Arithmetic, geometric representation, De Moivre's theorem, polynomials over the complex numbers, and loci.
These are the statements the Mathematics Extension 2 syllabus makes about Complex Numbers. Dux records each one against the questions you answer, so you can see which of them you have actually been tested on and which have never been asked.
The need for a number whose square is negative, the arithmetic of the complex field, and conjugates.
Addition, multiplication and division in Cartesian form, and the algebraic identities that follow from conjugates.
The Argand diagram, modulus and argument, and the geometric meaning of addition and multiplication.
Powers and roots of complex numbers in polar form, and the geometry of the nth roots of unity.
The conjugate root theorem, factorising real polynomials over the complex numbers, and the fundamental theorem of algebra.
Using complex numbers to establish trigonometric identities and to solve geometric problems.
Describing and sketching loci given by modulus and argument conditions, including circles, rays and regions.
Conjugate root pairs, factorising over the complex numbers, and the relationship between roots and coefficients.
HSC Mathematics Extension 2 - Trial Paper Shape is 100 marks in 180 minutes. Questions on Complex Numbers are not one shape repeated: the command word decides what the marker is looking for, and the mark value decides how much of it. These are the ones Mathematics Extension 2 uses.
Dux writes practice for this module in exactly these shapes, so the drill is the same task the paper sets — not a general knowledge quiz with the module name on it.
Complex Numbers is module 3 of 6 in Mathematics Extension 2. Students routinely lose marks because a question spans two modules, so the neighbouring modules are worth reading next to this one.
Dux diagnoses Mathematics Extension 2 dot point by dot point, then writes questions in the shape this course marks. Written answers come back marked against the bands.