Module 12 Synthesis & Exam Technique

Every Module 12 question can be tackled by choosing one of three lenses:

1. Algebraic (rectangular). $z = a + bi$. Use when adding, subtracting, equating real/imaginary parts, finding conjugates, or testing rectangular conditions like $\Re(z) = 1$.
2. Geometric (modulus & argument / polar). $z = r(\cos\theta + i\sin\theta)$. Use when multiplying, dividing, raising to powers, computing rotations, or working with $|z - z_0|$ and $\arg(z - z_0)$.
3. Geometric (loci on the Argand plane). Sketch first; describe the curve in words; only then write algebra. Useful whenever a condition involves $z$ as a moving point.

Loci library — memorise these:

• $|z - z_0| = r$ — circle, centre $z_0$, radius $r$.
• $|z - z_0| < r$ — open disc; $|z - z_0| \leq r$ — closed disc.
• $\arg(z - z_0) = \alpha$ — ray from $z_0$ (excluding $z_0$) at angle $\alpha$ to the positive real axis.
• $|z - z_1| = |z - z_2|$ — perpendicular bisector of segment from $z_1$ to $z_2$.
• $\Re(z) = a$ — vertical line $x = a$; $\Im(z) = b$ — horizontal line $y = b$.
• $|z| = a|z - z_0|$ (with $a \neq 1$) — circle (Apollonius); $a = 1$ gives perpendicular bisector instead.

Worked through the hook. $|z - 2| = |z + 2i|$ asks "which $z$ are equidistant from $2$ and $-2i$?" Geometrically, this is the perpendicular bisector of the segment from $(2, 0)$ to $(0, -2)$. The midpoint is $(1, -1)$; the segment has slope $1$, so the bisector has slope $-1$ and passes through $(1, -1)$: $y - (-1) = -1(x - 1)$, i.e. $y = -x$, i.e. $x + y = 0$, or in complex form $\Re(z) + \Im(z) = 0$.

Three forms: rectangular $a + bi$, polar $r(\cos\theta + i\sin\theta)$, exponential $r e^{i\theta}$ · +/− → rectangular; $\times$, $\div$, $^n$ → polar · $|z - z_0| = r$ circle; $\arg(z - z_0) = \alpha$ ray; $|z - z_1| = |z - z_2|$ perpendicular bisector · Sketch the Argand plane before doing algebra on a locus

Pause — copy the three-form summary (rectangular, polar, exponential), the operation-to-form mapping ($+/-$ → rect; $\times,\div,z^n$ → polar), and the three main locus types into your book.

We just saw the Module 12 framework: rectangular for $+/-$; polar for $\times, \div, z^n$; the loci $|z-z_0|=r$, $\arg(z-z_0)=\alpha$, $|z-z_1|=|z-z_2|$; always sketch first. That raises a question: what three habits differentiate a full-mark response from a near-miss in an HSC exam? This card answers it → state the form, sketch the Argand plane, and choose polar for products/powers and rectangular for sums.

Marking guides for HSC complex-number questions reward three habits beyond the algebra itself:

• State the form clearly. Write "$z = a + bi$ where $a, b \in \mathbb{R}$" or "$z = r(\cos\theta + i\sin\theta)$" at the top. This signals to the marker which approach you are taking, and prevents you from drifting between forms.
• Sketch the Argand plane for any locus. Even if the question doesn't say "sketch", drawing a small labelled diagram clarifies your thinking. Axes labelled $\Re(z)$, $\Im(z)$; fixed points $z_0$ marked; the locus curve drawn solid (or dashed if strict inequality).
• Conversion strategy. If the question gives you rectangular and asks for an argument or power, convert to polar first. If it gives you polar and asks for $\Re(z)$ or $\Im(z)$, expand back to rectangular. The "wrong form for the operation" is the single most common source of wasted marks.

State the form: "$z = a + bi$, $a, b \in \mathbb{R}$" or "$z = r(\cos\theta + i\sin\theta)$" · Sketch Argand plane for any locus or geometric condition · Polar form for products/quotients/powers; rectangular for sums/differences · $z_1 z_2$: multiply moduli, add arguments. $z_1/z_2$: divide moduli, subtract arguments

Pause — copy the three HSC habits (state form, sketch, choose polar/rect by operation) and the product/quotient rules ($z_1 z_2$: multiply moduli, add arguments; $z_1/z_2$: divide moduli, subtract arguments) into your book.