors. Express the first as the sum of a component parallel to the second and a component perpendicular to it, identifying each clearly.

The discriminator. The final line. VCAA pays a mark for writing the sum down, and the report records a significant number of candidates computing both resolutes correctly and never assembling them. The secondary discriminator is direction — the resolute of the first in the direction of the second, not the other way round.

The evidence. 2014 Exam 2 Section B Q3a at 41%, 5 marks, dist 14 / 4 / 8 / 10 / 23 / 41, average 3.5. [RPT14 E2]: "Many students made arithmetic errors finding the resolutes, and a significant number omitted the final line, where a was to be expressed as the sum of the two vector resolutes. A significant number of students unsuccessfully attempted to find the resolutes from first principles, instead of applying the standard formulas." 2009 Exam 1 Q3 at 38%; 2020 Exam 1 Q5b at 28%; 2013 Exam 2 Section B Q4b at 40% — [RPT13 E2]: "A few students had the resolutes the wrong way around."

Marks and placement. 4 marks, Examination 1.

Predicted band. 30–42%, anchored on 2014 Exam 2 Section B Q3a at 41% and 2013 Exam 2 Section B Q4b at 40%, discounted slightly for the absence of technology.

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D5.8 — Speed as a "show that", then its extremes

The design. A plane or space path with a trigonometric parameterisation. Part (a): show that the speed simplifies to a single expression, using identities supplied on the formula sheet. Part (b): hence state the maximum and minimum speeds and the times at which they occur.

The discriminator. Part (a) is a "show that" whose marks are in the identity work, not the destination. Part (b) discriminates on labelling — the archive records candidates producing both extreme values and not saying which is which.

The evidence. 2025 Exam 2 Section B Q4d at 35.56%, 3 marks — [RPT25 E2]: "Another 'show that' question which required working that shows the use of trigonometric identities that are given on the formula sheet." 2013 Exam 2 Section B Q5a at 38% — [RPT13 E2]: "Common errors included the omission of j, and not identifying which were the minimum and maximum speeds. Some students gave only one speed and neglected to say whether it was the minimum or the maximum." 2015 Exam 2 Section B Q4d at 39%; 2024 Exam 2 Section B Q4ciii at 48% — "some students forgot to take the square root of the square of the speed"; 2024 Exam 2 Section B Q4ci at 23%.

Marks and placement. 3 + 2 marks, Examination 2 Section B.

Predicted band. 30–42%, anchored on 2025 Exam 2 Section B Q4d at 35.56% and 2013 Exam 2 Section B Q5a at 38%.

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D5.9 — Every time at which position is perpendicular to velocity

The design. A plane path with a trigonometric parameterisation over a stated interval. Ask for all values of the parameter at which the position vector is perpendicular to the velocity vector.

The discriminator. Two steps, the second of which is where the archive says the marks go: forming and expanding the dot product is manageable, and then solving the resulting trigonometric equation over the stated domain without producing extra roots or dropping some is not.

The evidence. 2023 Exam 1 Q10d at 7%, 2 marks, dist 44 / 49 / 7 — 49% of the state scored exactly one of two, which is the signature of a design whose first step is reachable and whose second is not. [RPT23 E1]: "While many students realised that they needed to solve r·ṙ = 0, many were not able to get to the final result." 2025 Exam 1 Q5b at 30%, dist 33 / 37 / 30 — [RPT25 E1] records an extra root in a common incorrect response. 2012 Exam 2 Section B Q4b at 34% is the "show that" version, where the expansion must be spelled out; [RPT15 E2] on the analogous 2015 item: "Some students simply asserted that ṙ(t)·r̈(t) = 0."

Marks and placement. 2 marks, Examination 1.

Predicted band. 7–18%, anchored on 2023 Exam 1 Q10d at 7% and 2025 Exam 1 Q5b at 30%; the spread between those two is entirely a function of how many roots the trigonometric equation has in the interval.

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D5.10 — Closest approach with a periodic separation

The design. Two objects whose positions are vector functions of time, at least one of them periodic. Part (a): write down an expression for the distance between them at time t. Part (b): find the minimum distance and when it occurs, to a stated accuracy.

The discriminator. Forming the magnitude of the difference of the position vectors, not the difference of their magnitudes; and then locating the global minimum, where a periodic separation offers several local ones.

The evidence. 2017 Exam 2 Section B Q5cii at 15% — [RPT17 E2]: "Incorrect answers involving other locally minimum values were frequent." 2016 Exam 2 Section B Q4di at 27% and Q4dii at 29% — [RPT16 E2]: "Some students used an expression for the difference between position vector magnitudes." 2024 Exam 2 Section B Q4e at 24% — "This question part was often not attempted." 2007 Exam 2 Section B Q4d at 20%; 2022 Exam 2 Section B Q4c at 38%.

Marks and placement. 1 + 2 marks, Examination 2 Section B.

Predicted band. 18–30%, anchored on 2016 Exam 2 Section B Q4di/Q4dii at 27%/29%. The 1-mark "write down the expression" opener is what keeps it grading: 2017 Exam 2 Section B Q5ci demonstrates the pattern.

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D5.11 — Distance travelled versus displacement, on one path

The design. A path that reverses direction inside the stated interval. Part (a): the displacement over the interval. Part (b): the distance travelled. Part (c): the length of the path over a longer interval, as a definite integral, evaluated.

The discriminator. Three distinct objects with three distinct computations — a vector difference, a magnitude, and an integral of a magnitude — and terminals that must be in the parameter, not in the Cartesian variable. The archive shows both directions of confusion.

The evidence. 2022 Exam 2 Section B Q4d at 45% — [RPT22 E2]: "A number of incorrect student responses incorrectly found the straight-line distance between the endpoints of the travel. Some students used the Cartesian form of the curve to find the integrand, but very few of these used the correct limits, incorrectly using the time values." 2009 Exam 2 Section B Q3gi at 28% — "The most difficult part of this question for students was writing in the terminals for the definite integral, even though the values to use were given." 2008 Exam 2 Section B Q3ei at 19% and Q3eii at 8%. 2017 Exam 1 Q9b at 30% — [RPT17 E1]: "The most common error was using scalars throughout (finding distance rather than displacement)." 2025 Exam 2 Section B Q4g at 63.7% — [RPT25 E2]: "The definite integral was required."

Marks and placement. 1 + 2 + 2 marks, Examination 2 Section B.

Predicted band. 20–32% on the distance-travelled part, anchored on 2009 Exam 2 Section B Q3gi at 28% and 2022 Exam 2 Section B Q4d at 45%.

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D5.12 — Acceleration from velocity as a function of position, then the limit

The design. Velocity given as a rational or root function of displacement. Part (a): find the acceleration at a stated position. Part (b): find the value the velocity approaches as displacement grows without bound.

The discriminator. Part (a) turns on choosing between the two position-based acceleration forms — the archive shows one of them removing a square root that the other preserves — and on not using the time-based form at all. Part (b) turns on giving a value, not an inequality or a limit statement.

The evidence. 2023 Exam 1 Q3a at 35% and Q3b at 50% — [RPT23 E1] lists as a weakness "remembering to use an alternative form for acceleration", and on part (b): "Many students wrote [an inequality] for their answer." 2012 Exam 1 Q8 at 28% — [RPT12 E1]: "Many of those who used [v dv/dx] got into significant difficulty with square roots… those who used [d/dx(½v²)] were usually more successful, since squaring removed the square roots." 2024 Exam 1 Q9a at 32%; 2008 Exam 1 Q5a at 48% — [RPT08 E1]: "A large proportion thought that acceleration was given by a = dv/dx, despite a = v dv/dx being given on the formula sheet." 2009 Exam 2 Section B Q5c at 47% — "their answers were often poorly expressed with forms such as v < √(g/2) and v → √(g/2)."

Marks and placement. 2 + 1 marks, Examination 1.

Predicted band. 30–42%, anchored on 2023 Exam 1 Q3a at 35% and 2012 Exam 1 Q8 at 28%.

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2.6 Data analysis, probability and statistics

Statistics is the strongest-performing area in the subject on Examination 2 — median full-mark rate 67% across 2023–2025 written parts — and the worst on Examination 1, where 13 of 18 graded parts are separators and the median is 44%. [RPT21 E1] names it outright under areas of weakness: "poor understanding of hypothesis testing and confidence intervals."

The structural rule has held without exception since statistics entered the course: one Examination 1 question worth 3–5 marks, and Examination 2 Section B Question 6 — the last question on the paper — worth 8–10 marks in six to nine escalating parts. Thirteen sittings, thirteen Question 6s.

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D6.1 — A Type II error, stated in the language of the context

The design. A hypothesis test already completed across earlier parts. Then: state that the true mean is in fact a different value, and ask for the probability that the test reaches the wrong conclusion — phrased entirely in the words of the stem, with no statistical vocabulary at all. The tell is the phrase "is in fact".

The discriminator. Two stages that must be run against different centres: the critical sample mean is computed under the null hypothesis, and the probability is then computed under the true mean, using the same sample-mean standard deviation. The reports name both halves failing, plus the tail.

The evidence. 2021 Exam 2 Section B Q6e at 9%, 2 marks. 2018 Exam 2 Section B Q6f at 11%, 1 mark — [RPT18 E2]: "Only a small number of students attempted this question." 2023 Exam 2 Section B Q6g at 39%; 2024 Exam 2 Section B Q6c at 44% — [RPT24 E2]: "Common errors were: students sometimes did not find the critical value for X̄ when H₀ is true; students used the wrong tail." 2016 Exam 2 Section B Q6e at 46%; 2025 Exam 2 Section B Q6h at 53.65% — [RPT25 E2]: "Most students were able to find this Type II error if they were successful in part g."

Marks and placement. 2 marks, Examination 2 Section B Question 6, final part.

Predicted band. 10–30%. The band is wide because the archive shows the design's difficulty is almost entirely a function of whether the critical value is chained from a previous part (2025: 53.65%, with the critical value asked for separately at 62.95%) or has to be produced inside the same part (2021: 9%; 2018: 11%). Chain it and it grades; fold it in and it defeats.

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D6.2 — The largest sample for which a total stays within a capacity

The design. A capacity constraint on the total of n independent identically distributed variables. Ask for the largest n for which the probability of exceeding the capacity stays below a stated small value.

The discriminator. The total has standard deviation σ√n, not σ/√n. The parameter n appears in the mean and in the spread simultaneously, so the inequality has to be solved numerically or by trial. The archive's report names the exact misreading.

The evidence. 2021 Exam 2 Section B Q6a at 14%, 2 marks, dist 73 / 13 / 14. [RPT21 E2]: "Successful students used a trial-and-error approach or used a standardised value to solve for n. A common error was to approach this as a sampling problem." 2017 Exam 2 Section B Q6d at 16%, 3 marks — the same structure solving for the largest allowable population standard deviation rather than for n. 2021 Exam 2 Section B Q6b at 24%.

Marks and placement. 2 marks, Examination 2 Section B.

Predicted band. 12–25%, anchored on 2021 Exam 2 Section B Q6a at 14% and 2017 Exam 2 Section B Q6d at 16%. Note the dist: 73% scored zero. Adding a one-mark opener ("write down the mean and standard deviation of the total") converts it from defeating to grading.

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D6.3 — The difference of two sample means

The design. Two independent random samples from the same population, of stated and preferably different sizes. Ask for the probability that their means differ by less than a stated amount.

The discriminator. Two compounded rules: the variance of a difference is a sum, and each sample mean has variance divided by its own sample size. Then the two-sided reading of "differ by", which the 2019 report says almost nobody saw.

The evidence. 2019 Exam 2 Section B Q6b at 15%, 3 marks, dist 67 / 15 / 2 / 15. [RPT19 E2]: "Many students did not make a reasonable start to this question, or they were unable to correctly find the variance for the combined distributions. Very few students indicated an understanding that the difference between the samples could be negative and found Pr(X̄₁ − X̄₂ < 2) rather than correctly finding Pr(−2 < X̄₁ − X̄₂ < 2)." The same report named "working with linear combinations of sample means" as a weakness of the whole paper. The single-observation version, 2022 Exam 2 Section A Q18 at 42%, has a distractor at exactly half the correct answer — the one-tailed value — which took 16%.

Marks and placement. 3 marks, Examination 2 Section B.

Predicted band. 12–25% as a single three-mark part, anchored on 2019 Exam 2 Section B Q6b at 15%. The 2019 distribution is the archive's cleanest example of a cliff: 67% at zero, 15% at full, 2% in between. Split it — one mark for the distribution of the difference, two for the probability — and it should grade at 30–40%.

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D6.4 — The sample size needed for a stated change in interval width

The design. A confidence interval already computed in an earlier part. Ask how large a sample would be needed to decrease its width by a stated percentage, or by what factor the sample size must be multiplied.

The discriminator. Width is inversely proportional to the square root of the sample size, so a stated proportional reduction implies a squared change in n. The second discriminator is the English: "decrease the width by 60%" means the new width is 40% of the old.

The evidence. 2023 Exam 2 Section B Q6c at 28%, 1 mark — [RPT23 E2] names this construction as an area of weakness for the whole paper: "determining sample size to achieve a given change in the width of a confidence interval." 2017 Exam 2 Section A Q19 at 44%, where the four distractors are all plausible mis-squares. 2024 Exam 2 Section B Q6g at 46% — the margin-of-error version, where [RPT24 E2] closes with "As n is an integer, then n = 62". 2025 Exam 2 Section B Q6d at 53.11% — [RPT25 E2]: "Some responses rounded down… but this would have resulted in more than 1 mL."

Marks and placement. 1–2 marks, Examination 2 Section B.

Predicted band. 25–40%, anchored on 2023 Exam 2 Section B Q6c at 28% and 2017 Exam 2 Section A Q19 at 44%. This design is unusually efficient: a one-mark part that behaves like a three-mark part.

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D6.5 — The rejection region, not the critical value

The design. A two-tailed test at a stated significance level. Ask for the set of sample-mean values that would lead to the null hypothesis being rejected, correct to a stated accuracy.

The discriminator. Producing a region rather than a number. The archive measures this precisely: the state computes the critical value correctly and then does not write the inequality. The secondary discriminator is that a two-tailed test at a stated level uses half that level in each tail. Targets M6.

The evidence. 2021 Exam 2 Section B Q6d at 16%, 1 mark — [RPT21 E2], definitive: "Some students calculated [the critical value] but did not proceed to answer the question correctly as a range of values." 2019 Exam 2 Section B Q6f at 36% — [RPT19 E2]: "The most frequent incorrect response was 372.5, resulting from Pr(X̄ < x_c) = 0.05" on a test that was two-tailed. 2024 Exam 2 Section B Q6d at 58% — the version where both bounds are asked for by name, which is why it is twenty points higher. 2016 Exam 2 Section B Q6d at 43%; 2018 Exam 2 Section B Q6e at 48%.

Marks and placement. 2 marks, Examination 2 Section B.

Predicted band. 15–30%, anchored on 2021 Exam 2 Section B Q6d at 16% and 2019 Exam 2 Section B Q6f at 36%. Naming the two bounds in the stem, as 2024 did, moves it to 55–60% — the difference is entirely in whether the candidate is told the answer is a region.

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D6.6 — An inverse confidence interval, by hand

The design. Print a confidence interval and its level. Give two of the remaining quantities — the sample size, the population standard deviation, the sample mean — and ask for the others. Supply the normal quantile in the stem, since there is no technology.

The discriminator. The half-width is the quantile times the standard deviation over the root of the sample size, so recovering the standard deviation means multiplying back by that root — the step the distractor families in the multiple-choice versions are all built from. The secondary discriminator is exact arithmetic with a supplied quantile.

The evidence. 2016 Exam 1 Q2 at 17%, 3 marks — the by-hand confidence interval, with the instruction to "use an integer multiple of the standard deviation". 2021 Exam 1 Q3c at 56%, where the quantile is supplied and only the interval is wanted; [RPT21 E1]: "Students frequently used σ rather than σ/√n." The multiple-choice inversions run 62–65%: 2023 Exam 2 Section A Q20 at 63%, 2018 Exam 2 Section A Q18 at 62%, 2021 Exam 2 Section A Q18 at 65%. VCAA has already moved this design onto Examination 1 in the NHT series — 2025 NHT Exam 1 Q2 and 2026 NHT Exam 1 Q6, both 3 marks, both with the quantile supplied.

Marks and placement. 3 marks, Examination 1.

Predicted band. 20–35%, anchored between 2016 Exam 1 Q2 at 17% and the Section A inversions at 62–65%: removing technology from a 63% item and adding a second unknown should cost roughly thirty points.

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D6.7 — A weighted linear combination, solved backwards

The design. Two independent random variables with stated means and variances. The mean and variance of a weighted combination of them are given. Find the integer weights.

The discriminator. The mean equation is linear and the variance equation is quadratic, so the system has two solutions and the integrality condition in the stem is what discards one. The reports also record the state squaring the linear equation term by term, losing the cross term.

The evidence. 2018 Exam 1 Q4 at 43%, 4 marks, dist 5 / 8 / 8 / 36 / 43, average 3.0. [RPT18 E1]: "Common problems included failing to reject the non-integer solution and only stating the solution with minimal or no working… Algebraic errors were common, with some students having difficulty solving a quadratic equation. Quite a few students 'squared' both sides of the first equation to obtain 4a² + 4b² = 100." 2019 Exam 2 Section A Q19 at 75% is the technology-active version. 2024 Exam 1 Q6b at 37% is the forward version, where [RPT24 E1] records students "frequently forgetting to square either the cost values or the standard deviation".

Marks and placement. 4 marks, Examination 1.

Predicted band. 35–48%, anchored on 2018 Exam 1 Q4 at 43%. This is the best-shaped statistics separator in the archive by the §1.4 test: every mark band populated, 36% of the state at three of four, and an average of 3.0 out of 4.

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D6.8 — A random variable that is a non-linear function of another

The design. A geometric quantity expressed in terms of a normally distributed linear dimension — one part where the expression is affine in that dimension, one part where it is not. Ask for the mean and variance of each.

The discriminator. The variance rule applies to the affine case only; an additive constant changes the mean and not the variance; and in the non-affine case there is no rule at all, so the question must be re-read to see which quantity is actually random. The reports name this as a whole-paper weakness.

The evidence. 2019 Exam 1 Q3a at 89%, Q3b at 30%, Q3c at 50% — three parts of one question spanning sixty points. [RPT19 E1] on Q3b: "Students could use fractions to find [the variance]. Students who used this approach tended to score more highly than those using decimals… A number of students omitted the π from their answer." [RPT22 E2] names, as an area of weakness for the whole paper, "working with random variables that are functions of other variables." 2024 Exam 2 Section A Q20 at 48% is the scaling-then-summing version, where 25% chose the option that scales after summing.

Marks and placement. 1 + 2 + 2 marks, Examination 1.

Predicted band. 25–40% on the non-affine part, anchored on 2019 Exam 1 Q3b at 30% and Q3c at 50%.

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D6.9 — A scaled variable and a sum of variables, side by side

The design. Two parts that differ only in whether n observations are summed or one observation is multiplied by n. Ask for a probability in each case.

The discriminator. The standard deviations differ by a factor of the square root of n, and the archive shows the state conflating them in both directions. Placing the two cases adjacently makes the distinction the entire question rather than an incidental slip.

The evidence. 2022 Exam 1 Q3a at 31%, 2 marks, dist 59 / 9 / 31 — [RPT22 E1]: "A large number of students did not find the correct standard deviation… Students who successfully evaluated [it] often drew diagrams of the probability density function." 2021 Exam 2 Section B Q6b at 24% — [RPT21 E2]: "There was evidence of confusion between the correct sum of four random variables and incorrectly scaling a random variable by a factor of four." 2023 Exam 2 Section A Q19 at 67%, where 17% chose the option generated by multiplying the standard deviation by n instead of its square root.

Marks and placement. 2 + 2 marks, Examination 1.

Predicted band. 25–40%, anchored on 2022 Exam 1 Q3a at 31% and 2021 Exam 2 Section B Q6b at 24%; placing the two side by side should raise the second part relative to the archive instances, because the contrast is signposted.

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D6.10 — A conclusion that must contain three things

The design. After a p-value has been computed, ask the question in the stem's own words — whether a claim is correct, whether a process should be paused, whether a campaign succeeded — and require the reason to be given in terms of the p-value.

The discriminator. A full-mark conclusion contains three components and eight separate reports record each of them being omitted: the numerical comparison with both figures present, the decision about the null hypothesis, and the translation into the context. It is the only place in the area where the mark is awarded for prose.

The evidence. 2017 Exam 2 Section B Q6e at 45%, 2 marks; 2021 Exam 1 Q3bii at 49%; 2021 Exam 2 Section B Q6ciii at 55% — [RPT21 E2]: "Some students stated a correct conclusion but did not give a reason by referencing the p-value"; 2019 Exam 2 Section B Q6e at 59%; 2025 Exam 2 Section B Q6fii at 68.04% — [RPT25 E2]: "Responses needed to comment on the company's claim and also quote the significance level." [RPT18 E2] records the comparison stated backwards; [RPT21 E1] records candidates "confusing the p value with the significance level"; [RPT24 E2] records candidates who "did not fully answer the question regarding whether or not the machine should be paused".

Marks and placement. 2 marks, Examination 2 Section B.

Predicted band. 40–55%, anchored on 2017 Exam 2 Section B Q6e at 45%. This is deliberately a mid-band design: it separates the careful from the fast rather than the strong from the weak, and it is the cheapest available separator between a 42 and a 46.

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D6.11 — The expected value of a continuous random variable, by integration

The design. A probability density function on a closed interval, whose product with the variable requires partial fractions before it can be antidifferentiated. Part (a): "use integration to show that" the mean takes a stated value. Part (b): a sample-mean probability built on that mean, with the normal value supplied.

The discriminator. Mathematical Methods assumed knowledge arriving without warning inside a Specialist statistics question; then the partial fractions; then the "show that" working, since the answer is printed and only the derivation earns marks.

The evidence. 2025 Exam 1 Q4a at 37%, 3 marks, dist 32 / 23 / 9 / 37. [RPT25 E1]: "Students should know how to find the expected value (mean) of a continuous random variable. This question required partial fractions to be applied. A small number of students did not realise this… either because: the initial expression missed the t term on the numerator, or the coefficients for the partial fractions were incorrect, or the working towards the given answer was unclear." Q4b at 49%, which the mean feeds.

Marks and placement. 3 + 2 marks, Examination 1.

Predicted band. 30–42%, anchored on 2025 Exam 1 Q4a at 37%. The dist shows a working design: every band populated and a quarter of the state at one of three.

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D6.12 — The simulation count, with two numbers in play

The design. A stated number of samples, each of a stated size, with a confidence interval computed for each. Ask in how many of those intervals the population mean would be expected to lie.

The discriminator. Multiplying the confidence level by the number of samples, not by the sample size. Every stem of this type deliberately puts two numbers in play and the marker is checking which one was used. It is the study design's "use of simulation to illustrate variations in confidence intervals between samples" reduced to one mark.

The evidence. 2024 Exam 2 Section B Q6f at 51% — [RPT24 E2]: "Some students incorrectly used 50 rather than 40 as the number of samples." 2023 Exam 2 Section B Q6b at 58%; 2025 Exam 2 Section B Q6c at 76.91% — [RPT25 E2]: "Found by seeking 95% of 300."

Marks and placement. 1 mark, Examination 2 Section B.

Predicted band. 45–60%, anchored on 2024 Exam 2 Section B Q6f at 51%. The band is a function of how close the two numbers are: 2025's 300 samples of 25 is easy to disambiguate and scored 77%; 2024's 40 samples of 50 is not and scored 51%.

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3. Designs for the content the 2023 study design added and the archive barely covers

Twelve designs on material that is unambiguously in scope and has been examined once, never, or only in the Northern Hemisphere sitting. For each: what the study design permits, and what the specifications constrain.

The constraint that governs all of them is [SPEC], verbatim: "All of the content from the areas of study and the key knowledge and key skills that underpin the outcomes in Units 3 and 4 are examinable", set against "Each examination will conform to these specifications and will test a representative sample". Content can be absent from a paper without being out of scope — and a topic named in [SD] with no archive instance is a standing risk, not a settled exclusion.

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D7.1 — Proof by cases on residue classes

What the study design permits. "natural deduction and proof techniques: direct proofs using a sequence of direct implications, proof by cases, proof by contradiction, and proof by contrapositive", and an overview naming divisibility as a proof context.

What the specifications constrain. Divisibility is a context for proof, not content in its own right. 01-study-design.md and 02-discrete-proof.md both record that "Euclidean", "modular", "prime" and "greatest common divisor" appear nowhere in the Units 3 and 4 section of [SD], and that a corpus search for Euclidean algorithm|gcd|highest common factor across all 97 files returns zero hits; every modul hit is "modulus". A question cannot examine modular arithmetic for its own sake. A partition by remainder on division by a small integer is legitimate only if the cases are described in ordinary language — "an integer is of one of the forms …" — and the statement being proved is elementary.

The design. A divisibility or parity statement about an integer expression whose proof needs a three-way or four-way partition. Prove it, stating the partition and its exhaustiveness.

The discriminator and its evidence. The exhaustiveness sentence. 2008 Exam 1 Q8c at 36% shows the state both over-proving and under-proving a single-condition claim; 2024 Exam 1 Q2 at 65% shows that parametrising an integer is well within reach. 3–4 marks, Examination 1. Band 25–35%.

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D7.2 — A written proof by contradiction of a surd inequality

What the study design permits. "proof by contradiction" is named content, and [SAMPLE] Examination 1 Q5 is VCAA's own instance in exactly this shape.

What the specifications constrain. No irrationality proof has ever been set and 02-discrete-proof.md records that a search of all 97 corpus files for irrational returns zero hits. Irrationality is the canonical textbook illustration and remains the most likely unexamined contradiction question, but it should be treated as drill rather than as a prediction; the surd-inequality form has a VCAA exemplar and the irrationality form does not.

The design. An inequality between sums of surds. Instruct: use proof by contradiction. The negation of a strict inequality is non-strict, squaring is valid because both sides are positive, and the absurdity must be pushed to something visibly impossible and named as impossible.

The discriminator and its evidence. The negation, and the closing sentence. The only live relative is the recognition item 2024 NHT Exam 2 Section A Q1, which prints VCAA's own model proof of exactly this statement. The nearest graded anchors are the induction questions at 21% and 29%. 3 marks, Examination 1. Band 25–35%.

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D7.3 — De Moivre's theorem proved by induction

What the study design permits. [SD] Area of Study 3 names it verbatim: "De Moivre's theorem, proof for integral powers, powers and roots of complex numbers in polar form". Area of Study 1 adds: "The concepts, skills and processes from this area of study are to be applied in the other areas of study."

What the specifications constrain. Nothing excludes it. The compound-angle formulas are on the formula sheet, so the algebra of the step is fully supported. This is the single most conspicuous unexamined question in the whole design.

The design. Prove by mathematical induction that the nth power of a complex number in polar form has modulus raised to the n and argument multiplied by n, for positive integer n. Base case, assumption, step by multiplying by one more factor and expanding with the compound-angle identities, conclusion.

The discriminator and its evidence. The step is not an index law — it is one more multiplication followed by an expansion, which is exactly the failure [RPT23 E1] names on 2023 Exam 1 Q8 (21%): "Many students appeared to be thinking of index laws." VCAA has already set the inductive step alone as a two-mark "show that" — 2024 NHT Exam 2 Section B Q2a, no published percentage. 4 marks, Examination 1. Band 15–25%, below 2023 Exam 1 Q8's 21% only if the compound-angle expansion is required in full.

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D7.4 — Induction on a supplied recurrence or matrix power

What the study design permits. The Area of Study 1 overview names the contexts: "sequences and series including partial sums and partial products and related notations, complex numbers, matrices, vectors and calculus."

What the specifications constrain. Matrices and recurrence relations are Units 1 and 2 machinery. A Units 3 and 4 question must therefore supply the recurrence or the matrix in the stem and ask only for the proof; it cannot assume a technique for solving recurrences.

The design. Supply a first term and a recurrence, together with a closed form; prove the closed form by induction. The matrix variant supplies a two-by-two matrix and a closed form for its nth power.

The discriminator and its evidence. The step must quote the recurrence before substituting the assumption — the same "identify the defining operation" failure as D1.1. 2023 Exam 1 Q8 at 21%; 2025 Exam 1 Q7 at 29%. 4 marks, Examination 1. Band 25–35% for the recurrence version, which is more transparent than the derivative version; 18–26% for the matrix version.

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D7.5 — Quantifier negation, written

What the study design permits. "quantifiers 'for all' and 'there exists', examples and counter-examples" is a content dot point.

What the specifications constrain. 02-discrete-proof.md records that formal logic notation — conjunction, disjunction, negation and implication as symbols to be manipulated — is absent from [SD] and from every paper, and that all four live contrapositive items are set in English sentences. The item must be in words, not in symbols.

The design. A universally quantified implication about integers. Part (a): write its negation. Part (b): state what would have to be produced to establish that negation.

The discriminator and its evidence. Negating a universally quantified implication yields an existential conjunction, not another implication. [SAMPLE] Examination 2 Section A Q1 offers exactly that pair as two of its five distractors, alongside the correct contrapositive. The only measured evidence is 2024 Exam 2 Section A Q1 at 72% with 20% on the one-operation option. 1 + 1 marks, Examination 1. Band 30–45%.

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D7.6 — A reduction formula by parts

What the study design permits. "integration by parts" is a content dot point, new in 2023, and the formula is on the sheet.

What the specifications constrain. Symbolic reduction formulae are not named anywhere in [SD] and have no formula-sheet support. 09-nht-specialist.md flags them as one of the objects VCAA trials in NHT. The safe form is therefore the one already set: a multiple-choice item, or an extended-response part in which the reduction relation is supplied and only one application is required.

The design. A definite integral indexed by a positive integer. Ask, in multiple-choice form, for the relation between consecutive terms; or, written, for one application of parts with the boundary term evaluated.

The discriminator and its evidence. Evaluating the boundary term. 2023 Exam 2 Section A Q10 at 33% — the only live instance — has a distractor consisting of the unevaluated boundary term plus the reduced integral, which is exactly what a candidate who applies parts and stops produces. 2024 NHT Exam 2 Section A Q11 repeats the type. 1 mark, Examination 2 Section A. Band 28–40%.

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D7.7 — A non-linear asymptote, named

What the study design permits. Area of Study 2 names "graphs of rational functions of low degree, their asymptotic behaviour". Outcome 1 key knowledge says "the form of their sketch graphs and their key features, including linear asymptotes".

What the specifications constrain. The word linear in the key knowledge is VCAA's own hedge, and 03-functions-graphs.md records that where a function has a curved asymptote the live papers have asked only for the vertical ones — 2016 Exam 2 Section B Q1c, 2024 Exam 2 Section B Q1a — and that 2026 NHT Exam 2 Section B Q1e prints the instruction as "the straight-line asymptotes with their equations". The one place a non-linear asymptote has been asked for by name is 2026 NHT Exam 1 Q9, which asks for "the equation of the parabolic asymptote" of a cubic-over-linear rational function and then for the sketch.

The design. A rational function whose numerator exceeds its denominator in degree by two. Part (a): express it as a polynomial part plus a proper remainder. Part (b): state the equation of the curved asymptote. Part (c): sketch, labelling both asymptotes.

The discriminator and its evidence. Students who have drilled oblique asymptotes have a routine for degree difference one and none for degree difference two. The nearest graded anchors are the oblique-asymptote sketches: 2023 Exam 1 Q1b at 39%, 2008 Exam 1 Q1 at 19%, 2025 Exam 1 Q9c at 16%. 1 + 1 + 3 marks, Examination 1. Band 12–22% on the sketch. Set this only if the course has covered it; it is the clearest case in the document of a design that could defeat for the wrong reason.

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D7.8 — Continuity as a named object

What the study design permits. Nothing names continuity directly. It arrives through Area of Study 2's "graphs of simple quotient functions" and through the Outcome 1 key knowledge on key features, and it is examined as the value of a parameter that removes a discontinuity.

What the specifications constrain. 09-nht-specialist.md notes that "continuous" is not a term of art anywhere in the study design, and that 2026 NHT Exam 2 Section A Q2 is the instance that introduced it. A question may ask for the value that makes a hybrid continuous; it may not ask for a definition of continuity or for a limit argument.

The design. A hybrid function whose branches are a rational function with a removable discontinuity and a constant, or an inverse circular function. Find the value that makes it continuous, then sketch showing whether the point is filled or open.

The discriminator and its evidence. Evaluating the simplified rule at the excluded value rather than the original. 2025 Exam 1 Q9b at 46%; 2024 Exam 2 Section A Q2 at 48%; 2025 Exam 1 Q9c at 16% for the sketch with the hole. 1 + 3 marks, Examination 1. Band 30–45% for the value, 15–25% for the sketch.

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D7.9 — Pseudocode and algorithm tracing

What the study design permits. Outcome 1 key skill: "Interpret and apply algorithms … including the use of pseudocode". Outcome 2 key knowledge: "key elements of algorithm design, including sequencing, decision-making and repetition". Outcome 3 is defined in terms of computational thinking.

What the specifications constrain. Examination 1 assesses Outcome 1 only and [SPEC] excludes Outcome 3 from it, so a pseudocode item on Examination 1 must be pure interpretation with no technology dependence. The live instances are all on Examination 2: [SAMPLE] Examination 2 Section A Q2, 2023 Exam 2, and 2025 Exam 2 Q4.

The design. A short pseudocode block implementing a numerical method already in the course — Euler's method is the natural one, since its recursion is on the formula sheet. Ask what the block outputs, or which line would have to change to implement a stated variant.

The discriminator and its evidence. Tracing a loop by hand and matching it to the mathematics. 2023 Exam 2 Section A Q6 at 69% is Euler's method embedded in pseudocode, and its comparative ease is the point: the pseudocode itself is not the difficulty. The difficulty is in the variant question, which is where 2018 Exam 2 Section B Q3e at 25% and 2025 Exam 2 Section B Q3c at 38.93% put it. 2 marks, Examination 2. Band 35–50%.

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D7.10 — Simulation of the sampling distribution

What the study design permits. [SD] names simulation twice and both times as content: "simulation of repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of X̄", and "the use of simulation to illustrate variations in confidence intervals between samples".

What the specifications constrain. Simulation is technology-active, so Examination 2 only. The archive contains no question about the simulation process; the only live expression of the dot point is the count question (D6.12), which is arithmetic rather than simulation.

The design. Describe a simulation: a stated number of samples of a stated size drawn from a named non-normal parent. Part (a): state the mean and standard deviation the sample means would be expected to have. Part (b): explain what would change if the sample size were increased, and what would not. Part (c): the interval-count question.

The discriminator and its evidence. Part (b) — separating what depends on n from what does not — is the only place in the area where the answer is prose about a distribution. The closest measured relative is 2019 Exam 2 Section A Q20 at 70%, where the parent is uniform and the sample size is large. The count question runs 51–77%. 1 + 2 + 1 marks, Examination 2 Section B. Band 40–55% on part (b).

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D7.11 — The Type I / Type II error diagram

What the study design permits. "hypothesis test, relating the formulation, conduct, errors and results in terms of conditional probability."

What the specifications constrain. The one live attempt was invalidated: 2023 Exam 2 Section B Q6h shows pct = 100% in the data because, per [RPT23 E2], "Following the identification of an error in the question stimuli, this question was invalidated." That 100% is a full-marks-to-all award and is not evidence the question was easy. The NHT series has set the same idea successfully — 2023 NHT Exam 2 Section B Q6ci–cii asks for the sampling distribution under the true mean to be sketched and the error region shaded.

The design. Print two normal curves with the same spread, centred at the hypothesised and true means. Ask for the critical sample mean to be marked and the region representing the Type II error to be shaded, then for its probability.

The discriminator and its evidence. The error region is on the non-rejection side, measured under the true-mean curve — two independent choices. 2024 Exam 2 Section A Q19 at 68% is the definitional version, where 8% chose the Type I definition and 15% chose an option that is not a statistical statement at all. The computed versions run 9–54%. 2 marks, Examination 2 Section B. Band 30–45% for the diagram, on the evidence that drawing is easier than computing: compare 2024 Exam 2 Section B Q6c at 44% for the computation.

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D7.12 — Three planes and their geometric interpretation

What the study design permits. This is hidden scope and worth naming: Outcome 1 key knowledge includes "systems of equations with two and three variables and their geometric interpretation", which appears in no area-of-study dot point. It is what licenses three-plane problems.

What the specifications constrain. The live instances have asked only for a unique intersection point — 2025 Exam 2 Section B Q5a at 73.51%. The interpretation half of the key knowledge (the cases where three planes meet in a line, or in none) has never been examined.

The design. Three planes with one parameter. Part (a): the point of intersection for a generic parameter value. Part (b): the parameter values for which the system has no unique solution, and a description of the configuration in each case.

The discriminator and its evidence. Part (b) is a parameter-enumeration question dressed as geometry, and therefore inherits the failure rate of D2.1: 2021 Exam 2 Section B Q1di at 6%, 2024 Exam 2 Section B Q1di–iii at 25–27%. The reachable opener is 2025 Exam 2 Section B Q5a at 73.51%, and the vector machinery — the direction of a line of intersection as a cross product of normals — is measured at 2025 Exam 2 Section B Q5bi at 51.97%. 1 + 3 marks, Examination 2 Section B. Band 20–30% on part (b).

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4. Designs that cross areas of study

[SD] licenses this explicitly for proof — "The concepts, skills and processes from this area of study are to be applied in the other areas of study" — and implicitly everywhere else, since [SPEC] says each examination "will test a representative sample of the key knowledge and key skills from all outcomes". Examination 2 Section B is where crossing happens: 07-exam-craft.md documents the staircase structure in which a ten-mark question walks through two or three areas.

Ten designs. For each, the seam — the place where the two areas meet — is stated explicitly, because the seam is always the discriminator.

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X1 — A vector path carried into arc length and surface area

The design. A curve given as a position vector function of time on a stated interval. Part (a): the Cartesian equation and a sketch of the arc. Part (b): the speed, as a "show that". Part (c): the distance travelled, as a definite integral, evaluated. Part (d): the curve is rotated about an axis — the surface area, as a definite integral, evaluated.

The seam. The terminals. Part (c) integrates over the parameter; part (d) uses the same parameter but a different integrand and a multiplier that depends on which axis the rotation is about. A candidate who switches to Cartesian terminals anywhere in the chain loses everything downstream.

The evidence. 2022 Exam 2 Section B Q4d at 45% — [RPT22 E2]: "Some students used the Cartesian form of the curve to find the integrand, but very few of these used the correct limits, incorrectly using the time values." 2009 Exam 2 Section B Q3gi at 28% — "The most difficult part of this question for students was writing in the terminals." 2023 Exam 2 Section B Q3c at 38% for the surface area with ends; 2025 Exam 2 Section A Q9 at 49% for the parametric surface-area form. 3 + 3 + 2 + 3 marks, Examination 2 Section B. Band 25–35% across parts (c) and (d), anchored on the 28% and 38% instances.

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X2 — De Moivre proved by induction, then used

The design. Part (a): prove De Moivre's theorem for positive integer powers by mathematical induction. Part (b): hence write down the nth roots of a given complex number in polar form. Part (c): plot them and identify the polygon they form.

The seam. Part (b)'s word "hence". The theorem just proved is the tool for the roots, and the archive shows that a "hence" which crosses an area boundary is the one students ignore.

The evidence. For part (a): 2023 Exam 1 Q8 at 21% and 2024 NHT Exam 2 Section B Q2a, which sets the inductive step alone as a two-mark "show that". For parts (b) and (c): 2023 Exam 2 Section B Q2b at 61% and Q2c at 53% — [RPT23 E2]: "Omitting z = 1 was a common error." [RPT06 E2] on the plotting: "Common errors involved placing points on the wrong circle, using rays from the origin to show complex numbers, and not plotting all roots." 4 + 1 + 2 marks, Examination 1 (part a) or Examination 2 Section B (whole). Band 15–25% on part (a), 50–60% on (b) and (c).

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X3 — A distribution derived from a differential equation

The design. A quantity is modelled by a separable differential equation over a bounded interval, and its solution — suitably normalised — is presented as a probability density function. Part (a): solve the differential equation. Part (b): find the constant that normalises it. Part (c): use integration to find the mean. Part (d): a sample-mean probability with the normal value supplied.

The seam. Part (b). Nothing in either area of study tells the candidate that a density must integrate to one over its support; it is Mathematical Methods assumed knowledge, arriving in the middle of a Specialist question. Part (c) then requires the same antidifferentiation techniques as part (a) applied to a different integrand.

The evidence. 2025 Exam 1 Q4a at 37% is the live proof that VCAA will open a Specialist statistics question with a continuous-random-variable integral requiring partial fractions — [RPT25 E1]: "Students should know how to find the expected value (mean) of a continuous random variable." Q4b at 49%. The differential-equation half is anchored on 2016 Exam 1 Q10 at 14% and 2024 Exam 1 Q7 at 27%. 3 + 1 + 3 + 2 marks, Examination 1. Band 20–30% across parts (b) and (c).

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X4 — Induction proving a derivative formula, then a concavity claim

The design. Part (a): prove by induction a closed form for the nth derivative of a stated function. Part (b): hence determine, for a stated n, whether the graph of that derivative is concave up on an interval, with justification.

The seam. Part (b) uses the proved formula twice — once at the stated order and once two orders higher — and then requires the sign argument, not just the zero.

The evidence. Part (a) is 2023 Exam 1 Q8 at 21% exactly. Part (b) is the necessary-versus-sufficient family: 2020 Exam 1 Q6b at 9% — "Few students attempted to justify that a point of inflection occurred at this point" — and 2017 Exam 2 Section A Q10 at 6%. 4 + 2 marks, Examination 1. Band 15–25% on (a), 10–18% on (b).

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X5 — Roots of unity as position vectors

The design. The nth roots of unity, plotted. Part (a): list them. Part (b): treating them as position vectors in the plane, find the area of the polygon they form, using a vector method. Part (c): find the exact area of one of the circular segments cut off by a side.

The seam. Moving from the Argand plane to the coordinate plane. The same points are complex numbers in part (a) and position vectors in part (b), and the notational conventions differ — [RPT25 E2]: "If they are using coordinates, they must not have i in the coordinate."

The evidence. Part (b) is the cross-product area family: 2019 Exam 2 Section B Q4c at 22% — students "multiplied the lengths of two adjacent sides of the parallelogram as if they were finding the area of a rectangle" — and 2024 Exam 2 Section B Q5dii at 46%, where [RPT24 E2] records "omitting the division of the cross product by 2". Part (c) is the segment family: 2017 Exam 2 Section B Q4g at 24%. 1 + 2 + 2 marks, Examination 2 Section B. Band 25–35% on (b), 25–35% on (c).

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X6 — A parameter family of curves, then a volume

The design. A rational function with a parameter. Part (a): the asymptotes. Part (b): the parameter values for which there are no stationary points. Part (c): for one stated parameter value, the volume generated by rotating a bounded region about an axis, as a definite integral, evaluated.

The seam. Part (c)'s terminals are determined by part (a)'s asymptotes and by the intercepts, so a candidate who got the asymptotes wrong integrates over the wrong interval — and the volume formula is not on the formula sheet.

The evidence. Parts (a) and (b): 2021 Exam 2 Section B Q1b at 67% and Q1dii at 13%; 2024 Exam 2 Section B Q1di–iii at 25–27%. Part (c): 2022 Exam 2 Section B Q1di at 45% and Q1dii at 37% — [RPT22 E2]: "A significant number of responses incorrectly contained the integrand [the square of the difference], i.e. students stated the square of the difference rather than the difference of the squares." 2 + 2 + 3 marks, Examination 2 Section B. Band 15–25% on (b), 35–45% on (c).

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X7 — Linear dependence as a proof of coplanarity

The design. Four points in space. Part (a): show that the three displacement vectors from one of them form a linearly dependent set. Part (b): state, with a reason, what that establishes about the four points. Part (c): find the Cartesian equation of the plane containing them.

The seam. Part (b). A computation in Space and measurement becomes a claim requiring justification, which is Area of Study 1's territory — [SD] names "linear dependence and independence of a set of vectors and geometric interpretation" in one dot point, and the interpretation half has never been examined in writing.

The evidence. Part (a): 2008 Exam 1 Q3 at 16% — [RPT08 E1]: "A large number of students did not understand the concept of linear dependence"; 2021 Exam 1 Q6 at 26%. Part (c): 2023 Exam 1 Q9c at 56% — [RPT23 E1]: "Some arithmetic errors were seen, both in the calculation of the cross product and in the substitution of a point to find the Cartesian equation of the plane." 3 + 1 + 2 marks, Examination 1. Band 20–30% overall, with part (b) the cheapest mark and the most often omitted.

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X8 — The width of a confidence interval as a function of sample size

The design. Part (a): a confidence interval for a stated sample. Part (b): express the width of the interval as a function of the sample size. Part (c): sketch that function, labelling the horizontal asymptote and describing its behaviour. Part (d): the sample size needed to halve the width.

The seam. Part (c). The inverse-square-root relationship is a Functions-and-graphs object — a curve with a horizontal asymptote at zero — and no archive question has ever asked the state to draw it, which is why part (d) has been a reliable separator for a decade: candidates who have seen the graph do not make the "multiply by two" error.

The evidence. Part (d): 2023 Exam 2 Section B Q6c at 28%, named by [RPT23 E2] as a whole-paper weakness; 2017 Exam 2 Section A Q19 at 44%, whose four distractors are all mis-squares. Part (c) is anchored on the asymptote-labelling family: [RPT25 E2] on a 2025 sketch — "Many responses did not label the horizontal asymptote y = 0." 1 + 1 + 2 + 1 marks, Examination 2 Section B. Band 30–45% on (c), 25–40% on (d).

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X9 — Central acceleration, proved

The design. A position vector function tracing a circle at constant angular rate. Part (a): find the velocity and the acceleration. Part (b): show that the acceleration is a negative scalar multiple of the position vector, and interpret that geometrically. Part (c): show that the acceleration is perpendicular to the velocity.

The seam. Parts (b) and (c) are both "show that" statements about vectors, and both are marked on the connecting steps rather than the conclusion. [RPT15 E2]: "Some students simply asserted that ṙ(t)·r̈(t) = 0."

The evidence. 2007 Exam 1 Q9 at 18%, 3 marks, is this exact question. 2012 Exam 2 Section B Q4b at 34% is the perpendicularity half as a "show that"; 2023 Exam 1 Q10d at 7% is the version where the times must be solved for. 2 + 2 + 2 marks, Examination 1. Band 15–25%, anchored on 2007 Exam 1 Q9 at 18%.

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X10 — Two lines cannot meet: a proof by contradiction in space

The design. Two lines in space given in vector form. Part (a): prove by contradiction that they do not intersect — assume a common point, equate components, and derive an inconsistency. Part (b): hence find the shortest distance between them.

The seam. Part (a) turns a routine simultaneous-equations check into a proof, which requires the assumption to be stated and the inconsistency to be named. It is the cleanest available way to make Area of Study 1 do work inside Area of Study 5, and the technique is named content that has no live written instance.

The evidence. For part (a), the machinery is measured at 2025 Exam 1 Q2 at 59% and 2025 Exam 2 Section A Q18 at 49%, and the specific failure at [RPT25 E1]: "It was common for students to use the same parameter for both lines. This did not result in viable equations to solve." For part (b): 2024 Exam 1 Q10 at 14%. 3 + 3 marks, Examination 1. Band 25–35% on (a), 10–20% on (b).

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5. What the 2023 deletion of mechanics frees up

5.1 What was deleted, and how much of it there was

[SD]'s kinematics dot point restricts Area of Study 4 to "rectilinear motion of a single particle". 01-study-design.md §5.5.3 and 06-space-measurement.md §1.3 establish what that removed: connected particles and pulleys, friction and the coefficient of friction, statics and limiting equilibrium, resolution of forces, Newton's second law as a modelling step, resultant forces, momentum and impulse. The Mechanics block — momentum and the equation of motion — was deleted from the formula sheet at the same time, and the words "newton", "tension", "momentum" and "friction" appear in no 2023 November, 2025 November or 2026 NHT paper.

Roughly 40% of the Space-and-measurement marks in 2006–2022 were mechanics. The area's share of tagged November marks moved as follows:

5.2 Which separators went with it

Fifty-one separators disappeared, in three named groups:

Those are the two shallowest separator groups in the whole area. The groups that survive have lower medians: collisions and closest approach (27%), arc length and distance travelled (28%), velocity–time and multi-stage motion (30%), rectilinear kinematics and projectiles (32%), variable acceleration as a differential equation (32%). 06-space-measurement.md states the consequence plainly: "Losing the mechanics did not make this area easier; it removed the part of it the state was relatively best at."

The separator rate is flat across all three eras — 44.6%, 48.2%, 44.8% — so the redesign changed what is hard, not how much is hard.

5.3 Where the marks went

Calculus absorbed most of them. Its share of tagged marks rose from 24.6% (2006–2015) and 25.9% (2016–2022) to 27.8% (2023–2025), its highest ever, making it the single heaviest area of study for the first time. Statistics settled at 12.5–15.8% of the 120 marks. Discrete mathematics took 4.2%. And Space and measurement rebuilt itself to 31% by 2025 on the new cross-product, lines and planes content — [RPT24 E1]: "New topics from 2023 were again tested in 2024. In particular, proofs (Question 2), the cross product (Question 4b and Question 10) and lines in space (Question 10)."

5.4 What it implies for the shape of a modern Section B

The pre-2023 Section B reliably ended with a mechanics question — a staircase of equation-of-motion parts, typically Question 5, typically 10–13 marks. That slot is gone, and the archive shows what replaced it. The current Section B, in all three November papers of the design and across the NHT series, is:

Four consequences follow for anyone writing a modern paper.

First, the diagram marks have to be re-created somewhere else. Mechanics was the archive's main source of marks for drawing and labelling a configuration — [RPT13 E1]: "Where a force is resolved, students should not show the components as bold line segments with arrows. Instead, dashed line segments should be shown." The current design's natural homes for that kind of mark are the Argand diagram (open circles at ray endpoints, shaded intersections, corner points included), the three-dimensional configuration sketch for a skew-lines or plane problem — [RPT24 E1] explicitly credits candidates who "drew diagrams of skew lines and parallel planes to help motivate an appropriate formula" — and the direction-of-motion arrow on a path.

Second, the sign-discipline marks likewise. Mechanics generated them through force directions and inclined planes. They now live in exactly three places: the branch chosen after integrating a separable differential equation (2016 Exam 1 Q10 at 14%; 2024 Exam 1 Q7 at 27%), the sign of an area below an axis (2010 Exam 1 Q10 at 23%), and the direction of travel on a parametric path (2024 Exam 2 Section B Q4b at 41%).

Third, the multi-stage structure survives without the forces. The deleted material carried the archive's worst multi-stage items — 2019 Exam 2 Section B Q5d at 9% over two stages on an incline, 2016 Exam 2 Section B Q5e at 21% over three phases of flight. The structure is fully reproducible in current content: a filling-and-draining model that changes regime, a resisted motion whose model starts part-way through (2017 Exam 2 Section B Q2di at 17%), or a path whose parameterisation changes at a stated time.

Fourth, and most usefully, there is room for a sixth staircase. Mechanics occupied one whole Section B question. Its replacement — the lines-and-planes question — is the newest content in the design and the one with the thinnest archive: ten separators, median 39.9%, and only three years of live examples. That is where the design space is largest and where a trial-exam writer has least competition from published material.

5.5 On Examination 1

The forty marks are unchanged and the question count has stayed at 9–11. What has changed is density: 09-nht-specialist.md records 2025 NHT Exam 1 running to nine questions but thirty labelled parts, four of them one-mark "show that" items inside a single question, three of which the Assessment Guide marks "A1* ans. given" — the supplied answer is worth nothing and the whole mark hangs on the displayed reasoning. That density has no November precedent, and it is the most likely direction of travel: the mechanics marks have been redistributed as more parts, not as longer parts, which pushes the paper towards the one-mark and two-mark bands where the median full-mark rates are 62% and 50% rather than the three-mark band where it is 37%.

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6. How to use these

6.1 For a teacher writing a SAC or a trial examination

Budget the separators by mark value and position, not by topic. The base rates in §1.1 are the planning numbers: a three-mark written part is a separator 78% of the time and a four-mark part 84% of the time, before anything is done to make it hard. A paper built from six three-mark parts will produce a mean in the thirties without discriminating anywhere. The workable distribution across a 40-mark Examination 1 is roughly: twelve to sixteen marks at one mark each (median 62%), sixteen to twenty marks in two-mark parts (median 50%), and no more than three three-or-four-mark parts, of which at most one should be drawn from the sub-15% designs.

Check the predicted shape, not just the predicted band. §1.4 is the operative test. Before setting a design, ask what the mark distribution will look like. If the first mark and the last mark depend on the same step, the distribution will be bimodal and the item will defeat rather than grade. The archive's two model cases are worth holding side by side: 2018 Exam 1 Q4 at 43% with dist 5 / 8 / 8 / 36 / 43, and 2019 Exam 2 Section B Q6b at 15% with dist 67 / 15 / 2 / 15. The first is worth four marks of discrimination; the second is worth almost none.

Split anything predicted below 15%. Three archive pairs prove the point. 2019 Exam 2 Section B Q1e at 3% asks for a changed-variable integral in one part; 2014 Exam 1 Q5b at 48% asks for the same operation with the substitution supplied and the evaluation quarantined. 2021 Exam 2 Section B Q1dii at 13% asks for a parameter condition in one two-mark part; 2024 Exam 2 Section B Q1di–iii at 25–27% asks the same family of questions in three one-mark parts. 2021 Exam 2 Section B Q6e at 9% folds a critical value into a Type II error; 2025 Exam 2 Section B Q6g/Q6h at 62.95% and 53.65% asks for them separately.

Put the "show that" at the point of maximum dependency. This is VCAA's own stated design principle, from [RPT06 E2]: the instruction "is intended to help keep them on track and enable access to subsequent marks for later parts of the same question, even if the student could not 'show' a given result." A question with a hard step in the middle and no "show that" over it is a question whose last four marks are unreachable for most of the cohort.

Write the labelling instruction before you write the sketch. Every sketch design in §2.2 is marked against its instruction sentence, and the instruction sentence is where the marks are specified. 03-functions-graphs.md §3 catalogues VCAA's own wordings; reuse them verbatim rather than inventing.

For a SAC specifically, the constraint is different. School-assessed Coursework carries Outcomes 2 and 3, which the examinations under-assess: [SPEC] puts Examination 1 on Outcome 1 alone and Examination 2 "with an emphasis on Outcome 2". That makes the cross-area designs of §4 and the simulation design D7.10 more appropriate for a SAC than for a trial paper, and it makes the modelling-twist design D4.12 — which is the archive's purest test of Outcome 2 — the natural closing task.

Calibrate against the named anchor, not against your own sense of difficulty. Every band in this document rests on a specific archive question with a published percentage. If a trial-exam item written from one of these designs produces a cohort result far outside its band, the discrepancy is information: either the numbers did not close as cleanly as the archive instance's, or the cohort has drilled the construction.

6.2 For a student who wants to know what they are not prepared for

Learn the eight failure mechanisms, not the eighty-eight designs. §1.3 is the list, and every separator in the archive is one of them. Before writing anything, name which one the question in front of you is testing. If you cannot, you are answering from memory.

• M1 Verification is not proof. Never write the target on the first line; never manipulate both sides towards each other; never substitute values to check.
• M2 A necessary condition is not a sufficient one. A second derivative of zero is a candidate, not a conclusion.
• M3 When the answer could be a set, count. A modulus gives two; a cancellation condition gives several; a quartic gives four.
• M4 When the answer is a family, define the parameter. "For some integer k" is worth a mark on its own.
• M5 At a substitution, four things change. Write the substituted integral on a new line with its new limits before integrating.
• M6 Answer the object that was asked for — a region not a value, a vector not a magnitude, a domain not just a rule, coordinates not a parameter value.
• M7 A sketch is a checklist. Write out the required labels as a list before you draw.
• M8 The final part of a Section B question has a median full-mark rate of 28% — and a large part of that is that nobody reaches it. Budget 1.5 minutes per mark and protect the last two marks of every question.

The specific gaps, ranked by how likely they are to appear and how little practice material exists. The following are in scope under the current study design and have been examined once, never, or only in a Northern Hemisphere paper. If you have not written one of each, you are not prepared for the paper VCAA is entitled to set:

Practise the final parts first. The archive's own numbers make this the highest-value change to a revision routine: the first part of a multi-part question has a median full-mark rate of 71% and the final part 28%. Working past papers front-to-back spends most of the available time on the 71% material.

The three corrections with the best evidence behind them.

1. Write a concluding sentence on every proof and every "show that". [GUIDE] attaches a mark to the final line of every induction proof, and [RPT22 E1] records a 47% item lost partly to "incorrect conclusions drawn" after correct algebra.
2. After every argument computation, check the interval, and after every set answer, check the count. Six reports across fourteen years name the principal-value reduction; four name the missing second solution.
3. Draw the picture. Five separate reports recommend it for five different constructions — the quadrant of a complex number ([RPT07 E1], [RPT16 E1], [RPT17 E2], [RPT18 E1]), the shape of a rational inequality ([RPT20 E1], [RPT15 E1]), the configuration of two skew lines ([RPT24 E1]), the region under a normal curve ([RPT22 E1], [RPT25 E1]), and the order of the vertices of a quadrilateral ([RPT19 E2]). It costs thirty seconds and it is the single most frequently repeated piece of advice in twenty years of examination reports.

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Appendix — design index

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Sources: corpus/sm/questions.json (1,428 graded parts carrying a published percentage, of which 634 are separators, read 15 September 2026); corpus/sm/text/*.txt and corpus/sm/raw/* (papers, reports and assessment guides 2006–2026); research/sm/01-study-design.md through 09-nht-specialist.md. Every percentage in this document is VCAA's own published full-marks rate, verified against questions.json rather than transcribed from the area documents. Every design is a description; none is a question.