ATARMAxxing · Specialist Mathematics
QCE Specialist Mathematics Practice Exams with Worked Solutions
20 full-length papers · worked solutions for every question
The 20 practice exams inside the QCE Specialist Mathematics Mastery Pack, each set out like the real paper with a separate worked-solution guide. Open any paper to see what it covers.
- Unit 3 further complex numbers: De Moivre's theorem, roots of zn, conjugate root theorem, factorisation over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in 3D: cross product, area/volume, lines and planes, point-plane distance
- Unit 3 vector calculus: differentiation/integration of vector functions, projectile motion (cricket-ball trajectory context)
- Unit 3 further matrices: Gaussian elimination and nature of solutions
- Unit 4 integration techniques: inverse-trig integrals, by parts, partial fractions
- Unit 4 applications of integration: volumes of revolution, Simpson's rule, probability density functions
- Unit 4 rates of change and DEs: separable differential equations, related rates
- Unit 4 modelling motion: simple harmonic motion and resistive (variable-force) motion
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Paper 1 technology-free fluency (Sections A/B) and Paper 2 technology-active modelling (Sections C/D)
- Cricket-ball projectile/spin trajectory stimulus threaded through vector-calculus and motion items
- Unit 3 further complex numbers: De Moivre's theorem, nth roots, conjugate root theorem, polynomial factorisation over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in 3D: cross products, vector/Cartesian equations of lines and planes, distance formulae
- Unit 3 vector calculus: differentiation of vector functions, projectile and circular motion
- Unit 3 further matrices: Gaussian elimination for 3x3 linear systems
- Unit 4 integration techniques: inverse trig integrals, substitution, partial fractions, integration by parts
- Unit 4 applications of integral calculus: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Unit 4 rates of change and DEs: implicit differentiation, related rates, separable differential equations
- Unit 4 modelling motion: simple harmonic motion and variable forces
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Applied context throughout: designing a water reservoir dam, estimating water volume via solids of revolution and Simpson's rule
- Further complex numbers: De Moivre's theorem, powers/roots, conjugate root theorem, polynomial factorisation over C
- Mathematical induction: summation and divisibility proofs
- Vectors in 3D: lines, planes, cross products, distance formulae, spheres, projectile & circular motion via vector calculus
- Further matrices: Gaussian elimination for linear systems
- Integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integration: areas, volumes of revolution, Simpson's rule, probability density functions
- Rates of change & DEs: implicit differentiation, related rates, separable DEs, slope fields
- Modelling motion: simple harmonic motion applied to tide height, variable forces, ODE kinematics
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, required sample size
- Context: modelling tide height as SHM to predict safe/optimal harbour-navigation windows
- Statistical inference: sampling distribution of the sample mean, constructing and interpreting confidence intervals for a population mean (known sigma), and required sample size — set in a respiratory-illness clinical-trial recovery-time context
- Further complex numbers: De Moivre's theorem, powers and nth roots, polynomial factorisation over C, conjugate root theorem
- Mathematical induction: summation identities and divisibility proofs
- Vectors in three dimensions: dot and cross products, angle between vectors, lines/planes, spheres, distance
- Vector calculus and modelling motion: projectile motion, SHM, variable forces
- Further calculus: inverse-trig integrals, substitution, partial fractions, integration by parts, areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and differential equations: implicit and related rates, separable DEs, slope fields
- Further matrices: Gaussian elimination for systems of linear equations
- Paper 1 technology-free fluency and Paper 2 technology-active modelling across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%) cognitions
- Vectors in 3D: vector/Cartesian equations of lines and planes, cross products, concurrent vs skew roof-beam intersections, shortest-distance and point-to-plane formulae, spheres
- Further complex numbers: De Moivre's theorem, powers and roots, conjugate root theorem, factorisation over C
- Mathematical induction: summation and divisibility proofs
- Further matrices: Gaussian elimination for 3x3 linear systems
- Vector calculus and modelling motion: differentiation/integration of vector functions, projectile and circular motion, SHM, variable forces
- Integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integration: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and differential equations: implicit differentiation, related rates, separable DEs, slope fields
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Structural-engineering context throughout: roof-beam geometry, truss force systems, beam deflection sampling
- Unit 3 further complex numbers: De Moivre's theorem, modulus-argument form, conjugate root theorem, factorisation over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in 3D: dot/cross products, lines and planes, distance formulae, spheres
- Unit 3 vector calculus: differentiation/integration of vector functions, projectile motion
- Unit 3 further matrices: Gaussian elimination for linear systems
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Unit 4 applications of integration: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Unit 4 rates of change and DEs: implicit differentiation, related rates, separable differential equations, slope fields
- Unit 4 modelling motion: simple harmonic motion and variable forces
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Stimulus context: logistic population-growth model dP/dt = rP(1 - P/K) as a separable DE under resource constraints
- Cognitive spread across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%) with Paper 1 technology-free and Paper 2 technology-active
- Unit 3 further complex numbers: De Moivre's theorem, nth roots, conjugate root theorem, polynomial factorisation over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in 3D: cross products, lines/planes, distances, line-plane intersection
- Unit 3 vector calculus: differentiation/integration of vector functions, projectile and SHM kinematics
- Unit 3 further matrices: Gaussian elimination for 3x3 linear systems
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Unit 4 applications of integration: areas, volumes of revolution, Simpson's rule, probability density functions
- Unit 4 rates of change and differential equations: implicit differentiation, related rates, separable DEs, slope fields
- Unit 4 modelling motion: SHM and variable forces, ODE-based kinematics of a rocket with velocity-dependent thrust
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Rocket-launch stimulus: modelling vertical motion with variable thrust as a function of velocity using ODE techniques (Paper 2 Sections C and D)
- Cognitions: Select, Recall and use, Apply, Analyse and reason, Communicate across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%)
- Unit 3 further complex numbers: factorising z4+4 over C, conjugate root theorem, De Moivre for powers/roots and argument calculations (cryptography framing)
- Mathematical induction: summation and divisibility proofs
- Vectors in 3D: lines, planes, cross product, distance formulae, spheres
- Vector calculus: differentiation/integration of vector functions, projectile and circular motion
- Further matrices: Gaussian elimination for 3x3 linear systems
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integration: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and DEs: implicit differentiation, related rates, separable DEs, slope fields
- Modelling motion: SHM and ODE-based kinematics
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Unit 3 further complex numbers: De Moivre's theorem, powers and nth roots, conjugate root theorem and factorisation over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in 3D: cross product, scalar/vector equations of lines and planes, point-to-plane distance, spheres
- Unit 3 vector calculus: differentiating/integrating vector functions, projectile and circular motion
- Unit 3 further matrices: Gaussian elimination and parameter-dependent solution sets
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Unit 4 applications of integral calculus: area between curves as energy absorption of alloys, volumes of revolution, Simpson's rule, probability density functions
- Unit 4 rates of change and differential equations: implicit differentiation, related rates, separable DEs, slope fields
- Unit 4 modelling motion: simple harmonic motion, variable forces, ODE-based kinematics
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for mu, sample-size determination
- Materials-science stimulus thread: computing the area between stress-strain curves to compare toughness/energy absorption of two alloys
- Assessment objectives: Select, Recall and use, Apply, Analyse and reason, Communicate across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%)
- Unit 3 further complex numbers: De Moivre's theorem, roots of unity, polynomial factorisation over C, conjugate root theorem
- Signal-processing thread: proving Fourier harmonic-sum trig identities via the complex geometric series z=ei theta (Dirichlet kernel, odd-harmonic sum)
- Mathematical induction: summation and divisibility proofs
- Vectors in 3D: lines/planes, cross product, distance formulae, spheres
- Vector calculus: projectile and circular motion
- Further matrices: Gaussian elimination for linear systems
- Unit 4 integration techniques: inverse-trig, substitution, partial fractions, integration by parts
- Applications of integration: areas, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and differential equations: implicit/related rates, separable DEs, slope fields
- Modelling motion: SHM and variable forces
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Cognitions: Select, Recall and use, Apply, Analyse and reason, Communicate across simple familiar / complex familiar / complex unfamiliar
- Paper 1 (technology-free): De Moivre's theorem and complex arithmetic, conjugate root theorem, mathematical induction (summation and divisibility), 3D vectors (dot/cross product, lines/planes, distance, spheres), integration techniques (inverse trig, partial fractions, by parts, substitution), applications (areas, volumes of revolution, PDFs), differential equations (separable, implicit, related rates), SHM and projectile motion, Gaussian elimination, and statistical inference (sampling distribution of the mean, confidence intervals for mu, sample size).
- Paper 2 (technology-active): heavier computation and applied modelling on the same Units 3 & 4 topics, with the quality-control-in-manufacturing context (sampling distributions, batch sample sizes, 95% CI for mean component length) recurring across confidence-interval and sample-size items, plus Simpson's rule, volumes of revolution, logistic/separable DEs and slope fields, projectile and circular motion, and complex-number modelling.
- Cognitive spread: ~60% simple familiar, ~20% complex familiar, ~20% complex unfamiliar across Select, Recall-and-use, Apply, Analyse-and-reason and Communicate, with Band A model solutions showing full method, precise notation and contextual interpretation of results.
- Rates of change & separable differential equations applied to traffic-flow (Greenshields) modelling: setting up and solving dk/dt and dq/dk
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integral calculus: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Unit 3 further complex numbers: De Moivre's theorem, nth roots, factorisation over C, conjugate root theorem
- Mathematical induction: summation and divisibility proofs
- Vectors in 3D and vector calculus: cross products, lines/planes, projectile and circular motion
- Further matrices: Gaussian elimination for linear systems
- Modelling motion: SHM and variable-force/ODE kinematics
- Statistical inference: sampling distribution of the mean, confidence intervals for μ, and required sample size
- Paper 1 technology-free fluency (Sections A & B) and Paper 2 technology-active modelling (Sections C & D), spanning simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%)
- Vector calculus: differentiation of vector functions, velocity/speed/acceleration, projectile and circular/elliptical motion (astronomy satellite context)
- Further complex numbers: modulus-argument form, De Moivre's theorem, roots over C, conjugate root theorem, polynomial factorisation
- Mathematical induction: summation and divisibility proofs
- Vectors in 3D: lines and planes, cross product, distance formulae, spheres
- Further matrices: Gaussian elimination for linear systems
- Integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integration: areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and DEs: implicit differentiation, related rates, separable differential equations
- Modelling motion: simple harmonic motion, variable forces, ODE-based kinematics
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, required sample size
- Unit 4 integration techniques — partial-fraction decomposition of proper rational functions and integration to logarithmic form, anchored in a CT-dosimetry absorption-rate context
- De Moivre's theorem, conjugate root theorem and factorisation of polynomials over the complex field (Unit 3)
- Mathematical induction — summation and divisibility proofs (Unit 3)
- Vectors in three dimensions — cross products, scalar triple product, vector/Cartesian equations of lines and planes, distance formulae and spheres (Unit 3)
- Vector calculus and modelling motion — projectile motion, circular motion and simple harmonic motion (Units 3 & 4)
- Further calculus — inverse-trig integrals, substitution, integration by parts, implicit differentiation and related rates (Unit 4)
- Applications of integral calculus — areas between curves, volumes of revolution, Simpson's rule and probability density functions (Unit 4)
- Differential equations — separable DEs and slope fields (Unit 4)
- Statistical inference — sampling distribution of the mean, confidence intervals for a population mean and sample-size determination (Unit 4)
- Gaussian elimination for systems of linear equations (Unit 3)
- Paper 1 technology-free fluency and Paper 2 technology-active modelling across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%) cognitions
- Continuous probability density functions modelled on an ecological fish-weight survey (normalisation constant, mean, median, mode, variance, interval probabilities)
- Unit 4 integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Unit 4 applications of integration: areas between curves, volumes of revolution, Simpson's rule
- Unit 4 statistical inference: sampling distribution of the mean, confidence intervals for the population mean, required sample size
- Unit 4 rates of change and differential equations: separable DEs, related rates, implicit differentiation, logistic growth
- Unit 4 modelling motion: simple harmonic motion and ODE-based kinematics
- Unit 3 further complex numbers: modulus-argument form, De Moivre's theorem, conjugate root theorem, roots over C
- Unit 3 mathematical induction: summation and divisibility proofs
- Unit 3 vectors in three dimensions: dot/cross products, lines and planes, distance formulae, spheres, projectile and circular motion
- Unit 3 further matrices: Gaussian elimination for 3x3 linear systems
- QCAA cognitive spread: simple familiar (~60%), complex familiar (~20%), complex unfamiliar (~20%) across a technology-free Paper 1 and technology-active Paper 2
- Vectors in 3D: cross products, moment of a force (r×F), scalar triple product, lines/planes, distance formulae, spheres
- Further complex numbers: De Moivre's theorem, nth roots, conjugate root theorem, factorisation over C
- Mathematical induction: summation and divisibility proofs
- Vector calculus: differentiation/integration of vector functions, projectile and circular motion
- Further matrices: Gaussian elimination for linear systems
- Integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integration: areas, volumes of revolution, Simpson's rule, probability density functions
- Rates of change & DEs: implicit differentiation, related rates, separable DEs, slope fields
- Modelling motion: SHM and ODE-based kinematics
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Stimulus context throughout: bridge cable tension and the moment of a force vector about a support point
- Paper 1 (technology-free): De Moivre's theorem, conjugate root theorem, 3D vectors (dot/cross products, planes, distance), inverse-trig and partial-fraction integration, integration by parts, separable DEs, implicit differentiation, mathematical induction, and Gaussian elimination — all hand-computable.
- Paper 2 (technology-active): Gaussian elimination for the intersection of two planar ore seams and a drill line (geology/mining stimulus), volumes of revolution, related rates, Newton's law of cooling, simple harmonic motion, circular motion, area between curves, and statistical inference (sampling distribution, 95% CI for μ, sample size).
- QCAA EA marks-based standard across five assessment objectives (Select; Recall & use; Apply; Analyse & reason; Communicate) at simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%) levels, 120 marks total (Paper 1 = 60, Paper 2 = 60).
- Unit 4 integration techniques — inverse-trig integrals, substitution, partial fractions, and integration by parts (the paper's stimulus thread: present value of a continuous income stream with exponential decay, PV = ∫ R(t)e-rt dt)
- Unit 4 applications of integral calculus — areas between curves, volumes of revolution, Simpson's rule, probability density functions
- Unit 3 further complex numbers — De Moivre's theorem, conjugate root theorem, factorisation over C
- Unit 3 mathematical induction — summation identities and divisibility proofs
- Unit 3 vectors in three dimensions and vector calculus — cross products, distance to a plane, lines/spheres, differentiation of vector functions, projectile/circular motion
- Unit 3 further matrices — Gaussian elimination for 3x3 linear systems
- Unit 4 rates of change and differential equations — implicit differentiation, related rates, separable DEs, slope fields
- Unit 4 modelling motion — simple harmonic motion and variable-force/ODE kinematics
- Unit 4 statistical inference — sampling distribution of the mean, confidence intervals for μ, and sample-size determination
- Paper 1 technology-free fluency (Sections A/B) and Paper 2 technology-active modelling (Sections C/D), spanning simple familiar ~60%, complex familiar ~20%, complex unfamiliar ~20%
- Mathematical induction (summation and divisibility proofs) framed through a sports-science cumulative fatigue index
- Further complex numbers: De Moivre's theorem, powers and roots, conjugate root theorem, factorisation over C
- Vectors in 3D: cross products, lines and planes, point-plane distance, spheres, and vector calculus (projectile/circular motion)
- Further matrices: Gaussian elimination for systems of linear equations
- Integration techniques: inverse-trig integrals, substitution, partial fractions, integration by parts
- Applications of integral calculus: areas, volumes of revolution, Simpson's rule, probability density functions
- Rates of change and differential equations: implicit differentiation, related rates, separable DEs, slope fields
- Modelling motion: simple harmonic motion and variable forces
- Statistical inference: sampling distribution of the mean, confidence intervals for mu, sample size
- Paper 1 technology-free fluency and Paper 2 technology-active modelling across simple familiar, complex familiar and complex unfamiliar cognitions
- Rates of change and separable differential equations — Newton's law of cooling modelled as dT/dt = -k(T - Ts), solved by separation of variables
- The time constant tau = 1/k, its interpretation (excess temperature falls to 1/e of its initial value) and half-life reasoning
- Statistical inference — sampling distribution of the mean, z-based confidence intervals for a population mean mu, and minimum sample size
- Further complex numbers — De Moivre's theorem, roots over C, conjugate root theorem
- Mathematical induction — summation proofs with base case, inductive hypothesis and inductive step
- Vectors in 3D — cross product and area, distance from a point to a plane, and vector calculus (speed, projectile motion)
- Integration techniques and applications — inverse-trig integrals, partial fractions, integration by parts, areas, volumes of revolution, Simpson's rule, probability density functions
- Further matrices — Gaussian elimination for 3x3 linear systems
- Modelling motion — simple harmonic motion and ODE-based kinematics
- Paper 1 technology-free fluency (60 marks) and Paper 2 technology-active modelling (60 marks), across simple familiar (~60%), complex familiar (~20%) and complex unfamiliar (~20%) cognitions
Included in the QCE Specialist Mathematics Mastery Pack
20 full-length practice exams with worked solutions, 20 revision notes, 64 practice questions and 200 flashcards.
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