MATH40004 EFFECTIVE COVERAGE MATRIX REVISED 2
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: b8081ca4f9c49bd9a245f2591eb6d4c7b21f6c0057426ee9acafabe42b2ec952Source date: 2026-08-06
explanation and acceptance summary
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| MATH40004 seven-year ability units and effective-coverage audit - AUDITED_REVISED_2 | |||||||
| A= direct question type / entry-point coverage ;B= method-sequence coverage with practice transferring to unfamiliar formulations ;C= knowledge-label coverage only ;D= not covered. Effective coverage counts only A+B.2020 original question booklets and official solutions / marking materials are included in the final evidence directory . | |||||||
| Course code | MATH40004 | course name | Calculus and Applications | target examination | 2026 Summer Resit | final number of papers | 6 |
| formal-question ability units | 210 | Six earlier papers A+B | 185 | effective-coverage rate of the old six papers | 88.1% | revised version A+B | 207 |
| effective-coverage rate of the revised version | 98.6% | frequent / high-mark ability units | 187 | of which A+B | 187 | frequent / effective-coverage rate for high-mark abilities | 100.0% |
| treatment of the old six papers A Retain | 9 | B Revise | 8 | C Replace | 15 | D retire the entire question | 0 |
| key audit findings | |||||||
| 1 | although the old six papers nearly covered every knowledge-point label, A+B ability-unit coverage was only 88.1%, and 2025 the high-mark abilities of a unique global maximum and three types of series tests lacked effective practice . | ||||||
| 2 | the old version took 2026 question-number slots and method combinations copied across several papers ,P1/P4/P6 particularly clear ;P8 also represents the same anchoring, so is excluded from the final six-paper import package . | ||||||
| 3 | six revised papers A+B coverage 207/210=98.6%; all 187 frequent or high-mark ability units all reach A or B. | ||||||
| 4 | three units not reaching A/B are low-frequency specific contexts: third-order ODE system reformulation and two reciprocal-substitution integrals; listed explicitly under uncovered abilities without overclaiming . | ||||||
| 5 | accessible evidence for the course-method audit consists of official solutions, marking annotations and examiner comments; no accessible MATH40004 lecture notes/problem sheets/Tony answers were found, so no false claim of checking . | ||||||
| AUDITED_REVISED_2 evidence-chain completion | |||||||
| original filename | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf | ||||||
| source | the user in this MATH40004 provided in the course session; final ZIP included unchanged within 06_SOURCE_EVIDENCE/2020/. | ||||||
| content structure | 22 pages: cover and Q1-Q6 question booklet , 2020 Solutions and Marking Scheme, 2019-20 Examination Solutions, examiner comments . | ||||||
| SHA-256 | 143a8d4fdaaf7f49f781e464c0794cc7a29cd1f96d25440bc26dafa79164520b |
Seven-year ability-unit inventory
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| 2020-2026 complete inventory of formal-question ability units | |||||||||||||||||||||
| each row corresponds to one formal-examination subpart; beyond knowledge points it includes entry signal, first step, course-method sequence, tools and calculation / abstract / integration workload and estimated time . | |||||||||||||||||||||
| Competency unit ID | year | question number | mark allocation | Knowledge point | Knowledge point ID | question type | question recognition signal | first key step | complete course-method sequence | theorem / formula / tools | combination of knowledge points | calculation volume | degree of abstraction | degree of integration | normal completion time | official solution length | 2026 appeared in the main examination | best coverage in the old six papers | best coverage in the six revised papers | source location | official basis for difficulty |
| AU001 | 2020 | 1(a) | 4 | state rigorous definitions of continuity and differentiability | K01, K02 | definitions | the question asks for a definition / whether continuous or differentiable: first state the limit definition, then handle special points . | limit definition | state the definition → substitute into the difference quotient at the special point / Limit → squeeze or construct a sequence → obtain the conclusion . | ε-δ limits, difference quotients and squeezing | limits, continuity, differentiation and function analysis | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(a) | official marking :A category of basic definitions |
| AU002 | 2020 | 1(b) | 4 | Oscillating function at 0 continuity and differentiability at the point | K01, K02, K04 | definitions | the question asks for a definition / whether continuous or differentiable: first state the limit definition, then handle special points . | squeezing and piecewise differentiation | state the definition → substitute into the difference quotient at the special point / Limit → squeeze or construct a sequence → obtain the conclusion . | ε-δ limits, difference quotients and squeezing | limits, continuity, differentiation and function analysis | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | is | B | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(b) | official marking includes B/C |
| AU003 | 2020 | 1(c)(i) | 2 | parametric curve dy/dx | K06 | param_curve | parametric trajectory / arc length / tangents: first calculate dx/dt, dy/dt, then examine numerator and denominator . | dy/dt÷dx/dt | differentiate parametrically → axis intersections → horizontal / vertical tangent equations → recognise polar coordinates / arc-length integral . | parametric differentiation, polar coordinates and arc-length formula | limits, continuity, differentiation and function analysis | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(c)(i) | A category |
| AU004 | 2020 | 1(c)(ii) | 2 | intersections of the parametric curve with the axes | K06 | param_curve | parametric trajectory / arc length / tangents: first calculate dx/dt, dy/dt, then examine numerator and denominator . | zeros of trigonometric functions | differentiate parametrically → axis intersections → horizontal / vertical tangent equations → recognise polar coordinates / arc-length integral . | parametric differentiation, polar coordinates and arc-length formula | limits, continuity, differentiation and function analysis | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(c)(ii) | A category |
| AU005 | 2020 | 1(c)(iii) | 5 | horizontal / vertical tangents and graphical solution of a transcendental equation | K06, K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | tan t=-t and tan t=1/t | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | high | high | 8 minutes | approximately 0.64 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(c)(iii) | the official material includes C/D |
| AU006 | 2020 | 1(c)(iv) | 3 | sketch an Archimedean spiral trajectory | K07 | param_curve | parametric trajectory / arc length / tangents: first calculate dx/dt, dy/dt, then examine numerator and denominator . | convert to polar coordinates | differentiate parametrically → axis intersections → horizontal / vertical tangent equations → recognise polar coordinates / arc-length integral . | parametric differentiation, polar coordinates and arc-length formula | integration, geometry and physical modelling | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 1(c)(iv) | B category |
| AU007 | 2020 | 2(a) | 6 | improper integral e^{-x}/x^p the convergence range of | K14, K15 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | comparison test | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | in | high | 9 minutes | approximately 0.60 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 2(a) | A/B category |
| AU008 | 2020 | 2(b) | 6 | Fresnel existence of this type of improper integral | K15, K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | integration by parts and absolute-value estimates | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | in | high | 9 minutes | approximately 0.66 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 2(b) | B/C category |
| AU009 | 2020 | 2(c)(i) | 2 | sketch the region | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | function graph | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | B | B | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 2(c)(i) | A category |
| AU010 | 2020 | 2(c)(ii) | 2 | area of the region | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | definite integral | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | B | B | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 2(c)(ii) | A category |
| AU011 | 2020 | 2(c)(iii) | 4 | centroid of a plane region | K19, K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | reduce double integrals for moments to single integrals | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 2(c)(iii) | D category |
| AU012 | 2020 | 3(a)(i) | 3 | Taylor theorem and remainder | K08 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Lagrange remainder | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(a)(i) | A category |
| AU013 | 2020 | 3(a)(ii) | 3 | use e^x series approximation √e | K09, K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Maclaurin expansion | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(a)(ii) | B category |
| AU014 | 2020 | 3(a)(iii) | 4 | composite exponential limit | K09, K04 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | Taylor or L'Hospital | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | Taylor, power series and infinite series; limits, continuity, differentiation and function analysis | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | B | B | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(a)(iii) | D category |
| AU015 | 2020 | 3(b)(i) | 3 | periodically extend and sketch | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | periodic replication | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(b)(i) | A category |
| AU016 | 2020 | 3(b)(ii) | 5 | the sign function's Fourier sine series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | odd functions and orthogonality integrals | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | in | in | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(b)(ii) | B/C category |
| AU017 | 2020 | 3(b)(iii) | 2 | From Fourier deduce from the series Leibniz series | K24 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | x=π/2 | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 3(b)(iii) | A category |
| AU018 | 2020 | 4(a) | 6 | nonlinear second-order ODE reduce order and satisfy initial conditions | K38 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | let u=y' and regard as u(y) | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | low | high | 9 minutes | approximately 0.54 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 4(a) | A mainly this category |
| AU019 | 2020 | 4(b) | 5 | a shifted exponential function's Fourier transform | K27, K28 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | piecewise integration | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 4(b) | B category |
| AU020 | 2020 | 4(c)(i) | 3 | contains delta homogeneous solution of the equation | K33 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | index Ansatz | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 4(c)(i) | A category |
| AU021 | 2020 | 4(c)(ii) | 6 | use Fourier use a transform to find delta forcing ODE | K29, K31 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | delta and known transform pairs | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 4(c)(ii) | C/D category |
| AU022 | 2020 | 5(a)(i) | 5 | general solution of a two-dimensional linear system | K40 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | eigenvalue / eigenvector | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | in | 8 minutes | approximately 0.46 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 5(a)(i) | A category |
| AU023 | 2020 | 5(a)(ii) | 5 | saddle phase portrait, vector field and asymptotics for a specified initial value | K42, K40 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | eigendirection + Vector field | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 5(a)(ii) | B category |
| AU024 | 2020 | 5(a)(iii) | 3 | particular solution of a nonhomogeneous linear system | K43 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | exponential ansatz with undetermined coefficients | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 5(a)(iii) | A category |
| AU025 | 2020 | 5(b)(i) | 4 | parameter-dependent nonlinear ODE fixed points and stability | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | line and exponential curve | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 5(b)(i) | D category |
| AU026 | 2020 | 5(b)(ii) | 3 | bifurcation diagram and classification | K46 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | stable / unstable branch | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 5(b)(ii) | C/D category |
| AU027 | 2020 | 6(a) | 8 | second mixed partial derivatives of an implicit function and symmetry | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | total differential and chain rule | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | high | high | high | 12 minutes | approximately 0.88 pages, estimated from official-solution layout ) | is | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 6(a) | C/D category |
| AU028 | 2020 | 6(b)(i) | 4 | third-order constant-coefficient ODE homogeneous solution | K33 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | index Ansatz | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 6(b)(i) | A category |
| AU029 | 2020 | 6(b)(ii) | 5 | use variation of parameters for a third-order ODE particular solution | K36 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | A(x)e^{-2x} | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | No | A | A | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 6(b)(ii) | B category |
| AU030 | 2020 | 6(b)(iii) | 3 | third-order ODE rewrite as a first-order system | K39 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | y,u=y',w=y'' | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | B | D | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf · 6(b)(iii) | A category |
| AU031 | 2021 | 1(a) | 3 | oscillating piecewise-linear function at 0 continuous extension at the point | K01 | definitions | the question asks for a definition / whether continuous or differentiable: first state the limit definition, then handle special points . | Limit | state the definition → substitute into the difference quotient at the special point / Limit → squeeze or construct a sequence → obtain the conclusion . | ε-δ limits, difference quotients and squeezing | limits, continuity, differentiation and function analysis | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | B | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 1(a) | A category |
| AU032 | 2021 | 1(b) | 5 | sketch the piecewise-linear function and its behaviour near zero | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | piecewise linear | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | in | in | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | No | D | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 1(b) | B/C category |
| AU033 | 2021 | 1(c) | 6 | sketch the derivative and 0 not differentiable at the point | K02, K05 | definitions | the question asks for a definition / whether continuous or differentiable: first state the limit definition, then handle special points . | difference quotient at geometric nodes | state the definition → substitute into the difference quotient at the special point / Limit → squeeze or construct a sequence → obtain the conclusion . | ε-δ limits, difference quotients and squeezing | limits, continuity, differentiation and function analysis | in | in | high | 9 minutes | approximately 0.66 pages, estimated from official-solution layout ) | No | B | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 1(c) | C category |
| AU034 | 2021 | 1(d) | 6 | length of the derivative's graph | K07, K17 | param_curve | parametric trajectory / arc length / tangents: first calculate dx/dt, dy/dt, then examine numerator and denominator . | telescoping series | differentiate parametrically → axis intersections → horizontal / vertical tangent equations → recognise polar coordinates / arc-length integral . | parametric differentiation, polar coordinates and arc-length formula | integration, geometry and physical modelling | in | in | high | 9 minutes | approximately 0.60 pages, estimated from official-solution layout ) | is | D | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 1(d) | B category |
| AU035 | 2021 | 2(a)(i) | 4 | volume of a paraboloid-of-revolution bowl | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | disc method | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 2(a)(i) | A category |
| AU036 | 2021 | 2(a)(ii) | 4 | surface area of a surface of revolution | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | arc-length element | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 2(a)(ii) | A category |
| AU037 | 2021 | 2(b)(i) | 3 | surface density under a buoyancy condition | K19, K21 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | conservation / dimensions | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 2(b)(i) | D category |
| AU038 | 2021 | 2(b)(ii) | 3 | maximal surface density without sinking | K19, K21 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | buoyancy balance | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 2(b)(ii) | D category |
| AU039 | 2021 | 2(c) | 6 | physically feasible range for a variable-density fluid parameter | K19, K21 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | integration with variable density | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | is | B | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 2(c) | D category |
| AU040 | 2021 | 3(a)(i) | 2 | contains log conditions for existence of a power-type improper integral | K14, K15 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | L'Hospital/ powers dominate logarithms | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | low | high | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(a)(i) | A category |
| AU041 | 2021 | 3(a)(ii) | 4 | integral recurrence and find I5 | K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | expand the recurrence | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(a)(ii) | A category |
| AU042 | 2021 | 3(b) | 6 | sec² Riemann limit of the sum | K18 | riemann | contains 1/n limit of sums: identify the integration interval and actual mesh width . | Riemann and | Identify h=(b-a)/n → write hΣf(x_k) → take the limit as an integral → calculation . | Riemann definition of sums and basic integration | integration, geometry and physical modelling | in | in | high | 9 minutes | approximately 0.60 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(b) | A/B category |
| AU043 | 2021 | 3(c)(i) | 3 | sin(x²/π) derivative and sketching | K04, K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | Differentiate + Endpoint | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | low | high | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | C | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(c)(i) | A category |
| AU044 | 2021 | 3(c)(ii) | 3 | periodic even function Fourier coefficient formula | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | cosine series | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(c)(ii) | A category |
| AU045 | 2021 | 3(c)(iii) | 2 | Fourier convergence of the series and its derivative at endpoints | K23, K26 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | periodic extension | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | in | high | 3 minutes | approximately 0.34 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 3(c)(iii) | C category |
| AU046 | 2021 | 4(a)(i) | 4 | repeated-root linear ODE homogeneous solution and Wronskian | K33, K34 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | Wronskian | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 4(a)(i) | A category |
| AU047 | 2021 | 4(a)(ii) | 7 | piecewise forcing ODE initial-value problem | K35 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | undetermined coefficients + continuous matching | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | high | in | high | 10 minutes | approximately 0.74 pages, estimated from official-solution layout ) | No | D | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 4(a)(ii) | A/C Mixture |
| AU048 | 2021 | 4(b) | 4 | from the transform domain Taylor coefficients to find the third moment | K27, K49 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | Fourier transform moment formula | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace); multivariable calculus and implicit functions | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 4(b) | D category |
| AU049 | 2021 | 4(c) | 5 | use the energy theorem to evaluate a rational integral | K30 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | Plancherel | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | No | B | B | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 4(c) | B category |
| AU050 | 2021 | 5(a)(i) | 6 | linear system, line of fixed points and asymptotics for a specified initial value | K40, K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | zero eigenvalue and stable subspace | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | high | 9 minutes | approximately 0.54 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 5(a)(i) | A category |
| AU051 | 2021 | 5(a)(ii) | 3 | nonhomogeneous linear system | K43 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | undetermined coefficients | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | in | low | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 5(a)(ii) | C category |
| AU052 | 2021 | 5(b) | 6 | identify fixed points and types from a given phase portrait | K42, K45 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | local linearisation type | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits; nonlinear ODE and bifurcation | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 5(b) | D category |
| AU053 | 2021 | 5(c) | 5 | saddle-node bifurcation in a quartic one-dimensional system | K44, K45, K46 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | parameter branch | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 5(c) | B category |
| AU054 | 2021 | 6(a) | 8 | x^m y^n integrating factor and implicit solution | K47, K48 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | exactness condition | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | high | high | high | 12 minutes | approximately 0.88 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 6(a) | B/D Mixture |
| AU055 | 2021 | 6(b)(i) | 2 | zero contour | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | algebraic zero set | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 6(b)(i) | A category |
| AU056 | 2021 | 6(b)(ii) | 6 | stationary points and Hessian classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | Hessian trace / determinant | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | low | high | 9 minutes | approximately 0.54 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 6(b)(ii) | A category |
| AU057 | 2021 | 6(b)(iii) | 4 | consistency of contours, sign regions and classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | sign analysis | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf · 6(b)(iii) | C category |
| AU058 | 2022 | 1(a)(i) | 4 | flat function at 0 extension of the value and first two derivatives at the point | K03, K04 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | take logarithms / derivative | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | C | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(a)(i) | A category |
| AU059 | 2022 | 1(a)(ii) | 2 | derivatives of arbitrary order at 0 limit at the point | K03, K04 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | exponentials dominate powers | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | high | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | is | C | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(a)(ii) | B category |
| AU060 | 2022 | 1(b)(i) | 3 | inflection points and global extrema | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | function analysis | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | D | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(b)(i) | A category |
| AU061 | 2022 | 1(b)(ii) | 3 | function sketching | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | structural sketch | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | D | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(b)(ii) | B category |
| AU062 | 2022 | 1(c)(i) | 4 | x=1/t split an improper integral using substitution | K16, K14 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | reciprocal substitution | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(c)(i) | A/C Mixture |
| AU063 | 2022 | 1(c)(ii) | 4 | convergence of two improper integrals | K15 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | Limit + compare | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 1(c)(ii) | D category |
| AU064 | 2022 | 2(a) | 3 | mass and centroid formulas for a ring with linear density | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | polar-coordinate line integral | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(a) | A category |
| AU065 | 2022 | 2(b) | 2 | centroid of a uniform ring | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | sine and cosine integrals | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(b) | A category |
| AU066 | 2022 | 2(c)(i) | 2 | endpoint cases for an added arc | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | limiting case | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | in | low | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | No | B | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(c)(i) | B category |
| AU067 | 2022 | 2(c)(ii) | 3 | centroid formula after adding an arc | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | mass and moments | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(c)(ii) | C category |
| AU068 | 2022 | 2(c)(iii) | 3 | maximising the centroid and a transcendental equation | K05, K19 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | tanθ=π+θ | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis; integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(c)(iii) | B category |
| AU069 | 2022 | 2(d)(i) | 3 | N centroid formula for layered arcs | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | Parameter N | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | low | high | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(d)(i) | A category |
| AU070 | 2022 | 2(d)(ii) | 4 | large N asymptotically optimal angle and maximum centroid | K09, K11, K19 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | cubic dominant balance | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series; integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 2(d)(ii) | D category |
| AU071 | 2022 | 3(a)(i) | 2 | π/2 near cos the two terms of Taylor including remainder | K08, K09 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Lagrange remainder | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | high | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | B | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(a)(i) | A category |
| AU072 | 2022 | 3(a)(ii) | 3 | use Taylor Lower bound / control a singular function using an upper bound | K09, K15 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | local comparison | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series; integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | B | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(a)(ii) | B category |
| AU073 | 2022 | 3(a)(iii) | 3 | 0 near Taylor upper bound | K09, K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | upper bound on the interval | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | B | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(a)(iii) | C category |
| AU074 | 2022 | 3(a)(iv) | 4 | 1/√cos x existence and upper bound of an improper integral | K15 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | integrable singularity | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(a)(iv) | B/D Mixture |
| AU075 | 2022 | 3(b)(i) | 4 | piecewise even function Fourier series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | cosine series | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(b)(i) | A category |
| AU076 | 2022 | 3(b)(ii) | 4 | differentiate a known series to obtain an odd-function series and values at discontinuities | K23, K26 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | average of left and right limits | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 3(b)(ii) | A/D Mixture |
| AU077 | 2022 | 4(a)(i) | 4 | repeated-root homogeneous ODE and Wronskian | K33, K34 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | Wronskian | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | B | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 4(a)(i) | A category |
| AU078 | 2022 | 4(a)(ii) | 7 | Heaviside piecewise forcing ODE | K35 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | undetermined coefficients | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | high | high | high | 10 minutes | approximately 0.80 pages, estimated from official-solution layout ) | No | D | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 4(a)(ii) | D/C Mixture |
| AU079 | 2022 | 4(b)(i) | 4 | cosine transform of a compactly supported parabolic function | K27 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | even function | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 4(b)(i) | A category |
| AU080 | 2022 | 4(b)(ii) | 2 | use the energy theorem to obtain an integral identity | K30 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | squared modulus of the transform | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | low | in | high | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | No | B | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 4(b)(ii) | B category |
| AU081 | 2022 | 4(b)(iii) | 3 | from scaling / use symmetry to find another function's transform | K28 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | scaling + duality | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | D | B | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 4(b)(iii) | B category |
| AU082 | 2022 | 5(a)(i) | 7 | one-dimensional system with a singularity ODE fixed points and stability | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | fixed point / list singularities alongside | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | high | in | high | 10 minutes | approximately 0.74 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 5(a)(i) | A/C Mixture |
| AU083 | 2022 | 5(a)(ii) | 4 | bifurcation diagram with singularities and determination of no bifurcation | K46, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | criterion for unchanged stability | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 5(a)(ii) | D category |
| AU084 | 2022 | 5(a)(iii) | 2 | finite-time behaviour of initial values approaching a singularity | K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | singularity barrier | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | low | low | high | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 5(a)(iii) | A category |
| AU085 | 2022 | 5(b)(i) | 2 | check exactness | K47 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | M_y and N_x | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 5(b)(i) | A category |
| AU086 | 2022 | 5(b)(ii) | 5 | find an implicit solution using a single-variable integrating factor | K48 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | make exact | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 5(b)(ii) | B category |
| AU087 | 2022 | 6(a)(i) | 6 | zero-eigenvalue system, line of fixed points and general solution | K40, K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | centre / stable subspace | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | high | 9 minutes | approximately 0.54 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 6(a)(i) | A category |
| AU088 | 2022 | 6(a)(ii) | 3 | phase portrait | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | parallel trajectory lines | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 6(a)(ii) | B category |
| AU089 | 2022 | 6(a)(iii) | 2 | trajectory and asymptotics for a specified initial value | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | decay along eigencomponents | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | in | low | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 6(a)(iii) | B category |
| AU090 | 2022 | 6(b)(i) | 5 | first partial derivatives of an implicit function and maximum uncertainty | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | linear error propagation | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | high | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 6(b)(i) | C/A Mixture |
| AU091 | 2022 | 6(b)(ii) | 4 | second partial derivatives of an implicit function | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | for g(x,y,z(x,y)) Differentiate | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf · 6(b)(ii) | D category |
| AU092 | 2023 | 1(a)(i) | 4 | first and second derivatives of a composite function | K04 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | differentiate a composite function | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | C | B | MATH40004_Calculus and Applications_2023.pdf · 1(a)(i) | A category |
| AU093 | 2023 | 1(a)(ii) | 4 | e^{-x³} extrema and inflection points | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | sign analysis | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | in | low | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | No | D | B | MATH40004_Calculus and Applications_2023.pdf · 1(a)(ii) | A/B category |
| AU094 | 2023 | 1(a)(iii) | 2 | function sketching | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | qualitative sketch | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | low | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | No | D | B | MATH40004_Calculus and Applications_2023.pdf · 1(a)(iii) | B category |
| AU095 | 2023 | 1(b)(i) | 3 | Taylor theorem and local remainder | K08 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Lagrange remainder | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 1(b)(i) | A category |
| AU096 | 2023 | 1(b)(ii) | 4 | x log x in 1 two nonzero terms and a remainder at the point | K08, K09 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | substitute derivatives | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | B | B | MATH40004_Calculus and Applications_2023.pdf · 1(b)(ii) | A/B category |
| AU097 | 2023 | 1(b)(iii) | 3 | 1.1log1.1 rational approximation and error | K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | ξ control on an interval | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 1(b)(iii) | D category |
| AU098 | 2023 | 2(a)(i) | 4 | compare surface areas formed by rotating a parabola about two axes | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | without explicit integration | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 2(a)(i) | A/B category |
| AU099 | 2023 | 2(a)(ii) | 4 | volumes about both axes and find the critical L | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | compare powers | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 2(a)(ii) | B/C category |
| AU100 | 2023 | 2(a)(iii) | 2 | geometrically explain reversal of the volume comparison | K20 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | compare radii | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | low | high | low | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 2(a)(iii) | D category |
| AU101 | 2023 | 2(b)(i) | 5 | t and sinωt of Laplace transform | K32 | laplace | t≥0 transform or inverse transform: begin with the definition / known pairs and partial fractions . | Defining integral | Defining integral / known transform pair → partial fractions → inverse transform → initial-value or limit check . | Laplace definition, partial fractions and transform tables | integral transforms (Fourier/Laplace) | in | in | in | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 2(b)(i) | B/C category |
| AU102 | 2023 | 2(b)(ii) | 5 | inverse transform by partial fractions Laplace | K32 | laplace | t≥0 transform or inverse transform: begin with the definition / known pairs and partial fractions . | linear combination | Defining integral / known transform pair → partial fractions → inverse transform → initial-value or limit check . | Laplace definition, partial fractions and transform tables | integral transforms (Fourier/Laplace) | in | high | in | 8 minutes | approximately 0.64 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 2(b)(ii) | A/D Mixture |
| AU103 | 2023 | 3(a) | 4 | inductive proof of a finite sum/product identity | K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | double summation | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 3(a) | C/D category |
| AU104 | 2023 | 3(b)(i) | 4 | Fourier orthogonal coefficient formula for the integral of a square | K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | integral orthogonality | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 3(b)(i) | C category |
| AU105 | 2023 | 3(b)(ii) | 4 | Derive Parseval theorem | K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | energy identity | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 3(b)(ii) | B/D category |
| AU106 | 2023 | 3(c)-FS | 4 | even triangular function Fourier series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | cosine series | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 3(c)-FS | A category |
| AU107 | 2023 | 3(c)-Parseval | 4 | From Parseval sum reciprocals of fourth powers of odd integers | K24, K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | squared coefficients | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 3(c)-Parseval | A/B category |
| AU108 | 2023 | 4(a)(i) | 3 | e^{-a|x|} of Fourier transform | K27 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | evenness | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 4(a)(i) | A category |
| AU109 | 2023 | 4(a)(ii) | 5 | use convolution to find (4+ω²)^{-2} inverse transform | K28, K29 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | piecewise convolution integral | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | No | D | A | MATH40004_Calculus and Applications_2023.pdf · 4(a)(ii) | B category |
| AU110 | 2023 | 4(b)(i) | 6 | Euler–Cauchy homogeneous problem | K33, K34, K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | repeated-root basis and Wronskian | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | in | high | 9 minutes | approximately 0.66 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 4(b)(i) | A/C Mixture |
| AU111 | 2023 | 4(b)(ii) | 6 | variation of parameters / vary the functions to find a particular solution | K36, K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | integrate twice | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 4(b)(ii) | D/A Mixture |
| AU112 | 2023 | 5(a)(i) | 3 | general solution of a two-dimensional linear system | K40 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | real eigenvalue | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 5(a)(i) | A category |
| AU113 | 2023 | 5(a)(ii) | 4 | stable-node phase portrait and asymptotic direction | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | fast and slow decay | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 5(a)(ii) | C category |
| AU114 | 2023 | 5(a)(iii) | 3 | trajectory for a specified initial value | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | eigenline | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 5(a)(iii) | B category |
| AU115 | 2023 | 5(b)(i) | 4 | stability of two fixed points | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | Parameter r separate cases | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 5(b)(i) | A category |
| AU116 | 2023 | 5(b)(ii) | 3 | transcritical bifurcation | K46 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | transcritical | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | low | high | low | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 5(b)(ii) | B/D category |
| AU117 | 2023 | 5(b)(iii) | 3 | set of initial values causing divergence | K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | parameter regions | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | B | B | MATH40004_Calculus and Applications_2023.pdf · 5(b)(iii) | D category |
| AU118 | 2023 | 6(a) | 7 | find one depending only on x the integrating factor of | K47, K48 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | make exact | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | high | in | high | 10 minutes | approximately 0.68 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_2023.pdf · 6(a) | A/B category |
| AU119 | 2023 | 6(b)(i) | 2 | zero contour | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | three branches | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 6(b)(i) | A category |
| AU120 | 2023 | 6(b)(ii) | 6 | with a line of stationary points Hessian classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | trace / determinant | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 6(b)(ii) | B/D category |
| AU121 | 2023 | 6(b)(iii) | 5 | contours and consistency of classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | graphical interpretation | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | high | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_2023.pdf · 6(b)(iii) | C/A category |
| AU122 | 2024 | 1(a) | 2 | e^{-1/x}/x^p Limit | K03 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | take logarithms | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | low | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | is | D | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(a) | official B |
| AU123 | 2024 | 1(b)(i) | 2 | continuous extension | K01, K03 | definitions | the question asks for a definition / whether continuous or differentiable: first state the limit definition, then handle special points . | Limit | state the definition → substitute into the difference quotient at the special point / Limit → squeeze or construct a sequence → obtain the conclusion . | ε-δ limits, difference quotients and squeezing | limits, continuity, differentiation and function analysis | low | low | high | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | B | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(b)(i) | official A |
| AU124 | 2024 | 1(b)(ii) | 4 | piecewise differentiation and limits of derivatives of arbitrary order | K03, K04 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | exponentials dominate powers | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | C | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(b)(ii) | official A/B |
| AU125 | 2024 | 1(b)(iii) | 3 | inflection points and sketching | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | symmetry | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | D | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(b)(iii) | official A/B |
| AU126 | 2024 | 1(b)(iv) | 3 | divergence of an improper integral | K15, K09 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | and 1/x compare | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | integration, geometry and physical modelling ;Taylor, power series and infinite series | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(b)(iv) | official B |
| AU127 | 2024 | 1(c)(i) | 3 | rewrite a difference integral by reciprocal substitution | K16 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | substitution | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | B | D | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(c)(i) | official B |
| AU128 | 2024 | 1(c)(ii) | 3 | integrability after cancellation of the leading term in the difference | K09, K15 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | finite limit | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series; integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | B | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 1(c)(ii) | official C |
| AU129 | 2024 | 2(a) | 4 | tan of Riemann and | K18, K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | Riemann and | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 2(a) | official A/D |
| AU130 | 2024 | 2(b)(i) | 3 | tan x of Taylor expansion | K09 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Maclaurin | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | low | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 2(b)(i) | official A |
| AU131 | 2024 | 2(b)(ii) | 4 | centroid of a nonuniform thin rod | K19 | centroid_geometry | mass / centroid / solid of revolution: first draw an element and write mass, moment or shell / disc formula . | weighted integral | draw an infinitesimal element → total mass / area → moments about both axes → divide by the total → physical check . | area / volume / surface formula and mass moments | integration, geometry and physical modelling | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 2(b)(ii) | official A |
| AU132 | 2024 | 2(b)(iii) | 3 | use Taylor approximate centroid | K09, K11, K19 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | truncated approximation | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series; integration, geometry and physical modelling | low | low | high | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 2(b)(iii) | official A |
| AU133 | 2024 | 2(c) | 6 | logarithmic-spiral sketch, asymptotics and arc length | K07 | param_curve | parametric trajectory / arc length / tangents: first calculate dx/dt, dy/dt, then examine numerator and denominator . | parametric arc length | differentiate parametrically → axis intersections → horizontal / vertical tangent equations → recognise polar coordinates / arc-length integral . | parametric differentiation, polar coordinates and arc-length formula | integration, geometry and physical modelling | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 2(c) | official B/D |
| AU134 | 2024 | 3(a)(i) | 5 | sin(x/2) periodic extension Fourier series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | sine series | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | in | in | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(a)(i) | official A/C |
| AU135 | 2024 | 3(a)(ii) | 2 | convergence values at three points | K23 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | periodic left and right limits | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | in | low | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(a)(ii) | official A/B |
| AU136 | 2024 | 3(a)(iii) | 3 | use the series to find S | K24 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | select odd terms | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(a)(iii) | official C/D |
| AU137 | 2024 | 3(b)(i) | 2 | Parseval theorem | K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | orthogonal energy | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(b)(i) | official A |
| AU138 | 2024 | 3(b)(ii) | 2 | simplify the energy functional by integration by parts | K25, K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | periodic boundary terms vanish | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | Fourier series; integration, geometry and physical modelling | low | in | high | 3 minutes | approximately 0.34 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(b)(ii) | official C |
| AU139 | 2024 | 3(b)(iii) | 4 | for f' and f'' application Parseval | K25, K26 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | multiply coefficients by n | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(b)(iii) | official C/D |
| AU140 | 2024 | 3(b)(iv) | 2 | ensure E>0 of ε range | K25 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | minimum n=1 | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | high | low | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | No | B | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 3(b)(iv) | official D |
| AU141 | 2024 | 4(a)(i) | 4 | general solution of a two-dimensional linear system | K40 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | saddle eigenvalues | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 4(a)(i) | official A |
| AU142 | 2024 | 4(a)(ii) | 4 | saddle phase portrait and asymptotics | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | phase portrait | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 4(a)(ii) | official A/B |
| AU143 | 2024 | 4(b)(i) | 4 | cosine transform of the triangular hat function | K27 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | even function | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | in | low | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 4(b)(i) | official B/A |
| AU144 | 2024 | 4(b)(ii) | 4 | use the energy theorem to evaluate an integral | K30 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | Plancherel | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | B | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 4(b)(ii) | official C/D |
| AU145 | 2024 | 4(b)(iii) | 4 | use scaling and duality to find g the transform of | K28 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | transform properties | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | D | B | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 4(b)(iii) | official C/D |
| AU146 | 2024 | 5(a)(i) | 4 | fixed points and stability of a cubic one-dimensional system | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | sign diagram | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 5(a)(i) | official A/B |
| AU147 | 2024 | 5(a)(ii) | 4 | subcritical pitchfork bifurcation | K46 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | pitchfork | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | in | low | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 5(a)(ii) | official A/B |
| AU148 | 2024 | 5(b)(i) | 4 | third-order Euler–Cauchy change of variables | K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | x=e^z | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 5(b)(i) | official A/C/D |
| AU149 | 2024 | 5(b)(ii) | 8 | third-order Euler–Cauchy complete general solution | K33, K35, K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | undetermined coefficients | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | high | high | high | 12 minutes | approximately 0.88 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 5(b)(ii) | official A/B/C/D |
| AU150 | 2024 | 6(a)(i) | 2 | check nonexactness | K47 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | M_y,N_x | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 6(a)(i) | official A |
| AU151 | 2024 | 6(a)(ii) | 6 | x^m y^n integrating factor and implicit solution | K48 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | integrate after making exact | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | in | high | high | 9 minutes | approximately 0.72 pages, estimated from official-solution layout ) | is | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 6(a)(ii) | official D/A |
| AU152 | 2024 | 6(b)(i) | 1 | zero contour | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | line + parabola | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | low | low | low | 2 minutes | approximately 0.15 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 6(b)(i) | official A |
| AU153 | 2024 | 6(b)(ii) | 7 | stationary points and Hessian classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | trace / determinant | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | high | in | high | 10 minutes | approximately 0.74 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 6(b)(ii) | official A/B/C |
| AU154 | 2024 | 6(b)(iii) | 4 | contour diagram and consistency argument | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | graphical interpretation | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | No | A | A | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf · 6(b)(iii) | official B/C |
| AU155 | 2025 | 1(a) | 2 | definition of existence of an improper integral | K14 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | split the integral | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(a) | official A |
| AU156 | 2025 | 1(b)(i) | 4 | logx/(1+x²) the limit and zeros of | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | asymptotic comparison | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | low | low | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | No | D | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(b)(i) | official A |
| AU157 | 2025 | 1(b)(ii) | 6 | unique global maximum | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | without using a second derivative | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | in | in | high | 9 minutes | approximately 0.60 pages, estimated from official-solution layout ) | No | D | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(b)(ii) | official B/A |
| AU158 | 2025 | 1(b)(iii) | 3 | function sketching | K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | qualitative sketch | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | in | low | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | D | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(b)(iii) | official B |
| AU159 | 2025 | 1(c)(i) | 3 | convergence of an improper integral | K14, K15 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | -logx and logx/x² | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(c)(i) | official B/C |
| AU160 | 2025 | 1(c)(ii) | 2 | evaluate an integral by reciprocal substitution | K16 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | cancellation by symmetry | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | high | high | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | No | B | D | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 1(c)(ii) | official D |
| AU161 | 2025 | 2(a) | 6 | convergence tests for three types of infinite series | K12 | series | infinite series / iteration: first identify the applicable comparison, integral, ratio or fixed-point framework . | Parameter q separate cases | first check the necessary condition → choose a convergence test → separate parameter cases → handle the boundary separately . | compare / integral / Ratio / alternating /Dirichlet test | Taylor, power series and infinite series | in | in | high | 9 minutes | approximately 0.66 pages, estimated from official-solution layout ) | No | D | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 2(a) | official A/B/C |
| AU162 | 2025 | 2(b) | 5 | log power series and rational approximation of an integral | K09, K10, K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | error threshold | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | in | high | high | 8 minutes | approximately 0.64 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 2(b) | official A/D |
| AU163 | 2025 | 2(c)(i) | 4 | n repeated sin derivatives, zeros and extrema of a composite function | K04, K13 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | factor in the product derivative | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis ;Taylor, power series and infinite series | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | B | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 2(c)(i) | official D/C |
| AU164 | 2025 | 2(c)(ii) | 3 | iteration sin the inequality for | K13, K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | derivative sign | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | Taylor, power series and infinite series; limits, continuity, differentiation and function analysis | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 2(c)(ii) | official A/B |
| AU165 | 2025 | 2(c)(iii) | 2 | iteration sin Limit | K13 | series | infinite series / iteration: first identify the applicable comparison, integral, ratio or fixed-point framework . | F*=sinF* | first check the necessary condition → choose a convergence test → separate parameter cases → handle the boundary separately . | compare / integral / Ratio / alternating /Dirichlet test | Taylor, power series and infinite series | low | high | low | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 2(c)(iii) | official D |
| AU166 | 2025 | 3(a)(i) | 3 | binomial expansion (1-x²)^-1/2 | K09, K10 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | convergence domain | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | high | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(a)(i) | official A |
| AU167 | 2025 | 3(a)(ii) | 4 | integrate to obtain arcsin series | K09, K10 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | integrate a power series | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | in | in | high | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(a)(ii) | official B |
| AU168 | 2025 | 3(a)(iii) | 3 | arcsin(0.1) six-decimal approximation | K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | order of error | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | in | low | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(a)(iii) | official C |
| AU169 | 2025 | 3(b)(i) | 2 | even / odd function Fourier general formula | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | coefficient formula | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(b)(i) | official A |
| AU170 | 2025 | 3(b)(ii) | 5 | triangular hat function Fourier series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | integration by parts | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | low | in | 8 minutes | approximately 0.46 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(b)(ii) | official A |
| AU171 | 2025 | 3(b)(iii) | 3 | obtain a new one by translation and reflection Fourier series | K26 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | trigonometric expansion | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 3(b)(iii) | official D |
| AU172 | 2025 | 4(a)(i) | 3 | 1/(1+x²) of Fourier transform | K27, K28 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | Known e^{-|x|} | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 4(a)(i) | official B |
| AU173 | 2025 | 4(a)(ii) | 4 | cos of delta transforms and convolution | K29, K27 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | delta value | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | No | B | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 4(a)(ii) | official A/D |
| AU174 | 2025 | 4(a)(iii) | 2 | obtain an explicit convolution result by inverse transform | K29 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | linear inverse transform | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | low | high | low | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | No | D | B | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 4(a)(iii) | official D |
| AU175 | 2025 | 4(b)(i) | 6 | Second order Euler–Cauchy homogeneous solution and Wronskian | K33, K34, K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | repeated root +Wronskian | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | in | high | 9 minutes | approximately 0.60 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 4(b)(i) | official A/B |
| AU176 | 2025 | 4(b)(ii) | 5 | Euler–Cauchy particular solution | K35, K37 | scalar_ode | constant coefficients or Euler-Cauchy ODE: first solve the homogeneous characteristic equation and identify resonance in the forcing . | z Polynomial ×e^{2z} | homogeneous CF → Wronskian( if required ) → appropriate PI/ variation of parameters → initial values or piecewise matching → residual check . | characteristic equation , Wronskian, undetermined coefficients and variation of parameters , Euler-Cauchy | scalar linear ODE | in | in | high | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 4(b)(ii) | official A/C |
| AU177 | 2025 | 5(a)(i) | 4 | general solution of a nondiagonalisable linear system | K40, K41 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | generalised eigenvector | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 5(a)(i) | official A |
| AU178 | 2025 | 5(a)(ii) | 4 | degenerate unstable-node phase portrait | K42, K41 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | Jordan phase portrait | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 5(a)(ii) | official A |
| AU179 | 2025 | 5(a)(iii) | 3 | implicit equation of the trajectory family | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | separation of variables | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 5(a)(iii) | official C |
| AU180 | 2025 | 5(b)(i) | 5 | at most three fixed points and their stability | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | Parameter r sign diagram | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 5(b)(i) | official B |
| AU181 | 2025 | 5(b)(ii) | 4 | pitchfork and transcritical double bifurcation | K46 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | two types of bifurcation | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 5(b)(ii) | official D |
| AU182 | 2025 | 6(a)(i) | 4 | complex squaring map Jacobian | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | direct calculation | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | low | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | B | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 6(a)(i) | official B |
| AU183 | 2025 | 6(a)(ii) | 4 | inverse Jacobian partial-derivative identity | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | Cauchy-Riemann type structure | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 6(a)(ii) | official D |
| AU184 | 2025 | 6(b)(i) | 2 | circle and line zero contours | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | geometric zero set | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 6(b)(i) | official A |
| AU185 | 2025 | 6(b)(ii) | 6 | four stationary points Hessian classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | Hessian | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | low | high | 9 minutes | approximately 0.54 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 6(b)(ii) | official A |
| AU186 | 2025 | 6(b)(iii) | 4 | consistency of contour sign regions and classification | K50 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | graphical interpretation | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | in | high | 6 minutes | approximately 0.50 pages, estimated from official-solution layout ) | No | A | A | MATH40004 Calculus and Applications Q&S May 2025(1).pdf · 6(b)(iii) | official C/B |
| AU187 | 2026 | 1(a) | 2 | Taylor theorem and local remainder | K08 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | Lagrange remainder | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(a) | official 2 marks A |
| AU188 | 2026 | 1(b) | 3 | log x in 1 power series and convergence interval at the point | K09, K10 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | log(1+u) series | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | in | high | 4 minutes | approximately 0.36 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(b) | official 2A+1B |
| AU189 | 2026 | 1(c)(i) | 3 | g(x)=logx/(x²-1) limits at three points | K03, K05 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | endpoint asymptotics | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis | low | low | high | 4 minutes | approximately 0.30 pages, estimated from official-solution layout ) | is | D | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(c)(i) | official 3A |
| AU190 | 2026 | 1(c)(ii)-series | 2 | (x+1)g the local power series and radius of | K09, K10 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | power series | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | in | high | 3 minutes | approximately 0.28 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(c)(ii)-series | official 2B |
| AU191 | 2026 | 1(c)(ii)-approx | 3 | log3-log2 the two-digit rational approximation of | K11 | taylor | local approximation, error or power series requested: identify the expansion centre and known base series . | termwise truncation | write the base series / theorem → substitute or integrate / Differentiate → state the convergence domain → truncation → remainder bound . | Taylor theorem , Lagrange remainders and power-series endpoints | Taylor, power series and infinite series | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(c)(ii)-approx | official 2B+1D/ subsequent C/D |
| AU192 | 2026 | 1(c)(iii) | 3 | higher-derivative identity and g in 1 smoothness at the point | K04, K09 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | Leibniz simplify | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | limits, continuity, differentiation and function analysis ;Taylor, power series and infinite series | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | B | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(c)(iii) | official A/C/D Mixture |
| AU193 | 2026 | 1(c)(iv) | 4 | g existence of the improper integral of | K14, K15, K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | logx and logx/x² | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 1(c)(iv) | official A/C/D Mixture |
| AU194 | 2026 | 2(a)(i) | 2 | contains √ξ definite integral with the denominator | K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | logarithmic integral | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | D | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(a)(i) | official 2A |
| AU195 | 2026 | 2(a)(ii) | 3 | cylindrical leaking-water model and k the dimensions of | K21 | model | physical loss / conservation: first identify the state variable and dimensions of every inflow/outflow term . | orifice flow rate + porous-wall area | define the state variable → write the conservation rate → reduce dimension / substitution to make separable → initial value → limit and dimensional checks . | conservation, dimensions and separability ODE | integration, geometry and physical modelling | low | in | high | 4 minutes | approximately 0.42 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(a)(ii) | official A/B/C |
| AU196 | 2026 | 2(a)(iii) | 4 | find the water height and emptying time | K21, K17 | improper | unbounded integrand at endpoints or unbounded interval: first separate all problematic endpoints . | separation of variables | separate endpoints → find the local dominant term → compare / substitution / integration by parts → determine separately → combine . | comparison, substitution, integration by parts and local asymptotics | integration, geometry and physical modelling | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | B | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(a)(iii) | official A/B/D |
| AU197 | 2026 | 2(a)(iv) | 3 | k→0 limits, solution without a porous wall and physical comparison | K21, K03 | function_analysis | an explicit or composite function with extrema requested / inflection point / graph: first compute first and second derivatives and combine with endpoint limits . | L'Hospital+ physical interpretation | Differentiate → solve for critical points → sign table / Limit → second derivative or convexity/concavity → annotate the graph . | chain rule, first/second derivatives and monotonicity / convexity/concavity | integration, geometry and physical modelling; limits, continuity, differentiation and function analysis | low | high | high | 4 minutes | approximately 0.48 pages, estimated from official-solution layout ) | is | B | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(a)(iv) | official C/D |
| AU198 | 2026 | 2(b)(i) | 4 | step function Fourier series | K22 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | piecewise integration | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | in | in | low | 6 minutes | approximately 0.44 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(b)(i) | official A/B |
| AU199 | 2026 | 2(b)(ii) | 2 | convergence values at discontinuities | K23 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | substitute special points | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | high | low | 3 minutes | approximately 0.40 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(b)(ii) | official B/D |
| AU200 | 2026 | 2(b)(iii) | 2 | From Fourier obtain from the series π the alternating series of | K24 | fourier_series | periodic piecewise function: determine parity, compute coefficients, then handle jumps . | Leibniz series | determine parity → calculation a0,an,bn → write the series → take the average at jumps → use special points to obtain numerical series . | orthogonality, parity and averaging at jumps , Parseval | Fourier series | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 2(b)(iii) | official 2A |
| AU201 | 2026 | 3(a)(i) | 5 | general solution of a linear system with complex eigenvalues | K40, K41 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | e^t cos2t/sin2t | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 3(a)(i) | official 5B |
| AU202 | 2026 | 3(a)(ii) | 4 | unstable-spiral phase portrait, vector field and asymptotics | K42 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | determine rotation direction | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 3(a)(ii) | official 4A |
| AU203 | 2026 | 3(a)(iii) | 4 | particular solution of a nonhomogeneous linear system | K43 | linear_system | two-dimensional constant-coefficient system: first write the matrix and find the eigenstructure . | undetermined coefficients | eigenvalue / vector or Jordan sequence → real general solution → fixed-point stability → vectors on the axes → phase portrait / asymptotics → forcing PI. | eigendecomposition and complex eigenvalues , Jordan, phase portrait | linear ODE systems and phase portraits | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 3(a)(iii) | official 4D |
| AU204 | 2026 | 3(b)(i) | 2 | nonexactness check | K47 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | F_y≠G_x | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 3(b)(i) | official 2A |
| AU205 | 2026 | 3(b)(ii) | 5 | x^m y^n integrating factor and implicit solution | K48 | exact_ode | Mdx+Ndy=0: first compare M_y and N_x. | integrate the potential function | check nonexactness → assume an integrating factor and match powers / Solve ODE → integrate the potential function → cross-partial self-check . | exactness condition, integrating factor and potential function | exact equations and integrating factors | in | in | high | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 3(b)(ii) | official 3C+2A |
| AU206 | 2026 | 4(a)(i) | 5 | fixed points, singularities and stability of a one-dimensional system with singularities | K44, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | ry and y/(1+y) compare | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | in | high | 8 minutes | approximately 0.58 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 4(a)(i) | official 2A+3C |
| AU207 | 2026 | 4(a)(ii) | 4 | bifurcation diagram , transcritical and basin of attraction | K46, K45 | bifurcation | one-dimensional autonomous parameter system: factorise to find fixed points and draw the sign line . | r=1 transcritical bifurcation | find fixed points / singularity branch → sign line in each parameter interval → stability → branch diagram → classification / basin of attraction . | phase lines, stability derivatives and bifurcation diagrams | nonlinear ODE and bifurcation | in | high | high | 6 minutes | approximately 0.56 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 4(a)(ii) | official 4D |
| AU208 | 2026 | 4(b) | 5 | Airy of the equation Fourier integral solution | K31 | fourier_transform | function on the whole real line / convolution /ODE: first fix the transform convention and list differentiation, translation and multiplication x rules . | ω multiplication ↔i∂ω | write the transform pair → use linearity / translation / scaling / convolution / duality → solve in the transform domain if needed ODE → inverse transform . | transform table and translation / scaling / duality / convolution , Plancherel | integral transforms (Fourier/Laplace) | in | in | high | 8 minutes | approximately 0.52 pages, estimated from official-solution layout ) | is | B | B | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 4(b) | official 5B |
| AU209 | 2026 | 4(c)-Taylor | 2 | multivariable Taylor second-order general formula | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | quadratic form | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | low | low | low | 3 minutes | approximately 0.22 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 4(c)-Taylor | official 2A |
| AU210 | 2026 | 4(c)-implicit | 4 | first-order derivatives of an implicit function Taylor expansion | K49 | multivariable | implicit function /Jacobian/Hessian: First write F=0 or gradient /Hessian structure . | calculate using the implicit function theorem | write F/ gradient → find first partial derivatives → Hessian/ second-order chain rule → classification or Taylor → Figures / check by substitution . | implicit function theorem , Jacobian, Hessian, multivariable Taylor | multivariable calculus and implicit functions | in | low | high | 6 minutes | approximately 0.38 pages, estimated from official-solution layout ) | is | A | A | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf · 4(c)-implicit | official 4A |
disposition list for old practice questions
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| question-by-question disposition of the old eight practice papers | ||||||
| A Retain , B Revise , C Replace , D retired. The final six papers do not directly inherit old numbering; this table records the rationale for each old main question . | ||||||
| old paper | question number | main content of the old question | Knowledge point ID | Competency family | disposition conclusion | rationale for disposition |
| P1 | Q1 | Taylor/ power series / integrated improper-integral question, essentially replicating 2026 Q1 | K04, K08, K09, K10, K11, K14, K15 | improper, series, taylor | B Revise | retain the current structural-reinforcement function, but allow only paper 1 the paper is close to 2026; rewrite the function family and error-estimate sequence . |
| P1 | Q2 | physical loss model +Fourier series, replicating 2026 Q2 | K21, K22, K23, K24 | fourier_series, model | B Revise | the method sequence is valid, but the context and combination are excessively anchored; change to a spherical-droplet model and ramp function . |
| P1 | Q3 | complex-eigenvalue system + nonhomogeneous system + integrating factor, replicating 2026 Q3 | K40, K41, K42, K43, K47, K48 | exact_ode, linear_system | B Revise | as the sole 2026 retain the structural-reinforcement combination, changing the matrix / forcing /ODE. |
| P1 | Q4 | bifurcation +Fourier find ODE+ implicit function Taylor, replication 2026 Q4 | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | B Revise | only in paper 1 the paper retains the combination; the original multiple Airy repeated variation; recalibrate . |
| P2 | Q1 | binomial series and improper integrals | K09, K10, K11, K14, K15 | improper, series, taylor | C Replace | and several papers Q1 repeated method sequence; change to 2025 series tests / iteration entry point . |
| P2 | Q2 | centroid / geometry +Fourier/Parseval | K19, K20, K22, K25 | centroid_geometry, fourier_series, parseval | B Revise | high historical value, but clarify 2025 bridging with reduced fragmentation . |
| P2 | Q3 | Jordan system +Euler-Cauchy | K33, K34, K37, K40, K41, K42 | jordan, linear_system, scalar_ode | A Retain | direct coverage 2025 high-mark question entry points, included in the revised version after changing numbers and marks . |
| P2 | Q4 | bifurcation +Fourier transform +Hessian | K27, K30, K44, K45, K46, K50 | bifurcation, fourier_transform, hessian_contour, multivariable, parseval | B Revise | three historical families are effectively covered, but reorganise them as 2025 double bifurcation /Jacobian/Hessian. |
| P3 | Q1 | Taylor/log/ improper integral | K08, K09, K11, K14, K15 | improper, taylor | C Replace | and P1/P4/P6/P8 repeated; change to 2023-24 function analysis /Riemann and / curve . |
| P3 | Q2 | centroid +Fourier series | K19, K22, K23, K24 | centroid_geometry, fourier_series | A Retain | coverage 2022/2024 application entry point; rework as 2023-24 geometry /Laplace bridging . |
| P3 | Q3 | physical model +Fourier series | K21, K22, K23, K24 | fourier_series, model | B Revise | and 2026 Q2 same combination; retain the methods but distribute them across distinct historical roles . |
| P3 | Q4 | bifurcation +Fourier ODE+ multivariable Taylor | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | C Replace | typical 2026 anchored combination; change to 2023-24 system / transform /Euler-Cauchy. |
| P4 | Q1 | Taylor/ improper integral | K08, K09, K10, K11, K14, K15 | improper, series, taylor | C Replace | high duplication, for earlier definitions / no additional coverage of the parametric-curve gap . |
| P4 | Q2 | application model +Fourier | K21, K22, K23, K24 | fourier_series, model | C Replace | still centres on 2026 Q2, change to 2020-22 improper integral / centroid . |
| P4 | Q3 | Fourier series +Airy transform | K22, K23, K24, K31 | fourier_series, fourier_transform | C Replace | Airy label-only coverage through shifted variants does not fill earlier Taylor/ sign-function ability . |
| P4 | Q4 | linear system + integrating factor | K40, K42, K43, K47, K48 | exact_ode, linear_system | C Replace | repeated 2026 Q3, change to 2020-22 reduction of order /Green/ zero-eigenvalue line / implicit-function error . |
| P5 | Q1 | definition / parametric curve /Riemann and / Convergence | K01, K02, K06, K14, K15, K16, K18 | definitions, improper, param_curve, riemann | A Retain | varied historical entry points, meaningfully supplementing 2020-22 ability . |
| P5 | Q2 | geometric centroid / solid of revolution +Fourier derivative | K19, K20, K22, K23, K26 | centroid_geometry, fourier_series | A Retain | directly covers earlier geometry and derivative series . |
| P5 | Q3 | Laplace+ nonlinear reduction of order + variation of parameters + systematic | K32, K36, K38, K39 | laplace, scalar_ode | A Retain | less common but valuable abilities for rotation across papers . |
| P5 | Q4 | bifurcation + energy theorem +Hessian | K30, K44, K45, K46, K50 | bifurcation, fourier_transform, multivariable, parseval | A Retain | good cross-year mixing; split and redistribute into the revised version . |
| P6 | Q1 | log power series / iteration / improper integral | K09, K10, K11, K13, K14, K15 | improper, series, taylor | C Replace | and other Q1 repeated; finally moved to the mixed-challenge paper with a changed entry point . |
| P6 | Q2 | physical model +Fourier | K21, K22, K23, K24 | fourier_series, model | C Replace | too many identical combinations across six papers; change to 2022 arc centroid + derivative series . |
| P6 | Q3 | system + integrating factor | K40, K41, K42, K43, K47, K48 | exact_ode, linear_system | C Replace | and P1/P4/P8 overlap; change to centre / resonance and a single-variable integrating factor . |
| P6 | Q4 | bifurcation +Green function + implicit function Taylor | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | C Replace | continue copying 2026 Q4; change to a nonstandard singularity collision / translation Airy/ second-order spherical Taylor. |
| P7 | Q1 | function sketching / integral / parameter-dependent series | K05, K12, K14, K15 | function_analysis, improper, series | A Retain | effectively fills 2026 frequent function analysis not directly tested . |
| P7 | Q2 | 20 marks concentrated on Fourier transform properties | K27, K28, K29, K30 | fourier_transform, parseval | B Revise | valuable knowledge but overly concentrated per question, distorting time; distribute across bridging / historical paper . |
| P7 | Q3 | Heaviside ODE+Euler-Cauchy | K33, K34, K35, K37 | piecewise, scalar_ode, wronskian | A Retain | directly fills scalar ODE frequent gaps . |
| P7 | Q4 | Hessian contour lines | K50 | hessian_contour, multivariable | A Retain | fill 2021-25 continuous high-mark abilities, included in the gap-filling paper . |
| P8 | Q1 | Taylor/ iteration / improper integral | K04, K08, K09, K10, K11, K13, K14, K15 | improper, series, taylor | C Replace | replicate again 2026 Q1. |
| P8 | Q2 | conservation model +Fourier series | K21, K22, K23, K24 | fourier_series, model | C Replace | replicate again 2026 Q2. |
| P8 | Q3 | Jordan system + integrating factor | K40, K41, K42, K43, K47, K48 | exact_ode, jordan, linear_system | C Replace | replicate again 2026 Q3. |
| P8 | Q4 | singularity bifurcation +Green+ implicit function | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | C Replace | replicate again 2026 Q4; the original eight-paper compilation is no longer the final training source . |
Official examination question × Six earlier papers ABCD
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| formal-question ability units × Six earlier papers A/B/C/D Coverage matrix | |||||||||||||||||||||||||||||
| C indicates only identically named knowledge points and does not count as effective coverage. Old paper 7-8 papers audited separately but excluded from the old-six-paper baseline . | |||||||||||||||||||||||||||||
| Competency unit ID | year | question number | Knowledge point | P1Q1 | P1Q2 | P1Q3 | P1Q4 | P2Q1 | P2Q2 | P2Q3 | P2Q4 | P3Q1 | P3Q2 | P3Q3 | P3Q4 | P4Q1 | P4Q2 | P4Q3 | P4Q4 | P5Q1 | P5Q2 | P5Q3 | P5Q4 | P6Q1 | P6Q2 | P6Q3 | P6Q4 | Best | effective ? |
| AU001 | 2020 | 1(a) | state rigorous definitions of continuity and differentiability | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU002 | 2020 | 1(b) | Oscillating function at 0 continuity and differentiability at the point | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU003 | 2020 | 1(c)(i) | parametric curve dy/dx | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU004 | 2020 | 1(c)(ii) | intersections of the parametric curve with the axes | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU005 | 2020 | 1(c)(iii) | horizontal / vertical tangents and graphical solution of a transcendental equation | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU006 | 2020 | 1(c)(iv) | sketch an Archimedean spiral trajectory | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU007 | 2020 | 2(a) | improper integral e^{-x}/x^p the convergence range of | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU008 | 2020 | 2(b) | Fresnel existence of this type of improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU009 | 2020 | 2(c)(i) | sketch the region | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU010 | 2020 | 2(c)(ii) | area of the region | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU011 | 2020 | 2(c)(iii) | centroid of a plane region | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU012 | 2020 | 3(a)(i) | Taylor theorem and remainder | A | D | D | A | D | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU013 | 2020 | 3(a)(ii) | use e^x series approximation √e | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU014 | 2020 | 3(a)(iii) | composite exponential limit | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU015 | 2020 | 3(b)(i) | periodically extend and sketch | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU016 | 2020 | 3(b)(ii) | the sign function's Fourier sine series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU017 | 2020 | 3(b)(iii) | From Fourier deduce from the series Leibniz series | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU018 | 2020 | 4(a) | nonlinear second-order ODE reduce order and satisfy initial conditions | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU019 | 2020 | 4(b) | a shifted exponential function's Fourier transform | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU020 | 2020 | 4(c)(i) | contains delta homogeneous solution of the equation | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU021 | 2020 | 4(c)(ii) | use Fourier use a transform to find delta forcing ODE | D | D | D | A | D | D | D | A | D | D | D | B | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU022 | 2020 | 5(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU023 | 2020 | 5(a)(ii) | saddle phase portrait, vector field and asymptotics for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU024 | 2020 | 5(a)(iii) | particular solution of a nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU025 | 2020 | 5(b)(i) | parameter-dependent nonlinear ODE fixed points and stability | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU026 | 2020 | 5(b)(ii) | bifurcation diagram and classification | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU027 | 2020 | 6(a) | second mixed partial derivatives of an implicit function and symmetry | D | D | D | A | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU028 | 2020 | 6(b)(i) | third-order constant-coefficient ODE homogeneous solution | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU029 | 2020 | 6(b)(ii) | use variation of parameters for a third-order ODE particular solution | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU030 | 2020 | 6(b)(iii) | third-order ODE rewrite as a first-order system | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | B | is |
| AU031 | 2021 | 1(a) | oscillating piecewise-linear function at 0 continuous extension at the point | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU032 | 2021 | 1(b) | sketch the piecewise-linear function and its behaviour near zero | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU033 | 2021 | 1(c) | sketch the derivative and 0 not differentiable at the point | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU034 | 2021 | 1(d) | length of the derivative's graph | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU035 | 2021 | 2(a)(i) | volume of a paraboloid-of-revolution bowl | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU036 | 2021 | 2(a)(ii) | surface area of a surface of revolution | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | A | D | D | D | D | D | D | A | is |
| AU037 | 2021 | 2(b)(i) | surface density under a buoyancy condition | D | A | D | D | D | B | D | D | D | B | A | D | D | B | D | D | D | B | D | D | D | A | D | D | A | is |
| AU038 | 2021 | 2(b)(ii) | maximal surface density without sinking | D | A | D | D | D | B | D | D | D | B | A | D | D | B | D | D | D | B | D | D | D | A | D | D | A | is |
| AU039 | 2021 | 2(c) | physically feasible range for a variable-density fluid parameter | D | B | D | D | D | B | D | D | D | B | B | D | D | B | D | D | D | B | D | D | D | B | D | D | B | is |
| AU040 | 2021 | 3(a)(i) | contains log conditions for existence of a power-type improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU041 | 2021 | 3(a)(ii) | integral recurrence and find I5 | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU042 | 2021 | 3(b) | sec² Riemann limit of the sum | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU043 | 2021 | 3(c)(i) | sin(x²/π) derivative and sketching | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | C | No |
| AU044 | 2021 | 3(c)(ii) | periodic even function Fourier coefficient formula | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU045 | 2021 | 3(c)(iii) | Fourier convergence of the series and its derivative at endpoints | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU046 | 2021 | 4(a)(i) | repeated-root linear ODE homogeneous solution and Wronskian | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU047 | 2021 | 4(a)(ii) | piecewise forcing ODE initial-value problem | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU048 | 2021 | 4(b) | from the transform domain Taylor coefficients to find the third moment | D | D | D | A | D | D | D | A | D | D | D | B | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU049 | 2021 | 4(c) | use the energy theorem to evaluate a rational integral | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | B | is |
| AU050 | 2021 | 5(a)(i) | linear system, line of fixed points and asymptotics for a specified initial value | D | D | A | A | D | D | A | A | D | D | D | A | D | D | D | A | D | D | A | A | D | D | A | A | A | is |
| AU051 | 2021 | 5(a)(ii) | nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU052 | 2021 | 5(b) | identify fixed points and types from a given phase portrait | D | D | B | A | D | D | B | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | B | A | A | is |
| AU053 | 2021 | 5(c) | saddle-node bifurcation in a quartic one-dimensional system | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU054 | 2021 | 6(a) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU055 | 2021 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | A | A | is |
| AU056 | 2021 | 6(b)(ii) | stationary points and Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU057 | 2021 | 6(b)(iii) | consistency of contours, sign regions and classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU058 | 2022 | 1(a)(i) | flat function at 0 extension of the value and first two derivatives at the point | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | C | No |
| AU059 | 2022 | 1(a)(ii) | derivatives of arbitrary order at 0 limit at the point | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | C | No |
| AU060 | 2022 | 1(b)(i) | inflection points and global extrema | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU061 | 2022 | 1(b)(ii) | function sketching | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU062 | 2022 | 1(c)(i) | x=1/t split an improper integral using substitution | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU063 | 2022 | 1(c)(ii) | convergence of two improper integrals | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU064 | 2022 | 2(a) | mass and centroid formulas for a ring with linear density | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | A | D | D | D | D | D | D | A | is |
| AU065 | 2022 | 2(b) | centroid of a uniform ring | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU066 | 2022 | 2(c)(i) | endpoint cases for an added arc | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU067 | 2022 | 2(c)(ii) | centroid formula after adding an arc | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU068 | 2022 | 2(c)(iii) | maximising the centroid and a transcendental equation | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU069 | 2022 | 2(d)(i) | N centroid formula for layered arcs | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU070 | 2022 | 2(d)(ii) | large N asymptotically optimal angle and maximum centroid | B | D | D | D | B | A | D | D | B | A | D | D | B | D | D | D | D | A | D | D | B | D | D | D | A | is |
| AU071 | 2022 | 3(a)(i) | π/2 near cos the two terms of Taylor including remainder | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU072 | 2022 | 3(a)(ii) | use Taylor Lower bound / control a singular function using an upper bound | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | is |
| AU073 | 2022 | 3(a)(iii) | 0 near Taylor upper bound | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU074 | 2022 | 3(a)(iv) | 1/√cos x existence and upper bound of an improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU075 | 2022 | 3(b)(i) | piecewise even function Fourier series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU076 | 2022 | 3(b)(ii) | differentiate a known series to obtain an odd-function series and values at discontinuities | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU077 | 2022 | 4(a)(i) | repeated-root homogeneous ODE and Wronskian | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU078 | 2022 | 4(a)(ii) | Heaviside piecewise forcing ODE | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU079 | 2022 | 4(b)(i) | cosine transform of a compactly supported parabolic function | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU080 | 2022 | 4(b)(ii) | use the energy theorem to obtain an integral identity | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | B | is |
| AU081 | 2022 | 4(b)(iii) | from scaling / use symmetry to find another function's transform | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU082 | 2022 | 5(a)(i) | one-dimensional system with a singularity ODE fixed points and stability | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU083 | 2022 | 5(a)(ii) | bifurcation diagram with singularities and determination of no bifurcation | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU084 | 2022 | 5(a)(iii) | finite-time behaviour of initial values approaching a singularity | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU085 | 2022 | 5(b)(i) | check exactness | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU086 | 2022 | 5(b)(ii) | find an implicit solution using a single-variable integrating factor | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU087 | 2022 | 6(a)(i) | zero-eigenvalue system, line of fixed points and general solution | D | D | B | A | D | D | B | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | B | A | A | is |
| AU088 | 2022 | 6(a)(ii) | phase portrait | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU089 | 2022 | 6(a)(iii) | trajectory and asymptotics for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU090 | 2022 | 6(b)(i) | first partial derivatives of an implicit function and maximum uncertainty | A | D | D | A | D | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU091 | 2022 | 6(b)(ii) | second partial derivatives of an implicit function | D | D | D | A | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU092 | 2023 | 1(a)(i) | first and second derivatives of a composite function | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | C | No |
| AU093 | 2023 | 1(a)(ii) | e^{-x³} extrema and inflection points | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU094 | 2023 | 1(a)(iii) | function sketching | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU095 | 2023 | 1(b)(i) | Taylor theorem and local remainder | A | D | D | A | D | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU096 | 2023 | 1(b)(ii) | x log x in 1 two nonzero terms and a remainder at the point | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU097 | 2023 | 1(b)(iii) | 1.1log1.1 rational approximation and error | A | D | D | A | B | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | B | D | D | A | A | is |
| AU098 | 2023 | 2(a)(i) | compare surface areas formed by rotating a parabola about two axes | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU099 | 2023 | 2(a)(ii) | volumes about both axes and find the critical L | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU100 | 2023 | 2(a)(iii) | geometrically explain reversal of the volume comparison | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU101 | 2023 | 2(b)(i) | t and sinωt of Laplace transform | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | A | D | D | D | D | D | A | is |
| AU102 | 2023 | 2(b)(ii) | inverse transform by partial fractions Laplace | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU103 | 2023 | 3(a) | inductive proof of a finite sum/product identity | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU104 | 2023 | 3(b)(i) | Fourier orthogonal coefficient formula for the integral of a square | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU105 | 2023 | 3(b)(ii) | Derive Parseval theorem | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU106 | 2023 | 3(c)-FS | even triangular function Fourier series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU107 | 2023 | 3(c)-Parseval | From Parseval sum reciprocals of fourth powers of odd integers | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU108 | 2023 | 4(a)(i) | e^{-a|x|} of Fourier transform | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU109 | 2023 | 4(a)(ii) | use convolution to find (4+ω²)^{-2} inverse transform | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU110 | 2023 | 4(b)(i) | Euler–Cauchy homogeneous problem | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU111 | 2023 | 4(b)(ii) | variation of parameters / vary the functions to find a particular solution | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU112 | 2023 | 5(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU113 | 2023 | 5(a)(ii) | stable-node phase portrait and asymptotic direction | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU114 | 2023 | 5(a)(iii) | trajectory for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU115 | 2023 | 5(b)(i) | stability of two fixed points | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU116 | 2023 | 5(b)(ii) | transcritical bifurcation | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU117 | 2023 | 5(b)(iii) | set of initial values causing divergence | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | B | is |
| AU118 | 2023 | 6(a) | find one depending only on x the integrating factor of | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU119 | 2023 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | A | A | is |
| AU120 | 2023 | 6(b)(ii) | with a line of stationary points Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU121 | 2023 | 6(b)(iii) | contours and consistency of classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU122 | 2024 | 1(a) | e^{-1/x}/x^p Limit | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU123 | 2024 | 1(b)(i) | continuous extension | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU124 | 2024 | 1(b)(ii) | piecewise differentiation and limits of derivatives of arbitrary order | C | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | C | No |
| AU125 | 2024 | 1(b)(iii) | inflection points and sketching | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU126 | 2024 | 1(b)(iv) | divergence of an improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU127 | 2024 | 1(c)(i) | rewrite a difference integral by reciprocal substitution | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU128 | 2024 | 1(c)(ii) | integrability after cancellation of the leading term in the difference | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | is |
| AU129 | 2024 | 2(a) | tan of Riemann and | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU130 | 2024 | 2(b)(i) | tan x of Taylor expansion | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU131 | 2024 | 2(b)(ii) | centroid of a nonuniform thin rod | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU132 | 2024 | 2(b)(iii) | use Taylor approximate centroid | A | D | D | A | B | A | D | D | A | A | D | A | A | D | D | D | D | A | D | D | B | D | D | A | A | is |
| AU133 | 2024 | 2(c) | logarithmic-spiral sketch, asymptotics and arc length | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU134 | 2024 | 3(a)(i) | sin(x/2) periodic extension Fourier series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU135 | 2024 | 3(a)(ii) | convergence values at three points | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU136 | 2024 | 3(a)(iii) | use the series to find S | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU137 | 2024 | 3(b)(i) | Parseval theorem | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU138 | 2024 | 3(b)(ii) | simplify the energy functional by integration by parts | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU139 | 2024 | 3(b)(iii) | for f' and f'' application Parseval | D | A | D | D | D | B | D | D | D | A | A | D | D | D | A | D | D | A | D | D | D | D | D | D | A | is |
| AU140 | 2024 | 3(b)(iv) | ensure E>0 of ε range | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU141 | 2024 | 4(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU142 | 2024 | 4(a)(ii) | saddle phase portrait and asymptotics | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU143 | 2024 | 4(b)(i) | cosine transform of the triangular hat function | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU144 | 2024 | 4(b)(ii) | use the energy theorem to evaluate an integral | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | B | is |
| AU145 | 2024 | 4(b)(iii) | use scaling and duality to find g the transform of | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU146 | 2024 | 5(a)(i) | fixed points and stability of a cubic one-dimensional system | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU147 | 2024 | 5(a)(ii) | subcritical pitchfork bifurcation | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU148 | 2024 | 5(b)(i) | third-order Euler–Cauchy change of variables | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU149 | 2024 | 5(b)(ii) | third-order Euler–Cauchy complete general solution | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU150 | 2024 | 6(a)(i) | check nonexactness | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU151 | 2024 | 6(a)(ii) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU152 | 2024 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | A | A | is |
| AU153 | 2024 | 6(b)(ii) | stationary points and Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU154 | 2024 | 6(b)(iii) | contour diagram and consistency argument | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU155 | 2025 | 1(a) | definition of existence of an improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | A | is |
| AU156 | 2025 | 1(b)(i) | logx/(1+x²) the limit and zeros of | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU157 | 2025 | 1(b)(ii) | unique global maximum | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU158 | 2025 | 1(b)(iii) | function sketching | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU159 | 2025 | 1(c)(i) | convergence of an improper integral | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU160 | 2025 | 1(c)(ii) | evaluate an integral by reciprocal substitution | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU161 | 2025 | 2(a) | convergence tests for three types of infinite series | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU162 | 2025 | 2(b) | log power series and rational approximation of an integral | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU163 | 2025 | 2(c)(i) | n repeated sin derivatives, zeros and extrema of a composite function | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU164 | 2025 | 2(c)(ii) | iteration sin the inequality for | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | A | is |
| AU165 | 2025 | 2(c)(iii) | iteration sin Limit | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | A | is |
| AU166 | 2025 | 3(a)(i) | binomial expansion (1-x²)^-1/2 | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU167 | 2025 | 3(a)(ii) | integrate to obtain arcsin series | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU168 | 2025 | 3(a)(iii) | arcsin(0.1) six-decimal approximation | A | D | D | A | B | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | B | D | D | A | A | is |
| AU169 | 2025 | 3(b)(i) | even / odd function Fourier general formula | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU170 | 2025 | 3(b)(ii) | triangular hat function Fourier series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU171 | 2025 | 3(b)(iii) | obtain a new one by translation and reflection Fourier series | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | A | D | D | A | D | D | A | A | is |
| AU172 | 2025 | 4(a)(i) | 1/(1+x²) of Fourier transform | D | D | D | A | D | D | D | A | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU173 | 2025 | 4(a)(ii) | cos of delta transforms and convolution | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU174 | 2025 | 4(a)(iii) | obtain an explicit convolution result by inverse transform | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU175 | 2025 | 4(b)(i) | Second order Euler–Cauchy homogeneous solution and Wronskian | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | A | is |
| AU176 | 2025 | 4(b)(ii) | Euler–Cauchy particular solution | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU177 | 2025 | 5(a)(i) | general solution of a nondiagonalisable linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU178 | 2025 | 5(a)(ii) | degenerate unstable-node phase portrait | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU179 | 2025 | 5(a)(iii) | implicit equation of the trajectory family | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU180 | 2025 | 5(b)(i) | at most three fixed points and their stability | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU181 | 2025 | 5(b)(ii) | pitchfork and transcritical double bifurcation | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU182 | 2025 | 6(a)(i) | complex squaring map Jacobian | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | B | is |
| AU183 | 2025 | 6(a)(ii) | inverse Jacobian partial-derivative identity | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | D | A | D | D | D | D | D | D | B | A | is |
| AU184 | 2025 | 6(b)(i) | circle and line zero contours | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | A | A | is |
| AU185 | 2025 | 6(b)(ii) | four stationary points Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU186 | 2025 | 6(b)(iii) | consistency of contour sign regions and classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU187 | 2026 | 1(a) | Taylor theorem and local remainder | A | D | D | A | D | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU188 | 2026 | 1(b) | log x in 1 power series and convergence interval at the point | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU189 | 2026 | 1(c)(i) | g(x)=logx/(x²-1) limits at three points | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU190 | 2026 | 1(c)(ii)-series | (x+1)g the local power series and radius of | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
| AU191 | 2026 | 1(c)(ii)-approx | log3-log2 the two-digit rational approximation of | A | D | D | A | B | D | D | D | A | D | D | A | A | D | D | D | D | D | D | D | B | D | D | A | A | is |
| AU192 | 2026 | 1(c)(iii) | higher-derivative identity and g in 1 smoothness at the point | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | B | is |
| AU193 | 2026 | 1(c)(iv) | g existence of the improper integral of | A | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | B | D | D | D | A | D | D | D | A | is |
| AU194 | 2026 | 2(a)(i) | contains √ξ definite integral with the denominator | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU195 | 2026 | 2(a)(ii) | cylindrical leaking-water model and k the dimensions of | D | A | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | D | D | D | D | A | D | D | A | is |
| AU196 | 2026 | 2(a)(iii) | find the water height and emptying time | D | B | D | D | D | D | D | D | D | D | B | D | D | B | D | D | D | D | D | D | D | B | D | D | B | is |
| AU197 | 2026 | 2(a)(iv) | k→0 limits, solution without a porous wall and physical comparison | D | B | D | D | D | D | D | D | D | D | B | D | D | B | D | D | D | D | D | D | D | B | D | D | B | is |
| AU198 | 2026 | 2(b)(i) | step function Fourier series | D | A | D | D | D | B | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU199 | 2026 | 2(b)(ii) | convergence values at discontinuities | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU200 | 2026 | 2(b)(iii) | From Fourier obtain from the series π the alternating series of | D | A | D | D | D | D | D | D | D | A | A | D | D | B | A | D | D | A | D | D | D | B | D | D | A | is |
| AU201 | 2026 | 3(a)(i) | general solution of a linear system with complex eigenvalues | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU202 | 2026 | 3(a)(ii) | unstable-spiral phase portrait, vector field and asymptotics | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU203 | 2026 | 3(a)(iii) | particular solution of a nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | A | D | A | is |
| AU204 | 2026 | 3(b)(i) | nonexactness check | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU205 | 2026 | 3(b)(ii) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | D | A | is |
| AU206 | 2026 | 4(a)(i) | fixed points, singularities and stability of a one-dimensional system with singularities | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU207 | 2026 | 4(a)(ii) | bifurcation diagram , transcritical and basin of attraction | D | D | D | A | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | A | A | is |
| AU208 | 2026 | 4(b) | Airy of the equation Fourier integral solution | D | D | D | B | D | D | D | D | D | D | D | B | D | D | B | D | D | D | D | D | D | D | D | B | B | is |
| AU209 | 2026 | 4(c)-Taylor | multivariable Taylor second-order general formula | D | D | D | A | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU210 | 2026 | 4(c)-implicit | first-order derivatives of an implicit function Taylor expansion | A | A | D | A | A | D | D | D | A | A | A | A | A | D | A | D | D | D | D | D | A | D | D | A | A | is |
Official examination question × six revised papers ABCD
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| formal-question ability units × six revised papers A/B/C/D Coverage matrix | |||||||||||||||||||||||||||||
| final effective coverage =207/210; frequent and high-mark abilities =187/187. each column corresponds to a main question in a revised paper . | |||||||||||||||||||||||||||||
| Competency unit ID | year | question number | Knowledge point | P1Q1 | P1Q2 | P1Q3 | P1Q4 | P2Q1 | P2Q2 | P2Q3 | P2Q4 | P3Q1 | P3Q2 | P3Q3 | P3Q4 | P4Q1 | P4Q2 | P4Q3 | P4Q4 | P5Q1 | P5Q2 | P5Q3 | P5Q4 | P6Q1 | P6Q2 | P6Q3 | P6Q4 | Best | effective ? |
| AU001 | 2020 | 1(a) | state rigorous definitions of continuity and differentiability | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU002 | 2020 | 1(b) | Oscillating function at 0 continuity and differentiability at the point | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | A | D | D | C | D | D | D | A | is |
| AU003 | 2020 | 1(c)(i) | parametric curve dy/dx | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU004 | 2020 | 1(c)(ii) | intersections of the parametric curve with the axes | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU005 | 2020 | 1(c)(iii) | horizontal / vertical tangents and graphical solution of a transcendental equation | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU006 | 2020 | 1(c)(iv) | sketch an Archimedean spiral trajectory | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU007 | 2020 | 2(a) | improper integral e^{-x}/x^p the convergence range of | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU008 | 2020 | 2(b) | Fresnel existence of this type of improper integral | A | C | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU009 | 2020 | 2(c)(i) | sketch the region | D | D | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | D | D | D | B | is |
| AU010 | 2020 | 2(c)(ii) | area of the region | D | D | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | D | D | D | B | is |
| AU011 | 2020 | 2(c)(iii) | centroid of a plane region | D | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU012 | 2020 | 3(a)(i) | Taylor theorem and remainder | A | D | D | A | D | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU013 | 2020 | 3(a)(ii) | use e^x series approximation √e | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU014 | 2020 | 3(a)(iii) | composite exponential limit | B | D | D | D | B | D | D | D | B | B | D | D | B | D | B | D | B | D | D | D | B | B | D | D | B | is |
| AU015 | 2020 | 3(b)(i) | periodically extend and sketch | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU016 | 2020 | 3(b)(ii) | the sign function's Fourier sine series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU017 | 2020 | 3(b)(iii) | From Fourier deduce from the series Leibniz series | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU018 | 2020 | 4(a) | nonlinear second-order ODE reduce order and satisfy initial conditions | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | is |
| AU019 | 2020 | 4(b) | a shifted exponential function's Fourier transform | D | D | D | A | D | B | D | D | D | D | D | B | D | D | D | A | D | A | D | D | D | D | D | A | A | is |
| AU020 | 2020 | 4(c)(i) | contains delta homogeneous solution of the equation | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | A | D | A | D | D | D | D | D | D | A | is |
| AU021 | 2020 | 4(c)(ii) | use Fourier use a transform to find delta forcing ODE | D | D | D | A | D | B | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | A | is |
| AU022 | 2020 | 5(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU023 | 2020 | 5(a)(ii) | saddle phase portrait, vector field and asymptotics for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU024 | 2020 | 5(a)(iii) | particular solution of a nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | A | is |
| AU025 | 2020 | 5(b)(i) | parameter-dependent nonlinear ODE fixed points and stability | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU026 | 2020 | 5(b)(ii) | bifurcation diagram and classification | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU027 | 2020 | 6(a) | second mixed partial derivatives of an implicit function and symmetry | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | C | A | D | D | D | D | B | A | is |
| AU028 | 2020 | 6(b)(i) | third-order constant-coefficient ODE homogeneous solution | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | A | D | A | D | D | D | D | D | D | A | is |
| AU029 | 2020 | 6(b)(ii) | use variation of parameters for a third-order ODE particular solution | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU030 | 2020 | 6(b)(iii) | third-order ODE rewrite as a first-order system | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU031 | 2021 | 1(a) | oscillating piecewise-linear function at 0 continuous extension at the point | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU032 | 2021 | 1(b) | sketch the piecewise-linear function and its behaviour near zero | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU033 | 2021 | 1(c) | sketch the derivative and 0 not differentiable at the point | D | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU034 | 2021 | 1(d) | length of the derivative's graph | D | C | D | D | D | D | D | D | B | D | D | D | B | B | D | D | B | D | D | D | D | D | D | D | B | is |
| AU035 | 2021 | 2(a)(i) | volume of a paraboloid-of-revolution bowl | D | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | is |
| AU036 | 2021 | 2(a)(ii) | surface area of a surface of revolution | D | D | D | D | D | D | D | D | A | A | D | D | A | B | D | D | D | D | D | D | D | D | D | D | A | is |
| AU037 | 2021 | 2(b)(i) | surface density under a buoyancy condition | D | A | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | A | is |
| AU038 | 2021 | 2(b)(ii) | maximal surface density without sinking | D | A | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | A | is |
| AU039 | 2021 | 2(c) | physically feasible range for a variable-density fluid parameter | D | B | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | D | D | B | is |
| AU040 | 2021 | 3(a)(i) | contains log conditions for existence of a power-type improper integral | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU041 | 2021 | 3(a)(ii) | integral recurrence and find I5 | A | A | D | A | A | D | D | D | A | D | A | D | D | B | A | D | A | D | D | D | A | A | D | A | A | is |
| AU042 | 2021 | 3(b) | sec² Riemann limit of the sum | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | A | is |
| AU043 | 2021 | 3(c)(i) | sin(x²/π) derivative and sketching | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | C | D | D | D | B | is |
| AU044 | 2021 | 3(c)(ii) | periodic even function Fourier coefficient formula | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU045 | 2021 | 3(c)(iii) | Fourier convergence of the series and its derivative at endpoints | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU046 | 2021 | 4(a)(i) | repeated-root linear ODE homogeneous solution and Wronskian | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | A | D | A | D | D | D | D | D | D | A | is |
| AU047 | 2021 | 4(a)(ii) | piecewise forcing ODE initial-value problem | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU048 | 2021 | 4(b) | from the transform domain Taylor coefficients to find the third moment | D | D | D | A | D | B | D | B | D | D | D | B | D | D | D | A | D | B | B | D | D | D | D | A | A | is |
| AU049 | 2021 | 4(c) | use the energy theorem to evaluate a rational integral | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU050 | 2021 | 5(a)(i) | linear system, line of fixed points and asymptotics for a specified initial value | D | D | A | D | D | D | A | A | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | A | A | is |
| AU051 | 2021 | 5(a)(ii) | nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | A | is |
| AU052 | 2021 | 5(b) | identify fixed points and types from a given phase portrait | D | D | B | B | D | D | B | A | D | D | D | B | D | D | D | B | D | D | A | D | D | D | B | A | A | is |
| AU053 | 2021 | 5(c) | saddle-node bifurcation in a quartic one-dimensional system | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU054 | 2021 | 6(a) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU055 | 2021 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | D | A | B | D | D | D | D | A | is |
| AU056 | 2021 | 6(b)(ii) | stationary points and Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU057 | 2021 | 6(b)(iii) | consistency of contours, sign regions and classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU058 | 2022 | 1(a)(i) | flat function at 0 extension of the value and first two derivatives at the point | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | C | D | D | D | B | is |
| AU059 | 2022 | 1(a)(ii) | derivatives of arbitrary order at 0 limit at the point | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | C | D | D | D | B | is |
| AU060 | 2022 | 1(b)(i) | inflection points and global extrema | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU061 | 2022 | 1(b)(ii) | function sketching | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU062 | 2022 | 1(c)(i) | x=1/t split an improper integral using substitution | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU063 | 2022 | 1(c)(ii) | convergence of two improper integrals | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU064 | 2022 | 2(a) | mass and centroid formulas for a ring with linear density | D | D | D | D | D | D | D | D | A | A | D | D | A | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU065 | 2022 | 2(b) | centroid of a uniform ring | D | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU066 | 2022 | 2(c)(i) | endpoint cases for an added arc | D | D | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | A | D | D | A | is |
| AU067 | 2022 | 2(c)(ii) | centroid formula after adding an arc | D | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU068 | 2022 | 2(c)(iii) | maximising the centroid and a transcendental equation | D | D | D | D | D | D | D | D | B | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU069 | 2022 | 2(d)(i) | N centroid formula for layered arcs | D | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU070 | 2022 | 2(d)(ii) | large N asymptotically optimal angle and maximum centroid | B | D | D | D | B | D | D | D | B | A | D | D | D | A | B | D | D | D | D | D | B | A | D | D | A | is |
| AU071 | 2022 | 3(a)(i) | π/2 near cos the two terms of Taylor including remainder | B | D | D | D | B | D | D | D | B | B | D | D | D | D | B | D | D | D | D | D | B | B | D | D | B | is |
| AU072 | 2022 | 3(a)(ii) | use Taylor Lower bound / control a singular function using an upper bound | B | D | D | D | B | D | D | D | B | B | D | D | D | B | B | D | B | D | D | D | B | B | D | D | B | is |
| AU073 | 2022 | 3(a)(iii) | 0 near Taylor upper bound | B | D | D | D | B | D | D | D | B | B | D | D | D | D | B | D | D | D | D | D | B | B | D | D | B | is |
| AU074 | 2022 | 3(a)(iv) | 1/√cos x existence and upper bound of an improper integral | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU075 | 2022 | 3(b)(i) | piecewise even function Fourier series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | A | D | D | D | A | D | D | A | is |
| AU076 | 2022 | 3(b)(ii) | differentiate a known series to obtain an odd-function series and values at discontinuities | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU077 | 2022 | 4(a)(i) | repeated-root homogeneous ODE and Wronskian | D | D | D | D | D | D | A | D | D | D | D | B | D | D | D | D | D | B | D | D | D | D | D | D | A | is |
| AU078 | 2022 | 4(a)(ii) | Heaviside piecewise forcing ODE | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | A | is |
| AU079 | 2022 | 4(b)(i) | cosine transform of a compactly supported parabolic function | D | D | D | A | D | B | D | D | D | D | D | B | D | D | D | A | D | B | D | D | D | D | D | A | A | is |
| AU080 | 2022 | 4(b)(ii) | use the energy theorem to obtain an integral identity | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU081 | 2022 | 4(b)(iii) | from scaling / use symmetry to find another function's transform | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | B | is |
| AU082 | 2022 | 5(a)(i) | one-dimensional system with a singularity ODE fixed points and stability | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU083 | 2022 | 5(a)(ii) | bifurcation diagram with singularities and determination of no bifurcation | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU084 | 2022 | 5(a)(iii) | finite-time behaviour of initial values approaching a singularity | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU085 | 2022 | 5(b)(i) | check exactness | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU086 | 2022 | 5(b)(ii) | find an implicit solution using a single-variable integrating factor | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU087 | 2022 | 6(a)(i) | zero-eigenvalue system, line of fixed points and general solution | D | D | B | D | D | D | B | A | D | D | D | B | D | D | D | B | D | D | B | D | D | D | B | A | A | is |
| AU088 | 2022 | 6(a)(ii) | phase portrait | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU089 | 2022 | 6(a)(iii) | trajectory and asymptotics for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU090 | 2022 | 6(b)(i) | first partial derivatives of an implicit function and maximum uncertainty | A | D | D | A | D | D | D | B | A | D | D | D | D | D | A | A | D | C | A | D | D | D | D | A | A | is |
| AU091 | 2022 | 6(b)(ii) | second partial derivatives of an implicit function | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | C | A | D | D | D | D | B | A | is |
| AU092 | 2023 | 1(a)(i) | first and second derivatives of a composite function | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | C | D | D | D | B | is |
| AU093 | 2023 | 1(a)(ii) | e^{-x³} extrema and inflection points | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU094 | 2023 | 1(a)(iii) | function sketching | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU095 | 2023 | 1(b)(i) | Taylor theorem and local remainder | A | D | D | A | D | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU096 | 2023 | 1(b)(ii) | x log x in 1 two nonzero terms and a remainder at the point | B | D | D | D | B | D | D | D | B | B | D | D | D | D | B | D | D | D | D | D | B | B | D | D | B | is |
| AU097 | 2023 | 1(b)(iii) | 1.1log1.1 rational approximation and error | A | D | D | A | B | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | B | B | D | A | A | is |
| AU098 | 2023 | 2(a)(i) | compare surface areas formed by rotating a parabola about two axes | D | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | is |
| AU099 | 2023 | 2(a)(ii) | volumes about both axes and find the critical L | D | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | is |
| AU100 | 2023 | 2(a)(iii) | geometrically explain reversal of the volume comparison | D | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | A | is |
| AU101 | 2023 | 2(b)(i) | t and sinωt of Laplace transform | D | D | D | D | D | D | D | D | D | A | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU102 | 2023 | 2(b)(ii) | inverse transform by partial fractions Laplace | D | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU103 | 2023 | 3(a) | inductive proof of a finite sum/product identity | D | A | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | D | A | is |
| AU104 | 2023 | 3(b)(i) | Fourier orthogonal coefficient formula for the integral of a square | D | A | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | D | A | is |
| AU105 | 2023 | 3(b)(ii) | Derive Parseval theorem | D | A | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | D | A | is |
| AU106 | 2023 | 3(c)-FS | even triangular function Fourier series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU107 | 2023 | 3(c)-Parseval | From Parseval sum reciprocals of fourth powers of odd integers | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU108 | 2023 | 4(a)(i) | e^{-a|x|} of Fourier transform | D | D | D | A | D | B | D | D | D | D | D | B | D | D | D | A | D | B | D | D | D | D | D | A | A | is |
| AU109 | 2023 | 4(a)(ii) | use convolution to find (4+ω²)^{-2} inverse transform | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | B | D | A | D | D | D | D | D | D | A | is |
| AU110 | 2023 | 4(b)(i) | Euler–Cauchy homogeneous problem | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | D | D | B | D | D | D | D | D | D | A | is |
| AU111 | 2023 | 4(b)(ii) | variation of parameters / vary the functions to find a particular solution | D | D | D | D | D | D | B | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU112 | 2023 | 5(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU113 | 2023 | 5(a)(ii) | stable-node phase portrait and asymptotic direction | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU114 | 2023 | 5(a)(iii) | trajectory for a specified initial value | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU115 | 2023 | 5(b)(i) | stability of two fixed points | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU116 | 2023 | 5(b)(ii) | transcritical bifurcation | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU117 | 2023 | 5(b)(iii) | set of initial values causing divergence | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | B | B | is |
| AU118 | 2023 | 6(a) | find one depending only on x the integrating factor of | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU119 | 2023 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | D | A | B | D | D | D | D | A | is |
| AU120 | 2023 | 6(b)(ii) | with a line of stationary points Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU121 | 2023 | 6(b)(iii) | contours and consistency of classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU122 | 2024 | 1(a) | e^{-1/x}/x^p Limit | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU123 | 2024 | 1(b)(i) | continuous extension | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU124 | 2024 | 1(b)(ii) | piecewise differentiation and limits of derivatives of arbitrary order | C | D | D | D | C | D | D | D | B | D | D | D | B | D | D | D | B | A | D | D | C | D | D | D | A | is |
| AU125 | 2024 | 1(b)(iii) | inflection points and sketching | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU126 | 2024 | 1(b)(iv) | divergence of an improper integral | A | D | D | D | B | D | D | D | B | B | D | D | D | A | B | D | B | D | D | D | A | B | D | D | A | is |
| AU127 | 2024 | 1(c)(i) | rewrite a difference integral by reciprocal substitution | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU128 | 2024 | 1(c)(ii) | integrability after cancellation of the leading term in the difference | B | D | D | D | B | D | D | D | B | B | D | D | D | B | B | D | B | D | D | D | B | B | D | D | B | is |
| AU129 | 2024 | 2(a) | tan of Riemann and | D | C | D | D | D | D | D | D | A | D | D | D | D | B | D | D | A | D | D | D | D | D | D | D | A | is |
| AU130 | 2024 | 2(b)(i) | tan x of Taylor expansion | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU131 | 2024 | 2(b)(ii) | centroid of a nonuniform thin rod | D | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | A | D | D | A | is |
| AU132 | 2024 | 2(b)(iii) | use Taylor approximate centroid | A | D | D | A | B | D | D | D | A | A | D | D | D | A | A | D | D | D | D | D | B | A | D | A | A | is |
| AU133 | 2024 | 2(c) | logarithmic-spiral sketch, asymptotics and arc length | D | D | D | D | D | D | D | D | A | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU134 | 2024 | 3(a)(i) | sin(x/2) periodic extension Fourier series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU135 | 2024 | 3(a)(ii) | convergence values at three points | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU136 | 2024 | 3(a)(iii) | use the series to find S | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU137 | 2024 | 3(b)(i) | Parseval theorem | D | A | D | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | D | A | is |
| AU138 | 2024 | 3(b)(ii) | simplify the energy functional by integration by parts | D | A | D | D | D | D | D | D | D | D | A | D | D | B | D | D | B | D | D | D | D | A | D | D | A | is |
| AU139 | 2024 | 3(b)(iii) | for f' and f'' application Parseval | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU140 | 2024 | 3(b)(iv) | ensure E>0 of ε range | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU141 | 2024 | 4(a)(i) | general solution of a two-dimensional linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU142 | 2024 | 4(a)(ii) | saddle phase portrait and asymptotics | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU143 | 2024 | 4(b)(i) | cosine transform of the triangular hat function | D | D | D | A | D | B | D | D | D | D | D | B | D | D | D | A | D | B | D | D | D | D | D | A | A | is |
| AU144 | 2024 | 4(b)(ii) | use the energy theorem to evaluate an integral | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | B | D | D | D | D | D | D | B | is |
| AU145 | 2024 | 4(b)(iii) | use scaling and duality to find g the transform of | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | B | is |
| AU146 | 2024 | 5(a)(i) | fixed points and stability of a cubic one-dimensional system | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU147 | 2024 | 5(a)(ii) | subcritical pitchfork bifurcation | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU148 | 2024 | 5(b)(i) | third-order Euler–Cauchy change of variables | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU149 | 2024 | 5(b)(ii) | third-order Euler–Cauchy complete general solution | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | D | D | B | D | D | D | D | D | D | A | is |
| AU150 | 2024 | 6(a)(i) | check nonexactness | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU151 | 2024 | 6(a)(ii) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU152 | 2024 | 6(b)(i) | zero contour | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | D | A | B | D | D | D | D | A | is |
| AU153 | 2024 | 6(b)(ii) | stationary points and Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU154 | 2024 | 6(b)(iii) | contour diagram and consistency argument | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU155 | 2025 | 1(a) | definition of existence of an improper integral | A | D | D | D | D | D | D | D | D | D | D | D | A | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU156 | 2025 | 1(b)(i) | logx/(1+x²) the limit and zeros of | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU157 | 2025 | 1(b)(ii) | unique global maximum | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU158 | 2025 | 1(b)(iii) | function sketching | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | is |
| AU159 | 2025 | 1(c)(i) | convergence of an improper integral | A | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU160 | 2025 | 1(c)(ii) | evaluate an integral by reciprocal substitution | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | No |
| AU161 | 2025 | 2(a) | convergence tests for three types of infinite series | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | A | is |
| AU162 | 2025 | 2(b) | log power series and rational approximation of an integral | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU163 | 2025 | 2(c)(i) | n repeated sin derivatives, zeros and extrema of a composite function | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | D | D | D | B | is |
| AU164 | 2025 | 2(c)(ii) | iteration sin the inequality for | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | A | is |
| AU165 | 2025 | 2(c)(iii) | iteration sin Limit | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | A | is |
| AU166 | 2025 | 3(a)(i) | binomial expansion (1-x²)^-1/2 | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU167 | 2025 | 3(a)(ii) | integrate to obtain arcsin series | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU168 | 2025 | 3(a)(iii) | arcsin(0.1) six-decimal approximation | A | D | D | A | B | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | B | B | D | A | A | is |
| AU169 | 2025 | 3(b)(i) | even / odd function Fourier general formula | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU170 | 2025 | 3(b)(ii) | triangular hat function Fourier series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU171 | 2025 | 3(b)(iii) | obtain a new one by translation and reflection Fourier series | A | A | D | A | A | B | D | D | A | D | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU172 | 2025 | 4(a)(i) | 1/(1+x²) of Fourier transform | D | D | D | A | D | B | D | D | D | D | D | B | D | D | D | A | D | B | D | D | D | D | D | A | A | is |
| AU173 | 2025 | 4(a)(ii) | cos of delta transforms and convolution | D | D | D | D | D | B | D | D | D | D | D | B | D | D | D | B | D | B | D | D | D | D | D | D | B | is |
| AU174 | 2025 | 4(a)(iii) | obtain an explicit convolution result by inverse transform | D | D | D | D | D | B | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | D | B | is |
| AU175 | 2025 | 4(b)(i) | Second order Euler–Cauchy homogeneous solution and Wronskian | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | A | D | A | D | D | D | D | D | D | A | is |
| AU176 | 2025 | 4(b)(ii) | Euler–Cauchy particular solution | D | D | D | D | D | D | A | D | D | D | D | A | D | D | D | D | D | B | D | D | D | D | D | D | A | is |
| AU177 | 2025 | 5(a)(i) | general solution of a nondiagonalisable linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU178 | 2025 | 5(a)(ii) | degenerate unstable-node phase portrait | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU179 | 2025 | 5(a)(iii) | implicit equation of the trajectory family | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU180 | 2025 | 5(b)(i) | at most three fixed points and their stability | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU181 | 2025 | 5(b)(ii) | pitchfork and transcritical double bifurcation | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | A | is |
| AU182 | 2025 | 6(a)(i) | complex squaring map Jacobian | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | B | D | C | B | D | D | D | D | B | A | is |
| AU183 | 2025 | 6(a)(ii) | inverse Jacobian partial-derivative identity | D | D | D | B | D | D | D | A | A | D | D | D | D | D | D | B | A | C | B | D | D | D | D | B | A | is |
| AU184 | 2025 | 6(b)(i) | circle and line zero contours | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | D | A | B | D | D | D | D | A | is |
| AU185 | 2025 | 6(b)(ii) | four stationary points Hessian classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU186 | 2025 | 6(b)(iii) | consistency of contour sign regions and classification | D | D | D | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | D | A | D | D | D | D | A | is |
| AU187 | 2026 | 1(a) | Taylor theorem and local remainder | A | D | D | A | D | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | D | D | D | A | A | is |
| AU188 | 2026 | 1(b) | log x in 1 power series and convergence interval at the point | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU189 | 2026 | 1(c)(i) | g(x)=logx/(x²-1) limits at three points | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU190 | 2026 | 1(c)(ii)-series | (x+1)g the local power series and radius of | A | A | D | A | A | D | D | D | A | B | A | D | D | D | A | D | A | D | D | D | A | A | D | A | A | is |
| AU191 | 2026 | 1(c)(ii)-approx | log3-log2 the two-digit rational approximation of | A | D | D | A | B | D | D | D | A | D | D | D | D | D | A | D | D | D | D | D | B | B | D | A | A | is |
| AU192 | 2026 | 1(c)(iii) | higher-derivative identity and g in 1 smoothness at the point | B | D | D | D | B | D | D | D | B | B | D | D | B | D | B | D | B | D | D | D | B | B | D | D | B | is |
| AU193 | 2026 | 1(c)(iv) | g existence of the improper integral of | A | C | D | D | D | D | D | D | D | D | D | D | D | A | D | D | B | D | D | D | A | D | D | D | A | is |
| AU194 | 2026 | 2(a)(i) | contains √ξ definite integral with the denominator | D | C | D | D | D | D | D | D | D | D | D | D | D | B | D | D | B | D | D | D | D | D | D | D | B | is |
| AU195 | 2026 | 2(a)(ii) | cylindrical leaking-water model and k the dimensions of | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | is |
| AU196 | 2026 | 2(a)(iii) | find the water height and emptying time | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | B | D | D | D | D | D | D | D | B | is |
| AU197 | 2026 | 2(a)(iv) | k→0 limits, solution without a porous wall and physical comparison | D | B | D | D | D | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | is |
| AU198 | 2026 | 2(b)(i) | step function Fourier series | D | A | D | D | D | B | D | D | D | D | A | D | D | D | B | D | D | A | D | D | D | A | D | D | A | is |
| AU199 | 2026 | 2(b)(ii) | convergence values at discontinuities | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU200 | 2026 | 2(b)(iii) | From Fourier obtain from the series π the alternating series of | D | A | D | D | D | D | D | D | D | D | A | D | D | D | B | D | D | D | D | D | D | A | D | D | A | is |
| AU201 | 2026 | 3(a)(i) | general solution of a linear system with complex eigenvalues | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU202 | 2026 | 3(a)(ii) | unstable-spiral phase portrait, vector field and asymptotics | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | B | D | D | B | D | D | D | A | D | A | is |
| AU203 | 2026 | 3(a)(iii) | particular solution of a nonhomogeneous linear system | D | D | A | D | D | D | A | D | D | D | D | A | D | D | D | D | D | D | D | D | D | D | A | D | A | is |
| AU204 | 2026 | 3(b)(i) | nonexactness check | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU205 | 2026 | 3(b)(ii) | x^m y^n integrating factor and implicit solution | D | D | A | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | A | D | D | A | D | A | is |
| AU206 | 2026 | 4(a)(i) | fixed points, singularities and stability of a one-dimensional system with singularities | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU207 | 2026 | 4(a)(ii) | bifurcation diagram , transcritical and basin of attraction | D | D | D | B | D | D | D | A | D | D | D | D | D | D | D | D | D | D | C | D | D | D | D | A | A | is |
| AU208 | 2026 | 4(b) | Airy of the equation Fourier integral solution | D | D | D | B | D | D | D | D | D | D | D | D | D | D | D | B | D | D | D | D | D | D | D | B | B | is |
| AU209 | 2026 | 4(c)-Taylor | multivariable Taylor second-order general formula | D | D | D | A | D | D | D | B | D | D | D | D | D | D | D | A | D | C | A | D | D | D | D | B | A | is |
| AU210 | 2026 | 4(c)-implicit | first-order derivatives of an implicit function Taylor expansion | A | A | D | A | A | D | D | B | A | D | A | D | D | D | A | A | A | C | A | D | A | A | D | A | A | is |
design mapping for six revised papers
Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.
| roles and historical bases of the six revised papers | ||||||
| only P1 serves 2026 structural reinforcement ;P2-P6 all have a combination distinct from 2026 the main ability combination of . | ||||||
| revised paper | question number | role | main content | Knowledge point ID | Competency family | Historical basis |
| P1 | Q1 | 2026 structural reinforcement | 2026 structural reinforcement :arctan Taylor, error, higher derivatives and improper integrals | K04, K08, K09, K10, K11, K14, K15 | improper, series, taylor | 2020/2022/2023/2025/2026 Taylor and approximation ;2026 Difficulty unit Q1 |
| P1 | Q2 | 2026 structural reinforcement | spherical-droplet conservation model + ramp Fourier series | K17, K21, K22, K23, K24 | fourier_series, model | 2021/2024 physical modelling ;2020/2023/2026 Piecewise Fourier;2026 Difficulty unit Q2 |
| P1 | Q3 | 2026 structural reinforcement | complex-eigenvalue system / nonhomogeneous system + monomial integrating factor | K40, K41, K42, K43, K47, K48 | exact_ode, linear_system, phase_portrait | 2020-2026 system / phase portrait ;2021-2024/2026 integrating factor ;2026 Difficulty unit Q3 |
| P1 | Q4 | 2026 structural reinforcement | pitchfork+Fourier use a transform to find ODE+ implicit function Taylor | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | 2020-2026 bifurcation ;2020/2026 use a transform to find ODE;2020/2022/2025/2026 implicit function |
| P2 | Q1 | 2025 historical bridging | 2025 bridging: series tests , arcsin, iteration | K04, K09, K10, K11, K12, K13 | series, taylor | 2025 Q2 series and iteration ;2025 Q3 power-series approximation |
| P2 | Q2 | 2025 historical bridging | 2025 bridging :Fourier translation and reflection / duality / convolution | K22, K26, K27, K28, K29 | fourier_series, fourier_transform | 2025 Q3 Fourier translation ;2025 Q4 transform / convolution |
| P2 | Q3 | 2025 historical bridging | Euler-Cauchy/Wronskian+Jordan system | K33, K34, K35, K37, K40, K41, K42 | jordan, linear_system, scalar_ode, wronskian | 2025 Q4 Euler-Cauchy;2025 Q5 Jordan system |
| P2 | Q4 | 2025 historical bridging | double bifurcation +Jacobian/Hessian contour lines | K44, K45, K46, K49, K50 | bifurcation, hessian_contour, multivariable | 2025 Q5 double bifurcation ;2025 Q6 Jacobian/Hessian |
| P3 | Q1 | 2023-2024 Historical competencies | 2023-24: function analysis /Taylor/Riemann and / spiral | K04, K05, K07, K08, K09, K11, K18 | function_analysis, param_curve, riemann, taylor | 2023 Q1;2024 Q2 Riemann and / spiral |
| P3 | Q2 | 2023-2024 Historical competencies | solid of revolution / nonuniform centroid /Laplace | K09, K19, K20, K32 | centroid_geometry, laplace, taylor | 2023 Q2 geometry /Laplace;2024 Q2 centroid |
| P3 | Q3 | 2023-2024 Historical competencies | Fourier series /Parseval energy functional | K22, K23, K24, K25, K26 | fourier_series, parseval | 2023 Q3 Parseval;2024 Q3 energy functional |
| P3 | Q4 | 2023-2024 Historical competencies | saddle system /Fourier energy /Euler-Cauchy variation of parameters | K27, K30, K33, K36, K37, K40, K42 | fourier_transform, linear_system, parseval, scalar_ode | 2023 Q4 Euler-Cauchy/ convolution ;2024 Q4 system / energy |
| P4 | Q1 | 2020-2022 Historical competencies | 2020-22: definition / differentiability at a special point / parametric spiral | K01, K02, K04, K06, K07 | definitions, function_analysis, param_curve | 2020 Q1 definitions and parametric curves ;2021 Q1 continuity and differentiability |
| P4 | Q2 | 2020-2022 Historical competencies | parameter-dependent improper integral /Fresnel tail / centroid of a region | K14, K15, K17, K19, K20 | centroid_geometry, improper | 2020 Q2 improper integral / centroid ;2022 Q1 improper integral |
| P4 | Q3 | 2020-2022 Historical competencies | Taylor approximation / Symbol Fourier/ derivative series | K08, K09, K11, K22, K23, K24, K26 | fourier_series, taylor | 2020 Q3 Taylor/ Symbol Fourier;2022 Q3 derivative series |
| P4 | Q4 | 2020-2022 Historical competencies | nonlinear reduction of order /shifted Green/ zero-eigenvalue line / implicit-function error | K27, K28, K29, K31, K38, K40, K42, K49 | fourier_transform, linear_system, multivariable, scalar_ode | 2020 Q4/Q5/Q6;2022 Q6 |
| P5 | Q1 | check gaps across seven years | gap: flat function / integral recurrence /Riemann and / series | K03, K04, K12, K14, K15, K17, K18 | function_analysis, improper, riemann, series | 2021 Q3, 2022 Q1, 2024 Q1, 2025 Q2 |
| P5 | Q2 | check gaps across seven years | gap :Heaviside Piecewise ODE/Fourier Moment / energy | K27, K30, K33, K34, K35, K49 | fourier_transform, parseval, piecewise, scalar_ode, wronskian | 2021/2022 Piecewise ODE;2021 Fourier Moment / energy |
| P5 | Q3 | check gaps across seven years | gap: line of fixed points / read a phase portrait / second-order implicit derivatives | K40, K42, K45, K49 | linear_system, multivariable, phase_portrait | 2021/2022 zero-eigenvalue system ;2021 given phase portrait ;2020/2022 implicit function |
| P5 | Q4 | check gaps across seven years | gap: integrating factor /Hessian contour lines | K47, K48, K50 | exact_ode, hessian_contour, multivariable | 2021 Q6, 2023 Q6, 2024 Q6, 2025 Q6 |
| P6 | Q1 | seven-year mixed-transfer challenge | Mixture :log series / error / iteration / improper integral | K04, K09, K10, K11, K13, K14, K15 | improper, series, taylor | 2025 series / iteration + 2020/2023 approximation + improper integrals across years |
| P6 | Q2 | seven-year mixed-transfer challenge | mixed: asymptotics of arc centroids /Fourier derivative series | K09, K11, K19, K22, K23, K26 | centroid_geometry, fourier_series, taylor | 2022 Q2 arc centroid + 2022 Fourier derivative series |
| P6 | Q3 | seven-year mixed-transfer challenge | mixed: centre system / resonant forcing / single-variable integrating factor | K40, K42, K43, K47, K48 | exact_ode, linear_system, phase_portrait | 2024/2025 system phase portrait + 2020/2021 nonhomogeneous system + 2022 single-variable integrating factor |
| P6 | Q4 | seven-year mixed-transfer challenge | mixed: nonstandard singularity collision / translation Airy/ second-order implicit Taylor | K31, K44, K45, K46, K49 | bifurcation, fourier_transform, multivariable | 2022 singular dynamical system + 2024/2026 Airy + 2025/2026 multivariable Taylor |
Difficulty calibration
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| six papers and 2026 overall difficulty calibration | |||||
| A/B/C/D/M is a training-calibration label .2026 the official marking review shows A24, B15, C9, D12, also Mastery 20, total 80. | |||||
| examination paper | question count | total marks | minutes | difficulty budget | calibration notes |
| 2026 official paper | 4 | 80 | 120 | A24/B15/C9/D12/M20 | benchmark |
| revised version P1 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | 2026 structural reinforcement; calibrate steps and integration |
| revised version P2 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | 2025 bridging ;Jordan/Euler-Cauchy calibration of calculation volume |
| revised version P3 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | 2023-24 historical combination; proof / calibration of transform workload |
| revised version P4 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | 2020-22 historical combination; definition / curve / balance improper integrals |
| revised version P5 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | gap-filling paper; not using 2026 question positions as a template |
| revised version P6 | 4 | 80 | 120 | A24/B15/C9/D12/M20 | mixed transfer; unfamiliar entry points without exceeding the overall benchmark burden |
Not A_B coverage notes
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| the revised version did not reach A/B ability units of | |||||
| not falsely reported as covered; these units are not frequent or high-mark abilities across the seven years . | |||||
| Competency unit ID | year | question number | Knowledge point | best grade in the revised version | Not A/B reason |
| AU030 | 2020 | 6(b)(iii) | third-order ODE rewrite as a first-order system | D | low frequency / specific-context abilities not directly included because of six-paper capacity and avoidance of repetition; related knowledge still has C or coverage by neighbouring methods . |
| AU127 | 2024 | 1(c)(i) | rewrite a difference integral by reciprocal substitution | D | low frequency / specific-context abilities not directly included because of six-paper capacity and avoidance of repetition; related knowledge still has C or coverage by neighbouring methods . |
| AU160 | 2025 | 1(c)(ii) | evaluate an integral by reciprocal substitution | D | low frequency / specific-context abilities not directly included because of six-paper capacity and avoidance of repetition; related knowledge still has C or coverage by neighbouring methods . |
original evidence index
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| MATH40004 original course-evidence index - AUDITED_REVISED_2 | |||||||
| these PDF are formal examination materials provided by the user in this course session / solution materials; reports and matrices cannot substitute for original evidence. Final main ZIP included under original filenames within 06_SOURCE_EVIDENCE. | |||||||
| year | original filename | source | Page count | content type | SHA-256 | Final ZIP relative path | verification notes |
| 2020 | MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf | the user in this MATH40004 provided in the course session | 22 | question booklet + official solutions / marking + examiner comments ;2020 evidence-chain completion | 143a8d4fdaaf7f49f781e464c0794cc7a29cd1f96d25440bc26dafa79164520b | 06_SOURCE_EVIDENCE/2020/MATH40004MATH40011_Calculus and Applications_Q&S_May_2020(3).pdf | opened, pages counted, and rendered pages spot-checked |
| 2021 | MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf | the user in this MATH40004 provided in the course session | 18 | question booklet + official solutions / marking + examiner comments | dd4c6787242e8f80496649701189f403b0098744bb3337f100a6fed851ad9080 | 06_SOURCE_EVIDENCE/2021/MATH40004_2021_Calculus and Applications_Exam Q&S May 2021.pdf | opened and included SHA list |
| 2022 | MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf | the user in this MATH40004 provided in the course session | 25 | question booklet + official solutions / marking + examiner comments; filename May2002 is a naming typo; the cover states 2022 | b7d524a98eb0a41e18459f39dd1bdab62d8d6814f9654ce41b0c63d4fbc8897a | 06_SOURCE_EVIDENCE/2022/MATH40004_2022_Calculus and Applications_EXAM_Q&S&Comments_May2002(1).pdf | opened and included SHA list |
| 2023 | MATH40004_Calculus and Applications_2023.pdf | the user in this MATH40004 provided in the course session | 22 | question booklet + official solutions / marking + examiner comments | f8d9e7b539f21694a5124f5ee6fa24aba7739cdfce30f1589666ad023a412dfb | 06_SOURCE_EVIDENCE/2023/MATH40004_Calculus and Applications_2023.pdf | opened and included SHA list |
| 2024 | MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf | the user in this MATH40004 provided in the course session | 29 | question booklet + official solutions / marking + examiner comments | b5b40fb4c8335a675ffb62795be872c2e675d6eb9270f6364c563cbb859a30d5 | 06_SOURCE_EVIDENCE/2024/MATH40004_Calculus and Applications_Q&S_May 2024(1).pdf | opened and included SHA list |
| 2025 | MATH40004 Calculus and Applications Q&S May 2025(1).pdf | the user in this MATH40004 provided in the course session | 22 | question booklet + official solutions / marking + examiner comments | 508e54a8847e5fca5affd3864822d293a7f32cad2288b759d63732fd868a5aa1 | 06_SOURCE_EVIDENCE/2025/MATH40004 Calculus and Applications Q&S May 2025(1).pdf | opened and included SHA list |
| 2026 | MATH40004 Calculus and Applications Q&S May 2026 (1).pdf | the user in this MATH40004 provided in the course session | 18 | question booklet + official solutions / marking ;2026 difficulty-calibration benchmark | 2fb045b5e1fcfe05775a6d60ca5d60c87b8b3ccca5c1080eeb87d748ee05c25b | 06_SOURCE_EVIDENCE/2026/MATH40004 Calculus and Applications Q&S May 2026 (1).pdf | opened and included SHA list |