§-Specialist Mathematics syllabus

Specialist Mathematics syllabus, dot point by dot point

Every dot point in the VIC Specialist Mathematics study design, 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: Calculus

Unit overview →

How do we differentiate a relation that is not written as $y = f(x)$ , and how does the second derivative reveal concavity and points of inflection?

Implicit differentiation of relations defined by equations in

x

and

y

, the second derivative and its use to determine concavity and points of inflection, and the analysis of curves using first and second derivative information

7 min read →

How do we describe lines and planes in three dimensions using vectors, and how do we move between vector, parametric and Cartesian forms?

Vector equations of lines and planes in three dimensions, their parametric and Cartesian forms, the use of a direction vector for a line and a normal vector for a plane, and the determination of intersections and the angle between a line and a plane

6 min read →

How do we add, scale and take products of vectors in two and three dimensions, and what do the scalar and vector products tell us geometrically?

Vectors in two and three dimensions in

\mathbf{i}, \mathbf{j}, \mathbf{k}

form, magnitude and unit vectors, the scalar (dot) product and the angle between vectors, vector projection, and the use of the scalar product to test for perpendicular and parallel vectors

6 min read →

How does De Moivre's theorem let us compute powers and roots of complex numbers, and how do the roots arrange themselves on the Argand plane?

De Moivre's theorem

(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)

for integer

n

, its use in finding powers and the

n

distinct

n

th roots of a complex number, and the factorisation of polynomials over the complex numbers

6 min read →

How are the inverse circular functions defined by restricting the domains of sine, cosine and tangent, and what are the resulting domains, ranges and graphs?

The inverse circular functions

\arcsin

\arccos

and

\arctan

, the domain restrictions needed to define them, their domains, ranges and graphs, and the evaluation of exact values and composite expressions

7 min read →

How do the reciprocal, absolute value of the function, and absolute value of the variable transformations reshape a known graph, and what features are preserved?

The transformations

y = \frac{1}{f(x)}

y = |f(x)|

and

y = f(|x|)

applied to a known graph

y = f(x)

, the effect on intercepts, asymptotes, turning points and symmetry, and the sketching of the resulting curves

7 min read →

How do we identify the asymptotes, intercepts and key features of rational functions, and how do reciprocal and absolute value transformations reshape a graph?

Graphs of rational functions including reciprocal functions, the location of vertical, horizontal and oblique asymptotes, and the effect of reciprocal and modulus transformations on the shape and key features of a graph

7 min read →

What are the main methods of mathematical proof beyond induction, and when is each appropriate for establishing or refuting a statement?

Methods of proof including direct proof, proof by contrapositive, proof by contradiction, and the use of a single counterexample to disprove a universal statement, together with the language of quantifiers and implication

6 min read →

How is the cross product of two three-dimensional vectors defined, and how does it give a perpendicular vector, an area and a test for parallel vectors?

The vector (cross) product of two three-dimensional vectors, its definition in component form, the geometric meaning of its direction and magnitude, and its applications to finding a normal vector, the area of a parallelogram or triangle, and testing for parallel vectors

6 min read →

How do we solve polynomial equations completely over the complex numbers, and how does the conjugate root theorem constrain the roots of a real polynomial?

Solution of polynomial equations over the complex numbers, the fundamental theorem of algebra, the conjugate root theorem for polynomials with real coefficients, and the full factorisation of real polynomials into linear and irreducible quadratic factors

6 min read →

How does the principle of mathematical induction let us prove a statement true for every positive integer, and what are the essential parts of a valid induction proof?

The principle of mathematical induction and its use to prove propositions about positive integers, including the base step, the inductive assumption and the inductive step, applied to summation formulas, divisibility results and inequalities

6 min read →

How are complex numbers represented and manipulated in Cartesian and polar form, and how does the Argand plane connect the two?

Arithmetic and algebra of complex numbers in Cartesian form

z = a + bi

and polar form

z = r\,\mathrm{cis}\,\theta

, the modulus and argument, conjugates, and representation of complex numbers and their operations on the Argand plane

6 min read →

How do equations and inequalities involving the modulus and argument of a complex number describe lines, circles, rays and regions on the Argand plane?

Description and sketching of subsets of the complex plane defined by conditions on modulus and argument, including circles

|z - z_0| = r

, perpendicular bisectors

|z - a| = |z - b|

, rays

\arg(z - z_0) = \alpha

, and the regions defined by the corresponding inequalities

6 min read →

How are ellipses and hyperbolas described by their Cartesian equations, and how do parametric equations trace out a curve as a parameter varies?

The equations and key features of ellipses and hyperbolas, including centre, vertices, axes and asymptotes, and the description of curves by parametric equations together with conversion between parametric and Cartesian forms

6 min read →

Unit 4: Calculus

Unit overview →

What are the derivatives of the inverse circular functions, and how do we use the chain rule to differentiate composite expressions involving arcsin, arccos and arctan?

Differentiation of the inverse circular functions

\arcsin

\arccos

and

\arctan

, the standard derivative results, the use of the chain rule for composite forms, and the related standard antiderivatives

7 min read →

How do we test a claim about a population mean using a sample, and what does the p value tell us about the strength of the evidence?

Hypothesis testing for a population mean, the null and alternative hypotheses, one-tailed and two-tailed tests, the test statistic and its

p

value, the comparison with a significance level, the decision and its interpretation, and the meaning of Type I and Type II errors

6 min read →

How do the mean and variance of linear combinations of random variables behave, and how do we use a sample mean to test a hypothesis about a population mean?

Linear combinations of independent random variables and their mean and variance, the distribution of the sample mean

\bar{X}

, the construction of confidence intervals for a population mean, and hypothesis testing for the mean using a

p

value

7 min read →

How do momentum and impulse describe changes in motion, and how do we analyse connected bodies that share a common acceleration?

Momentum

p = mv

and impulse as the change in momentum, the impulse-momentum relationship, and the analysis of connected particles such as bodies linked by a string over a pulley or in contact, which share a common acceleration

7 min read →

How do we relate the rates of change of two connected quantities, and how does the chain rule link the rate we want to the rate we know?

7 min read →

How does a slope field picture the solutions of a differential equation, and how does Euler's method generate a numerical approximation to a particular solution?

Slope (direction) fields as a representation of a first-order differential equation, the sketching of solution curves on a slope field, and Euler's method for the numerical approximation of a solution from an initial condition with a chosen step size

6 min read →

How do we differentiate and integrate a vector function of time, and how does this describe the position, velocity and acceleration of a particle moving in a plane or space?

Vector functions of a real variable, the differentiation and integration of a position vector

\mathbf{r}(t)

to obtain velocity and acceleration, the speed as the magnitude of velocity, and the application to motion in two and three dimensions

6 min read →

How is the sample mean distributed across repeated samples, and why does the central limit theorem make it approximately normal?

The distribution of the sample mean

\bar{X}

as a random variable, its mean and standard deviation (the standard error), the effect of sample size, and the central limit theorem giving the approximate normality of

\bar{X}

for large samples

6 min read →

How does calculus describe the motion of a particle along a line, and how do we move between position, velocity and acceleration including when acceleration depends on velocity or position?

Application of calculus to rectilinear motion, the relationships between position, velocity and acceleration including the forms

a = \frac{\mathrm{d}v}{\mathrm{d}t} = v\frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x}(\tfrac12 v^2)

, and the use of these to analyse motion with variable acceleration

7 min read →

How do we set up and solve first-order differential equations, and what does a solution curve through a given initial condition represent?

Formulation and solution of first-order differential equations including those solvable by direct integration and by separation of variables, the use of initial conditions to find particular solutions, and the interpretation of solutions in modelling contexts

7 min read →

Which techniques let us integrate functions that the standard antiderivatives cannot handle directly, and how do we choose between substitution, partial fractions and trigonometric methods?

Antidifferentiation techniques including integration by substitution, the use of partial fractions, trigonometric identities and inverse-trigonometric standard forms, and the evaluation of definite integrals using these techniques

8 min read →

How does integration give the length of a curve and the surface area generated when a curve is rotated, in both Cartesian and parametric settings?

The use of definite integrals to find the arc length of a curve and the surface area of a solid of revolution, in Cartesian form

y = f(x)

and in parametric form

x = x(t)

y = y(t)

, and the setting up of the appropriate integral

7 min read →

How do Newton's laws relate the forces on a body to its acceleration, and how do we resolve forces to analyse motion and equilibrium?

Newton's laws of motion, the resultant of forces acting on a particle, the resolution of forces into components, the relationship

\mathbf{F} = m\mathbf{a}

, and the analysis of equilibrium and of motion under constant forces including weight, normal reaction and friction

6 min read →

How do we build an approximate confidence interval for a population mean from a sample, and what does the confidence level actually mean?

Construction and interpretation of approximate confidence intervals for a population mean using the sample mean and standard error, the choice of confidence level and its

z

value, the effect of sample size on the interval width, and the correct interpretation of a confidence interval

6 min read →

How does integration give the volume of a solid formed by rotating a region about an axis, and how do we choose between disc, washer and shell setups?

The use of definite integrals to find the volume of a solid of revolution generated by rotating a region about the

x

-axis or

y

-axis, using the disc and washer (annulus) methods, and the setting up of the appropriate integral

7 min read →