§-Specialist Mathematics syllabus

WA · SCSA← Specialist Mathematics

Specialist Mathematics syllabus, dot point by dot point

Every dot point in the WA Specialist Mathematics syllabus, in syllabus order, with a focused answer for each. Open any dot point for the answer, exam-style questions and links to related points.

Unit 3

Unit overview →

How do we restrict the circular functions so that their inverses are genuine functions, and what do those inverses look like?

Define the inverse circular functions with their restricted domains and ranges and sketch their graphs

How can we raise a complex number to a high integer power without expanding the binomial?

State and apply de Moivre's theorem to evaluate integer powers of complex numbers and derive trigonometric identities

How do we extend the real numbers to solve every polynomial equation?

Represent complex numbers in Cartesian and polar form, perform arithmetic, and apply de Moivre's theorem

How does plotting a complex number reveal its size and direction through modulus and argument?

Represent complex numbers on the Argand plane and find modulus and argument, converting to polar form

Given the graph of y equals f of x, how do we sketch the graph of its reciprocal one over f of x?

Sketch the reciprocal of a function, relating zeros to asymptotes and turning points to turning points

How do transformations and structure determine the shape of a graph?

Sketch rational functions, reciprocal and modulus graphs, and use transformations and asymptotic behaviour

How do the zeros of the numerator and denominator control the shape of a rational function's graph?

Sketch graphs of rational functions, identifying intercepts, vertical, horizontal and oblique asymptotes

How do the two absolute value transformations reshape a graph, and how do we solve modulus equations?

Sketch graphs of the modulus functions y equals the absolute value of f of x and y equals f of the absolute value of x, and solve modulus equations

How do equations and inequalities in z carve out lines, circles and regions in the Argand plane?

Describe and sketch subsets of the complex plane defined by equations and inequalities in modulus and argument

How do we differentiate and integrate the harder functions of Specialist?

Differentiate inverse trig functions, use implicit differentiation, and integrate rational functions, partial fractions and trig forms

How do we describe the position of a moving particle by a vector function and differentiate it to get velocity and acceleration?

Use vector functions of time and differentiate them to find velocity, speed and acceleration along a path

How do vectors describe direction, length and angle in space?

Use 3D vectors with the dot product and cross product to find lengths, angles, projections and areas

How does admitting complex roots let every polynomial factor completely, and how do real polynomials behave?

Factorise polynomials over the complex numbers using the conjugate root theorem and the fundamental theorem of algebra

How does the dot product measure the angle between two three-dimensional vectors and project one onto another?

Compute the scalar (dot) product of vectors in three dimensions and use it for angles, perpendicularity and projections

How do we build a vector perpendicular to two given vectors, and why does its length measure area?

Compute the vector (cross) product in three dimensions and use it to find perpendicular vectors and areas

Why does every nonzero complex number have exactly n distinct nth roots, evenly spaced on a circle?

Find the nth roots of a complex number and the nth roots of unity, and represent them on the Argand plane

Why does multiplying complex numbers rotate and scale, and how does polar form make this obvious?

Multiply and divide complex numbers in polar form, interpreting the result as rotation and scaling

How do the four arithmetic operations work once we admit the number i with i squared equal to negative one?

Perform addition, subtraction, multiplication and division of complex numbers in Cartesian form using the conjugate

How does a vector equation with a parameter trace out a curve, and how do we convert it to cartesian form?

Write vector and parametric equations of curves and convert between vector, parametric and cartesian forms

Unit 4

Unit overview →

Why is the sample mean approximately normal even when the population is not?

State the central limit theorem and use it to compute probabilities for the sample mean

How does the sample mean vary from sample to sample, and what are its mean and standard deviation?

Describe the sampling distribution of the sample mean, including its mean and the standard error

How do we turn one sample mean into an interval estimate for the population mean, and what does the confidence level mean?

Construct and interpret confidence intervals for a population mean and find the required sample size

How do matrices encode and combine linear transformations of the plane?

Use 2x2 matrices for arithmetic, determinants and inverses, and as linear transformations of the plane

How do definite integrals measure the area enclosed between two curves, even when they cross?

Find the area between curves using definite integration, accounting for intersection points and which curve is on top

How does proving a base case and an inductive step establish a statement for every natural number?

Prove statements by mathematical induction, including summation, divisibility and inequality results

How do the exponential and logistic differential equations model unrestricted and limited growth?

Set up and solve the exponential growth-decay equation and the logistic equation and interpret their solutions

How does a field of small line segments reveal the family of solutions to a differential equation without solving it?

Interpret and sketch slope (direction) fields for first-order differential equations and sketch solution curves

How do we split a rational function into simpler fractions so that each piece integrates to a logarithm?

Resolve a rational function into partial fractions and integrate each term

When a differential equation factors into a function of x times a function of y, how do we solve it by separating the variables?

Solve separable first-order differential equations and apply an initial condition to find the particular solution

How do sample means let us make confident statements about a population mean?

Use the distribution of the sample mean and the central limit theorem to build confidence intervals for a population mean

How does integrating the area of circular cross-sections give the volume of a solid of revolution?

Find the volume of a solid generated by rotating a region about the x-axis or y-axis using the disc method

How do vectors describe lines and planes and the relationships between them?

Write vector and parametric equations of lines and planes and find intersections, distances and angles

How do trigonometric identities convert products and powers of sine and cosine into integrable forms?

Integrate trigonometric functions by first applying double-angle, Pythagorean and product identities

How does reversing the chain rule let us integrate functions built by composition?

Integrate by substitution, changing the variable and the limits to evaluate definite and indefinite integrals

Which techniques unlock the integrals and applications of Specialist Unit 4?

Integrate using substitution, partial fractions and trig identities, and apply integration to volumes and differential equations

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