UniGeniusHSC Maths Extension 2

Course map

The nature of proof

  • The language of proofYear 12 · syllabus outcome ME2-12-01
    • Write negations, converses and contrapositives, and use quantifiers correctly
    • Prove statements directly and by contradiction
    • Test statements with examples and counterexamples
    LessonPractice
  • Proving inequalitiesYear 12 · syllabus outcome ME2-12-01
    • Prove inequalities from squares, the triangle inequality and AM-GM
    • Prove inequalities in geometry and with calculus
    • Use the squeeze theorem
    LessonPractice
  • Further mathematical inductionYear 12 · syllabus outcome ME2-12-01
    • Prove identities and inequalities by induction
    • Prove calculus results and results about recursive formulas by induction
    • Write the base case, inductive step and conclusion in full
    LessonPractice

Further work with vectors

  • Vector equations of lines and curvesYear 12 · syllabus outcome ME2-12-02
    • Write and use vector equations of lines in two and three dimensions
    • Decide whether lines are parallel, intersect or are skew
    • Move between vector and Cartesian equations of curves, circles and spheres
    LessonPractice
  • Vectors and geometryYear 12 · syllabus outcome ME2-12-02
    • Use the properties of the scalar product
    • Prove and use the Cauchy-Schwarz inequality
    • Prove geometric results with vectors
    LessonPractice

Introduction to complex numbers

  • Complex numbers: arithmetic and polar formYear 12 · syllabus outcome ME2-12-03
    • Calculate with complex numbers in Cartesian form, including division and square roots
    • Find the modulus and principal argument and move between Cartesian and polar form
    • Prove and use the modulus, argument and conjugate identities
    LessonPractice
  • Complex equations, de Moivre and rootsYear 12 · syllabus outcome ME2-12-03
    • Solve quadratics with complex coefficients and use the conjugate root theorem
    • Use de Moivre's theorem for powers and trigonometric identities
    • Find nth roots of complex numbers and place them on the complex plane
    LessonPractice
  • Complex numbers in geometryYear 12 · syllabus outcome ME2-12-03
    • Interpret addition and multiplication of complex numbers as vectors, rotations and enlargements
    • Prove geometric results with complex numbers
    • Describe lines, circles, rays and regions in the complex plane
    LessonPractice

Further integration

  • Trigonometric integrals and substitutionYear 12 · syllabus outcome ME2-12-04
    • Use products as sums and the t-formulas in equations and integrals
    • Integrate by substitution, given or chosen
    • Evaluate definite integrals after a substitution, changing the limits
    LessonPractice
  • Partial fractions, parts and recurrencesYear 12 · syllabus outcome ME2-12-04
    • Integrate rational functions with partial fractions and by completing the square
    • Integrate by parts, more than once where needed
    • Derive and use recurrence formulas, and join techniques
    LessonPractice

Applications of calculus to mechanics

  • Forces and motion without resistanceYear 12 · syllabus outcome ME2-12-05
    • Use the forms of acceleration and Newton's laws
    • Resolve forces, including on smooth inclined planes
    • Solve problems with a single smooth pulley
    LessonPractice
  • Simple harmonic motionYear 12 · syllabus outcome ME2-12-05
    • Prove that a motion is simple harmonic
    • Find amplitude, period, centre, speed and acceleration
    • Use v^2 = n^2(A^2 - (x - c)^2) and the displacement equations
    LessonPractice
  • Resisted motion and projectilesYear 12 · syllabus outcome ME2-12-05
    • Derive and use equations for rectilinear and vertical resisted motion
    • Find terminal velocity, greatest height and time to greatest height
    • Analyse projectiles with resistance proportional to the speed
    LessonPractice

The examination

  • The examinationYear 12
    • Know the paper: sections, marks, time, the reference sheet and calculators
    • Write Section II answers that show the reasoning the marking rewards
    • Know that school-based assessment is set by your school
    LessonPractice

NSW HSC Mathematics Extension 2, a Year 12 course. Mathematics Advanced and Extension 1 are assumed knowledge. School-based assessment is set by your school, so it is described in a lesson rather than practised here.