Further Pure Mathematics 1

Board Coverage

Board	Paper	Notes
AQA	Paper 2	Complex numbers, matrices, series, proof
Edexcel	FP1	Complex numbers, matrices, polar coords, proof by induction
OCR (A)	Pure Core 1	Complex numbers, matrices, polar coords, hyperbolic
CIE (9709)	Paper 3	Complex numbers, polars, further calculus, induction

:::info This content sits at the transition between single A-Level Mathematics and Further Mathematics. You must be confident with all Core Pure topics before tackling these.

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1. Complex Numbers

1.1 Argand Diagram

A complex number $z = x + iy$ is represented as the point $(x, y)$ on the Argand diagram (a modified Cartesian plane where the horizontal axis is the real axis and the vertical axis is the imaginary axis).

• The modulus $|z| = r = \sqrt{x^2 + y^2}$ is the distance from the origin to $(x, y)$.
• The argument $\arg(z) = \theta$ is the angle from the positive real axis, measured anticlockwise. Principal argument: $-\pi < \theta \leq \pi$.

$z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}$

1.2 Modulus-Argument Form

Multiplication: $z_1 z_2 = r_1 r_2\,e^{i(\theta_1 + \theta_2)}$. Moduli multiply; arguments add.

Division: $\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2}\,e^{i(\theta_1 - \theta_2)}$. Moduli divide; arguments subtract.

Modulus properties:

$|z_1 z_2| = |z_1|\,|z_2| \qquad |z_1 + z_2| \leq |z_1| + |z_2| \qquad |z^n| = |z|^n$

1.3 de Moivre’s Theorem

For integer $n$:

$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

Equivalently: $(e^{i\theta})^n = e^{in\theta}$.

Applications:

• Raising complex numbers to large powers
• Trigonometric identities (equating real and imaginary parts)

1.4 Roots of Unity

The $n$th roots of unity are the solutions to $z^n = 1$:

$z_k = e^{2\pi i k / n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k = 0, 1, 2, \ldots, n-1$

They lie on the unit circle at vertices of a regular $n$-gon. The sum of all $n$th roots of unity is $0$.

Example. The cube roots of unity are $1$, $\omega = e^{2\pi i/3}$, and $\omega^2$. They satisfy $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.

1.5 Loci in the Complex Plane

Circle: $|z - z_0| = r$ represents a circle centred at $z_0$ with radius $r$.

Perpendicular bisector: $|z - z_1| = |z - z_2|$ is the perpendicular bisector of the segment joining $z_1$ and $z_2$.

Half-line (ray): $\arg(z - z_0) = \alpha$ is a half-line from $z_0$ making angle $\alpha$ with the positive real axis.

Region inequalities: $|z - z_0| < r$ is the interior of a circle; $\arg(z - z_0) < \alpha$ is a region bounded by a half-line.

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2. Matrices

2.1 Matrix Operations

Addition: $A + B$ elementwise (same dimensions required).

Scalar multiplication: $kA$ multiplies every entry.

Matrix multiplication: $(AB)_{ij} = \sum_k A_{ik} B_{kj}$. Number of columns of $A$ must equal number of rows of $B$. As a general principle, $AB \neq BA$.

Identity: $I_n$ has $1$s on the diagonal and $0$s elsewhere. $AI = IA = A$.

Determinant of $2 \times 2$:

$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

Inverse of $2 \times 2$:

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

A matrix has an inverse if and only if its determinant is non-zero (it is non-singular).

2.2 Determinant of $3 \times 3$

Expand by cofactors along any row or column. Along row 1:

$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix}$

2.3 Inverse of $3 \times 3$

Form the matrix of cofactors, transpose it (giving the adjugate), then divide by the determinant:

$A^{-1} = \frac{1}{|A|}\text{adj}(A)$

Solving $A\mathbf{x} = \mathbf{b}$: $\mathbf{x} = A^{-1}\mathbf{b}$ (unique solution when $|A| \neq 0$).

2.4 Transformations in 2D

A matrix $M$ represents a linear transformation. The image of point $(x, y)$ is $M\begin{pmatrix} x \\ y \end{pmatrix}$.

Transformation	Matrix
Reflection in $y = x$	$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
Rotation $\theta$ anticlockwise	$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Enlargement scale factor $k$	$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$
Reflection in $x$-axis	$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Area scale factor $= |\det(M)|$.

2.5 Transformations in 3D

$3 \times 3$ matrices represent transformations in 3D space. Key examples:

• Reflection in $xy$-plane: $\begin{pmatrix} 1&0&0\\0&1&0\\0&0&-1 \end{pmatrix}$
• Rotation about the $z$-axis: extend the 2D rotation matrix to $3 \times 3$ with a $1$ in position $(3,3)$

Volume scale factor $= |\det(M)|$.

2.6 Invariant Points and Lines

An invariant point satisfies $M\mathbf{x} = \mathbf{x}$, i.e. $(M - I)\mathbf{x} = \mathbf{0}$.

An invariant line through the origin satisfies $M\mathbf{x} = \lambda\mathbf{x}$ for some scalar $\lambda$. This means finding the eigenvectors of $M$.

A line invariant as a set means every point on the line maps to some point on the same line (not necessarily itself).

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3. Further Calculus

3.1 Integration by Parts

$\int u\,dv = uv - \int v\,du$

Choosing $u$ and $dv$: Use the LIATE priority: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Choose $u$ from higher priority.

Repeated integration by parts is needed when the integrand includes products like $x^n e^{ax}$, $x^n \sin ax$, or $x^n \cos ax$.

3.2 Standard Integrals

$\int \frac{1}{x^2 + a^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C \qquad \int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{-1}\frac{x}{a} + C$

$\int \frac{1}{x^2 - a^2}\,dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$

3.3 First Order Differential Equations

Separable form: $\dfrac{dy}{dx} = f(x)g(y)$. Rearrange: $\dfrac{1}{g(y)}\,dy = f(x)\,dx$, then integrate both sides.

Integrating factor method for $\dfrac{dy}{dx} + P(x)y = Q(x)$:

$\text{Integrating factor } \mu = e^{\int P(x)\,dx}$

$y \cdot \mu = \int Q(x)\mu\,dx + C$

General solution: contains an arbitrary constant $C$.

Particular solution: found by substituting initial conditions to determine $C$.

3.4 Second Order Differential Equations

Homogeneous equation: $a\dfrac{d^2y}{dx^2} + b\dfrac{dy}{dx} + cy = 0$

Auxiliary equation: $am^2 + bm + c = 0$

Case 1 — Distinct real roots $m_1, m_2$:

$y = Ae^{m_1 x} + Be^{m_2 x}$

Case 2 — Repeated root $m$:

$y = (Ax + B)e^{mx}$

Case 3 — Complex roots $\alpha \pm i\beta$:

$y = e^{\alpha x}(A\cos\beta x + B\sin\beta x)$

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4. Polar Coordinates

4.1 Converting Between Cartesian and Polar

A point with Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$:

$x = r\cos\theta \qquad y = r\sin\theta \qquad r = \sqrt{x^2 + y^2} \qquad \tan\theta = \frac{y}{x}$

4.2 Sketching Polar Curves

Given $r = f(\theta)$:

1. Determine the range of $\theta$ (often $0 \leq \theta \leq 2\pi$ or a subset).
2. Find where $r = 0$ and maxima of $r$.
3. Compute a table of $(r, \theta)$ values at key angles ($0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, $3\pi/2$, $2\pi$).
4. Plot each point and join smoothly.

Common curves:

Equation	Shape
$r = a$	Circle, radius $a$, centre at origin
$r = a\theta$	Archimedean spiral
$r = a(1 + \cos\theta)$	Cardioid
$r^2 = a^2\cos 2\theta$	Lemniscate

4.3 Area Enclosed by a Polar Curve

$A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$

For a full curve traced once as $\theta$ goes from $\alpha$ to $\beta$, substitute the full range.

Sector area between two values $\theta_1$ and $\theta_2$:

$A = \frac{1}{2}\int_{\theta_1}^{\theta_2} r^2\,d\theta$

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5. Hyperbolic Functions

5.1 Definitions

$\cosh x = \frac{e^x + e^{-x}}{2} \qquad \sinh x = \frac{e^x - e^{-x}}{2} \qquad \tanh x = \frac{\sinh x}{\cosh x}$

Key values: $\cosh 0 = 1$, $\sinh 0 = 0$, $\tanh 0 = 0$.

5.2 Identities

$\cosh^2 x - \sinh^2 x = 1$ $1 - \tanh^2 x = \text{sech}^2\, x$ $\coth^2 x - 1 = \text{cosech}^2\, x$

Compound angle (Osborn’s rule): Replace every $\sin^2$ with $-\sinh^2$ in a standard trig identity.

$\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y$ $\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y$

5.3 Calculus of Hyperbolic Functions

$\frac{d}{dx}\sinh x = \cosh x \qquad \frac{d}{dx}\cosh x = \sinh x \qquad \frac{d}{dx}\tanh x = \text{sech}^2\, x$

$\int \cosh x\,dx = \sinh x + C \qquad \int \sinh x\,dx = \cosh x + C$

5.4 Inverse Hyperbolic Functions

$\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right) \qquad \cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right)\text{ for } x \geq 1$

$\tanh^{-1} x = \frac{1}{2}\ln\frac{1+x}{1-x}\text{ for } |x| < 1$

Derivatives:

$\frac{d}{dx}\sinh^{-1} x = \frac{1}{\sqrt{1+x^2}} \qquad \frac{d}{dx}\cosh^{-1} x = \frac{1}{\sqrt{x^2-1}}$

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6. Proof

6.1 Proof by Induction (Advanced)

Steps:

1. Base case: Show the statement holds for $n = n_0$ (in most cases $n = 1$).
2. Inductive hypothesis: Assume the statement holds for $n = k$.
3. Inductive step: Show that if it holds for $n = k$, it also holds for $n = k + 1$.
4. Conclusion: By the principle of mathematical induction, the statement holds for all $n \geq n_0$.

Common applications:

• Summation formulas: $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$
• Divisibility: proving $3^{2n} - 1$ is divisible by 8
• Inequalities: $2^n > n^2$ for $n \geq 5$
• De Moivre’s theorem for positive integers
• Matrix powers: $A^n$ for a given matrix $A$

6.2 Proof by Contradiction

1. Assume the negation of the statement to be proved.
2. Derive a logical contradiction.
3. Conclude the original statement must be true.

Classic examples:

• $\sqrt{2}$ is irrational
• There are infinitely many primes
• $\log_2 3$ is irrational

6.3 Counterexamples

A single counterexample is sufficient to disprove a universal claim.

Example. “All prime numbers are odd.” Counterexample: $2$ is prime and even.

Strategy: For disproof, try small values, boundary cases, and special cases to find a counterexample.

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7. Common Mistakes

1. Wrong principal argument. $\arg(z)$ must satisfy $-\pi < \theta \leq \pi$. Points in the third and fourth quadrants need negative arguments.
2. Forgetting $r$ when multiplying complex numbers in polar form. $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$, not $e^{i(\theta_1+\theta_2)}$.
3. Matrix multiplication is not commutative. $AB \neq BA$ as a general principle. Always maintain order.
4. Incorrect area formula for polar curves. It is $\frac{1}{2}\int r^2\,d\theta$, not $\int r\,d\theta$.
5. Confusing trig and hyperbolic identities. The key difference: $\cosh^2 x - \sinh^2 x = 1$ (minus, not plus). Use Osborn’s rule carefully.
6. Incomplete induction step. You must explicitly show how the $n = k$ hypothesis is used to establish the $n = k + 1$ case. Directly writing the $n = k + 1$ expression is not enough.
7. Incorrect auxiliary equation roots. For repeated roots, the solution is $(Ax + B)e^{mx}$, not $Ae^{mx} + Be^{mx}$ (which is only valid for distinct real roots).

Worked Examples

Example 1: Solving a Second Order Differential Equation

Problem: Solve $\dfrac{d^2y}{dx^2} - 5\dfrac{dy}{dx} + 6y = 0$ given $y(0) = 1$ and $y'(0) = 5$. Solution: Auxiliary equation: $m^2 - 5m + 6 = 0$, giving $m = 2$ or $m = 3$ (distinct real roots). $y = Ae^{2x} + Be^{3x}$ $y(0) = A + B = 1$. $y' = 2Ae^{2x} + 3Be^{3x}$, so $y'(0) = 2A + 3B = 5$. Solving: $B = 3$, $A = -2$. Solution: $y = 3e^{3x} - 2e^{2x}$.

Example 2: Polar Curve Area

Problem: Find the area enclosed by one loop of the curve $r = a\sin 2\theta$. Solution: One loop is traced as $\theta$ goes from $0$ to $\pi/2$ (where $r$ returns to zero). $A = \frac{1}{2}\int_0^{\pi/2} a^2\sin^2 2\theta\,d\theta = \frac{a^2}{2}\int_0^{\pi/2}\frac{1 - \cos 4\theta}{2}\,d\theta = \frac{a^2}{4}\left[\theta - \frac{\sin 4\theta}{4}\right]_0^{\pi/2} = \frac{a^2\pi}{8}$

Common Pitfalls

• Confusing the principal argument: For points in the third and fourth quadrants, the argument must be given as a negative angle (e.g., $-3\pi/4$, not $5\pi/4$).
• Matrix multiplication order: $AB \neq BA$. When solving $A\mathbf{x} = \mathbf{b}$, compute $A^{-1}\mathbf{b}$, not $\mathbf{b}A^{-1}$.
• Wrong polar area formula: Use $\frac{1}{2}\int r^2\,d\theta$, not $\int r\,d\theta$. The extra $r/2$ factor is frequently missed.

Summary

Further Pure 1 covers complex numbers (modulus-argument form, de Moivre, roots of unity, loci), matrices (operations, determinants, transformations, invariants), further calculus (integration by parts, standard integrals, first and second order ODEs), polar coordinates, hyperbolic functions, and proof techniques (induction, contradiction, counterexamples). Mastery of these topics is essential before progressing to Further Pure 2.

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