ExamCram — Every Skill. Exam Ready. Fast.

Explore the skills · QCAA

Specialist Mathematics

QCAA External Exam Revision · Units 3 & 4

Explore the subject and its skills. Full lessons, examples and practice require subject access. Accounts and purchasing are not open yet.

Find a skill

    Further complex numbers

    • Prove and apply modulus and argument propertiesRevision skill
    • Use De Moivre’s theorem for integral powersRevision skill
    • Determine and represent roots of unityRevision skill
    • Determine and represent roots of complex numbersRevision skill
    • Apply the factor and remainder theoremsRevision skill
    • Apply the conjugate root theoremRevision skill
    • Solve polynomial equations over the complex numbersRevision skill

    Mathematical induction and trigonometric proofs

    • Construct proofs by mathematical inductionRevision skill
    • Interpret and manipulate sigma notationRevision skill
    • Prove summation results by inductionRevision skill
    • Prove divisibility results by inductionRevision skill
    • Prove De Moivre’s theorem for positive integer powersRevision skill
    • Derive and apply multiple-angle trigonometric identitiesRevision skill

    Vectors in two and three dimensions

    • Represent vectors in two and three dimensionsRevision skill
    • Use vector magnitude, direction and componentsRevision skill
    • Apply vector arithmetic and the section formulaRevision skill
    • Use the scalar product to find anglesRevision skill
    • Test vectors for parallelism and perpendicularityRevision skill
    • Calculate scalar and vector projectionsRevision skill
    • Model displacement, force and relative velocity with vectorsRevision skill
    • Construct vector proofs in geometryRevision skill
    • Use sphere equations and parametric vector curvesRevision skill
    • Use vector, parametric and Cartesian equations of linesRevision skill
    • Use cross products and equations of planesRevision skill

    Vector calculus

    • Interpret position vectors and particle pathsRevision skill
    • Derive Cartesian equations from vector pathsRevision skill
    • Distinguish intersecting paths from colliding particlesRevision skill
    • Differentiate vector functionsRevision skill
    • Integrate vector functions and apply initial conditionsRevision skill
    • Model straight-line, projectile and circular motion with vectorsRevision skill

    Further matrices

    • Calculate determinants and inverses of matricesRevision skill
    • Classify singular and non-singular matricesRevision skill
    • Solve matrix equationsRevision skill
    • Model and solve systems of linear equations with matricesRevision skill
    • Use Gaussian elimination and augmented matricesRevision skill
    • Interpret unique, inconsistent and dependent systems geometricallyRevision skill
    • Apply dominance and Leslie matricesRevision skill

    Integration techniques

    • Integrate using trigonometric identitiesRevision skill
    • Establish and use the integral of secant squaredRevision skill
    • Integrate by substitutionRevision skill
    • Recognise and evaluate logarithmic integralsRevision skill
    • Use inverse trigonometric functions and their derivativesRevision skill
    • Evaluate standard inverse trigonometric integralsRevision skill
    • Integrate using partial fractionsRevision skill
    • Integrate by partsRevision skill

    Applications of integral calculus

    • Calculate areas between curvesRevision skill
    • Calculate volumes of revolution about the x-axisRevision skill
    • Calculate volumes of revolution about the y-axisRevision skill
    • Approximate definite integrals using Simpson’s ruleRevision skill
    • Use the exponential probability density function, mean and standard deviationRevision skill
    • Calculate probabilities and quantiles with exponential modelsRevision skill

    Rates of change and differential equations

    • Differentiate implicit relationsRevision skill
    • Solve differential equations of the form dy/dx = f(x)Revision skill
    • Solve differential equations of the form dy/dx = g(y)Revision skill
    • Solve separable differential equationsRevision skill
    • Interpret and construct slope fieldsRevision skill
    • Model logistic growth, cooling and radioactive decayRevision skill

    Modelling motion

    • Apply momentum, force and Newton’s lawsRevision skill
    • Resolve concurrent forces in non-equilibrium systemsRevision skill
    • Use alternative differential forms of accelerationRevision skill
    • Model simple harmonic motionRevision skill
    • Model vertical motion with and without resistanceRevision skill
    • Model motion on a smooth inclined planeRevision skill

    Statistical inference

    • Interpret the sample mean as a random variableRevision skill
    • Use the sampling distribution of the sample meanRevision skill
    • Apply the central limit theorem to sample meansRevision skill
    • Model probabilities involving sample meansRevision skill
    • Calculate and interpret confidence intervals for meansRevision skill
    • Calculate and interpret margin of errorRevision skill
    • Relate confidence, sample size, margin of error and interval variabilityRevision skill
    • Distinguish population-value intervals from confidence intervals for a meanRevision skill
    Explore another subject →