Space and measurement

VCE Specialist Mathematics Units 3 & 4 · Area of Study 5 · Compiled 15 September 2026

For candidates targeting a study score of 45+. Every claim is tied to a source. Where the corpus is damaged, garbled or silent, that is stated instead of being filled in.

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Contents

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0. Sources, method, and what the numbers mean

0.1 pct and "separator"

pct is the percentage of the state that scored full marks on that part — not the average. A 3-mark part with pct = 30 may still have had an average of 1.6/3. A separator is any part with pct ≤ 50: fewer than half the state got it clean. For a 45+ candidate the separators are the entire game, because everything above 70% is already priced into the raw score you need.

Distribution of pct across the 531 Space-and-measurement parts that carry a percentage:

243 of 531 parts (46%) are separators. Exam 2 Section B is where the area bites: the median extended-response part in this area is itself a separator.

0.2 Weight of the area, by year

Share of tagged November marks carried by topic == "Space and measurement" [QJSON]:

Era averages: 2006–2015: 40.5%. 2016–2022: 34.3%. 2023–2025: 26.2%. [SDOC] §1.8 quotes 21% for the current design. That figure is reproduced exactly by the 2023 paper pair (25 of 120 marks) on the tagging snapshot used here; the three-year average is higher because 2024 and 2025 restored the area's weight. The area rebuilt to 31% by 2025 as VCAA leaned into the new planes-and-lines content. Treat 25–32% as the planning figure, i.e. 30–38 marks of 120.

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.

0.3 Known corpus defects affecting this area

• The 2024 November papers and the 2025 NHT papers have no text layer. Documents_exams_mathematics_2024_2024specmaths1-w.pdf, ...2024specmaths2-w.pdf, 2025-05_2025-NHT-specialistmaths1.pdf and ...2.pdf are image-only; pymupdf returns zero characters. No 2024 November wording is quoted verbatim anywhere below. Those questions are described from [RPT24 E1] and [RPT24 E2] only, and the reports' own equations are stripped (VCAA writes them as OMML, which does not survive DOCX text extraction).
• The 2011 papers extracted with a CID-shifted font encoding — every codepoint offset, all spaces lost. Consequently [QJSON] carries collided refs for 2011 Exam 2 Section B (Q2b–2b appears twice, Q4d–4d five times) and one part-level dist that sums to 46 rather than 100. 2011 refs below are reproduced exactly as the corpus holds them and are flagged.
• Topic tagging is machine-assigned and imperfect. A small number of parts tagged Space and measurement are unambiguously Calculus: 2006 Exam 2 Section B Q4a–Q4e (slope field, Euler's method), 2021 Exam 1 Q9ci–9cii (a derivative and an area), 2025 Exam 2 Section B Q3a/3e/3g (a salt-tank mixing differential equation), and one 2011 concentration part. They are listed in §4 for completeness with honest descriptions, and are not used to build question types.
• questions.json was being regenerated by another process during compilation (mtime 15:34). The 544-part snapshot used here was taken before the rewrite and the mark/percentage fields are unaffected.

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1. What the study design puts in this area

1.1 The content, verbatim

[SD] overview of Area of Study 5:

"In this area of study students cover the arithmetic and algebra of vectors; linear dependence and independence of a set of vectors; proof of geometric results using vectors; vector representation of curves in the plane and their parametric and Cartesian equations; vector kinematics in one, two and three dimensions; vector, parametric and Cartesian equations of lines and planes."

Topic 1 — Vectors. [SD], verbatim:

This topic includes:
- addition and subtraction of vectors and their multiplication by a scalar, position vectors
- linear dependence and independence of a set of vectors and geometric interpretation
- magnitude of a vector, unit vector, the orthogonal unit vectors [i, j, k]
- resolution of a vector into rectangular components
- scalar (dot) product of two vectors, deduction of dot product for the [i, j, k] vector system and its use to find scalar resolute and vector resolute
- vector (cross) product of two vectors in three dimensions, including the determinant form
- parallel and perpendicular vectors
- vector proofs of simple geometric results, such as 'the diagonals of a rhombus are perpendicular', 'the medians of a triangle are concurrent' and 'the angle subtended by a diameter in a circle is a right angle'.

Source note, following [SDOC] §1.8.1: the symbols in square brackets are embedded MathType images in VCAA's DOCX and extract as nothing. Reconstructing them as i, j, k is an inference from the formula sheet, which uses exactly that notation with an under-tilde.

Topic 2 — Vector and Cartesian equations. [SD], verbatim:

This topic includes:
- vector equations and parametric equations of curves in two or three dimensions involving a parameter (and the corresponding Cartesian equation in the two-dimensional case)
- vector equation of a straight line, given the position of two points, or equivalent information, in both two and three dimensions
- vector cross product, normal to a plane and vector, parametric and Cartesian equations of a plane.

Topic 3 — Vector calculus. [SD], verbatim:

This topic includes:
- position vector as a function of time and sketching the corresponding path given the function, including circles, ellipses and hyperbolas in Cartesian or parametric forms
- the positions of two particles each described as a vector function of time, and whether their paths cross or if the particles meet
- differentiation and anti-differentiation of a vector function with respect to time and applying vector calculus to motion in a plane and in three dimensions.

1.2 The Outcome 1 key knowledge and key skills that live in this area

[SD] attaches key knowledge and key skills to the outcomes, not to areas of study ([SDOC] §1.1). The Outcome 1 items that bear on Space and measurement, verbatim:

Key knowledge:

• geometric interpretation of vectors in the plane and of complex numbers in the complex plane
• definition and properties of vectors, vector operations, the geometric representation of vectors and the geometric interpretation of linear dependence and independence of a set of vectors
• standard contexts for the application of vectors to the motion of a particle and to geometric problems
• techniques for solving kinematics problems in one, two and three dimensions
• the vector product and methods determining vector equations of lines and planes
• systems of equations with two and three variables and their geometric interpretation

Key skills:

• perform operations on vectors and interpret them geometrically
• apply vectors to motion of a particle and to geometric problems
• solve kinematics problems using a variety of techniques
• formulate problems which require solutions with systems of linear equations

Two of these are hidden scope — they appear in no area-of-study dot point:

• "systems of equations with two and three variables and their geometric interpretation" is what licenses three-plane intersection problems (unique point, line, no solution). 2025 Exam 2 Section B Q5a is exactly that: "Find the point of intersection of the three planes" (pct = 73.51) [PAPERS].
• "the vector product and methods determining vector equations of lines and planes" states, in one clause, the whole 2023 expansion.

1.3 What the 2023 redesign did to this area

Three things happened simultaneously, and all three are visible in the archive.

(a) The cross product, planes and lines in space came IN. [RPT23 E1], verbatim: "New topics tested in 2023 included integration by parts (Question 5), area of a surface of revolution (Question 7), proof by induction (Question 8) and planes (Question 9)." [RPT24 E1], verbatim: "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)."

(b) Mechanics went OUT. [SD]'s kinematics dot point restricts the calculus area to "rectilinear motion of a single particle". [SDOC] §5.5 establishes what that deleted: 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, and momentum. The words "newton", "tension", "momentum" and "friction" appear in no 2023 November, 2025 November or 2026 NHT paper [PAPERS].

(c) The formula sheet moved with the content. Three changes, all verifiable in the corpus text:

The friction line F ≤ μN is in Documents_exams_mathematics_2012_2012specmath2-w.txt line 1042 and absent from the 2022 sheet [PAPERS]. The constant-acceleration block v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u+v)t first appears on the 2023 November sheet; the 2023 NHT sheet is still the old one ([SDOC] §5.6).

What this means for drilling the archive. Roughly 40% of the Space-and-measurement marks in 2006–2022 are mechanics. Those questions are still excellent training in sign discipline, diagram discipline and equation-of-motion discipline — but as exam preparation for the current design they are out of scope. The clean rule:

Pre-2023 questions that are now the best practice available are precisely the vector-calculus ones: 2016 Exam 2 Section B Q4 (two ships), 2017 Exam 2 Section B Q5 (boat and jet ski), 2018 Exam 2 Section B Q4 (two yachts), 2020 Exam 2 Section B Q4 (aeroplane and drone), 2022 Exam 2 Section B Q4 (minigolf ball). None of them touch forces, and all five are essentially reusable today.

1.4 What the exams actually test — counts

The 544 tagged parts carry 878 marks and split by paper as follows [QJSON]:

Part sizes: 308 one-mark parts, 155 two-mark, 67 three-mark, 11 four-mark, 3 five-mark. The area is delivered in small pieces, which is why a single misread — a scalar written for a vector, a wrong integration limit, an unstated second solution — costs a whole part rather than one method mark inside a long question.

The 243 separators, grouped as in §4 (the grouping is the author's; every ref and percentage is VCAA's):

Two readings of that table matter. First, the largest single block of separator mass sits in families that survive the redesign untouched: variable acceleration as a differential equation (35) and the dot-product / resolute / angle cluster (28). Second, the groups with the lowest medians — collisions and closest approach (27%), arc length and distance travelled (28%), velocity-time and multi-stage motion (30%) — are all current content. Losing the mechanics did not make this area easier; it removed the part of it the state was relatively best at.

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2. The complete catalogue of question types

Fifty types: forty-two set out in full below, plus eight deleted mechanics types tabulated compactly at the end of the section. For each: a name, VCAA's literal wording quoted from a real paper, what it is really testing, the standard method, at least three archive instances with ref and pct, typical marks, and the traps the reports name.

All of T1–T42 are examinable under the 2023– design. T1–T24 are vectors, cross product, lines and planes; T25–T34 are vector calculus and paths; T35–T42 are rectilinear kinematics, gravity and projectiles. T43–T50 are the deleted mechanics; they are catalogued because they are roughly 40% of the archive's marks in this area and because the reports' diagnoses on them are the clearest statements VCAA has ever published about sign discipline and diagram discipline.

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T1. Vector between two points

Wording — 2013 Exam 1 Q3a: "Find AB in the form ai + bj + ck." 2010 Exam 1 Q3a: the same, from two given position vectors.

Really testing — that AB = OB − OA = b − a, in that order, and that the answer is written as a vector, not an ordered triple.

Method — subtract tail from head. Write i, j, k explicitly.

Traps — [RPT10 E1]: "A small number found BA instead of AB and others gave the answer as (−2, 1, 2) rather than in the form requested." [RPT13 E1]: "Some students gave their answer as an ordered triple without indicating that it was a vector rather than a point." The 35% on 2019 Exam 2 Section B Q4a is the same skill made hard by vertex order: [RPT19 E2] — "A significant proportion of students did not correctly consider the order of the vertices of the parallelogram and consequently set AB = DC… A diagram could assist to avoid this error."

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T2. Magnitude, and distance between two points in three dimensions

Wording — 2012 Exam 2 Section A Q16: "The distance between the points P(−2, 4, 3) and Q(1, −2, 1) is". 2010 Exam 2 Section A Q16: "The square of the magnitude of the vector d = 5i + j − 10k is".

Really testing — |r| = √(x² + y² + z²), and reading whether the question wants |r| or |r|².

Method — components, square, sum, root — and stop where the question stops.

Traps — [RPT13 E1] on Q3c: "Other typical errors included giving 38, which is the square of the length, and finding the lengths of all three sides but not stating which was the length of the hypotenuse." 2015 Exam 1 Q3 is the compound version — integrate r'(t) first, then take the magnitude: [RPT15 E1] lists "forgetting to include a constant (vector) of integration" as the dominant error.

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T3. Unit vector in the direction of a given vector

Wording — 2014 Exam 1 Q1a: "Find the unit vector in the direction of a." 2013 Exam 2 Section B Q4a: "Find a unit vector in the direction of b."

Really testing — dividing by the magnitude, and surd arithmetic under exam pressure.

Method — â = a/|a|. Rationalise only if the form demands it.

Traps — [RPT14 E1]: "The most common errors involved finding the magnitude to be √6 or 6 but not finding the unit vector. Some students made arithmetic errors and, finding the magnitude to be √5 or √7, obtained the correct unit vector but then incorrectly rationalised the denominators." [RPT19 E2]: "students who were otherwise successful in Question 4d. did not always give the required unit vector."

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T4. Unit vector perpendicular to given vectors

Wording — 2006 Exam 2 Section A Q16: "A unit vector perpendicular to 5i + j − 2k is". 2019 Exam 2 Section B Q4d: "Show that 6i + 2j + 5k is perpendicular to both AB and AD, and hence find a unit vector that is perpendicular to the base of the pyramid." 2023 Exam 2 Section A Q14: "Let a = i + j, b = i − j and c = i + 2j + 3k. If n is a unit vector such that a·n = 0 and b·n = 0, then c·n is equal to".

Really testing — before 2023: verifying perpendicularity by dot product and normalising. Since 2023: the cross product is now the intended route.

Method — n = (a × b)/|a × b|. Pre-2023 route: set n = xi + yj + zk, solve a·n = 0, b·n = 0 up to scale, normalise.

Traps — all three are separators, which is remarkable for a mechanical task. 2023 Exam 2 Section A Q14 is the sharpest: there are two unit vectors perpendicular to both a and b, so c·n has two values, and the item is built to see whether you notice.

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T5. Resolution of a vector into rectangular components

Wording — 2007 Exam 2 Section A Q16: "A force of magnitude 18 newtons acts on a body at an angle of 150° in the anticlockwise direction to the vector i. A vector representation of this force could be". 2021 Exam 2 Section A Q11: "Let i be a unit vector pointing east and let j be a unit vector pointing north. A group of hikers travels 5 km in the direction south 30° west and then north for 10 km. The position vector a of the group of hikers with respect to the starting point is".

Really testing — converting a bearing or an angle to i–j components with correct signs.

Method — draw the vector, read off (|v|cos θ) i + (|v| sin θ) j with θ measured from the positive i direction; for compass bearings, convert to that convention first.

Traps — sign of the component in the second and third quadrants; "south 30° west" is 30° from south, not from west.

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T6. Linear dependence — find the parameter

Wording — 2008 Exam 1 Q3: "Find m so that a, b and c form a linearly dependent set of vectors." 2016 Exam 1 Q5b: "Find the value of d if the vectors are linearly dependent." 2022 Exam 2 Section A Q11: "Consider the vectors a = 2i + 3j + pk, b = i + 2j − qk and c = −3i + 2j + 5k, where p and q are real numbers. If these vectors are linearly dependent, then".

Really testing — that c = αa + βb has a solution; equivalently that the 3×3 determinant of components is zero.

Method — write c = αa + βb, equate components to get three equations in α, β and the parameter, solve. The determinant route is faster and VCAA accepts it: [RPT19 E1] — "A number of students successfully evaluated a determinant in order to find the value."

Traps — [RPT08 E1] on the 16% outing: "A large number of students did not understand the concept of linear dependence." [RPT16 E1] notes the shortcut worth knowing: "Some insightful solutions were seen using the fact that 2a + b eliminated j."

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T7. Linear independence — find the values for which the set is independent

Wording — 2021 Exam 1 Q6: "Consider the three vectors a = i + 6j − 3k, b = 2i − 8j + 5k and c = 3i − 2j + (1 − p²)k, where p is a real constant. Find the values of p for which the three vectors are linearly independent."

Really testing — the negation. You solve for dependence and then complement the answer set — and you must write the complement.

Method — solve the dependence condition for p; answer p ∈ R \ {those values}.

Traps — [RPT21 E1], verbatim: "Most students realised that they first needed to write down and solve a system of linear equations in order to find the values of p for which the set of vectors were linearly dependent. 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." One third of the state got 3/4 on that question. That single missing sentence is the difference between a 43 and a 46.

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T8. Collinearity and division of a segment in a ratio

Wording — 2008 Exam 2 Section A Q17: "P, Q and R are three collinear points with position vectors p, q and r respectively, where Q lies between P and R. If QR = ½ PQ, then r is equal to". 2010 Exam 2 Section B Q1ai: "Find MA" (M the midpoint).

Really testing — expressing an unknown position vector through a chain of displacements.

Method — r = q + QR = q + ½(q − p). Always route through the origin.

Traps — sign reversal. [RPT10 E2]: "the most common error being one of sign reversal: BA = b − a." These are the cheapest marks in the whole area; drop none of them.

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T9. Scalar (dot) product to test or impose perpendicularity

Wording — 2008 Exam 2 Section A Q14: "If the vectors a = mi + 4j + 3k and b = mi + mj − 4k are perpendicular, then". 2014 Exam 1 Q1c: find m such that a stated dot-product condition holds. 2022 Exam 2 Section A Q12: "Consider the vectors u(x) = cosec(x)i − 3j and v(x) = cos(x)i + j. If u(x) is perpendicular to v(x), then possible values for x are".

Really testing — a·b = 0 ⟺ a ⊥ b, and then solving whatever equation that produces.

Method — form a·b, set to zero, solve; check the domain.

Traps — [RPT14 E1]: "many equated the dot product to 1 or −1. Sign errors were quite common when rearranging the equation to find m." The cosec(x) item is a domain trap: x = 0, π are excluded.

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T10. Angle between two vectors

Wording — 2022 Exam 1 Q6a: "Find the cosine of the acute angle between the vectors a = 2i − 3j + 6k and b = i + 2j − 2k." 2011 Exam 2 Section A Q12: "The angle between the vectors 3i + 6j − 2k and 2i − 2j + k, correct to the nearest tenth of a degree, is". 2014 Exam 2 Section A Q15: "If θ is the angle between a = 3i + 4j − k and b = i − 4j + 3k, then cos(2θ) is".

Really testing — cos θ = (a·b)/(|a||b|) — and then reading precisely what is asked for: cos θ, θ, cos 2θ, sin 2θ, the acute angle, or the obtuse one.

Method — dot product over product of magnitudes. For cos 2θ use 2cos²θ − 1; for sin 2θ use 2 sin θ cos θ with sin θ = +√(1 − cos²θ) when θ is taken acute.

Traps — [RPT24 E1] on Q4a: "While many students successfully found that [cos θ], not all were able to find the correct angle (in degrees or radians) between the vectors." [RPT19 E2]: "Some students who would otherwise have been successful did not explicitly answer the question and instead found an approximate value of the angle" — when VCAA asks for cos θ, give cos θ. [RPT14 E1] on the 42% Q1b: "Common errors included not using the dot product or direction cosines, but instead using a 'tan' argument from a right-angled triangle."

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T11. Angle at a vertex — ∠ABC from three position vectors

Wording — 2006 Exam 2 Section B Q2b: "Use a vector method to find the cosine of ∠ADC, the angle between DA and DC." 2015 Exam 2 Section A Q17: "Points A, B and C have position vectors … The cosine of angle ABC is equal to". 2017 Exam 1 Q5: "Relative to a fixed origin, the points B, C and D are defined respectively by the position vectors b = i − j + 2k, c = 2i − j + k and d = ai − 2j, where a is a real constant. Given that the magnitude of angle BCD is π/3, find a."

Really testing — that the angle at B is between BA and BC — both starting at B — not between the position vectors a and c.

Method — form the two displacement vectors out of the vertex, then dot product.

Traps — 2017 Exam 1 Q5 at pct = 11 (dist 25/5/45/13/11, average 1.8/4) is the hardest pure dot-product item in the archive. [RPT17 E1]: "The most common errors involved finding the dot product of two position vectors rather than the vectors [out of the vertex]". [RPT06 E2] on the 16% part: "Few managed to show ∠ABC and ∠ADC were supplementary. Most resorted instead to finding numerical approximations of the two angles."

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T12. Scalar resolute

Wording — 2010 Exam 2 Section A Q15: "The scalar resolute of a = 3i − k in the direction of b = 2i − j + 2k is". 2018 Exam 2 Section A Q14: "The scalar resolute of a = 3i − 2k in the direction of b = −i + 2j + 3k is". 2026 NHT Exam 2 Section A Q15: "Let u = ai + 3j − 5k, where a ∈ R, and v = 2i + 2j − k. If the scalar resolute of u in the direction of v is 2, then the value of a is".

Really testing — a·b̂ is a number. The answer has no i, j, k in it.

Method — a·b / |b|.

Traps — the scalar resolute is signed. [RPT18 NHT E1]: "The scalar resolute can take two values" when the question fixes only a magnitude. 2024 Exam 2 Section A Q14 at 36% is the item that mixes scalar and vector resolute in one stem; [RPT24 E2]'s own comment reads: "Vector resolute of [ ] in the direction of [ ] so hence [ ]. Scalar resolute [ ] or [ ] in the direction of [ ]" — the equations are stripped by the DOCX extraction, but the structure of the comment shows both objects were in play.

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T13. Vector resolute (the parallel component)

Wording — 2016 Exam 1 Q5a: "Find the vector resolute of a in the direction of b." 2012 Exam 2 Section A Q17: "If u = 2i − 2j + k and v = 3i − 6j + 2k, the vector resolute of v in the direction of u is". 2021 Exam 2 Section A Q13: "The scalar resolute of vector a in the direction of vector b is ±4. If b = 3i, the vector resolute of a in the direction of b is".

Really testing — (a·b̂)b̂ is a vector parallel to b. Two b̂'s, not one.

Method — (a·b/|b|²) b. Never divide by |b| only once.

Traps — [RPT16 E1], verbatim and comprehensive: "This question was well answered by students who knew what a vector resolute was and used the correct formula. Some found the scalar resolute and several had an incorrect formula for the vector resolute (sometimes not using the unit vector, occasionally finding the vector resolute in the direction of a)."

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T14. Perpendicular component, and "express a as the sum of two vector resolutes"

Wording — 2014 Exam 2 Section B Q3a, the canonical five-mark version: "Express a as the sum of two vector resolutes, one of which is parallel to b and the other of which is perpendicular to b. Identify clearly the parallel vector resolute and the perpendicular vector resolute." 2020 Exam 1 Q5b: "Find the component of a that is perpendicular to b." 2009 Exam 1 Q3: give the parallel and perpendicular parts.

Really testing — a = (a·b̂)b̂ + [a − (a·b̂)b̂], and then actually writing the sum down.

Method — compute the parallel resolute; subtract it from a for the perpendicular one; write the final line a = (…) + (…).

Traps — [RPT14 E2], verbatim: "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." [RPT13 E2]: "A few students had the resolutes the wrong way around, finding the resolute of b in the direction of a."

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T15. Inverse resolute — the resolute is given, find the parameter

Wording — 2020 Exam 1 Q5a: "Let a = 2i − 3j + k and b = i + mj − k, where m is an integer. The vector resolute of a in the direction of b is −(11/18)(i + mj − k). Find the value of m." 2016 Exam 2 Section A Q11: "If the scalar resolute of a in the direction of b is √74/273 …, then λ equals". 2018 NHT Exam 1 Q2 (3 marks): "Let a = 3i − 2j + mk and b = 2i − j + 3k, where m ∈ R. Find the value(s) of m such that the magnitude of the vector resolute of a parallel to b is equal to √14."

Really testing — running the resolute formula backwards into a quadratic, then filtering solutions against a stated condition ("where m is an integer", "where n is a positive real number").

Method — equate the scalar multiple, cross-multiply, solve the quadratic, then apply the stated constraint to discard a root.

Traps — [RPT20 E1]: "Students who factorised to solve the quadratic equation were generally more successful than those who used the quadratic formula." The constraint sentence ("where m is an integer") is not decoration — it is how you choose between the two roots.

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T16. Vector proof of a geometric result

Wording — 2015 Exam 1 Q1b: "Show that the diagonals of the rhombus OABC are perpendicular." 2022 Exam 1 Q6b: "OPQ is a semicircle of radius a with equation y = √(a² − (x − a)²). P(x, y) is a point on the semicircle. (i) Express the vectors OP and QP in terms of a, x, y, i and j … (ii) Hence, using the vector scalar (dot) product, determine whether OP is perpendicular to QP." 2010 Exam 2 Section B Q1dii: "Use a vector method to show that OQ is perpendicular to AB."

Really testing — the study design's own three named results: rhombus diagonals, concurrent medians, angle in a semicircle. The method is always the same and the marks are for the connecting steps, not the conclusion.

Method — name the two vectors in terms of the given basis; take the dot product; simplify to zero using the defining property of the figure (equal side lengths for a rhombus, |OP| = a for a circle); state the conclusion in words.

Traps — [RPT06 E2]: "Many students just stated that AC and BD were perpendicular without demonstrating or explicitly saying why." [RPT10 E2] is the deeper one: students "working with the general case" wrote |b| = |a| = 1 to force the dot product to zero — i.e. they assumed the very thing they were proving. And 2013 Exam 2 Section B Q4eiii at pct = 10 (dist 90/10) is a one-mark notation question: [RPT13 E2] — "The majority of students had scalar 0 as the result of the vector calculation, rather than the vector 0. This is a conceptual error — a sum of vectors is a vector."

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T17. Vector (cross) product — computation

Wording — 2026 NHT Exam 2 Section B Q4ci: "Find the cross product D₃(t) × D₄(t)." 2026 NHT Exam 2 Section B Q4cii: "Evaluate this cross product when t = 1 and explain what this result means in terms of the locations of Drone 3 and Drone 4." Sample [SD]-era Exam 2 Section A Q3 asks students to identify the correct 3×3 determinant.

Really testing — the determinant expansion supplied on the formula sheet, and — new in the 2026 NHT paper — the interpretation: a × b = 0 means the vectors are parallel, so the two drones are collinear with the origin.

Method — write the i j k determinant with the two vectors as rows two and three; expand with alternating signs. On Exam 2, CAS does it: [RPT24 E2] — "Students should be aware that their CAS technology can help them find cross products accurately."

Traps — [RPT23 E1]: "Most students realised that a cross product could be used to find a vector perpendicular to the plane. 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." The middle term of the determinant carries a minus sign; that is the single most common by-hand slip.

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T18. Area of a triangle or parallelogram

Wording — 2023 Exam 1 Q9e: "AB and AD are adjacent sides of a parallelogram. Find the area of this parallelogram." 2019 Exam 2 Section B Q4c: "Find the area of the base of the pyramid." 2010 Exam 1 Q3c: "Hence find the area of triangle OAB." 2026 NHT Exam 2 Section B Q5a: "Using a vector method, find the area of the triangle with vertices P₁, P₂ and P₃. Give your answer in the form a√b/c, where a, b, c ∈ Z⁺ and c = a + 1."

Really testing — |a × b| for the parallelogram, ½|a × b| for the triangle. Pre-2023 the only route was ½|a||b| sin θ after finding cos θ.

Method — cross product, magnitude, halve for a triangle. If the question says "hence", you must use the previous part's cos θ and sin θ = √(1 − cos²θ).

Traps — [RPT19 E2] is brutal: "A frequent issue here was the 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." [RPT10 E1]: "A large proportion of the cohort ignored the word 'hence' and attempted to find the area by another means, which did not attract any marks." [RPT24 E2] on Q5dii: "omitting the division of the cross product by 2 to find the area; incorrectly assuming the triangle was either isosceles or right-angled."

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T19. Cartesian equation of a plane

Wording — 2023 Exam 1 Q9c: "Hence find the equation of the plane in Cartesian form." 2024 NHT Exam 2 Section B Q5b: "Find the Cartesian equation of the plane P." 2026 NHT Exam 2 Section B Q5b: "The points P₁, P₂ and P₃ lie in the plane Π₁. Find the Cartesian equation of this plane." Sample Exam 1 Q12: "The vectors a = 2i + 3j − k and b = 4i − 2j + 3k lie in a plane that passes through the point (3, 2, 1). Find the Cartesian equation of this plane."

Really testing — normal first, point second. n·(r − r₀) = 0 becomes ax + by + cz = d with d = n·r₀.

Method — get a normal (cross product of two spanning vectors, or of two displacement vectors between three given points); write ax + by + cz = d; substitute the known point to find d.

Traps — substituting the wrong point; forgetting that a plane through three points needs two displacement vectors from the same vertex. 2024 NHT Exam 2 Section A Q15 asks only for "A normal vector to the plane that contains the points (2, −1, −1), (3, 1, 1) and (−1, −1, 2)" — normal-only variants exist and are one mark.

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T20. Vector and parametric equations of a line in space

Wording — 2024 Exam 2 Section B Q5a: (from [RPT24 E2]) students had to write a vector equation of a line; "A common error was to write the vector rather than the vector equation of the line." Sample Exam 1 Q15a: "Find the vector equation of the line through the points A(3, 1, −1) and B(5, 2, −6)." 2023 Exam 2 Section B Q5d: "Write down an equation of the line L in parametric form."

Really testing — r(t) = r₁ + t·d with d = B − A; and the difference between a vector, a vector equation, and a set of parametric equations.

Method — point plus parameter times direction. Parametric form: split into x(t), y(t), z(t). Both forms are on the current formula sheet.

Traps — [RPT24 E2] on Q5a: "Some students did not use the correct notation. A common error was to write the vector rather than the vector equation of the line." [RPT25 E2] on Q5bi: "Several responses used CAS to find the solution to the two equations of the planes, but left the answer in parametric form or wrote it as the vector equation of a line. The response should have then identified the direction vector to answer the question."

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T21. Intersections — line with plane, line with line, plane with plane

Wording — 2025 Exam 2 Section A Q18: "The lines given by r₁(λ) = 2i + rj + 3k + λ(i + j + 4k) and r₂(μ) = i + sk + μ(i − j − k) intersect at the point (4, 3, t), where λ, μ ∈ R and r, s and t are real constants. The values of r, s and t respectively are". 2026 NHT Exam 2 Section A Q19: "Which one of the following vector equations describes the line of intersection of the planes given by x − 2y + z = 3 and 2x + y − z = 5?" 2025 Exam 2 Section B Q5a: "Find the point of intersection of the three planes." Sample Exam 1 Q13b: "Find the point of intersection of the line given by r = 2i + 5k + t(2i + 4j − 3k) with the plane given by 2x − 2y + z = 6."

Really testing — (a) two lines need two different parameters; (b) a line meets a plane by substituting the parametric coordinates into the Cartesian equation; (c) two planes meet in a line whose direction is n₁ × n₂.

Method — line ∩ plane: substitute x(t), y(t), z(t) into ax + by + cz = d, solve for t, back-substitute. Line ∩ line: set r₁(λ) = r₂(μ), three equations in two unknowns, solve two and check the third. Plane ∩ plane: direction = n₁ × n₂; find one point by setting a convenient variable to zero.

Traps — [RPT25 E1] on the two-lines question states the error precisely: "It was common for students to use the same parameter for both lines. This did not result in viable equations to solve. In this case, students were ineligible for full marks." [RPT25 E2] on Q18: "Knowing the lines intersect at the point (4, 3, t) allows simultaneous equations to be developed and solved" — the given point is the gift; use it first.

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T22. Angle between two planes, and between a line and a plane

Wording — 2023 Exam 2 Section A Q18: "What value of k, where k ∈ R, will make the following planes perpendicular? Π₁: 2x − ky + 3z = 1, Π₂: 2kx + 3y − 2z = 4." 2025 Exam 2 Section A Q15: "Consider the two planes described by the equations 2x + 2y + z = 2 and ax + 4z = 1… The angle between the two planes is cos⁻¹(2/3). The value of a satisfies the equation". 2026 NHT Exam 1 Q11: "Consider the two planes 2x + y + z = 4 and px + (p + 1)y + 2z = 1… Given that the acute angle between the two planes is 60°, find all possible values of p." 2023 Exam 2 Section B Q5c: "At what acute angle does the line given by r(t) = 3i + 2j + 4k + t(i − 2j + 2k), t ∈ R, intersect the plane Π? Give your answer in degrees correct to the nearest degree." Sample Exam 1 Q15b: "Find the sine of the angle that this line makes with the plane given by x + 2y − z = 9."

Really testing — the angle between planes is the angle between their normals; the angle between a line and a plane is the complement of the angle between the line's direction and the normal. That is why VCAA asks for the sine.

Method — planes: cos θ = |n₁·n₂|/(|n₁||n₂|). Line and plane: sin θ = |d·n|/(|d||n|).

Traps — [RPT23 E2] on Q5c: "A significant number of students did not proceed beyond finding the angle of [the normal]." That is the whole trap, stated by VCAA. Sine for a line-and-plane, cosine for plane-and-plane.

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T23. Shortest distances in three dimensions

Wording — 2025 Exam 2 Section B Q5c: "Find the shortest distance from the point (1, 1, 2) to the plane P₃." 2025 Exam 2 Section B Q5dii: "Find all values of m for which the shortest distance between plane P₁ and the plane of the form 6x + 27z = m is 23/(3√85)." 2023 Exam 2 Section B Q5e: "Find the shortest distance from the origin to the plane Π." 2023 Exam 2 Section B Q5b: "Find the shortest distance from point B to the line segment AC." 2024 Exam 1 Q10: distance between two skew lines (from [RPT24 E1]). 2026 NHT Exam 2 Section B Q4b: "Determine the shortest distance between Drone 1 and Drone 2, and the time when this occurs."

Really testing — a family of formulas none of which is on the formula sheet. Point to plane: |n·(P − P₀)|/|n|. Point to line: |AP × d|/|d|. Parallel planes: apply the point-to-plane formula to any point of one. Skew lines: |(a₂ − a₁)·(d₁ × d₂)|/|d₁ × d₂|.

Method — identify which of the four configurations you have, then pick the formula. Draw it if in doubt.

Traps — [RPT24 E1] on the 14% skew-lines question, verbatim: "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." [RPT25 E2] on Q5dii: "Many responses did not demonstrate that the modulus needed to be used and consequently only one of the solutions was found." A distance equation |f(m)| = k has two solutions; 2025 Exam 2 Section B Q5dii's answer is m = 1 and m = 47.

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T24. Parallel planes and parallel lines — establishing the relation

Wording — 2025 Exam 2 Section B Q5di: "Consider a family of planes, Λ, with equation 6x + 27z = m, where m ∈ N. Show that the plane P₁ is parallel to each member of Λ." 2026 NHT Exam 2 Section B Q5ci: "Show that the plane Π₃ given by d − 1.5z = x − ½y, where d ∈ R, is parallel to the plane Π₂." 2018 Exam 2 Section A Q12: "If |a + b| = |a| + |b| and a, b ≠ 0, which one of the following is necessarily true?"

Really testing — "parallel" means normals are scalar multiples, and you must say so.

Method — write both normals; exhibit the scalar k with n₂ = k n₁; state the conclusion.

Traps — [RPT25 E2]: "Many responses did not identify that they were working with the normal vectors of the planes and many mixed up the multiple, writing [1/3] rather than 3." 2018 Exam 2 Section A Q12 at 36% is the abstract cousin: the triangle inequality is an equality exactly when the vectors are parallel and in the same direction; four of the five distractors are true-sounding.

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T25. Cartesian equation of a path from a vector function

Wording — 2023 Exam 1 Q10b: "Show that the Cartesian equation of the path of the particle is (x − 2)² + (y − 1)² = 9." 2013 Exam 1 Q7a: "Show that the Cartesian equation of the path of the particle is x²/16 − y²/4 = 1." 2020 Exam 2 Section B Q4bi: "Use r_A(t) to show that the Cartesian equation of the path of the aeroplane is given by (x−450)²/22500 + (y−400)²/40000 = 1." 2011 Exam 2 Section A Q13: "The position of a particle at time t is given by … The Cartesian equation of the path of the particle is".

Really testing — eliminating t using a Pythagorean or double-angle identity, and recognising the conic. [SD] names "circles, ellipses and hyperbolas".

Method — isolate cos/sin/sec/tan of the parameter from x and from y; apply cos² + sin² = 1 or sec² − tan² = 1; tidy. If the result is given, every algebraic step must appear.

Traps — [RPT12 E2]: "A frequent error was (y − 6400)² in the Cartesian equation and a less frequent error was the omission of the 6400 altogether. A number of students used the velocity vector and eliminated t instead of using the position vector." [RPT08 E2] on the 32% instance: "A significant number of students gave only the positive square root of the right side if they chose to make y the subject."

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T26. Sketching the path — domain, direction, endpoints

Wording — 2016 Exam 2 Section B Q4b: "Sketch and label the path of each ship on the axes below. Show the direction of motion of each ship with an arrow." 2020 Exam 2 Section B Q4bii: "Sketch the path of the aeroplane on the axes provided below. Label the position of the aeroplane when t = 0, using coordinates, and use an arrow to show the direction of motion." 2025 Exam 2 Section B Q4b: "On the graph above, draw an arrow from the point (9, 0) to indicate the direction of motion of the particle." 2025 Exam 2 Section B Q4f: "On the graph on page 18, trace the path of the particle for t ∈ [0, π]."

Really testing — that a vector function traces only part of the conic, over the t-domain given, in a definite direction, with endpoints.

Method — convert the t-domain to an x-domain; plot the endpoints and mark them with coordinates; put in the arrow; keep the curve smooth.

Traps — [RPT06 E1] is the classic: "A proportion of students ignored the restriction on the domain. Of those who did restrict the domain, many used the given t-values rather than converting to Cartesian using 2 ≤ x ≤ 6… Students need to be reminded that the instruction 'sketch' does [require endpoints]." [RPT16 E2]: "Many students did not take note of when the vector functions applied and consequently plotted the paths over incorrect domains." [RPT25 E2] on Q4f: "Some students did not take care to finish their drawing at the point with coordinates (5, 4)."

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T27. Velocity and acceleration by differentiating r(t)

Wording — 2012 Exam 1 Q9a: "Find the velocity of the particle at time t." 2009 Exam 2 Section B Q3d: find ṙ(t). 2020 Exam 2 Section B Q1bii: "Find the velocity, v, in m s⁻¹, of the particle when t = π."

Really testing — component-wise differentiation, the product and chain rules inside components, and keeping i, j, k attached.

Method — differentiate each component; keep the basis vectors; substitute only at the end.

Traps — [RPT20 E2] on the 35% part: "Many students gave the (scalar) magnitude of the velocity rather than the required velocity." [RPT09 E2]: "The most common error was the omission of some of i, j, k from the expression." [RPT15 E2] lists among the paper's weaknesses: "dropping of i, j, k in a vector question, whereby an expression ends up as a mixture of vector and scalar quantities."

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T28. Speed, maximum speed and minimum speed

Wording — 2016 Exam 1 Q8a: "Find an expression for the speed, in metres per second, of the body at time t." 2022 Exam 2 Section B Q4bii: "Find the minimum speed of the ball, in metres per second, and the time, in seconds, at which this minimum speed occurs." 2013 Exam 2 Section B Q5a: minimum and maximum speeds. 2025 Exam 2 Section B Q4d: "Show that the speed of the particle, in m s⁻¹, at time t can be expressed as √(125 − 100 cos(3t/2))." Sample Exam 1 Q16: "Find the time at which the minimum speed occurs and calculate the minimum speed."

Really testing — speed is |ṙ(t)|, a scalar; then a max/min of a function of t, usually via a Pythagorean identity rather than calculus.

Method — differentiate, square-and-add components, simplify with cos² + sin² = 1 or a compound-angle identity, then read off the extremes of the surviving trigonometric term.

Traps — [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." [RPT15 E2] on Q4d: "Some students could not simplify √((−5π/6 sin(πt/30))² + (5π/6 cos(πt/30))²) using the Pythagorean identity." [RPT24 E2] on Q4ciii: "The common error was that some students forgot to take the square root of the square of the speed."

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T29. Perpendicularity conditions on r, v and a

Wording — 2012 Exam 2 Section B Q4b: "Find the acceleration of the space station, and show that its acceleration is perpendicular to its velocity." 2023 Exam 1 Q10d: "Find all values of t for which the position vector of the particle, r(t), is perpendicular to its velocity vector, ṙ(t)." 2025 Exam 2 Section A Q16: "For what value of n is the particle's acceleration perpendicular to its velocity when t = ½?" 2015 Exam 2 Section B Q4c: "Show that the velocity of the helicopter is perpendicular to its acceleration."

Really testing — setting a dot product of two differentiated vectors to zero, and spelling out the expansion rather than asserting the result.

Method — differentiate as needed; form the dot product; expand term by term; simplify to zero (or solve for the parameter). For r·ṙ = 0, note that this is equivalent to d/dt(|r|²) = 0.

Traps — [RPT15 E2]: "Some students simply asserted that ṙ(t)·r̈(t) = 0 [without showing it]." [RPT23 E1] on the 7% part: "While many students realised that they needed to solve r·ṙ = 0, many were not able to get to the final result." [RPT25 E1] on Q5b: "A common incorrect response to the equation was [an extra root] in addition to [the correct ones]" — solving a trigonometric or polynomial equation over the stated domain is where this type is actually lost.

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T30. Integrating a vector function of time

Wording — 2007 Exam 1 Q6a: find r(t) from ṙ(t) and an initial position. 2015 Exam 1 Q3: "The velocity of a particle at time t seconds is given by ṙ(t) = (4t − 3)i + 2t j − 5k … Find the distance of the particle from the origin in metres when t = 2, given that r(0) = i − 2k." 2025 Exam 2 Section A Q17: "The acceleration vector of a particle that starts from rest is given by a(t) = 4cos(2t)i − 10sin(2t)j + 6e^{2t}k… The velocity vector of the particle, v(t), is given by".

Really testing — the vector constant of integration, and the initial condition that pins it.

Method — antidifferentiate component-wise, add c (a vector), substitute the initial condition, solve for c.

Traps — [RPT07 E1]: "the most common error being to omit a constant (vector) of integration c. This fortuitously happened to be the zero vector, but students who did not show a constant of integration could not get full marks." [RPT25 E2] on Q17: "Care needs to be taken to include a constant vector of integration when integrating the acceleration vector to find the velocity. Students could use a definite integral." [RPT20 E2] on Q15: "Use a = dv/dt then antidifferentiate twice to find the position vector."

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T31. Collision versus paths crossing

Wording — 2016 Exam 2 Section B Q4a: "Show that the two ships will not collide, clearly stating your reason." 2018 Exam 2 Section B Q4b: show the yachts do not collide. 2021 Exam 1 Q9bi: "Show that the particles A and B will collide"; Q9bii: "Hence, find the coordinates of the point of collision." 2025 Exam 1 Q5a: "Show that for collision to occur when t = 1, the value of c is −4." 2017 Exam 2 Section B Q5d: "Assuming the vessels would collide shortly after starting, find the time of the collision and the value of a."

Really testing — [SD]'s own distinction: "whether their paths cross or if the particles meet". Paths crossing needs two independent parameters; collision needs the same t.

Method — collision: set r_A(t) = r_B(t) component-wise and check that the same t satisfies both/all equations. Paths crossing: set r_A(s) = r_B(t) with two parameters, or intersect the Cartesian equations.

Traps — [RPT21 E1] on Q9bii: "Some students mistakenly gave the point of collision as [the parameter value], confusing the parameter with the x-value of the point of collision." [RPT18 E2] on Q4c (34%): "Missing the condition that t ≥ 0 led, incorrectly, to two points being provided. Many otherwise correct responses were not expressed in the required form, as coordinates correct to three decimal places." [RPT17 E2] on Q5d: "Many went on to give decimal approximations rather than supplying the exact forms."

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T32. Closest approach — minimising the distance between two moving objects

Wording — 2016 Exam 2 Section B Q4di: "Find the value of t, correct to three decimal places, when the ships are closest"; Q4dii: "Find the minimum distance between the two ships, in kilometres, correct to two decimal places." 2017 Exam 2 Section B Q5ci: "Write down an expression for the distance between the jet ski and the boat at any time t"; Q5cii: "Find the minimum distance separating the boat and the jet ski." 2026 NHT Exam 2 Section B Q4b: "Determine the shortest distance between Drone 1 and Drone 2, and the time when this occurs. Give the distance in kilometres and the time in hours, each correct to three decimal places." 2007 Exam 2 Section B Q4d: closest approach of an aircraft to a beacon.

Really testing — building d(t) = |r₂(t) − r₁(t)| and minimising it — on Exam 2, numerically.

Method — form the difference of position vectors (not of magnitudes), take its magnitude, minimise. Minimising d² is equivalent and avoids the square root. The "r·v = 0" shortcut works when one object is fixed.

Traps — [RPT16 E2]: "Some students used an expression for the difference between position vector magnitudes" — i.e. |r₂| − |r₁| instead of |r₂ − r₁|. [RPT17 E2] on the 15% part: "Incorrect answers involving other locally minimum values were frequent" — a periodic d(t) has many local minima and the question wants the global one. [RPT24 E2] on Q4e: "This question part was often not attempted."

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T33. Arc length and distance travelled along a path

Wording — 2009 Exam 2 Section B Q3gi: "Write down a definite integral of the form ∫√(a + bt²) dt which represents the distance travelled by the child"; Q3gii: "Find the distance travelled by the child, correct to the nearest tenth of a metre." 2022 Exam 2 Section B Q4d: "How far does the ball travel during the first four seconds after passing through O?" 2025 Exam 2 Section B Q4g: "Find the length of the path traced in part f." 2018 Exam 1 Q10: "The distance d metres that the particle travels along the curve in three-quarters of a second is given by d = ∫₀^{3/4} √(at² + bt + c) dt. Find a, b and c, where a, b, c ∈ Z." 2023 Exam 1 Q10c: "If the distance travelled along the curve from A to B is 4π/3, find a."

Really testing — distance = ∫|ṙ(t)| dt with correct time limits, and the algebra of simplifying √((dx/dt)² + (dy/dt)²).

Method — differentiate, square, add, simplify, integrate over the t-interval. Never integrate over x-limits when the parameter is t.

Traps — this is the lowest-scoring family in the area (median pct 28). [RPT18 E1] on the 2% question: "Most students recognised that the arc length formula needed to be applied, but some had difficulty differentiating arcsin(t) + t√(1 − t²)… Many students had difficulty simplifying (dx/dt)² + (dy/dt)²." [RPT09 E2]: "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 in Question 3b." [RPT22 E2] on Q4d: "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." [RPT25 E2] on Q4g: "Several responses only included the answer without stating how it was found. The definite integral was required."

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T34. Direction of motion, angle of elevation, and periodicity of a path

Wording — 2015 Exam 2 Section B Q4aii: find the angle of elevation. 2007 Exam 2 Section B Q4c: angle of descent of an aircraft. 2010 Exam 1 Q8c: find tan θ where θ is the angle the direction of motion makes with the positive x direction. 2015 Exam 2 Section B Q4b / 2009 Exam 2 Section B Q3c: the period of the motion. 2025 Exam 2 Section B Q4c: "Find the value of t for which the particle will first return to its starting point."

Really testing — reading geometry out of components: elevation from the vertical component over the horizontal magnitude; direction of travel from ṙ, not r; period from the lowest common multiple of the component periods.

Method — elevation: tan(angle) = (vertical component)/√(sum of squares of horizontal components). Direction: tan θ = ẏ/ẋ. Period: LCM of the component periods, exact form.

Traps — [RPT15 E2]: "A significant number of students seemed not to know what 'angle of elevation' meant. A number found the complementary angle — the angle with the vertical. Others found angles made with the i or j directions. A small number of students tried to find the angle of elevation using the velocity vector." [RPT10 E1]: "Few realised that this was an application of the chain rule and could be solved using tan θ = dy/dx = ẏ(t)/ẋ(t)." [RPT09 E2]: "Few students realised that consideration of the period of the i and j components of the motion was needed. A common answer was 6 s." [RPT25 E2] on Q4c: "Finding the lowest common multiple of the two periods … was the simplest method… Many responses quoted the answer as a decimal rather than exact form."

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T35. Constant-acceleration formulae

Wording — 2022 Exam 2 Section A Q14 and, reworded, 2025 Exam 2 Section A Q12: "A particle moving in a straight line with constant acceleration has a velocity of 7 m s⁻¹ at point A and 17 m s⁻¹ at point B. The velocity of the particle, in metres per second, at the midpoint of AB is". 2010 Exam 2 Section A Q19: "An object is moving in a northerly direction with a constant acceleration of 2 m s⁻²… The exact initial velocity of the object could have been". 2015 Exam 2 Section B Q5c: "v² = 2 × 0.902 × 30".

Really testing — choosing the formula that avoids the unknown you do not want, and — in the midpoint item — realising that the velocity at the midpoint of the distance is √((u² + v²)/2), not (u + v)/2.

Method — list u, v, a, s, t; pick the formula omitting the quantity you neither know nor want. All four formulas are on the current sheet.

Traps — the same midpoint item was set in 2022 and again in 2025 and the state failed it both times (28%, then 33%). [RPT25 E2]'s comment is one line: "Use the constant acceleration formulas to find the velocity at the midpoint." [RPT12 E2]: "Consistency of signs in the constant acceleration formula was a challenge for many. Few students demonstrated any understanding of the concept of displacement."

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T36. Motion under gravity — vertical projection

Wording — 2025 Exam 2 Section A Q13: "From an open window, a person projects a ball vertically up using an outstretched arm… The point of projection of the ball is 50 m above the ground and its velocity of projection is 20 m s⁻¹. The time, in seconds, it takes for the ball to reach the tray of a truck that is 1 m above the ground directly below the point of projection is closest to". 2023 Exam 2 Section A Q13: "A tourist in a hot air balloon, which is rising vertically at 2.5 m s⁻¹, accidentally drops a phone over the side when the phone is 80 metres above the ground. Assuming air resistance is negligible, how long in seconds, correct to two decimal places, does it take for the phone to hit the ground?" 2013 Exam 2 Section A Q19 and 2018 Exam 2 Section A Q17 are the same question with different numbers.

Really testing — sign convention. The dropped object keeps the balloon's upward velocity; the displacement is negative; g is negative if up is positive.

Method — declare "take up as positive", write u, a = −9.8, s = −(height), solve the quadratic, discard the negative root.

Traps — [RPT25 E2], verbatim and quotable to students: "Constant acceleration formulas may be used. However, care must be taken with the signs. Taking upwards as positive then: …" The distractors are always built from (a) forgetting the initial upward velocity, (b) the sign of s, (c) the other root of the quadratic.

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T37. Projectiles given as a vector function

Wording — 2023 Exam 2 Section A Q16: "A student throws a ball for his dog to retrieve. The position vector of the ball, relative to an origin O at ground level t seconds after release, is given by r_B(t) = 5t i + 7t j + (15t − 4.9t² + 1.5)k…" 2008 Exam 2 Section A Q13: "A cricket ball is hit from an origin at ground level so that its position vector at time t is given by r(t) = 15t i + (20t − 5t²) j… When the cricket ball reaches its maximum height, its position vector is". 2021 Exam 2 Section B Q4a: "Show that the path of the rear wheels of the car, while in the air, is given in Cartesian form by y = x tan θ − 4.9x²/(u² cos²θ)." 2013 Exam 2 Section B Q5c: projectile off a ramp.

Really testing — reading max height from the vertical component (ż = 0), range from z = 0, and — in the harder versions — eliminating t to get the trajectory in terms of u and θ.

Method — treat the vertical component as a quadratic in t; set its derivative to zero for maximum height, the function itself to zero for landing. For the Cartesian trajectory, t = x/(u cos θ).

Traps — 2023 Exam 2 Section A Q16 at 32% is the distance travelled vs displacement trap in projectile clothing. [RPT23 E2]'s own comment: "Max. height = 13 m; 2 × 13 = 26 m. Ball thrown from a height of 1.5 m, so total vertical distance travelled is 26.0 − 1.5 = 24.5 m." [RPT13 E2]: "The major error with this question related to consistency of signs. A number of students broke the motion into two stages, causing more work for themselves."

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T38. Acceleration from velocity as a function of position — v dv/dx

Wording — 2023 Exam 1 Q3a: "A particle moves along a straight line. When the particle is x m from a fixed point O, its velocity, v m s⁻¹, is given by v = (3x + 2)/(2x − 1), where x > 1. Find the acceleration of the particle, in m s⁻², when x = 2." 2012 Exam 1 Q8: "The velocity, v m/s, of a body when it is x metres from a fixed point O is given by v = 2x/(1 + x²). Find an expression for the acceleration of the body in terms of x in simplest form." 2019 Exam 2 Section A Q16: "…v = e^x sin(x). The acceleration of the particle, in m s⁻², can be expressed as". 2011 Exam 2 Section A Q20, 2020 Exam 2 Section A Q17, 2006 Exam 2 Section A Q12: identical structure.

Really testing — that when v is given as a function of x, acceleration is v dv/dx or d/dx(½v²) — not dv/dt, and not dv/dx.

Method — a = v · dv/dx. When v is messy, a = d/dx(½v²) is usually less work, especially with square roots.

Traps — [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." [RPT12 E1]: "This question was answered reasonably well by most students who recognised the need to use v dv/dx or d/dx(½v²). Many of those who used the former got into significant difficulty with square roots… those who used the latter were usually more successful." [RPT23 E1] lists as an area of weakness: "remembering to use an alternative form for acceleration." [RPT24 E1] on Q9a: "Many students incorrectly assumed that the speed detection device would be activated by the car travelling at a speed greater than 40 km/h" — a units-and-interpretation trap on top of the calculus.

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T39. Motion as a differential equation — "show that", then solve

Wording — 2010 Exam 1 Q2a: "Show that the acceleration of the body is given by dv/dt = (v − 4)/2"; Q2b: solve it. 2015 Exam 2 Section B Q5di: show the required dx/dv form; Q5dii: "show that 1.4x = −v − 7 logₑ(7 − v) + 7 logₑ(7)". 2018 Exam 2 Section B Q5c: obtain x = −v/2 − 4.9 logₑ((g − 2v)/g) … in the required form. 2021 Exam 2 Section B Q4d: "Show that v in terms of s is given by v = (180s)^{1/3}." 2022 Exam 1 Q8: "A body moves in a straight line so that when its displacement from a fixed origin O is x metres, its acceleration, a, is −4x m s⁻²… Find v in terms of x for this interval."

Really testing — choosing the acceleration form that makes the variables separable, inverting the derivative when needed, carrying the constant of integration, and choosing the correct branch/sign.

Method — decide whether you want t or x as the independent variable; write a = dv/dt (for t) or a = v dv/dx (for x); separate; integrate; apply the initial condition before rearranging; choose the sign consistent with the physical situation.

Traps — [RPT10 E1] on Q2b: "Many students correctly found that |v − 4| = 4e^{t/2}, but mistakenly assumed that this meant v − 4 = 4e^{t/2} when the correct equation was 4 − v = 4e^{t/2} to satisfy [the initial condition]." [RPT09 E1]: "A large number of students made a common fraction-reciprocal error, going from dv/dx = (v² − 3)/v to dx/dv = v/(v² − 3)… [inverted wrongly]." [RPT22 E1] on Q8: "A number of students chose the incorrect sign for their final answer." [RPT15 E2] on Q5dii: "In this 'show that' question, many students did not include the constant [of integration] or lack of detail showing its evaluation."

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T40. Terminal velocity and limiting behaviour

Wording — 2017 Exam 2 Section B Q2c: "Find the limiting (terminal) velocity, in m s⁻¹, that the skydiver would reach." 2009 Exam 2 Section B Q5c: same. 2023 Exam 1 Q3b: "Find the value that the velocity of the particle approaches as x becomes very large."

Really testing — setting a = 0 and solving, or taking a limit of a rational function; and expressing the answer as a single exact value, not an inequality.

Method — terminal velocity: solve a(v) = 0, take the physically attainable root. Limiting velocity in x: divide numerator and denominator by the highest power.

Traps — [RPT09 E2]: "their answers were often poorly expressed with forms such as v < √(g/2) and v → √(g/2)" — VCAA wants the value. [RPT23 E1]: "Many students wrote [an inequality] for their answer." [RPT17 E2]: "Correct solutions were generally obtained by setting a = 0. Some answers were not given in exact form."

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T41. Definite integrals for time and distance in resisted motion

Wording — 2017 Exam 2 Section B Q2di: "Write down an expression involving a definite integral that gives the time taken for the skydiver to reach a speed of 30 m s⁻¹"; Q2e: "Write down an expression involving a definite integral that gives the distance through which the skydiver falls to reach a speed of 30 m s⁻¹. Find this distance." 2018 Exam 2 Section B Q5ei: the same structure. 2015 Exam 2 Section B Q5div: definite integral for a time. 2008 Exam 2 Section B Q2d: t = ∫ 40/(180 − 3v) dv.

Really testing — t = ∫ dv/a(v) and x = ∫ v dv/a(v), with limits in v, and — repeatedly — whether the motion described started at the beginning or part-way through.

Method — from a = dv/dt, t = ∫_{v₀}^{v₁} dv/a(v). From a = v dv/dx, x = ∫_{v₀}^{v₁} v dv/a(v). Set the lower limit to the velocity at the moment the stated model starts applying.

Traps — [RPT17 E2] on the 17% part is the definitive statement: "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." [RPT18 E2] on Q5ei: "An incorrect integrand, typically the reciprocal of the correct integrand, was common." [RPT15 E2]: "Some attempted to find the time using direct integration instead of a definite integral."

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T42. Velocity–time graphs and multi-stage motion

Wording — 2009 Exam 2 Section B Q1b: "Calculate the distance travelled by the car during the first nine seconds of its motion"; Q1d: average speed; Q1f: the multi-stage catch-up. 2012 Exam 2 Section B Q3ei: "Write down the expressions for the distance travelled by the car during each of the three stages of its motion"; Q3eii: "Find the total distance travelled from when the car starts to accelerate to when it comes to rest." 2016 Exam 2 Section B Q5e: total time of flight of a rocket across three phases. 2019 Exam 2 Section B Q5d: two-stage motion up an incline.

Really testing — treating each phase separately, carrying end-of-phase velocity and position into the next phase, and summing correctly.

Method — tabulate: for each stage, u, v, a, s, t. Solve stage by stage. Sum distances, not velocities.

Traps — [RPT09 E2]: "A large number of students equated part of the area under the first section of the graph to 20t, and did not take account of the several-stage motion of the car. A large number of students solved the problem to the point of getting the correct time, but did not complete the last step to find the distance travelled." [RPT16 E2]: "Inconsistent signs caused some difficulty" and students "added only some parts of the motion to get the total time." [RPT19 E2] lists as a paper-level weakness: "dealing with motion under constant acceleration over two stages on an inclined plane."

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The deleted mechanics types — T43 to T50

These eight types generated 40% of the area's marks for seventeen years and were removed in November 2023 ([SDOC] §5.5). They are listed compactly, because their reports remain the best available commentary on force diagrams and sign discipline, and because they are still live for anyone working pre-2023 papers.

"Impulse" is never used. A search for impulse across all 97 corpus files returns zero hits [PAPERS]. VCAA has only ever set this concept as "the change in momentum", and it was always computed as m(v₂ − v₁) — never as ∫F dt.

[RPT18 E1] on T50: the momentum item is a vector question — "knowing that the position vector needed to be differentiated in order to find the velocity vector and hence the momentum of a particle" was listed as an area of strength, yet pct was 31, because students returned a magnitude instead of a vector, or differentiated once too few times.

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3. The standard wordings

VCAA reuses a small number of sentences almost verbatim. Learning the sentence is learning the task.

3.1 The resolute wordings — three distinct sentences

Two further variants exist and are pure traps: - "The vector resolute of a = 2i − j + 3k that is perpendicular to b = i + j − k is" (2019 NHT Exam 2 Section A) — the perpendicular one is still called a vector resolute. - "The magnitude of the component of the force F that acts in the direction d" (2020 Exam 2 Section A Q14, pct 76) and "The component of the force F = ai + bj … in the direction of the vector w = i + j" (2015 Exam 2 Section A Q15, pct 49) — "component" here means the resolute, and whether it is the scalar or the vector one is settled by the word "magnitude".

3.2 "Show that the acceleration … is given by"

Five live instances in the corpus [PAPERS]:

• 2008 Exam 2 Section B Q2a: "Show that the acceleration of the skier is 4.5 m/s²." (pct 91)
• 2010 Exam 1 Q2a: "Show that the acceleration of the body is given by dv/dt = (v − 4)/2." (pct 92)
• 2017 NHT Exam 2 Section B Q5b: "Show that the acceleration of the 5 kg mass is 4 m s⁻²."
• 2019 NHT Exam 2 Section B Q5dii: "Show that the acceleration of the pallet down the plane is given by g(5 − t)/(t + 290) m s⁻², for t ∈ [0, 5)."
• 2022 Exam 2 Section B Q5a: "Show that the acceleration of the object is given by (8 − k) m s⁻²." (pct 72)

What it demands. Not the answer — the answer is printed. It demands the chain: a labelled force or model statement, an equation of motion or an acceleration form, and the algebra connecting them. [RPT16 E2] on the rocket: "As this was a 'show that' question, students needed to show clear progress from an equation of motion to the required differential equation. Some students did not include the weight force in the initial equation; drawing a diagram showing forces could have benefited these students." [RPT25 E2] generalises it: "'Show that' questions require clear steps of working that lead to the given answer."

The cognate wordings behave identically: "Show that the speed of the particle … can be expressed as" (2025 Exam 2 Section B Q4d, pct 35.56 — [RPT25 E2]: "Another 'show that' question which required working that shows the use of trigonometric identities that are given on the formula sheet"), and "Show that the Cartesian equation of the path of the particle is" (T25).

3.3 "Find the distance travelled by the particle"

Six instances, and the wording is always precise about which distance:

• 2009 Exam 2 Section B Q1b: "Calculate the distance travelled by the car during the first nine seconds of its motion." (pct 76)
• 2009 Exam 2 Section B Q1c: "Calculate, correct to the nearest 0.1 m, the distance travelled by the car while it is decelerating." (pct 71)
• 2009 Exam 2 Section B Q3gi: "Write down a definite integral of the form ∫√(a + bt²) dt which represents the distance travelled by the child." (pct 28)
• 2012 Exam 2 Section B Q3eii: "Find the total distance travelled from when the car starts to accelerate to when it comes to rest." (pct 30)
• 2012 Exam 2 Section A Q18: "The distance travelled by the particle in the first 2 seconds of its motion is given by".
• 2022 Exam 2 Section B Q4d: "How far does the ball travel during the first four seconds after passing through O?" (pct 45)

What it demands. For motion along a curve, ∫|ṙ(t)| dt. For rectilinear motion that changes direction, ∫|v| dt — split at the turning points. Never |r(t₂) − r(t₁)|, which is displacement. [RPT07 E2] on Q4e (pct 18): "A common erroneous approach was to work out |r(60)| − |r(0)| instead of |r(60) − r(0)|" — and even the corrected version is the displacement, which was what that particular question wanted. Read which one is being asked.

3.4 The i–j–k conventions

VCAA sets the basis explicitly, in one of four standard sentences, and the sentence tells you what the components mean:

Consequences that are examined: "height above ground" is the k (or j) component only, never |r|. [RPT09 E2] on Q3a: "A common error was to find |r(0)| rather than the magnitude of its vertical component." [RPT15 E2] on Q4ai: "A number of students attempted to solve |r(t)| = 60, failing to realise that only the k component was 60." "Lands" means the vertical component equals zero.

Notation in the written papers uses an under-tilde (a̰) for vectors and an over-arrow for displacements between named points; the corpus text extraction drops both, which is why archived question text often reads as scalars. VCAA does penalise notation: [RPT07 E2] — "Lack of tildes indicating what should be vectors was common"; [RPT13 E2] lists among the paper's weaknesses "lack of proper vector notation, in particular the confusion of scalar 0 with null vector 0."

3.5 Other recurring sentences

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4. The separators — all 243

Every Space and measurement part with pct ≤ 50, grouped by type. Format: ref — pct% — one-line description. Marks in brackets where >1.

4.1 Vector algebra, magnitude, unit vectors, linear dependence (18)

• 2006 Exam 2 Section A Q16 — 49% — unit vector perpendicular to 5i + j − 2k
• 2006 Exam 2 Section A Q20 — 50% — three coplanar forces in equilibrium: which relation holds
• 2007 Exam 2 Section A Q15 — 37% — a vector perpendicular to the line 3x + 2y + 1 = 0
• 2008 Exam 1 Q3 — 16% [3] — find m for a linearly dependent set
• 2008 Exam 2 Section A Q17 — 43% — collinear points, find r from QR = ½PQ
• 2009 Exam 2 Section A Q17 — 30% — relation among three vectors drawn at 120°
• 2011 Exam 1 Q9ci — 41% [2] — construct the third vector for a dependence argument (2011 text CID-garbled)
• 2016 Exam 2 Section A Q14 — 41% — 3-4-5 string geometry, which tension relation
• 2018 Exam 2 Section A Q12 — 36% — |a + b| = |a| + |b| implies what
• 2019 Exam 2 Section B Q4a — 35% [2] — fourth vertex of a parallelogram from three given vertices
• 2020 Exam 1 Q5b — 28% — the component of a perpendicular to b
• 2021 Exam 1 Q6 — 26% [4] — values of p for linear independence (the complement is the mark)
• 2021 Exam 2 Section A Q13 — 46% — vector resolute given the scalar resolute is ±4
• 2023 Exam 2 Section A Q14 — 48% — unit n with a·n = b·n = 0; find c·n (two answers)
• 2023 Exam 2 Section A Q15 — 18% — if the sum of two unit vectors is a unit vector, find |a − b|
• 2024 Exam 2 Section A Q12 — 50% — vector item; no text layer for the 2024 November paper
• 2024 Exam 2 Section A Q16 — 43% — vector item; [RPT24 E2] comment reads only "is the only solution"
• 2025 Exam 2 Section A Q14 — 48% — |a·b| = |a × b| ⟹ the angle between a and b

4.2 Dot product, angles at a vertex, resolutes (28)

• 2006 Exam 2 Section B Q2c — 16% [2] — cos ∠ABC, then show ∠ABC and ∠ADC supplementary
• 2006 Exam 2 Section B Q2d — 20% [3] — show ∠APC = 2∠ADC via cos and a double-angle identity
• 2009 Exam 1 Q3 — 38% [3] — components of a parallel and perpendicular to b
• 2012 Exam 1 Q9d — 13% [2] — angle between −√3 i + j and r(t) at t = 0
• 2013 Exam 2 Section B Q4b — 40% [3] — parallel and perpendicular resolutes with surds
• 2013 Exam 2 Section B Q4d — 44% [2] — angle between two vectors, one decimal place, degrees
• 2014 Exam 1 Q1b — 42% [2] — angle a vector makes with a coordinate direction
• 2014 Exam 2 Section B Q3a — 41% [5] — express a as the sum of two vector resolutes
• 2015 Exam 1 Q9c — 16% [2] — acute angle between two tangents (vector method or tan(A − B))
• 2015 Exam 2 Section A Q15 — 49% — the component of F = ai + bj in the direction of i + j
• 2015 Exam 2 Section A Q17 — 48% — cos ∠ABC from three position vectors
• 2016 Exam 2 Section B Q4c — 24% [2] — obtuse angle between two ships' paths, via velocity vectors
• 2017 Exam 1 Q5 — 11% [4] — ∠BCD = π/3; find a (hardest vector item in the archive)
• 2020 Exam 1 Q5a — 44% [3] — vector resolute given; solve the resulting quadratic for integer m
• 2022 Exam 2 Section B Q4a — 35% [2] — angle the initial velocity makes with the forward direction
• 2024 Exam 1 Q4a — 45% [2] — angle between two vectors in degrees or radians
• 2024 Exam 2 Section A Q14 — 36% — scalar resolute and vector resolute in one stem
• 2008 Exam 2 Section B Q2eii — 40% [2] — resolve forces on a waterskier in two directions
• 2008 Exam 2 Section B Q2eiii — 28% [2] — solve the pair for tan θ to three decimals
• 2008 Exam 2 Section B Q2eiv — 27% — hence the tension T
• 2010 Exam 2 Section B Q1dii — 49% [3] — show OQ ⊥ AB by dot product (specific or general case)
• 2013 Exam 2 Section B Q4eiii — 10% — the vector sum is the null vector 0, not scalar 0
• 2014 Exam 2 Section B Q3biii — 37% [2] — equate coefficients of a and c to find λ and μ
• 2014 Exam 1 Q8b — 40% — show tan θ = sec θ from the two resolved equations
• 2019 Exam 2 Section B Q4d — 49% [3] — show perpendicular to both edges, hence the unit normal
• 2022 Exam 1 Q6bii — 47% [3] — angle in a semicircle by the scalar (dot) product
• 2025 Exam 1 Q5b — 30% [2] — velocities at right angles at the moment of collision
• 2012 Exam 2 Section B Q4b — 34% [3] — show acceleration perpendicular to velocity, terms spelled out

4.3 Cross product, planes, lines and distances in three dimensions (10)

• 2010 Exam 1 Q3c — 30% [3] — hence the area of triangle OAB from cos θ
• 2019 Exam 2 Section B Q4c — 22% [2] — area of a parallelogram (not a rectangle)
• 2019 Exam 2 Section B Q4e — 2% [2] — volume of a pyramid; height via a scalar resolute (dist 96/2/2)
• 2023 Exam 2 Section B Q5c — 50% [2] — acute angle at which a line meets a plane (sine, not cosine)
• 2024 Exam 1 Q10 — 14% [3] — distance between two skew lines (dist 59/14/13/14)
• 2024 Exam 2 Section B Q5b — 50% [3] — work with the given line equation, not part (a)'s
• 2024 Exam 2 Section B Q5dii — 46% [2] — area of a triangle: cross product ÷ 2, correct spanning vectors
• 2025 Exam 2 Section A Q18 — 49% — two lines intersecting at (4, 3, t): find r, s, t
• 2025 Exam 2 Section B Q5dii — 33.79% [3] — values of m for a given distance between parallel planes (modulus ⟹ two answers)
• 2006 Exam 2 Section A Q14 — 50% — direction of motion of a particle at t = 9

4.4 Paths: Cartesian equation, sketching, periodicity (19)

• 2006 Exam 1 Q7b — 31% [2] — sketch a parabolic path over the correct restricted domain with endpoints
• 2008 Exam 2 Section B Q3aii — 32% — Cartesian equation y² = x²(1 − x²) (both square roots)
• 2008 Exam 2 Section B Q3b — 19% [2] — sketch that relation
• 2008 Exam 2 Section B Q3c — 10% [2] — how many times through the origin (the y component decides)
• 2009 Exam 2 Section B Q3c — 40% — period of the combined i and j motion
• 2011 Exam 2 Section B Q2d–2d — 41% [3] — sketch an ellipse with exact axis intercepts (2011 ref collision)
• 2012 Exam 2 Section B Q4d — 46% [2] — Cartesian path of the space station, (y + 6400)²
• 2014 Exam 1 Q2b — 42% [2] — sketch a parabola with correct domain, symmetry and labelled points
• 2015 Exam 2 Section B Q4b — 43% — the period of the horizontal motion
• 2016 Exam 2 Section B Q4b — 36% [3] — sketch both ships' paths over their own domains, with arrows
• 2017 Exam 2 Section A Q12 — 49% — identify the path of a particle in the Cartesian plane
• 2020 Exam 2 Section B Q4a — 47% [3] — maximum speed of the aeroplane
• 2021 Exam 1 Q9aii — 41% — Cartesian equation in the first quadrant; justify the positive root
• 2023 Exam 1 Q3b — 50% — limiting velocity as x → ∞
• 2025 Exam 2 Section B Q4d — 35.56% [3] — show speed = √(125 − 100 cos(3t/2))
• 2012 Exam 2 Section B Q4a — 49% — height above the surface when directly overhead
• 2018 Exam 2 Section B Q4d — 25% [2] — set up and solve the speed inequality over 0 ≤ t ≤ 5
• 2017 Exam 2 Section B Q5bii — 48% — coordinates of the boat at the matching-speed time
• 2020 Exam 2 Section B Q1bii — 35% [2] — the velocity (a vector), not its magnitude

4.5 Vector calculus: velocity, acceleration, speed (22)

• 2007 Exam 1 Q9 — 18% [3] — show a = −ω²r for a circularly moving particle
• 2007 Exam 2 Section B Q4c — 11% [2] — angle of descent from the velocity vector
• 2010 Exam 1 Q8b — 37% [3] — speed at t = 3π/2 via the product rule
• 2008 Exam 2 Section B Q3d — 31% [2] — speed as the magnitude of the velocity vector, at the correct time
• 2010 Exam 1 Q8c — 20% — tan θ = ẏ/ẋ for the direction of motion
• 2012 Exam 1 Q9b — 38% [2] — speed at t = 1 in exact surd form
• 2013 Exam 1 Q7c — 29% [2] — speed with sec and tan derivatives
• 2012 Exam 2 Section B Q4c — 34% [2] — speed of the space station in km/h, nearest integer
• 2013 Exam 2 Section B Q5a — 38% [3] — minimum and maximum speed, correctly labelled
• 2013 Exam 2 Section B Q5b — 15% [2] — when acceleration is zero: t = 6n, n ∈ Z⁺
• 2014 Exam 2 Section A Q19 — 49% — maximum magnitude of the net force from v(t)
• 2022 Exam 2 Section B Q4bii — 49% [2] — minimum speed of the ball and the time at which it occurs
• 2015 Exam 2 Section B Q4aii — 33% [2] — angle of elevation
• 2015 Exam 2 Section B Q4d — 39% [2] — speed via the Pythagorean identity, keeping the k component
• 2016 Exam 1 Q8c — 26% [3] — maximum magnitude of the net force (differentiate twice, then maximise)
• 2019 Exam 2 Section A Q15 — 35% — describe the motion when a = λj and v = u i
• 2023 Exam 1 Q10d — 7% [2] — all t with r ⊥ ṙ (dist 44/49/7)
• 2024 Exam 2 Section B Q4ci — 23% — the square of the speed, in terms of the given parameter
• 2024 Exam 2 Section B Q4ciii — 48% — take the square root to get the speed
• 2009 Exam 2 Section B Q3e — 50% [2] — speed at t = 14, to the stated accuracy
• 2009 Exam 2 Section B Q3f — 29% [2] — show the magnitude of the acceleration is constant
• 2020 Exam 2 Section A Q15 — 38% — position after time under two constant forces (integrate twice)

4.6 Collisions, closest approach, meeting (11)

• 2007 Exam 2 Section B Q4d — 20% [2] — r·v = 0 for closest approach to a beacon
• 2012 Exam 2 Section B Q4e — 29% [3] — both times at which |r(t)| = 1000
• 2016 Exam 2 Section B Q4di — 27% [2] — time of closest approach, three decimal places
• 2016 Exam 2 Section B Q4dii — 29% — the minimum distance itself
• 2017 Exam 2 Section B Q5cii — 15% — global minimum distance, not a local one
• 2017 Exam 2 Section B Q5d — 27% [3] — collision time and the value of a, in exact form
• 2018 Exam 2 Section B Q4c — 34% [2] — intersection coordinates with t ≥ 0, three decimal places
• 2018 Exam 2 Section B Q4e — 28% [2] — the period during which |r_B − r_A| ≤ 0.2, in minutes
• 2022 Exam 2 Section B Q4c — 38% [3] — minimum distance from the ball to the hole
• 2024 Exam 2 Section B Q4e — 24% [2] — form the distance expression and minimise numerically
• 2025 Exam 1 Q5c — 25% — equal acceleration magnitudes at collision; reject the extra roots

4.7 Arc length and distance travelled along a path (13)

• 2007 Exam 2 Section B Q4e — 18% [2] — total distance flown = speed × time along a straight path
• 2007 Exam 2 Section B Q5d — 38% [3] — a difference of two definite integrals with different limits
• 2007 Exam 2 Section B Q5e — 15% [3] — catch-up equation combining an integral and a constant-speed term
• 2007 Exam 2 Section B Q5f — 8% — solve it numerically to the nearest second
• 2008 Exam 2 Section B Q3ei — 19% — set up the arc-length integral with correct terminals
• 2008 Exam 2 Section B Q3eii — 8% — evaluate it
• 2009 Exam 2 Section B Q3gi — 28% [2] — the integral in the prescribed form ∫√(a + bt²) dt
• 2009 Exam 2 Section B Q3gii — 28% — evaluate to the nearest 0.1 m
• 2012 Exam 2 Section B Q3ei — 34% [2] — expressions for all three stages of the motion
• 2012 Exam 2 Section B Q3eii — 30% — total distance travelled
• 2018 Exam 1 Q10 — 2% [5] — find a, b, c in the arc-length integrand (dist 35/22/24/17/0/2)
• 2022 Exam 2 Section B Q4d — 45% [2] — distance travelled in the first four seconds
• 2023 Exam 1 Q10c — 36% — arc length 4π/3 along a circle ⟹ the parameter a

4.8 Velocity–time graphs and multi-stage motion (7)

• 2009 Exam 2 Section B Q1d — 49% — average speed over a multi-stage journey
• 2009 Exam 2 Section B Q1f — 18% [3] — catch-up time and the distance, across stages
• 2012 Exam 2 Section A Q19 — 44% — v = x ⟹ x(t) with the right branch of the exponential
• 2014 Exam 2 Section A Q22 — 49% — read a velocity–time graph
• 2016 Exam 2 Section B Q5e — 21% [3] — total flight time across three phases (dist 44/27/7/21)
• 2019 Exam 2 Section B Q5d — 9% [5] — two-stage motion up an incline (dist 54/26/4/6/2/9)
• 2011 Exam 2 Section B Q2d–2d — 30% [3] — landing position and distance to the target (2011 ref collision)

4.9 Rectilinear kinematics, gravity, projectiles (19)

• 2009 Exam 2 Section A Q19 — 48% — time of flight for a 45° projectile, exact
• 2009 Exam 2 Section B Q5g — 12% [3] — combine vertical fall with horizontal drift by Pythagoras
• 2011 Exam 2 Section B Q2b–2b — 6% [2] — maximum height, exact form, not rounded (2011 ref collision)
• 2012 Exam 2 Section B Q5b — 49% [2] — s = ut + ½at² with consistent signs
• 2012 Exam 2 Section B Q5c — 23% [2] — find the launch speed u with the correct total time
• 2013 Exam 2 Section B Q5c — 29% [2] — projectile off a ramp; sign consistency
• 2013 Exam 2 Section B Q5d — 39% — horizontal range from the correct horizontal velocity component
• 2016 Exam 2 Section A Q16 — 46% — horizontal range of a cricket ball on level ground
• 2016 Exam 2 Section B Q5d — 39% [2] — maximum height once the propulsion stops (acceleration now constant)
• 2017 Exam 1 Q9b — 30% [3] — displacement after 10 s, as a vector
• 2018 Exam 2 Section A Q17 — 48% — camera dropped from an ascending balloon
• 2021 Exam 2 Section B Q4b — 40% [2] — minimum launch speed to clear a point
• 2021 Exam 2 Section B Q4c — 6% [3] — θ and u for the trajectory to join the track smoothly (dist 80/9/5/6)
• 2021 Exam 2 Section B Q4e — 3% [3] — braking distance to locate point W (dist 87/10/1/3)
• 2022 Exam 2 Section A Q14 — 28% — velocity at the midpoint of AB
• 2023 Exam 2 Section A Q13 — 45% — phone dropped from a rising balloon
• 2023 Exam 2 Section A Q16 — 32% — total vertical distance travelled, allowing for the release height
• 2025 Exam 1 Q3c — 28% [2] — compare two particles' displacements after 3 s
• 2025 Exam 2 Section A Q12 — 33% — velocity at the midpoint of AB

4.10 Variable acceleration and motion as a differential equation (35)

• 2006 Exam 2 Section A Q12 — 35% — a when v = sin(2x)
• 2006 Exam 2 Section B Q3cii — 18% [3] — express x as an integral in v, with correct terminals
• 2006 Exam 2 Section B Q3ciii — 20% — evaluate it
• 2008 Exam 1 Q5a — 48% [2] — acceleration from v(x) using v dv/dx
• 2008 Exam 1 Q5b — 34% [3] — solve dx/dt = −x² for x(t)
• 2008 Exam 2 Section A Q22 — 45% — t as an integral of 1/f(v)
• 2008 Exam 2 Section B Q2d — 30% [2] — definite integral for the time under linear resistance
• 2009 Exam 1 Q7 — 23% [4] — v dv/dx separable; reciprocal error and partial fractions
• 2009 Exam 2 Section B Q5c — 47% — terminal velocity as a value, not an inequality
• 2007 Exam 2 Section B Q5b — 31% — the limiting speed of the car from v = 20 − 2tan⁻¹(t) (an inequality, not an equation)
• 2009 Exam 2 Section B Q5d — 49% — time to reach a stated velocity, exact log form
• 2009 Exam 2 Section B Q5e — 41% [2] — distance by integrating v(t), constant of integration included
• 2009 Exam 2 Section B Q5f — 30% [2] — distance using v dv/dx with velocity limits
• 2010 Exam 1 Q2b — 23% [4] — solve dv/dt = (v − 4)/2; choose the branch consistent with v(0)
• 2010 Exam 2 Section A Q22 — 40% — the integral form relating ∫F dx to the change in ½mv²
• 2012 Exam 1 Q8 — 28% [3] — acceleration from v(x); d/dx(½v²) is the clean route
• 2012 Exam 2 Section B Q5dii — 49% [2] — time at which v = 0
• 2012 Exam 2 Section B Q5diii — 37% [3] — distance by integrating v over that interval
• 2015 Exam 2 Section A Q22 — 42% — time to maximum height with ẍ = −(9.8 + 0.1v²)
• 2015 Exam 2 Section B Q5di — 30% [2] — invert and rearrange into the required dx/dv form
• 2015 Exam 2 Section B Q5div — 12% [3] — the definite integral for the time
• 2017 Exam 2 Section B Q2c — 48% — terminal velocity
• 2017 Exam 2 Section B Q2di — 17% [2] — the definite integral for time, from 19.6, not from 0
• 2017 Exam 2 Section B Q2dii — 25% — evaluate it
• 2017 Exam 2 Section B Q2e — 16% [3] — the definite integral for distance, plus the first-stage 19.6 m
• 2018 Exam 2 Section B Q5c — 23% [2] — solve v dv/dx and present in the required form
• 2018 Exam 2 Section B Q5d — 26% — evaluate
• 2018 Exam 2 Section B Q5ei — 41% — definite integral for time (not the reciprocal integrand)
• 2018 Exam 2 Section B Q5eii — 41% — evaluate
• 2019 Exam 2 Section A Q16 — 49% — acceleration when v = e^x sin(x)
• 2020 Exam 2 Section B Q5e — 21% [4] — resisted motion after the string breaks; v dv/dx
• 2021 Exam 2 Section B Q4d — 31% [2] — show v = (180s)^{1/3} from a = 60/v
• 2022 Exam 1 Q8 — 37% [4] — a = −4x; find v(x) and choose the correct sign
• 2023 Exam 1 Q3a — 35% [2] — acceleration at x = 2 from v = (3x + 2)/(2x − 1)
• 2024 Exam 1 Q9a — 32% — show the speed threshold is exceeded, using the right acceleration form

4.11 Forces: diagrams, resolving, equilibrium (21)

• 2006 Exam 1 Q4b — 42% [2] — tension by Pythagoras or resolution
• 2006 Exam 2 Section B Q3bii — 34% [2] — resolve parallel and perpendicular to the direction of motion
• 2006 Exam 2 Section B Q3biii — 42% — the lift-to-drag ratio from those two equations
• 2007 Exam 2 Section A Q20 — 43% — correct force diagram for a box sliding up a rough incline
• 2008 Exam 1 Q7 — 32% [3] — resolve or use Lami's theorem for two strings
• 2008 Exam 2 Section A Q15 — 36% — magnitude of the sum of two perpendicular-ish forces
• 2012 Exam 1 Q4b — 44% — maximum T before the crate leaves the floor (N = 0)
• 2014 Exam 1 Q8ai — 44% [2] — resolve to get T₁ cos θ = 5g and T₁ = 49 sec θ
• 2011 Exam 1 Q7b–7b — 41% [3] — tension found by resolving vertically / triangle of forces (2011 ref collision)
• 2014 Exam 1 Q8c — 25% [3] — maximum θ and the justification that both strings break
• 2014 Exam 2 Section A Q18 — 45% — four forces on a smooth plane
• 2014 Exam 2 Section B Q5bi — 46% [2] — force diagram distinguishing the two normals and two frictions
• 2015 Exam 2 Section A Q16 — 23% — which vector equation relates T₁, T₂, W in equilibrium
• 2016 Exam 1 Q1c — 48% [2] — magnitude of the tension
• 2017 Exam 2 Section A Q17 — 37% — magnitude and direction of a resultant
• 2018 Exam 2 Section B Q5a — 44% — force diagram labelling the resistance as 20v, not v
• 2019 Exam 1 Q9b — 10% [3] — resolve both directions; double-angle identity to the given result
• 2019 Exam 2 Section A Q17 — 50% — three coplanar forces in equilibrium
• 2021 Exam 2 Section B Q5a — 46% [2] — the m₁–m₂ relation for constant speed down a smooth plane
• 2022 Exam 2 Section A Q15 — 26% — which statement about a mass on an incline is true
• 2022 Exam 2 Section A Q16 — 17% — the cosine-rule equation for three forces in equilibrium

4.12 Equations of motion, connected particles, friction (29)

• 2006 Exam 2 Section B Q3ci — 38% — write the equation of motion (signs on every term)
• 2007 Exam 2 Section A Q22 — 45% — equation of motion for a jet with resistance kv² per kg
• 2007 Exam 2 Section B Q3c — 42% [3] — connected particles with one mass on an incline
• 2007 Exam 2 Section B Q3e — 14% [4] — limiting equilibrium in both directions with friction
• 2010 Exam 2 Section B Q2d — 35% [2] — new coefficient of friction after the tension changes
• 2010 Exam 2 Section B Q2e — 36% [3] — acceleration down the plane, then velocity after 3 m
• 2011 Exam 2 Section B Q4b–4b — 6% [2] — add the two equations of motion (2011 ref collision; dist sums to 46)
• 2011 Exam 2 Section B Q4d–4d — 5% [2] — solve for θ when the acceleration is zero (2011 ref collision)
• 2011 Exam 2 Section B Q4d–4d — 41% [3] — friction term and resolution on a double incline
• 2011 Exam 2 Section B Q4d–4d — 25% [4] — limiting equilibrium both ways; distinguish m₁ from m₂
• 2012 Exam 1 Q4c — 40% [3] — T for the crate to be on the point of moving
• 2013 Exam 2 Section A Q22 — 38% — the equation of motion for a parachutist with kv²
• 2013 Exam 2 Section B Q5e — 23% [3] — friction on an incline; distance to stop
• 2013 Exam 2 Section B Q5f — 34% — minimum coefficient of friction to prevent sliding
• 2014 Exam 2 Section B Q5biii — 36% [3] — time at which two blocks collide (displacements, not velocities)
• 2016 Exam 2 Section B Q5a — 39% — equation of motion ⟹ the given differential equation
• 2017 Exam 1 Q9a — 45% [2] — magnitude of the constant force from the change in velocity
• 2018 Exam 1 Q1b — 41% [3] — magnitude and direction of the acceleration, using the total mass
• 2019 Exam 2 Section A Q14 — 36% — tension in a three-mass connected system
• 2019 Exam 2 Section B Q5ci — 39% — the inequality on θ for motion to occur
• 2020 Exam 2 Section A Q20 — 43% — spring-balance reading in a descending lift
• 2020 Exam 2 Section B Q5bii — 45% — the values of k for which the system moves
• 2020 Exam 2 Section B Q5d — 48% [2] — speed at the midpoint in terms of k
• 2021 Exam 2 Section A Q16 — 48% — how the acceleration changes when the incline angle doubles
• 2021 Exam 2 Section B Q5bii — 24% [2] — acceleration in terms of m₁, m₂, μ
• 2021 Exam 2 Section B Q5c — 11% [2] — distance up the plane after the string goes slack
• 2021 Exam 2 Section B Q5d — 3% [3] — time to return to P, friction now reversed (dist 87/8/2/3)
• 2022 Exam 1 Q5b — 48% — the braking force R for constant velocity
• 2022 Exam 2 Section B Q5e — 35% [3] — motion once the stated forces no longer act

4.13 Momentum (1)

• 2018 Exam 1 Q6 — 31% [3] — change in momentum between two times (a vector)

4.14 Mis-tagged parts (10)

These carry topic == "Space and measurement" in [QJSON] but are Calculus questions. Listed for completeness (see §0.3).

• 2006 Exam 2 Section B Q4ci — 21% — second derivative of an implicitly defined relation
• 2006 Exam 2 Section B Q4cii — 8% [2] — verify a point of inflection (sign change, not just y'' = 0)
• 2006 Exam 2 Section B Q4d — 28% — sketch a solution curve on a slope field
• 2006 Exam 2 Section B Q4e — 16% [2] — two steps of Euler's method
• 2007 Exam 2 Section A Q12 — 27% — related rates with an angle of elevation
• 2011 Exam 2 Section B Q4d–4d — 29% — write down a concentration expression (2011 ref collision)
• 2021 Exam 1 Q9ci — 15% [2] — show a derivative result using the product and chain rules
• 2021 Exam 1 Q9cii — 22% [2] — hence find an area
• 2025 Exam 2 Section B Q3a — 20.43% — compare initial and incoming salt concentrations
• 2025 Exam 2 Section B Q3g — 18.05% — time at which the concentration reaches a value

4.15 Which types separate most

Two different questions, two different answers.

By volume — where the separators actually are. That ranking is the §1.4 table: variable acceleration as a differential equation (35 separators), connected particles and friction (29, now deleted), the dot-product cluster (28), vector calculus on r(t) (22).

By severity — where a separator is likely to be a deep one. Median pct of the separators inside each group, lowest first:

The individually hardest parts in the whole area, by pct:

Six of those 21 are pure mechanics and are dead: 2011 Exam 2 Section B Q4d-4d, 2011 Exam 2 Section B Q4b-4b, 2019 Exam 1 Q9b, 2019 Exam 2 Section B Q5d, 2021 Exam 2 Section B Q5c and 2021 Exam 2 Section B Q5d.

Fifteen are still live, and they cluster in two places. Six are distance-travelled, arc-length or resisted-motion integrals: 2018 Exam 1 Q10, 2008 Exam 2 Section B Q3eii, 2007 Exam 2 Section B Q5f, 2015 Exam 2 Section B Q5div, 2009 Exam 2 Section B Q5g and 2021 Exam 2 Section B Q4e. Three are angle-between-vectors items: 2017 Exam 1 Q5, 2012 Exam 1 Q9d and 2007 Exam 2 Section B Q4c. The remaining six are one-off geometric or algebraic traps — the pyramid volume, the null vector, r ⊥ ṙ, the skew-line distance, the maximum height in exact form, and the smooth-join condition. Those two clusters are where the ceiling of this area sits.

4.16 What the reports say went wrong — the five recurring diagnoses

1. A required form or accuracy was ignored. [RPT16 E2]: "the instruction to give the answer correct to three decimal places must be followed to gain full marks." [RPT17 E2]: "The time of collision and the value of a in Question 5d. were occasionally given as decimal approximations rather than the required (by default) exact form." [RPT25 E2]: "Exact answers are expected unless told otherwise."
2. A 'show that' was asserted rather than shown. [RPT15 E2]: "omission of the constant of integration or lack of detail showing its evaluation, particularly in a 'show that' question." [RPT15 E2] again: "Some students simply asserted that [the dot product was zero]."
3. A vector was reported as a scalar, or a scalar as a vector. [RPT20 E2]: "Many students gave the (scalar) magnitude of the velocity rather than the required velocity." [RPT13 E2]: "the confusion of scalar 0 with null vector 0." [RPT17 E1]: "leaving the force as a vector and not finding its magnitude."
4. Only one of two answers was given. [RPT25 E2]: "Many responses did not demonstrate that the modulus needed to be used and consequently only one of the solutions was found." [RPT24 E1]: "Some students considered only [one case] and so did not find both values." [RPT14 E1]: "Some students did not give both values of m."
5. The model's starting point was misidentified. [RPT17 E2]: "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." [RPT16 E2]: "The most common misconception arising in this question was not realising that acceleration was now constant, and some students proceeded to use their equation for the displacement obtained in Question 5c."

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5. What makes a hard one hard

5.1 Sign conventions for direction

The single most-cited error in the entire report corpus for this area. It appears in three forms.

(a) You did not declare the positive direction. [RPT15 E1]: "Typical mistakes involved sign errors with a conflict in the direction assigned to be positive." [RPT14 E1]: "There was inconsistency in the use of positive direction in resolving forces." The fix is mechanical: write "take up as positive" (or "take down the plane as positive") as the first line, and then be ruthless about it — g becomes −9.8, the displacement of an object below the origin becomes negative, and a retarding force gets a minus sign.

(b) You dropped a sign because the answer looked wrong. [RPT07 E1] records this exactly: "some decided that they must have made an error and dropped the negative sign." A negative acceleration is not an error; it is information.

(c) You picked the wrong branch after integrating. [RPT10 E1]: "Many students correctly found that |v − 4| = 4e^{t/2}, but mistakenly assumed that this meant v − 4 = 4e^{t/2} when the correct equation was 4 − v = 4e^{t/2}." [RPT22 E1]: "A number of students chose the incorrect sign for their final answer." [RPT21 E1]: "A common error was for students to neglect to justify the choice of sign for the path of the particle in the first quadrant." The absolute value from ∫dv/(v − a) must be resolved using the initial condition, and the resolution must be written down.

5.2 Displacement versus distance travelled

Three distinct objects, and VCAA asks for each of them:

[RPT17 E1] on 2017 Exam 1 Q9b: "The most common error was using scalars throughout (finding distance rather than displacement)." [RPT22 E2] on Q4d: "A number of incorrect student responses incorrectly found the straight-line distance between the endpoints of the travel." [RPT23 E2] on Section A Q16: the ball rises 13 m, falls 13 m, so the distance travelled is 26 m less the 1.5 m launch height, 24.5 m — while the displacement is −1.5 m. Only 32% of the state separated those.

5.3 Scalar versus vector resolute

The three-line test that costs nothing and saves marks:

1. Does the answer have i, j, k in it? If the question said "scalar resolute", it must not.
2. Did you divide by |b| once or twice? Scalar resolute: once. Vector resolute: twice (i.e. (a·b/|b|²)b).
3. Which vector is the direction? "Resolute of a in the direction of b" projects a onto b. [RPT13 E2]: "A few students had the resolutes the wrong way around."

And a fourth, for the perpendicular one: the perpendicular resolute is a minus the parallel resolute, and 2014 Exam 2 Section B Q3a shows that VCAA will award a mark purely for writing the final sum a = (parallel) + (perpendicular).

5.4 Choosing the right form of acceleration

The formula sheet gives four: a = d²x/dt² = dv/dt = v dv/dx = d/dx(½v²). The choice is dictated by what the given data is a function of, and what you are asked for:

[RPT12 E1] on 2012 Exam 1 Q8 (28%): "Many of those who used [v dv/dx] got into significant difficulty with square roots and the associated algebraic simplification. On the other hand, those who used [d/dx(½v²)] were usually more successful, since squaring removed the square roots." That is a free technique upgrade.

The three most expensive misreads: using dv/dx instead of v dv/dx; using a constant-acceleration formula when the acceleration is not constant ([RPT12 E2]: "One error… was the use of constant acceleration formulas for this part"); and failing to notice that the acceleration becomes constant partway through a multi-stage question ([RPT16 E2]).

5.5 Drawing and labelling a correct force diagram

Relevant only for pre-2023 practice, but the reports' standard is worth internalising for any labelled diagram:

• Show only forces that act on the body. [RPT06 E1]: "many invented imaginative forces such as normal and friction forces." [RPT09 E2]: "Common errors with the diagram involved forces not acting on the device, and extra forces being introduced."
• Include every force that does act. [RPT18 E1]: "omitting the normal reaction force on the 8 kg mass."
• Distinguish separate bodies' forces. [RPT14 E2]: "A large number of students did not distinguish between the different normal reactions and different friction forces acting on each block."
• Draw components as dashed lines. [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."
• Label with the symbols the question supplies. [RPT09 E2]: "A number of students avoided using the labels for the forces given in the question." [RPT18 E2]: "Many students labelled the resistance force incorrectly as v, rather than 20v."
• Idiosyncratic notation is penalised. [RPT18 E1]: "Examples of idiosyncratic notation were observed. This was most apparent in the labelling of the force diagram in Question 1. In some cases the student's intent was not clear and full marks were not awarded."

5.6 Units, accuracy and form

• g = 9.8, always. Every paper's front page: "Take the acceleration due to gravity to have magnitude g m s⁻², where g = 9.8."[RPT09 E1]: "A large number of students were unable to evaluate50g; the most common errors were 49 and 500 (despiteg = 9.8being printed in the instructions)."[RPT16 E1]: "A few usedg = 10`."
• Do not round mid-question. [RPT06 E2]: "The major error in this question was using the rounded answer from part ii. An exact answer for t, or at least one with more accuracy than one decimal place, was needed." [RPT15 E2]: "The most common error was to use the rounded value of 425 instead of 250g sin(10°), when three decimal place accuracy was required."
• Answer in the unit asked. [RPT18 E2]: "Responses in terms of hours, rather than minutes, were given by a number of students who did not respond to the specifics of the question." [RPT24 E1]: "not all were able to find the correct angle (in degrees or radians)."
• Answer in the form asked. [RPT13 E1]: giving √38 when the length was wanted, or the length when the square was wanted. [RPT19 E2]: giving θ when cos θ was wanted.
• Coordinates are coordinates. [RPT25 E2] on Q4a: "A common error was not presenting the answer in coordinate form." [RPT17 E2]: "Some answers were not given in coordinate form."

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6. The worked method sheet — 12 highest-yield types

Ranked by (marks in the area) × (separator rate). Each entry gives the by-hand (Exam 1) method and the technology-active (Exam 2) method, because the two papers reward different routes.

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M1. Vector resolutes

By hand. Compute a·b and |b|² as integers if you can. Then - scalar resolute = a·b/|b| - vector resolute = (a·b/|b|²)·b — leave b un-normalised and put the whole scalar in front. This avoids rationalising twice. - perpendicular resolute = a − (a·b/|b|²)b - final line: a = (parallel) + (perpendicular).

With CAS. dotP(a,b)/norm(b) and dotP(a,b)/dotP(b,b)*b. Define a and b as lists first — [RPT25 E2]: "Students must make sure variables are defined if they are being used in formulas."

Checks. The perpendicular resolute must satisfy (a − p)·b = 0. The two resolutes must add to a.

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M2. Angle between two vectors, at a vertex, or between a line and a plane

By hand. cos θ = (a·b)/(|a||b|). For an angle at vertex B, first write BA = a − b and BC = c − b. For a line and a plane, sin θ = |d·n|/(|d||n|) — the complement. For two planes, cos θ = |n₁·n₂|/(|n₁||n₂|).

For cos 2θ: 2cos²θ − 1. For sin 2θ: 2 cos θ √(1 − cos²θ) with the positive root when θ is acute.

With CAS. arccos(dotP(a,b)/(norm(a)*norm(b))) in degree or radian mode as the question specifies. [RPT13 E2]: "Some students gave the answer in radians."

Checks. Acute answer requested? Take |a·b|. Obtuse requested? Do not.

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M3. Linear dependence and independence

By hand. Write c = αa + βb. Three component equations, two unknowns plus the parameter. Solve two, substitute into the third — that equation is the dependence condition. Alternative: set the 3×3 determinant of the nine components to zero.

Independence is the complement. Write p ∈ R \ {…}. This sentence is worth a mark (2021 Exam 1 Q6).

With CAS. det([[...],[...],[...]]) = 0 and solve.

Checks. Look for a shortcut first — [RPT16 E1]: "2a + b eliminated j."

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M4. Cross product, area and normal vectors

By hand. Write the determinant with i j k on the top row. Expand: i(y₁z₂ − y₂z₁) − j(x₁z₂ − x₂z₁) + k(x₁y₂ − x₂y₁). The middle term carries the minus sign — the formula sheet writes it as (x₂z₁ − x₁z₂)j with the sign absorbed, which is the safer form to copy.

• Area of a parallelogram on a and b: |a × b|.
• Area of a triangle: ½|a × b|.
• Normal to a plane through A, B, C: AB × AC.

With CAS. crossP(a,b). [RPT24 E2] explicitly recommends it.

Checks. (a × b)·a = 0 and (a × b)·b = 0. If the area came out as |a||b| you multiplied side lengths — the 2019 error.

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M5. Equations of planes and lines in space

Plane, Cartesian. Normal n = (a, b, c) from a cross product; write ax + by + cz = d; substitute one known point for d.

Plane, vector/parametric. r(s,t) = r₀ + s·u + t·v where u, v span the plane. Both forms are on the sheet.

Line, vector. r(t) = r₁ + t(r₂ − r₁). Parametric: three scalar equations.

Line ∩ plane. Substitute x(t), y(t), z(t) into the plane's Cartesian equation; solve for t; back-substitute.

Line ∩ line. Use two different parameters. Set r₁(λ) = r₂(μ); solve two components; verify the third. [RPT25 E1] — using the same parameter makes you "ineligible for full marks".

Two planes ∩. Direction = n₁ × n₂; a point by setting one variable to a convenient value and solving the 2×2 system.

Checks. Substitute the answer back into every original equation.

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M6. Shortest distances in three dimensions

Memorise all four — none is on the formula sheet.

Draw the configuration first. [RPT24 E1] records that the students who drew "diagrams of skew lines and parallel planes to help motivate an appropriate formula" were the ones who succeeded on a 14% question.

When the distance is given and a parameter is wanted, the equation is |f(m)| = k, so solve both f(m) = k and f(m) = −k. 2025 Exam 2 Section B Q5dii has answers m = 1 and m = 47; two-thirds of the state found at most one.

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M7. Cartesian equation of a path, and sketching it

By hand. Isolate the trigonometric function from each component: - x = a cos t, y = b sin t ⟹ x²/a² + y²/b² = 1 (ellipse) - x = a sec t, y = b tan t ⟹ x²/a² − y²/b² = 1 (hyperbola) - x = f(t), y = g(t) polynomial ⟹ solve for t and substitute - A sin t cos t product needs sin 2t = 2 sin t cos t; a sin²t needs sin²t = ½(1 − cos 2t).

Sketching. Convert the t-domain to an x-range; plot and label the endpoints with coordinates; draw the direction arrow by evaluating ṙ at one convenient t; keep the curve smooth.

With CAS. Graph parametrically with the t-range set to the question's domain, then transfer shape and endpoints to the printed axes. [RPT25 E2]: "students can sketch the function on their CAS calculator and set the domain, range and scale to match those provided in the question."

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M8. Velocity, acceleration and speed from r(t)

By hand. Differentiate component-wise; keep i, j, k on every line. Speed = |ṙ| = √(ẋ² + ẏ² + ż²); simplify with cos² + sin² = 1 before substituting.

Max/min speed. Simplify to a single trigonometric term, e.g. √(125 − 100 cos(3t/2)), then read off: maximum where the cosine is −1, minimum where it is +1. Only use calculus if the expression will not collapse.

Perpendicularity. ṙ·r̈ = 0 — expand every product and show the cancellation; do not assert it.

With CAS. Define r(t) as a list; v(t) := d/dt(r(t)); norm(v(t)); fMin/fMax over the domain.

Checks. Speed is a scalar. Velocity is a vector. Read the question again before writing the final line.

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M9. Collisions, meeting and closest approach

Collision. Set the components equal and solve for a single t. If the i equation gives t = 5/9 and the j equation gives t = 13/2, the particles do not collide, and saying so with those two values is the mark.

Paths crossing. Two parameters s and t, or intersect the two Cartesian equations.

Closest approach. d(t) = |r₂(t) − r₁(t)|. Minimise d² to avoid the root. On Exam 2: fMin(norm(r2(t)-r1(t)), t) over the stated domain, then check it is the global minimum by graphing — [RPT17 E2] records "incorrect answers involving other locally minimum values".

Special case. For one moving object and one fixed point, closest approach is where (r − p)·ṙ = 0.

Checks. Domain restrictions (t ≥ 0) and the stated accuracy.

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M10. Arc length and distance travelled

distance = ∫_{t₁}^{t₂} √((dx/dt)² + (dy/dt)²) dt (add (dz/dt)² in three dimensions). This is on the formula sheet in its parametric form only — the Cartesian arc-length integral was removed in 2023 ([SDOC] §1.7.1).

By hand. Differentiate, square, add, and simplify before integrating — VCAA constructs these so that the radicand collapses. 2018 Exam 1 Q10 wanted only a, b, c in ∫₀^{3/4}√(at² + bt + c) dt: the whole question was the simplification, and 35% of the state scored zero.

With CAS. Build the integrand from the defined r(t), integrate numerically over the time limits.

Checks. Limits in t, never in x. If the motion is rectilinear and reverses, split the integral at the turning points and add the magnitudes.

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M11. Variable acceleration — the decision tree

1. What is v (or a) a function of — t, x, or v?
2. What is wanted — a, v, x, or t?
3. Pick the acceleration form from the table in §5.4.
4. Separate, integrate, and apply the initial condition before rearranging.
5. Resolve the modulus from ∫dv/(v − k) using the initial condition, and write down which branch you took.

By hand shortcut. If v contains a square root or a quotient, use a = d/dx(½v²) — squaring first removes the root ([RPT12 E1]).

With CAS. deSolve(v'=f(v) and v(0)=v0, t, v). [RPT25 E2]: "Use the DEsolve functionality on CAS… Alternatively, use separation of variables to solve the differential equation manually."

"Show that" discipline. Every intermediate line, the constant of integration, and its evaluation. [RPT15 E2]: "many students did not include the constant."

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M12. Constant acceleration, gravity and multi-stage motion

Set-up, every time. Write the positive direction; then a table of u, v, a, s, t for each stage.

Formula choice. Pick the one omitting the quantity you neither know nor want: v = u + at (no s) · s = ut + ½at² (no v) · v² = u² + 2as (no t) · s = ½(u + v)t (no a). All four are on the current sheet.

Gravity. Up positive ⟹ a = −9.8. An object dropped from a rising balloon keeps the balloon's upward initial velocity. An object below the release point has negative displacement.

Midpoint of a distance. v_mid = √((u² + v²)/2) — from v² = u² + 2as applied over half the distance. Not (u + v)/2, which is the mean over time. This item has now been set twice (2022, 2025) and failed twice.

Multi-stage. End-of-stage velocity is the next stage's u; end-of-stage position accumulates. Check whether the acceleration changes at each boundary — [RPT16 E2]: "not realising that acceleration was now constant".

Checks. Discard the negative root of a time quadratic. Verify the units. Verify the answer is physically plausible — [RPT18 E1] lists "not checking that the answer was reasonable" among the paper's weaknesses.

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Appendix — the 2023+ formula sheet, Space and measurement entries

Reproduced from 2025-11_2025-SpecialistMaths2.txt [PAPERS]. What is not listed here is not supplied and must be known.

Kinematics a = d²x/dt² = dv/dt = v dv/dx = d/dx(½v²) v = u + at · s = ut + ½at² · v² = u² + 2as · s = ½(u + v)t

Vectors in two and three dimensions r(t) = x(t)i + y(t)j + z(t)k · |r(t)| = √(x(t)² + y(t)² + z(t)²) ṙ(t) = dr/dt = (dx/dt)i + (dy/dt)j + (dz/dt)k vector scalar product: r₁·r₂ = |r₁||r₂|cos θ = x₁x₂ + y₁y₂ + z₁z₂ vector cross product: the i j k determinant, expanded as (y₁z₂ − y₂z₁)i + (x₂z₁ − x₁z₂)j + (x₁y₂ − x₂y₁)k vector equation of a line: r(t) = r₁ + t r₂; parametric equations of a line vector equation of a plane: r(s,t) = r₀ + s r₁ + t r₂; parametric equations of a plane; Cartesian equation ax + by + cz = d

Not on the sheet — and therefore to be memorised: unit vector; scalar resolute; vector resolute; perpendicular resolute; area of a triangle or parallelogram from a cross product; distance from a point to a plane; distance from a point to a line; distance between two skew lines; angle between two planes; angle between a line and a plane; the Cartesian arc-length integral (deleted in 2023).