Pure Mathematics Problem-Solving Strategies

Paper 1 is dominated by Further Pure Mathematics — complex numbers, matrices, further algebra, further calculus, further vectors, and proof by induction. This lesson provides targeted strategies, common pitfalls, and worked approaches for the most frequently examined question types.

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Complex Numbers: Key Strategies

Argand Diagram Loci

The most frequently examined complex number topic is loci in the Argand diagram:

Locus	Geometric meaning
$	z - z_1
$	z - z_1
$\arg(z - z_1) = \theta$arg(z−z1​)=θ	Half-line from $z_1$z1​ at angle $\theta$θ to the positive real axis
$	z - z_1

Strategy: Always sketch the locus first. Label the centre, radius, and key points. For combined loci (e.g., $|z - 3| = |z - i|$∣z−3∣=∣z−i∣ and $\arg(z) = \pi/4$arg(z)=π/4), find the intersection by solving simultaneously.

De Moivre's Theorem

$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$(cosθ+isinθ)n=cosnθ+isinnθ