Mathematics Extension 2

All 20 practice exams

1. Exam 1 — Complex numbers & proof (De Moivre, nth roots, regions, induction, contradiction) — ~37 marks; Further integration (parts, partial fractions, reduction, substitution) — ~20 marks; Vectors & mechanics (3D lines, dot product, SHM, resisted motion, projectiles) — ~43 marks
2. Exam 2 — Complex numbers & proof (roots of unity, polynomial conjugates, AM–GM/inequalities, induction, contradiction) — ~28 marks; Further integration (trig substitution, completing the square, reduction, by parts) — ~18 marks; Vectors & mechanics (spheres, 3D geometry proof, skew lines, SHM from rest, resisted horizontal motion, projectiles) — ~54 marks
3. Exam 3 — Complex numbers — roots of unity, De Moivre, regions & loci — ~24 marks; Further integration — substitution, partial fractions, by parts, reduction — ~20 marks; Mechanics, vectors & proof — SHM, resisted motion, 3D lines, induction & inequalities — ~46 marks
4. Exam 4 — Complex numbers — quadratics with complex coefficients, nth roots, geometry of rotation — ~22 marks; Further integration & calculus by induction — quadratic denominators, reduction, d/dx(xⁿ) — ~22 marks; Mechanics, vectors & proof — resisted horizontal motion, projectiles, spheres, triangle & AM–GM inequalities — ~46 marks
5. Exam 5 — Complex numbers (regions, roots of complex numbers, quadratics with complex coefficients, geometry) — ~30 marks; Proof & further integration (AM–GM, triangle inequality, divisibility induction, completing the square, partial fractions) — ~36 marks; Vectors & mechanics (spheres, 3D geometric proof, horizontal resisted motion, SHM from a spring) — ~34 marks
6. Exam 6 — Complex numbers (exponential form, De Moivre identities, quadratics with complex coefficients, loci) — ~30 marks; Further integration & proof (reduction formula, partial fractions with quadratic factor, induction inequality, contradiction) — ~36 marks; Vectors & mechanics (3D dot-product geometry, projectile in a resisting medium, resisted vertical motion upward) — ~34 marks
7. Exam 7 — Complex numbers — De Moivre powers, square roots of a + ib, conjugate-root polynomials, loci & regions, geometry of iz — ~32 marks; Further integration — t-substitution, trig substitution, parts (x arctan x), partial fractions with a repeated factor, reduction by parts — ~28 marks; Proof, vectors & mechanics — contradiction with a sum, induction (inequality & calculus), 3D lines & spheres, resisted horizontal motion, SHM from a graph, projectile through two points — ~40 marks
8. Exam 8 — Complex numbers — Cartesian arithmetic & conjugates, roots of z^4 = -16/16x^4 = 1, polar/exponential conversion, |z - a| = k|z - b| circle locus, sum of roots of unity series — ~32 marks; Further integration — quadratic-denominator (ln + arctan split), trig (sin^2) substitution x = sin u, partial fractions over a difference of squares, reduction for integral cos^n / x^n e^x — ~28 marks; Proof, vectors & mechanics — triangle/AM-GM inequality, induction (divisibility 7 | 8^n - 1 type & geometric sum), 3D dot-product angle/perpendicularity, SHM v^2 = n^2(a^2 - x^2) energy, vertical resisted motion to terminal speed — ~40 marks
9. Exam 9 — Complex numbers — square roots, De Moivre & multiple-angle identities, roots of unity, Apollonius loci, roots of z³ = −8i — ~27 marks; Further integration — partial fractions, completing the square, by-parts & xⁿeˣ reduction — ~20 marks; Proof, vectors & mechanics — induction (divisibility/sigma), 3D lines & spheres, SHM, resisted vertical & linear motion, AM–GM and symmetric inequalities — ~53 marks
10. Exam 10 — Complex numbers — real & complex quadratics, z⁴ = −16, cube roots of unity, regions, perpendicular-bisector loci & annular sectors — ~23 marks; Further integration — substitution (u and trig), tanⁿ reduction formula — ~20 marks; Proof, vectors & mechanics — induction (factorial inequality, d/dx xⁿ), cube geometry by vectors, resisted falling motion (tanh & exponential), projectile range, triangle inequality, irrationality & SHM — ~57 marks
11. Exam 11 — Complex Numbers (De Moivre, roots of unity, loci) — ~26 marks; Further Integration (reduction, partial fractions, by parts) — ~22 marks; Mechanics (SHM, resisted vertical motion, projectiles) — ~20 marks
12. Exam 12 — Vectors (3D lines, intersection, angle, distance) — ~22 marks; Complex Numbers (loci, nth roots, polynomial conjugate roots) — ~24 marks; Further Integration & Mechanics (trig substitution, resisted motion, SHM) — ~24 marks
13. Exam 13 — Complex numbers — exponential form, De Moivre for sin/cos powers, nth roots of a complex number, loci & regions — ~30 marks; Further integration & proof — t-substitution, partial fractions with repeated factor, reduction by parts, induction (recursive & inequality), contradiction — ~36 marks; Vectors & mechanics — 3D lines (closest approach), spheres, SHM from a graph, resisted vertical motion downward, projectile with unknown angle — ~34 marks
14. Exam 14 — Complex numbers — quadratics with complex coefficients, real polynomials with conjugate roots, roots of unity sum identity, ray/half-plane regions — ~30 marks; Further integration & proof — trig substitution x = a sinθ, partial fractions with irreducible quadratic, induction (sigma & divisibility), AM–GM and triangle inequality — ~36 marks; Vectors & mechanics — spheres & tangency, vector proof of a geometric theorem, horizontal resisted motion, projectile range on an incline — ~34 marks
15. Exam 15 — Complex numbers (exponential form, nth roots of a complex number, conjugate-root polynomials, |z−a|=k|z−b| Apollonius locus) — ~32 marks; Proof & further integration (∀/∃ language, contrapositive, AM–GM, even-n divisibility induction, reduction formula for ∫tanⁿx, completing the square) — ~34 marks; Vectors & mechanics (3D median/Apollonius proof, skew lines & angle, terminal velocity downward, SHM established from a v²–x relation) — ~34 marks
16. Exam 16 — Complex numbers (De Moivre for cos 5θ, sum of nth roots of unity, |z−a|=|z−b| perpendicular-bisector locus, complex-coefficient quadratic with a repeated root) — ~32 marks; Proof & further integration (proof by contradiction for ∛2, induction on a first-order recurrence, secⁿ reduction by parts, partial fractions with a repeated linear factor) — ~33 marks; Vectors & mechanics (sphere & tangent-line condition via dot product, perpendicularity, projectile clearing a wall (two angles), SHM time to a directed centre-crossing) — ~35 marks
17. Exam 17 — Complex numbers — De Moivre to derive a sin/cos 5θ identity, cube roots of a complex number, real-polynomial conjugate roots, square roots & quadratics — ~30 marks; Further integration & proof — partial fractions with a repeated linear factor, ∫xⁿeˣ reduction formula, completing-the-square inverse-tan, induction for the binomial theorem, contradiction for irrationality of log₂5 — ~35 marks; Vectors & mechanics — sphere–line intersection in 3D, vector proof that the diagonals of a rhombus are perpendicular, resisted motion on a horizontal line (v as a function of x), SHM established from x = a sin(nt+α) — ~35 marks
18. Exam 18 — Complex numbers — square roots of a complex number, sum of the nth roots of unity being zero, geometry of multiplication by i (rotation), region from |z−1| ≤ |z+i| — ~27 marks; Further integration & proof — trig substitution x = a tan θ, reduction formula for ∫cosⁿx, AM–GM applied to a minimisation, triangle inequality with geometric interpretation, induction that 2ⁿ > n² for n ≥ 5 — ~37 marks; Vectors & mechanics — vector equation of a line & point of closest approach to a point, dot-product proof that an angle in a semicircle is 90°, resisted vertical motion upward (first power of speed), projectile in a resisting medium (full motion) — ~36 marks
19. Exam 19 — Complex numbers — roots of unity to factorise/sum, Euler-form power reduction, geometry of iz rotation, polar division, region Re(z) > Im(z) — ~30 marks; Proof & further integration — contrapositive & converse logic, irrationality by contradiction, geometric induction (polygon exterior angles), recurrence by induction, by-parts with ln, partial fractions — ~36 marks; Vectors & mechanics — 3D line intersection & point test, dot-product angle, linear resisted motion (resistance ∝ v) with v(x), projectile onto an inclined plane — ~34 marks
20. Exam 20 — Complex numbers — complex-coefficient quadratic by square roots, De Moivre power of a complex number, exponential-form product, locus |z−a| = |z−b| & argument wedge, parallelogram law — ~31 marks; Proof & further integration — implication vs equivalence, triangle inequality, induction proof of the binomial theorem, reduction formula for ∫xⁿe^{ax}, trig substitution — ~36 marks; Vectors & mechanics — vector proof that the diagonals of a parallelogram bisect each other, sphere & midpoint, SHM established from x = a cos(nt+α) with energy, resisted vertical motion upward (resistance ∝ v²) — ~33 marks

All 20 revision notes

• The language of proof: implication, converse, contrapositive and proof by contradiction
• Proving inequalities from a > b and the non-negativity of squares
• The triangle inequality and the AM–GM inequality for two numbers
• Induction for series in sigma notation and for non-standard base cases or steps
• Divisibility and inequality proofs by induction
• Induction in calculus, geometry, the binomial theorem and recursive formulae
• Vectors in three dimensions: components, magnitude, the dot product and spheres
• Vector equations of lines: r = a + λb, parallel and perpendicular lines, and points on a line
• Complex numbers in Cartesian form: arithmetic, the conjugate, square roots and the Argand plane
• Polar and exponential form: modulus, argument, Euler's formula and the geometry of multiplication
• De Moivre's theorem: induction proof, powers and trigonometric identities
• Quadratics and polynomials over the complex numbers, and the conjugate root theorem
• Geometry of complex arithmetic: vectors, the parallelogram law and rotation by i
• nth roots of unity, nth roots of a complex number, and regions of the Argand plane
• Integration by substitution and rational functions with a quadratic denominator
• Partial fractions and integrating rational functions
• Integration by parts and reduction (recurrence) formulae
• Simple harmonic motion: equations, the defining property and graphs
• Straight-line motion with variable acceleration and resisted motion
• Projectile motion: parametric equations, the Cartesian path and resisted projectiles