Additional Mathematics Formula Book

Section 1: Algebra, Sequences and Series¶

A. Algebra¶

Remainder Theorem: When $f(x)$ is divided by $(x - a)$, the remainder is $f(a)$.

Factor Theorem: $(x - a)$ is a factor of $f(x)$ if and only if $f(a) = 0$.

B. Quadratics¶

For $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Vertex form $a(x + h)^2 + k$, vertex at $(-h,\, k)$:

$h = \frac{b}{2a} \qquad k = c - ah^2$

Discriminant $\Delta = b^2 - 4ac$:

Vieta's formulas for roots $\alpha$ and $\beta$ of $ax^2 + bx + c = 0$:

$\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a}$

Forming a new quadratic from known sum $S$ and product $P$:

$x^2 - Sx + P = 0$

Multiply through by a constant if needed so that $a, b, c \in \mathbb{Z}$.

C. Inequalities¶

Quadratic inequalities: find the critical values (roots), sketch the parabola, read the solution set from the graph.

Rational inequalities $\dfrac{f(x)}{g(x)} \lessgtr k$: rearrange to get zero on one side, find critical values (roots and excluded values), apply a sign diagram.

D. Surds, Indices, and Logarithms¶

Surd rules:

$\sqrt{a} \times \sqrt{b} = \sqrt{ab} \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \qquad \sqrt{a} \times \sqrt{a} = a$

Rationalise using the conjugate: $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$.

Laws of indices:

If $a^m = a^n$ and $a \neq 0,\, 1$, then $m = n$.

Logarithm definition: $a^x = b \iff \log_a b = x \quad (a > 0,\; a \neq 1,\; b > 0)$

Key values: $\log_a 1 = 0$, $\log_a a = 1$.

Laws of logarithms:

To solve $a^x = b$: apply $\log$ to both sides and use the power law: $x = \dfrac{\log b}{\log a}$.

Linearisation:

E. Sequences and Series¶

Sigma notation: $\displaystyle\sum_{r=1}^{n} U_r$ means the sum of $U_r$ from $r = 1$ to $r = n$.

Arithmetic sequences ($a$ = first term, $d$ = common difference, $l$ = last term, $n$ terms):

$T_n = a + (n-1)d \qquad S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$

Geometric sequences ($a$ = first term, $r$ = common ratio):

$T_n = ar^{n-1} \qquad r = \frac{T_{n+1}}{T_n} \qquad S_n = \frac{a(1 - r^n)}{1 - r}$

Sum to infinity (converges when $\lvert r \rvert < 1$ only):

$S_\infty = \frac{a}{1 - r}$

Section 2: Coordinate Geometry, Vectors, and Trigonometry¶

A. Coordinate Geometry¶

For two points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1} \qquad d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$

Parallel lines: equal gradients. Perpendicular lines: $m_1 m_2 = -1$.

Equation of a line: $y - y_1 = m(x - x_1)$

Circle equations:

The tangent at a point on a circle is perpendicular to the radius at that point.

B. Vectors¶

For $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$:

$\lvert\mathbf{v}\rvert = \sqrt{x^2 + y^2} \qquad \hat{\mathbf{v}} = \frac{\mathbf{v}}{\lvert\mathbf{v}\rvert}$

Dot product:

$\begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{pmatrix} c \\ d \end{pmatrix} = ac + bd$

Angle between vectors:

$\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\lvert\mathbf{a}\rvert\lvert\mathbf{b}\rvert}$

• $\mathbf{a} \cdot \mathbf{b} = 0$: vectors are perpendicular.
• $\mathbf{b} = k\mathbf{a}$ for some scalar $k$: vectors are parallel.
• Three points $A$, $B$, $C$ are collinear if $\overrightarrow{AB} = k\overrightarrow{AC}$.

Direction of a vector: $\theta = \arctan\!\left(\dfrac{y}{x}\right)$, then adjust for the correct quadrant (add $180^\circ$ in Q2 or Q3).

C. Trigonometry¶

Conversion:

$\theta^\circ \times \frac{\pi}{180} \to \text{rad} \qquad \theta\;\text{rad} \times \frac{180}{\pi} \to {}^\circ$

Arc and sector ($\theta$ in radians):

$l = r\theta \qquad A_{\text{sector}} = \frac{1}{2}r^2\theta \qquad A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$

Exact values:

CAST rule: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4) are positive.

Graph properties:

Pythagorean identities:

$\sin^2\theta + \cos^2\theta = 1 \qquad \tan^2\theta + 1 = \sec^2\theta$

Compound angle formulas:

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Double angle formulas:

$\sin 2A = 2\sin A\cos A$

$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$

$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

Section 3: Introductory Calculus¶

A. Differentiation¶

Chain rule (let $u = g(x)$, $y = f(u)$):

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Product rule ($y = uv$):

$\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$

Quotient rule ($y = \dfrac{u}{v}$):

$\frac{dy}{dx} = \frac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}$

Stationary points: set $f'(x) = 0$, solve for $x$, find $y$.

Tangent and normal at $(x_0, y_0)$: tangent gradient $m_T = f'(x_0)$; normal gradient $m_N = -\dfrac{1}{m_T}$.

Connected rates: $\dfrac{dy}{dt} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}$

B. Integration¶

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + c \qquad (n \neq -1)$

$\int (ax + b)^n\,dx = \frac{(ax + b)^{n+1}}{a(n+1)} + c$

$\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax) + c \qquad \int \cos(ax)\,dx = \frac{1}{a}\sin(ax) + c$

Area under a curve between $x = a$ and $x = b$:

$A = \int_a^b y\,dx$

Area between two curves ($f(x) \geq g(x)$ on $[a,b]$):

$A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx$

Volume of revolution about the $x$-axis:

$V = \pi\int_a^b y^2\,dx$

Kinematics¶

Displacement $s$, velocity $v$, acceleration $a$, time $t$:

$v = \frac{ds}{dt} \qquad a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

$s = \int v\,dt \qquad v = \int a\,dt$

SUVAT equations (uniform acceleration only; $u$ = initial velocity, $s$ = displacement):

$v = u + at \qquad s = ut + \tfrac{1}{2}at^2 \qquad v^2 = u^2 + 2as \qquad s = \tfrac{1}{2}(u + v)t$

Particle is stationary when $v = 0$. Direction changes when $v$ changes sign.

Section 4: Probability and Statistics¶

A. Data Representation and Analysis¶

Mean:

$\bar{x} = \frac{\sum x}{n} \quad \text{(ungrouped)} \qquad \bar{x} = \frac{\sum fx}{\sum f} \quad \text{(grouped)}$

Variance and standard deviation:

$S^2 = \frac{\sum(x - \bar{x})^2}{n} \qquad S = \sqrt{\frac{\sum(x - \bar{x})^2}{n}} \quad \text{(ungrouped)}$

$S^2 = \frac{\sum f(x - \bar{x})^2}{\sum f} \quad \text{(grouped)}$

Measures of spread:

$\text{Range} = x_{\max} - x_{\min} \qquad \text{IQR} = Q_3 - Q_1 \qquad \text{Semi-IQR} = \frac{Q_3 - Q_1}{2}$

Skewness:

B. Probability Theory¶

$P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}} \qquad P(A') = 1 - P(A) \qquad 0 \leq P(A) \leq 1$

Two non-zero-probability events cannot be simultaneously mutually exclusive and independent.

Tree diagrams: multiply along branches for intersection probabilities; add across branches for union. The syllabus restricts trees to two initial branches.

Venn diagrams: restricted to two sets; the four regions ($A$ only, $B$ only, $A \cap B$, neither) must sum to $P(S) = 1$.