§-Section

TASSpecialist Mathematics← Specialist Mathematics syllabus

Unit 3

14 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do equations and inequalities in z describe lines, circles and regions on the Argand diagram?

Sketch curves and regions in the complex plane defined by modulus and argument conditions.

Loci from modulus and argument conditions, circles, perpendicular bisectors, rays and shaded regions on the Argand diagram, with worked examples for TCE Mathematics Specialised Unit 3.

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How do we sketch rational functions by finding their intercepts and asymptotes?

Sketch rational functions, identifying vertical, horizontal and oblique asymptotes.

Sketching rational functions using intercepts, vertical asymptotes, horizontal and oblique asymptotes via division, and sign analysis, with worked examples for TCE Mathematics Specialised Unit 3.

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How do translations, dilations and reflections build complex graphs from simple ones?

Sketch graphs using transformations and the addition of ordinates.

Translations, dilations and reflections of graphs, the order of transformations, and addition of ordinates, with worked examples for TCE Mathematics Specialised Unit 3.

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How do complex numbers extend the real number system and let us solve every polynomial?

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

Cartesian and polar forms of complex numbers, modulus-argument arithmetic, De Moivre's theorem and roots, for TCE Mathematics Specialised Unit 3.

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How does the graph of one over f of x relate to the graph of f?

Sketch the reciprocal of a function, relating its features to those of the original function.

Sketching y equals one over f of x from the graph of f, using zeros, asymptotes, turning points and sign, with worked examples for TCE Mathematics Specialised Unit 3.

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How do we sketch and solve problems with the absolute value function?

Sketch graphs involving absolute value and solve absolute value equations and inequalities.

The absolute value function, sketching y equals the modulus of f of x, and solving absolute value equations and inequalities, with worked examples for TCE Mathematics Specialised Unit 3.

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How does plotting complex numbers on the Argand diagram reveal the geometry of arithmetic?

Represent complex numbers and their operations geometrically on the Argand diagram.

Plotting complex numbers, geometric meaning of addition, multiplication, conjugates and modulus, and modulus and argument inequalities, for TCE Mathematics Specialised Unit 3.

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How do we combine functions and reverse them while keeping domain and range correct?

Form composite and inverse functions, determining the correct domain and range of each.

Composition of functions, the existence of inverses, finding inverse functions and restricting domains, with attention to domain and range, for TCE Mathematics Specialised Unit 3.

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How do we describe position, direction, lines and planes in three-dimensional space using vectors?

Use 3D vectors with dot and cross products to find angles, projections, lines and planes.

Three-dimensional vectors: magnitude, dot and cross products, angles, scalar projection, and equations of lines and planes for TCE Mathematics Specialised Unit 3.

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How do we add, multiply, divide and conjugate complex numbers reliably in Cartesian form?

Perform complex arithmetic in Cartesian form and use conjugate properties to divide and simplify.

Adding, multiplying and dividing complex numbers in Cartesian form, conjugate properties, and realising denominators, with worked examples for TCE Mathematics Specialised Unit 3.

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How does allowing complex roots let us factorise every polynomial completely?

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

The fundamental theorem of algebra, conjugate root theorem, and factorising real and complex polynomials into linear and quadratic factors, for TCE Mathematics Specialised Unit 3.

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What are the nth roots of unity and why do they lie equally spaced on the unit circle?

Find the nth roots of unity and use their symmetry and algebraic properties.

Solving z to the n equals one, the geometry of equally spaced roots on the unit circle, and the sum and product properties of roots of unity, for TCE Mathematics Specialised Unit 3.

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How can we prove a statement is true for every positive integer using induction?

Prove statements for all positive integers using the principle of mathematical induction.

The principle of mathematical induction, the base case and inductive step, and proofs for sums, divisibility and inequalities, with worked examples for TCE Mathematics Specialised Unit 3.

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How do we differentiate relations that are not given as explicit functions of x?

Apply implicit and parametric differentiation and use related rates to solve problems.

Implicit differentiation, parametric differentiation and related rates of change, with fully worked examples and common pitfalls for TCE Mathematics Specialised.

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