The sentences VCAA reuses every year
The wording bank
VCAA does not invent new phrasings. The same question stems return year after year, and for several of them the examiner reports prescribe the answer almost word for word. Learn these as fixed text and the marks follow.
How to read this page
Anything set in italic against the red rule is VCAA's own words — question stems, examiner commentary, prescribed model answers. Everything else is this site's explanation. For the command words themselves (“determine”, “show that”, “write down”) and the rules on rounding and working, see exam craft.
Functions, relations and graphs
Full functions, relations and graphs reference
VCAA reuses sentences. Each one carries a specific obligation, and the reports penalise students who answer a neighbouring question instead. These are quoted from the papers, with the years in which the exact form appears.
3.1 Sketching
| Wording | Years | What it demands |
|---|---|---|
| "Sketch the graph of f. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation." | 2016 E1 Q3a; near-identical 2012 E2 Q2a, 2021 E1 Q4a, 2023 E1 Q3a, 2023 NHT E1 Q6b, 2024 NHT E1 Q3b | Every intercept as an ordered pair, every asymptote as an equation (x = 2, not 2), asymptotes drawn dashed. A number instead of a coordinate scores zero for that feature. |
| "Label the endpoints with their coordinates." | 2013 E1 Q9b, 2018 E1 Q3b, 2022 NHT E1 Q4b, 2025 E1 Q3c | Closed dot, coordinates written, and the curve stops there. 2012 report: "Some students indicated the coordinates of an 'included' endpoint in square brackets. Only round brackets are to be used for coordinates." |
| "Label the coordinates of the turning points and x-intercepts." | 2026 NHT E1 Q4b; 2025 E2 Q1b ("labelling the stationary points and axial intercepts with their coordinates") | Turning points and intercepts. Omitting either loses a mark. |
| "Label any stationary points with their coordinates, correct to two decimal places." | 2006 E2 Q?c, 2008 E2 Q1ci, [SAMP] E2 Q1a ("correct to three decimal places") |
Technology-active. Give exactly the requested number of places — 2008 report: answers "correct to only one decimal place when the question asked for two" were not accepted. |
| "On the same set of axes sketch the graph of y = |f(x)|." / "On the axes above, sketch the graph of g." | 2006 E1 Q7a, 2009 E2 Q1b, 2019 E1 Q4b, 2022 E1 Q6a | Trace over the parts that do not move; only the reflected parts are new. 2009 report: "Students were required to draw over the middle section of the original graph (this should have been clearly indicated)." |
| "Sketch the graph of the inverse function y = f⁻¹(x) on the axes above. Label any endpoints and axial intercepts with their coordinates." | 2023 E1 Q7b | Reflect in y = x; the domain of f⁻¹ is the range of f, so the sketch must stop in the right place. |
3.2 Key features
| Wording | Years | What it demands |
|---|---|---|
| "State the range of f." | 2006 E1 Q7b, 2007 E2 Q4b, 2012 E2 Q2bii, 2017 E1 Q7a, 2019 E1 Q8c, 2021 E1 Q3a, 2022 E2 Q4a, 2023 E1 Q7a, 2025 E1 Q3a | A set, in interval or set-difference notation. Not an inequality in y. 2025 report: "incorrectly stating the range of values in terms of y (as in −1 ≤ y ≤ 3) is not acceptable notation." |
| "State the maximal domain of g and the range of g over its maximal domain." | 2019 NHT E1 Q4a; "Find the maximal domain of f" 2022 E1 Q5b; "State D, the maximal domain of f" 2023 NHT E1 Q6a; "State the maximal domain for f(x)" 2024 NHT E1 Q3a | The largest set of x for which the rule is defined — solve the constraint, answer with ∪, never ∩. |
| "Write down the amplitude and period of the function." | 2006 E1 Q4a, 2010 E1 Q4a; "Find the period and amplitude" 2014 E2 Q1a; "State the period and the amplitude of g" 2021 NHT E2 Q1a | Two numbers. period = 2π/n. The most common single error in the archive is 2π or 6π for a period. |
| "The period and range of this function are respectively…" | 2015, 2016, 2017, 2018 NHT, 2019, 2024 NHT E2 Q1 | The Section A opener, most years. Free marks. |
| "State the interval for which the graph of f is strictly decreasing." | 2009 E2 Q1a; "Find the largest interval of x values for which h is strictly decreasing" 2023 E2 Q3e; "State the maximal domain over which f is strictly increasing" 2022 E2 Q4bii | A closed interval at a stationary point. 2023 report: "Round brackets were often seen; these were incorrect as the largest interval of x values was required, which included the interval endpoints." |
| "State the set of values for which the gradient of the hill is strictly decreasing." | 2019 E2 Q2b (3%) | Not where the function is decreasing — where f′ is decreasing. The 2019 report devotes a general-comments bullet to this distinction. |
3.3 Transformations
| Wording | Years | What it demands |
|---|---|---|
| "State a sequence of two transformations that maps the graph of y = f(x) to the graph of y = h(x)." | 2007 E2 Q3di, 2015 E2 Q4c, 2022 NHT E2 Q2bi ("Write a sequence of two transformations"), 2024 NHT E1 Q9b | Exactly the stated number of transformations, in a valid order, each named with axis and direction. |
| "Write down a possible sequence of three transformations to map from g to h." | 2025 E2 Q1e | "Possible" means more than one answer is accepted — but yours must work. |
| "Complete a possible sequence of transformations to map f to g." followed by a printed first line | 2023 E2 Q5a | The printed line fixes part of the order. |
| "State a sequence of two transformations, a dilation followed by a translation, that will map the graph of y = h(t) to the graph of y = p(t)." … "State a sequence of two transformations, a translation followed by a dilation …" | 2026 NHT E2 Q2 | The same map, both orders. The translation constant differs between them. This is the clearest statement VCAA has made that order changes the parameters. |
| "Describe the transformation that maps the graph of g₁ onto the graph of g_k." | 2017 E2 Q4gi, Q4gii; 2018 E2 Q3b; 2021 NHT E2 Q2dii; 2023 NHT E2 Q1e | A single transformation, in words. Dilation wording must be "from the x-axis"/"from the y-axis" or "parallel to the axis" — "in the y-axis" was explicitly marked wrong in 2017. |
| "Describe the translation that maps the graph of y = f(x) onto the graph of y = g(x)." | 2014 E2 Q5b | Naming the translation ("a translation of 1 unit in the negative direction of the x-axis") — the direction is the mark. |
| "The following sequence of transformations is applied to the graph of the function g … Find i. the rule of the image of g … ii. the domain of the image of g after the sequence of transformations has been applied." | 2013 E1 Q9c | Apply in the printed order. The image gets a new name. The domain moves too. |
| "The transformation T : R² → R², T([x, y]) = [a 0; 0 b][x, y] + [c, d] maps the graph of … onto the graph of …. State the values of a, b, c and d." | 2006, 2008, 2009, 2010, 2012, 2014, 2016, 2017, 2018, 2019, 2020 | Retired. Matrix representation left the study design in 2023 and the papers in 2021. |
3.4 Inverses and composites
| Wording | Years | What it demands |
|---|---|---|
| "Find the rule and domain of f⁻¹, the inverse function of f." | 2008 E1 Q10a, 2016 E2 Q4bi, 2017 E2 Q4b, 2019 NHT E1, 2022 E2 Q4d | Rule and domain. 2007 report, as policy: "When the inverse function is asked for, the domain must be given. Students will be penalised in the future if the domain is left out." |
| "State the rule and domain of f⁻¹." | 2018 E1 Q5 | Same, phrased as a single 3-mark instruction. |
| "Determine the equation and the domain for the inverse function f⁻¹." | 2023 E1 Q7c | "Equation" — write f⁻¹(x) = …, not just an expression. |
| "The function f : D → R with rule … will have an inverse function for …" / "Find the largest value of a such that when g is restricted to the domain (−∞, a] it has an inverse function." | 2006, 2008, 2010, 2013, 2014, 2019 NHT, 2021 NHT, 2025 E1 Q5b | One-to-one. The endpoint is a stationary point and is included. |
| "Explain why p is not a one-to-one function." | 2021 E2 Q3c | A reason, not a definition: fails the horizontal line test / is many-to-one / two x-values share some y-value. |
| "Find the rule for h, where h(x) = f(g(x))." / "State the rule of g(f(x))." | 2016 E1 Q5ai, 2019 E1 Q9a/Q9c, 2017 NHT E1 Q7a, 2021 NHT E1 Q1ai | Substitution only. Do not expand. |
"State the domain and range of h." (where h is the composite) |
2016 E1 Q5aii | Domain from the inner function; range by sketching f over ran g. |
| "Find the largest possible value of c such that the range of g is a subset of the domain of f." | 2017 E1 Q7bi | The existence condition ran g ⊆ dom f, run backwards. |
| "Determine the maximal domain, D, such that g ∘ h exists." | [SAMP] E1 Q2c |
The same condition in the 2023 sample paper — this is the form VCAA signposted for the current design. |
| "Find the range of g and, hence, show that f(g(x)) is defined for all x ∈ (−0.8, 1.8)." | 2026 NHT E2 Q4fi | The existence condition asked as a "show that". |
3.5 Parameters, families and proof
| Wording | Years | What it demands |
|---|---|---|
| "Find the values of d such that the graph of y = f(x + d) has i. one positive x-axis intercept ii. two positive x-axis intercepts." | 2014 E2 Q5c | An interval, with brackets decided by testing the critical position itself. |
| "Find the values of a for which p has only one stationary point." / "State the values of b ∈ R for which the graph of y = f(x) + b has no x-intercepts." | 2018 E2 Q1hi, Q1b | Same family logic; the second is 65%, the first 18% — the difference is whether the critical value is read off a printed graph or must be derived. |
| "Find the value(s) of k for which this system has no real solutions." | 2025 E2 Q4 (Section A) | Cross-references the Algebra area; the technique (discriminant / coincident lines) is the same family reasoning. |
| "Show that …" | 63 occurrences, 17 years | The answer is printed. Marks are for the route. Never start from "LHS = RHS". Never verify with one numerical case. |
| "Hence, or otherwise, …" | 2023 E1 Q5b, 2017 NHT E1, 2026 NHT E2 | "Hence" is a hint that the previous part's answer is the fast route; "or otherwise" means an independent method is accepted but will cost time. |
| "Give your answer correct to two decimal places." vs silence | throughout | Silence means exact. 2025 Exam 1 instruction page: "In all questions where a numerical answer is required, an exact value must be given unless otherwise specified." |
| "State…" vs "Find…" | throughout | Both require the answer; neither excuses you from working when the part is worth more than one mark. 2017 E2 Q2h: "Some students wrote 7 minutes without showing any working. As indicated in the instructions on the examination, for questions worth more than one mark, appropriate working must be shown." |
Algebra, number and structure
Full algebra, number and structure reference
VCAA reuses sentences. These are the recurring ones, with the years they appear and exactly what each demands.
| Wording | Years seen | What it demands |
|---|---|---|
| "Solve the equation … for x" | Every Exam 1, 2007–2025 | All solutions, exactly, with no domain unless one is stated. If the equation has an implied domain (logs, radicals), you must apply it yourself. |
| "… for x ∈ [a, b]" / "where a ≤ x ≤ b" | 2007, 2008, 2009, 2010, 2013, 2014, 2015, 2018, 2019, 2025 | Transform the domain before solving. List every solution in range and no others. |
| "have no solution for" | 2006, 2014 | Determinant zero and inconsistent. Answer is a value or values. |
| "have infinitely many solutions for" | 2008, 2010, 2022, 2024 NHT | Determinant zero and consistent. Usually one value. |
| "have a unique solution only for" / "provided" | 2007, 2009, 2024 NHT | Determinant non-zero. Answer is a set complement, R \ {…}. |
| "Find the value(s) of k for which this system has no real solutions" | 2025 | The 2023-design phrasing. "Value(s)" signals that the count is not obvious. |
| "has no real roots when" / "has two real solutions" | 2017, 2019, 2023 | Discriminant. Answer is an inequality or an interval. |
| "Find all values of k such that …" | 2016, 2023, 2025 | Every value, usually an interval or union. |
| "exactly one / exactly three solutions" | 2018, 2025 | A boundary case: tangency, or a turning point meeting a line. |
| "the number of solutions to" / "the number of tangents" | 2019, 2023 | A count, obtained by sketching. |
| "Show that …" | 2011, 2016, 2017, 2020, 2023, 2024, 2025 | The answer is given; the marks are entirely for the steps. On Exam 2 this forbids a bare CAS line. |
| "Verify that …" | 2023, 2025, [SAMP] |
Substitute and demonstrate. One mark, and working is still required. |
| "Express … in the form …" | 2018, 2020, 2025 | The stated shape is a specification. Check the letter constraints (∈ Z⁺, "integers"). |
| "in terms of r / a / k" | 2008, 2010, 2014, 2015, 2016, 2019, 2022, 2023 | A literal answer. A number scores zero. |
| "Hence, or otherwise, …" | 2017, 2019, 2025 | "Hence" is a route, not decoration. The previous part is the intended method and usually the only tractable one. |
| "Give your answer correct to two decimal places" | 2010, 2013, 2015, 2018, 2020, 2023, 2025 | Round at the end only. Two decimal places means two — not one, not exact. |
| "an exact value must be given unless otherwise specified" | Exam 1 instruction page, every year | The default. Surds, fractions, log_e(…) and π stay symbolic. |
| "In questions where more than one mark is available, appropriate working must be shown" | Exam 1 instruction page, every year | Invoked by the reports whenever an answer-only response loses marks. |
| "Find all possible values of n" | 2023 | General solution; k ∈ Z must appear. |
| "for all x ∈ R" / "for all k > 1" | 2016, 2020 | A proof or a bounding argument, not a check of examples. |
| "Apply Newton's method, with an initial estimate of x₀ = …" | 2023, 2025, 2024 NHT, [SAMP] |
The [FS] formula, iterated, to the stated accuracy. |
| "Consider the algorithm below … In order, the values printed by the algorithm are" | 2025, 2024 NHT, [SAMP] |
Trace table. Watch the loop exit. |
| "Using the trapezium rule and n trapeziums of equal width" | 2024, 2024 NHT, [SAMP] |
The [FS] formula. Integration is explicitly not acceptable. |
Two more, from the front matter of every paper, that the reports invoke constantly:
"In all questions where a numerical answer is required, an exact value must be given unless otherwise specified." — Exam 1 instruction page
"A correct answer scores 1; an incorrect answer scores 0. Marks will not be deducted for incorrect answers." — Exam 2 Section A instruction page
Calculus
These are VCAA's recurring sentences, quoted from the papers, with the years they appear and what each one demands. Command terms matter: [RPT] 2024 Exam 1 general comments states it directly —
"Students should be familiar with the specific terminology used in mathematics. Command terms such as 'verify', 'show that' and 'hence' have precise meanings, and it is important that students understand what approach is needed to correctly answer the question… If a particular method is specified, the student's work should clearly demonstrate that the required method has been applied."
3.1 "Find the average rate of change of … between … and …"
Years in the papers: 2007, 2010, 2012, 2013, 2014, 2015, 2016, 2017 (Nov and NHT), 2018, 2019, 2021, 2022, 2023, 2025, 2026 NHT, and [SAMP] Exam 1 Q5a.
Variants: - Bare calculation — "Calculate the average rate of change of f between x = −π/3 and x = π/6." (2016 Exam 1 Q6a) - With units — "Find the average rate of change of the amount of drug X in the bloodstream, in milligrams per hour…" (2018 Exam 2 Q2b); "in metres per minute" (2023 Exam 2 Q2c) - Multiple choice, abstract — "The average rate of change of the function with rule f (x) = x³ − x + 1 between x = 0 and x = 3 is…" (2007 Exam 2 MCQ); "…the average rate of change of f(x) with respect to x on the interval…" (2012 Exam 2 MCQ) - Multiple choice, graphical — "Over which of the following time intervals did the daily price undergo the greatest average rate of change?" (2025 Exam 2 MCQ 11) - Reversed, with a parameter — "The average rate of change of the function with the rule f (x) = x² − 2x over the interval [1, a]…" (2017 Exam 2 MCQ 20) - Algebraic proof — "Show, using algebra, that the average rate of change of f over the interval [a, b] is given by …" (2026 NHT Exam 2 Q2e.i, 3 marks)
What it demands: (f(b) − f(a))/(b − a). Nothing else. Two evaluations, one subtraction, one division — no calculus. The graphical variant is answered by laying a ruler along each chord; [RPT] 2025 MCQ 11 says exactly that: "Use a ruler to draw line segments for each option. Then, select the line segment with the steepest gradient."
3.2 "Find the average value of f over the interval …"
Years: 2006, 2008, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 NHT, 2018 NHT, 2019 (Nov and NHT), 2020, 2021, 2022 (Nov and NHT), 2024 NHT, 2025, 2026 NHT, [SAMP] Exam 1 Q5b.
Variants: - Direct — "Find the average value of g between x = 0 and x = 2." (2025 Exam 2 Q1d) - Reversed for a parameter — "The average value of f over the interval [−2, p] is zero." (2015 Exam 2 MCQ); "The average value of f (x) = x² − 2x over the interval [1, a] is 13." (2018 NHT Exam 2 MCQ); "The average value of g on the interval [−1, 1] is 31/12. Find all possible values of a." (2013 Exam 1 Q6); "The average value of the function, f, over 0 ≤ x ≤ m is 1/3. Find the value of m." (2024 NHT Exam 1) - Optimised — "Consider the average value of the function f over the interval x ∈ [0, k], where k ∈ [0, 2]. Find the value of k that results in the maximum average value." (2022 Exam 1 Q8c) - Of a derivative — "Find the average value of the derivative function g′(x) between x = π/8 and x = π/6." (2022 Exam 2 Q5d) - Graph-identification MC — "Let h be a function with an average value of 2 over the interval [0, 6]. The graph of h over this interval could be…" (2013 Exam 2 MCQ 15) - Under transformation — "Consider a continuous function f : R → R, which has an average value of 2 over the interval [−1, 1]. Let g : R → R, g(x) = mf (nx) + c, where m, n and c are positive real numbers. Which one of the following must be true for all possible values of m, n and c?" (2026 NHT Exam 2 MCQ 15) - From a graph, with no rule — "Use features of the graph in part a. to find the average value of f between x = −π and x = …" (2017 NHT Exam 1)
What it demands: (1/(b − a))∫ₐᵇ f(x)dx. When the interval or the value is unknown, that same expression becomes an equation. When the function is only drawn, the integral is a geometric area.
The pairing is deliberate. [SAMP] Exam 1 Question 5 puts average rate of change in part a and average value in part b — same interval [−1, 1], same function f(x) = 2 − x². 2016 Exam 1 Question 6 does exactly the same thing (pct 32 and 16). VCAA is testing the distinction, not the computation.
3.3 "The graph of f has a stationary point of inflection at …"
Years: 2009, 2010, 2014 (all multiple choice), then nothing until 2024 NHT, 2025 Exam 1 and 2025 Exam 2 — the gap matching the topic's removal from the 2016–2022 study design.
Variants:
- MC distractor — "The graph of f has a stationary point of inflection where x = 4." (2010 Exam 2 MCQ); "C. there is a stationary point of inflection." (2009 Exam 2 MCQ)
- Given as data — "Given that f has a stationary point of inflection at (−2, 2) and an axial intercept at …" (2024 NHT Exam 2, 2 marks)
- State it — "State the coordinates of the stationary point of inflection for the graph of y = f (x)g(x)." (2025 Exam 1 Q7d.i, 1 mark)
- Demonstrate it — "Complete the following gradient table with appropriate values of x and g′(x) to show that g has a stationary point of inflection." (2025 Exam 2 Q1c, 2 marks)
- As a transformation target — "Let h be the result after applying a sequence of transformations to g, such that h has a stationary point of inflection at (1, 0) and a local maximum at (−1, 1)." (2025 Exam 2 Q1e, 3 marks)
- Non-stationary — "Find the coordinates of the point of inflection for h, correct to two decimal places." (2023 Exam 2 Q3d, 1 mark); "Find the coordinates of the point of inflection of f." ([SAMP] Exam 2)
What it demands: where it is given, it is data — use it to build two equations (f′ = 0 and f = value). Where it must be found, Exam 2 wants a technology answer to the stated accuracy; Exam 1 wants the repeated-factor recognition (a factor (x − a)³ gives a stationary point of inflection at x = a). Where VCAA wants it shown, a gradient table with the same sign on both sides is the complete answer.
3.4 "Find the area of the region bounded by …"
Years: 2007, 2008, 2012, 2013, 2018, 2021 NHT, 2024 NHT, 2026 NHT; plus the near-identical "Find the total area of…" in 2006, 2012, 2013, 2015, 2018, 2019, 2020, 2021, 2022 NHT, 2023, 2026 NHT, and "Find the area enclosed by…" in 2016.
Variants, and what each boundary list means:
- "…the curve y = f (x), the line x = π/12, the line x = π/6 and the x-axis" (2026 NHT Exam 1 Q5c) — two vertical boundaries given; the terminals are handed to you.
- "…the graph of g and the x-axis" (2021 NHT Exam 1, 2 marks) — the terminals are the x-intercepts; you must find them.
- "The area of the region bounded by the curve with equation y = kx², where k is a positive constant, the x-axis [and x = 9] is 27" (2007 Exam 1 Q10) — reversed: solve for k.
- "…the y-axis, the x-axis, the curve y = e^(2x) and the line x = C" (2008 Exam 1 MCQ) — four boundaries, one of them a parameter.
- "…the line segment CA, the x-axis, the line x = 1 and the line x = a" (2008 Exam 2 Q2c.i) — a straight-line boundary, so a trapezium beats an integral.
- "Find the area of the region bounded by the graph of f and the line segment ST." (2013 Exam 1 Q10c) — area between a curve and a chord.
- "Let A be the area of the region bounded by the curves y = f (x), y = g (x) and the line x = 2." (2018 Exam 1 Q8) — two curves plus a vertical cut.
- "Find the total area of the regions bounded by the tangent l and y = f (x)." (2018 Exam 2 Q2e) — a curve and its own tangent.
- "Find, in terms of p, the area of the region bounded by the function f, the line …" (2024 NHT Exam 2) — parameterised.
- "Find the total area of the regions bounded by the graph of y = g(x) and the line L, between the stationary points of g." (2026 NHT Exam 2, 3 marks) — a curve and the chord joining its own turning points.
What it demands: identify every boundary, deduce the terminals, decide which function is upper, split at every crossing. The word total is the signal that the region crosses an axis or the other curve and must be split.
3.5 "Find the equation of the tangent to the graph of y = f(x) at the point where x = …"
Years: 2006, 2012, 2013, 2014, 2015, 2016, 2018 (Nov and NHT), 2019 (Nov and NHT), 2021 (Nov and NHT), 2022 (Nov and NHT), 2023, 2024 NHT, 2025, 2026 NHT, [SAMP] Exam 2 — effectively every year.
What it demands: an equation, in the form y = mx + c or y − y₁ = m(x − x₁). A gradient alone scores zero. [RPT] 2019 Exam 2 Q5a: "An equation was required." [RPT] 2016 Exam 2 Q1b: "Some students did not write an equation." The error runs both ways, so read the noun: [RPT] 2025 Exam 1 Q1b: "Some students proceeded further to find the equation of the tangent line instead of only finding the gradient of the tangent at x = 8, as required. Students are reminded to carefully read the question and answer what is required."
3.6 "State the set of values for which … is strictly increasing / strictly decreasing"
Years: 2009, 2019, 2021 NHT, 2022 (Nov and NHT), 2023 (Nov and NHT), 2026 NHT, [SAMP] Exam 2.
Variants:
- Of the function — "Find the largest interval of x values for which h is strictly decreasing. Give your answer correct to two decimal places." (2023 Exam 2 Q3e); "State the interval for which the graph of f is strictly decreasing." (2009 Exam 2 Section B Q1a); "State the maximal domain over which f is strictly increasing." (2022 Exam 2)
- Of the gradient function — "Find the interval of x for which the gradient function of the ramp is strictly increasing." ([SAMP] Exam 2 Section B Q2b.ii); "State the set of values for which the gradient of the hill is strictly decreasing." (2019 Exam 2 Q2b)
- Classification — "When t ≥ 4, is the function C₂ strictly increasing, strictly decreasing or neither?" (2023 NHT Exam 2); "State if p and q are each strictly increasing, strictly decreasing or neither." (2022 NHT Exam 2)
- With a parameter — "Find all values of k such that g is strictly decreasing for x ≥ 1." (2026 NHT Exam 2)
What it demands: read the subject noun first. "the function" → solve f′ ⋛ 0. "the gradient function", "the gradient of the hill" → solve f″ ⋛ 0, i.e. find where f′ itself is increasing. Then use square brackets at finite endpoints.
3.7 "Show that …", "Verify that …", "Hence, …"
Years: every year.
[RPT] 2024 Exam 1 Q7b.iii is the definitive statement on verify:
"This question required students to 'hence, verify […]' so it was not appropriate to attempt to use a calculus technique. Students are reminded that the word 'verify' means to demonstrate or check the truth of a statement, so it was not sufficient to merely discuss the interval in terms of general positive or negative tendencies without referring to specific values and showing the 'check' had been completed."
[RPT] 2020 Exam 1 general comments on show that: "'Show that …' questions require a reasoned argument. Remember the answer is given and students are required to provide detailed progression to the answer given." [RPT] 2016 Exam 2 Q3c adds the failure mode: "Some students incorrectly started with the given expression with no explanation of its origin."
[RPT] 2017 Exam 1 Q2b on hence: "Students generally were not able to form an integral from their previous answer, ignoring the 'hence' instruction."
What each demands: Show that — a forward derivation ending at the printed result, with no step omitted. Verify — substitute the specific values and display the check; no calculus if the question says "hence". Hence — you must use the previous part; an independent method can score zero even when correct.
3.8 "In all questions where a numerical answer is required, an exact value must be given unless otherwise specified"
Printed on the instruction page of every Exam 1 and Exam 2 since 2006. The word "exact" appears in the report commentary for 30 of the 450 Calculus parts — and 24 of those 30 are separators.
[RPT] 2024 Exam 2 general comments: "In all questions where a numerical answer is required, students should give an exact value unless otherwise specified. Some students gave approximate rather than exact answers to Questions 1c.ii, 1d.i, 2f.ii, 5b.iii and 5b.iv."
[RPT] 2010 Exam 2 Q3e: "Exact answers were required for x and T." [RPT] 2019 Exam 2 Q5g: "Some students rounded their answer to 18°. An exact answer was required." [RPT] 2025 Exam 2 Q4g.iii: "Exact answers were required. However, some students gave only approximate answers." [RPT] 2009 Exam 2 Q4c.ii: "An exact answer was required."
The converse is equally punished. [RPT] 2013 Exam 2 Q3g: "Some students left their answer in exact form" where a rounded value was asked for; [RPT] 2025 Exam 2 Q4g.ii: "Some students gave their answer in exact form, not correct to two decimal places as required by the question."
3.9 "Write down a definite integral that gives …"
Years: 2018 (Exam 1 and Exam 2), 2020, 2023, 2024; plus "Use a definite integral to …" in 2021 and 2025.
What it demands: the expression, not the number — and complete notation. [RPT] 2025 Exam 1 general comments: "An integral or integration statement must be accompanied by a 'dx' … and [the definite integral] has the [lower terminal] written as a subscript." [RPT] 2023 Exam 2 general comments explains why the type exists at all:
"Students are allowed to use their technology to find the area between two curves using the bounded area function, but they must show some relevant working if the question is worth more than one mark. Often, questions require that a definite integral is written down, such as in Question 1c.ii."
3.10 The accuracy instruction
"correct to two decimal places", "correct to three decimal places", "to the nearest square metre", "to the nearest whole number" appear on most Exam 2 Calculus parts. [RPT] 2015 Exam 2 Q1e: "A significant number of students did not give their answer correct to three decimal places." [RPT] 2019 Exam 2 Q2e.ii: "Other students rounded their answers incorrectly, giving (11.11, 8.94)" instead of (11.12, 8.95). [RPT] 2012 Exam 2 Q4f: "Some students rounded off too early in their calculations."
The rule: carry full precision through every intermediate step; round once, at the end, to the stated accuracy.
Data analysis, probability and statistics
Full data analysis, probability and statistics reference
These are the fixed phrases. Each has a meaning that does not change from year to year, and each is a place where marks are routinely lost.
| Wording | Years it appears | What it demands |
|---|---|---|
| "correct to four decimal places" | 2008, 2009, 2011, 2013, 2015, 2016 (×4), 2017 (×6), 2018 (×2), 2019 (×8), 2020, 2021 NHT (×5), 2023 (×5), 2024 NHT, 2025 (×2) | Exactly four. Not three, not five, and not rounded from an intermediate value you already rounded. 2008 [RPT]: "Some students gave the answer correct to only three decimal places and others rounded incorrectly, giving 0.1677." 2009 [RPT]: "Students must ensure that they work to more decimal places than what is required in the answer." |
| "correct to three decimal places" | 2015, 2016, 2018, 2020, 2021, 2022, 2023 (NHT), 2025, 2026 NHT | Same rule. 2016's Q3d and 2018's Q4a both use it. |
| "Give your answer correct to two decimal places" | 2016, 2023, 2025 | Common for physical quantities (cm, minutes). |
| "In all questions where a numerical answer is required, an exact value must be given unless otherwise specified" | Every paper, on the instruction page | The default is exact. A probability question that does not say "correct to n places" wants 11/32, not 0.34375. 2022 [RPT]: "In Question 3ai. some students gave 0.0313 as their answer when the exact answer was 0.03125." 2016 [RPT]: "Approximate answers are usually required in probability questions" — usually, not always; read the line. |
| "Given that A, find the probability that B" | 2006, 2007, 2008, 2009, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2022, 2023, 2024, 2025, 2026 NHT — every single year | Pr(B\|A) = Pr(A ∩ B)/Pr(A). The condition goes in the denominator. Form the intersection explicitly before dividing. 2006 [RPT]: "The denominator was usually correct but the answer to part b. was often placed in the numerator." |
| "given that they have already queued for one minute" / "given that it lives for at least two weeks" | 2019, 2023 | A continuous condition. {already queued one minute} = {T > 1}. Never {T = 1}. |
| "at least one" / "at least two" / "more than three of seven" | 2016, 2017, 2019, 2020, 2023, 2025, 2026 NHT | Translate to an inequality on the integer count before touching the calculator. "At least one" = 1 − Pr(0). |
| "Do not use a normal approximation" | 2016 (Q3d), 2019 (Q4fiv), 2021 Exam 1 (Q6c) |
The question is about P̂ for a small n; VCAA is blocking the shortcut. Use the binomial. Conversely, 2026 Exam 2 NHT Section B Q3bv says "Use a normal approximation to find the minimum value of n" — the first time the archive asks for one explicitly. |
| "Show that …" | 2010, 2012, 2016, 2020, 2021, 2022, 2023, 2024, 2025 | The answer is printed. The marks are entirely for the derivation. 2025 [RPT]: "It was not sufficient to verify the solutions … by substitution." 2023 [RPT]: "students were expected to be explicit and clear with their workings, and to arrive at the expected result in a logical, step-by-step manner." |
| "Appropriate working must be shown" (instruction page: "In questions where more than one mark is available, appropriate working must be shown") | Every paper | For a binomial or normal part, stating the distribution with its parameters is the working. 2012 [RPT]: "Identifying the distribution with the correct parameters is sufficient working." 2023 [RPT]: "students were expected to identify and write down the n and p values for the binomial distribution, not just the answer." |
| "Express your answer in the form …" | 2016, 2017, 2019, 2020, 2021, 2025 | The form is a mark. a/2ᵇ with a, b ∈ Z means you must factorise 4096. |
| "Interpret the confidence interval …" / "Explain why this confidence interval suggests …" | 2018 (Nov), 2019 NHT, 2022 NHT | Name the interval's two endpoints, name the comparison value, and say whether the value lies inside. 2018 [RPT]: "The confidence interval needed to be referred to in the answer." |
| "Find the value of p̂ that was used to obtain this approximate 95% confidence interval" | 2017, 2018, 2018 NHT, 2023 | Midpoint of the interval. |
| "Determine the sample size used in the calculation of this confidence interval" | 2018 NHT, 2019, 2019 NHT, 2023, 2024 NHT, 2025 | Half-width = z√(p̂(1−p̂)/n), solve for n, report an integer. |
| "Use z = 2 to approximate the 95% confidence interval" | 2023 Exam 1 | A technology-free concession. Use 2, not 1.96. |
| "Let P̂ be the random variable that represents the sample proportion of …" | 2017, 2018, 2019, 2021, 2022, 2023, 2025, 2026 NHT | The signal that a binomial is about to be asked for in proportion language. |
| "Correct mathematical notation should be used… calculator syntax is not acceptable" | 2008, 2009, 2011, 2016 | ∫₂³ f(x)dx, not ∫(f(x), x, 2, 3). 2008 [RPT]: "Students should not use calculator syntax in their responses; for example, [0.84, 0.64; 0.16, 0.36]^8[1; 0]."* |
| "correct to the nearest integer" | 2013, 2019, 2023 | Applies to counts and to percentages (the 2023 confidence-level question). |