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QCE · QCE Units 3 & 4

Specialist Mathematics

Induction, complex numbers, vectors, further calculus and statistical inference — full Paper 1 (tech-free) + Paper 2 (tech-active) practice EAs with worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
Allofficial past papers

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Sample revision note

De Moivre's theorem and roots of unity

Statement and Proof of De Moivre's Theorem

De Moivre's theorem states that for any complex number in polar form and any integer n:

(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)

More generally, if z = r(cos θ + i sin θ) then zn = rn(cos nθ + i sin nθ). The theorem is fundamental to Unit 3 Further Complex Numbers in QCAA Specialist Mathematics and must be proved rigorously.

Proof for positive integers by mathematical induction:

Base case (n = 1): (cos θ + i sin θ)1 = cos θ + i sin θ = cos(1·θ) + i sin(1·θ). True.

Inductive step: Assume the result holds for some positive integer k, i.e., (cos θ + i sin θ)k = cos(kθ) + i sin(kθ). Consider n = k + 1:

  • (cos θ + i sin θ)k+1 = (cos θ + i sin θ)k · (cos θ + i sin θ)
  • = [cos(kθ) + i sin(kθ)] · (cos θ + i sin θ)    (inductive hypothesis)
  • = cos(kθ)cos θ − sin(kθ)sin θ + i[cos(kθ)sin θ + sin(kθ)cos θ]
  • = cos(kθ + θ) + i sin(kθ + θ)    (compound angle identities)
  • = cos((k+1)θ) + i sin((k+1)θ)

The result holds for n = k + 1. By the principle of mathematical induction, the theorem holds for all positive integers.

Extension to n = 0: (cos θ + i sin θ)0 = 1 = cos 0 + i sin 0. Consistent.

Extension to negative integers: For n = −m, where m is a positive integer:

  • (cos θ + i sin θ)−m = 1/[cos(mθ) + i sin(mθ)]    (using positive integer case)
  • Multiply numerator and denominator by the conjugate: = [cos(mθ) − i sin(mθ)] / [cos2(mθ) + sin2(mθ)]
  • = cos(mθ) − i sin(mθ) = cos(−mθ) + i sin(−mθ) = cos(nθ) + i sin(nθ) ✓

The theorem therefore holds for all n ∈ ℤ. Note carefully: for rational n = p/q, De Moivre's theorem yields one valid value, not the complete set of roots. This distinction is essential when finding nth roots.

Multiplying and Dividing in Polar Form: The Geometric Basis

The inductive proof above rests on the product rule for complex numbers in polar form. Understanding this geometrically clarifies why De Moivre's theorem takes the form it does.

If z1 = r1(cos α + i sin α) and z2 = r2(cos β + i sin β), then:

  • z1z2 = r1r2[cos(α + β) + i sin(α + β)]

That is: moduli multiply, arguments add. Repeated multiplication by the same number z multiplies the modulus by r each time and adds θ to the argument each time. After n multiplications: modulus becomes rn, argument becomes nθ. This is De Moivre's theorem, expressed geometrically as repeated rotation and scaling on the Argand diagram.

Worked example — Simplification using De Moivre's theorem:

Evaluate (1 + i)8.

  • Convert to polar form: |1 + i| = √2, arg(1 + i) = π/4.
  • So 1 + i = √2(cos π/4 + i sin π/4).
  • By De Moivre: (√2)8(cos(8 · π/4) + i sin(8 · π/4)) = 16(cos 2π + i sin 2π) = 16(1 + 0i) = 16.

Worked example — Division:

Simplify (cos(π/5) + i sin(π/5))7 / (cos(π/5) + i sin(π/5))3.

  • = (cos(π/5) + i sin(π/5))4 = cos(4π/5) + i sin(4π/5).

In Cartesian form, cos(4π/5) = −cos(π/5) and sin(4π/5) = sin(π/5), using supplementary angle identities. These values are not standard unit-circle values, so the polar form is the preferred final answer unless a decimal approximation is requested. Always reduce the power using index laws before applying De Moivre's theorem.

Sample exam question

Paper 1 (technology-free). A cricket analyst models two influences on a ball as vectors a = 2i - j + 3k and b = i - 2k. The magnitude of the cross product a x b is:

  • 3√2
  • 3√6
  • √66
  • 9
Show the worked answer

Answer: B

a x b = (2,-1,3)x(1,0,-2) = (2, 7, 1); |a x b| = √(4+49+1) = √54 = 3√6.

All 20 practice exams

  1. Exam 1 — Unit 3 further complex numbers: De Moivre's theorem, roots of z^n, 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
  2. Exam 2 — 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
  3. Exam 3 — 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
  4. Exam 4 — 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
  5. Exam 5 — 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
  6. Exam 6 — 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
  7. Exam 7 — 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
  8. Exam 8 — Unit 3 further complex numbers: factorising z^4+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
  9. Exam 9 — 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
  10. Exam 10 — 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=e^{i theta} (Dirichlet kernel, odd-harmonic sum); Mathematical induction: summation and divisibility proofs
  11. Exam 11 — 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.
  12. Exam 12 — 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
  13. Exam 13 — 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
  14. Exam 14 — 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)
  15. Exam 15 — 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
  16. Exam 16 — 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
  17. Exam 17 — 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).
  18. Exam 18 — 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
  19. Exam 19 — 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)
  20. Exam 20 — 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

All 20 revision notes

  • De Moivre's theorem and roots of unity
  • Factorising polynomials over the complex field
  • Proving trig identities using De Moivre's theorem
  • Mathematical induction for sums and divisibility
  • Lines and planes in three-dimensional space
  • Cross product and geometric applications in 3D
  • Equations of spheres and 3D regions
  • Calculus of vector-valued functions
  • Projectile motion and uniform circular motion via vectors
  • Solving systems of linear equations using Gaussian elimination
  • Integration by parts and partial fraction decomposition
  • Inverse trig integrals and integration by substitution
  • Volumes of solids of revolution (disk and shell methods)
  • Areas between curves and PDF applications
  • Separable differential equations and slope field analysis
  • Implicit differentiation and related rates of change
  • Simple harmonic motion via differential equations
  • Variable forces and motion with non-constant acceleration
  • Distribution of sample means and confidence intervals for μ
  • Interpreting and evaluating statistical inference results