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

Common applications:

1. Finding powers of complex numbers: convert to modulus-argument form first.
2. Finding $n$n-th roots: $z^n = w$zn=w has $n$n roots equally spaced around a circle of radius $|w|^{1/n}$∣w∣1/n.
3. Expressing $\cos n\theta$cosnθ or $\sin n\theta$sinnθ in terms of powers of $\cos\theta$cosθ and $\sin\theta$sinθ.

Strategy for roots: Write $w = re^{i\alpha}$w=reiα and then $z_k = r^{1/n} e^{i(\alpha + 2k\pi)/n}$zk​=r1/nei(α+2kπ)/n for $k = 0, 1, \ldots, n-1$k=0,1,…,n−1.

Exam Tip: When finding the $n$n-th roots of a complex number, you MUST find all $n$n roots. A common error is to find only one. State the arguments of all roots and sketch them on an Argand diagram.

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Matrices: Key Strategies

Eigenvalues and Eigenvectors

Finding eigenvalues: solve $\det(A - \lambda I) = 0$det(A−λI)=0.

For a $2 \times 2$2×2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$A=(ac​bd​):

$\lambda^2 - (a + d)\lambda + (ad - bc) = 0$λ2−(a+d)λ+(ad−bc)=0

$\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$λ2−tr(A)λ+det(A)=0

Strategy: Use the trace and determinant to write down the characteristic equation quickly.

For each eigenvalue $\lambda$λ, solve $(A - \lambda I)\mathbf{v} = \mathbf{0}$(A−λI)v=0 to find the eigenvector.

Matrix Transformations

Matrix	Transformation
$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$(cosθsinθ​−sinθcosθ​)	Rotation by $\theta$θ anticlockwise
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$(10​0−1​)	Reflection in the x-axis
$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$(−10​01​)	Reflection in the y-axis
$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$(k0​0k​)	Enlargement scale factor $k$k

Strategy: For combined transformations, multiply matrices in the correct order: if transformation $A$A is applied first, then $B$B, the combined matrix is $BA$BA (not $AB$AB).

The Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic equation. If $\lambda^2 - p\lambda + q = 0$λ2−pλ+q=0 is the characteristic equation of $A$A, then $A^2 - pA + qI = 0$A2−pA+qI=0.

Application: Use this to express $A^{-1}$A−1 in terms of $A$A and $I$I: $A^{-1} = \frac{1}{q}(pI - A)$A−1=q1​(pI−A).

Exam Tip: The Cayley-Hamilton theorem is frequently used in "show that" questions. State the characteristic equation, substitute the matrix, and verify each entry.

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