A complete, syllabus-aligned formula book for CSEC Additional Mathematics. Each formula sits in its proper section with a short, clear note on when and why to reach for it. Built to be searched, scanned, and trusted in the final stretch.
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Formulas
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Syllabus-Aligned
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For Every Student
Kerwin Springer · The Student Hub
How to read this book
Given Printed on the CSEC formula sheet
Memorize Not on the sheet — commit it to memory
Optional Helpful but not required by the syllabus
01
Section One
Algebra, Sequences & Series
The foundation of CSEC Additional Mathematics. Master the algebraic toolkit — quadratics, polynomials, indices, logarithms, sequences — and the rest of the course follows naturally.
Quadratics
A quadratic takes the form $ax^2 + bx + c = 0$ where $a \ne 0$. You will see it everywhere: algebra, parabolas, motion problems, and inside trigonometric equations.
Quadratic FormulaMemorize
$$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
UseWhen the quadratic does not factorise cleanly, or when exact roots are required.
DiscriminantMemorize
$$D = b^2 - 4ac$$
Reads$D > 0$: two distinct real roots $D = 0$: one real repeated root $D < 0$: no real roots
UseFor problems involving symmetric facts about the roots — sums, products, reciprocals.
Build a Quadratic from its RootsMemorize
$$x^2 - (\alpha + \beta)\,x + \alpha\beta = 0$$
NoteThe middle coefficient is the negative of the sum. A small sign — a frequent slip.
Completed Square FormMemorize
$$f(x) = a(x + h)^2 + k\,,\quad h = \dfrac{b}{2a}\,,\quad k = c - \dfrac{b^2}{4a}$$
Vertex$(-h,\;k)$ — the minimum if $a>0$, the maximum if $a<0$.
AxisThe axis of symmetry is the vertical line $x = -h$.
Polynomial Division, Factor & Remainder Theorems
Two theorems that let you handle cubics and quartics without grinding through long division every time.
Factor TheoremMemorize
$$f(a) = 0 \;\;\Longrightarrow\;\; (x - a) \text{ is a factor of } f(x)$$
UseTo test whether a value is a root of a polynomial. Substitute and check for zero.
Remainder TheoremMemorize
$$f(a) = r \;\;\Longrightarrow\;\; r \text{ is the remainder of } f(x) \div (x - a)$$
NoteFor divisor $(ax + b)$ evaluate at $f\!\left(-\dfrac{b}{a}\right)$.
Long Division Layout
$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$. The cycle is Divide → Multiply → Subtract → Bring down, repeated until the remainder has lower degree than the divisor.
Equations Reducible to a Quadratic
Many high-degree or surd equations collapse to a clean quadratic after one good substitution.
The SubstitutionMemorize
$$\text{Let } u = f(x) \;\;\Longrightarrow\;\; au^2 + bu + c = 0$$
TriggerYou see the shape $\big(f(x)\big)^2 + b\,f(x) + c = 0$. Solve for $u$, then back-substitute for $x$.
Both are geometric progressions in disguise. Interest grows the principal; depreciation shrinks it. Same machinery, opposite direction.
Compound InterestMemorize
$$A = P\!\left(1 + \dfrac{r}{100}\right)^{\!T}$$
Where$A$ = final amount, $P$ = principal, $r$ = rate (%) per period, $T$ = number of periods.
DepreciationMemorize
$$A = P\!\left(1 - \dfrac{r}{100}\right)^{\!T}$$
Where$A$ = depreciated value, $P$ = original value, $r$ = rate of decline (%), $T$ = number of periods.
02
Section Two
Coordinate Geometry, Vectors & Trigonometry
Where geometry meets algebra. Lines, circles, vectors, and trigonometric ratios all describe shape and direction with equations — and CSEC tests them as a connected toolkit.
Straight Lines
GradientMemorize
$$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$
MeaningRate of change of $y$ with respect to $x$ between two points.
Three Equivalent Forms of a Straight Line
Slope–Intercept
$$y = mx + c$$
UseWhen you have the gradient $m$ and $y$-intercept $c$ directly.
Point–Gradient
$$y - y_1 = m(x - x_1)$$
UseWhen you have one point on the line and the gradient.
General Form
$$ax + by + c = 0$$
UseThe form most often required in the final answer.
MethodAverage the $x$-coordinates and the $y$-coordinates separately.
Perpendicular BisectorMemorize
Procedure
Find the midpoint of the segment.
Find the gradient of the segment.
Take the perpendicular gradient $\;m_\perp = -\dfrac{1}{m}$.
Apply point–gradient form using the midpoint and $m_\perp$.
Circles
CSEC gives circles in two forms. Standard form shows the centre and radius directly; general form is the form you'll most often have to convert from.
Standard FormGiven on Exam Sheet
$$(x - a)^2 + (y - b)^2 = r^2$$
Centre$(a,\;b)$ — read straight from the equation.
Radius$r$ — the square root of the right-hand side.
General FormGiven on Exam Sheet
$$x^2 + y^2 + 2fx + 2gy + c = 0$$
Centre$(-f,\;-g)$
Radius$r = \sqrt{f^2 + g^2 - c}$
Conversion Shortcut
Compare coefficients directly: $2f$ is the coefficient of $x$, $2g$ is the coefficient of $y$, $c$ is the constant. Then centre and radius come straight from the formulas above — no completing the square required.
Circle from Diameter EndpointsMemorize
Centre = midpoint of the two endpoints.
Radius = half the distance between them.
Substitute into the standard form $(x - a)^2 + (y - b)^2 = r^2$.
Tangent & Normal to a Circle
A radius drawn to a point on the circle is perpendicular to the tangent at that point. That single fact powers every question on this topic.
Compute the gradient of the radius to the given point.
Tangent gradient is the negative reciprocal.
Apply point–gradient form.
Normal at a PointMemorize
$$m_{\text{normal}} = m_{\text{radius}}$$
Key insightThe normal to a circle is the radius extended. It passes through the centre. Use point–gradient form with the point on the circle.
Line–Circle Intersection
MethodMemorize
From the linear equation, express $y$ in terms of $x$.
Substitute into the circle equation.
Solve the resulting quadratic.
Back-substitute to find the $y$-coordinates.
Reading the Discriminant
$D > 0$ → two intersection points · $D = 0$ → tangent (one point of contact) · $D < 0$ → the line misses the circle.
Vectors
A vector carries both magnitude and direction. CSEC works exclusively in two dimensions and accepts either column form or $\mathbf{i}, \mathbf{j}$ form.
Two Equivalent NotationsMemorize
$$\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} = x\mathbf{i} + y\mathbf{j}$$
Where$\mathbf{i} = \begin{pmatrix}1\\0\end{pmatrix}$ along the $x$-axis, $\mathbf{j} = \begin{pmatrix}0\\1\end{pmatrix}$ along the $y$-axis.
Magnitude (Modulus)Given on Exam Sheet
$$|\mathbf{v}| = \sqrt{x^2 + y^2}$$
IdeaPythagoras on the components. This is the vector's length.
Once you find the principal (acute) angle, use CAST to locate the other solutions in the range:
Quadrant 2: $180° - \theta$ · Quadrant 3: $180° + \theta$ · Quadrant 4: $360° - \theta$.
UseTo eliminate $\tan\theta$ from an equation, or to simplify expressions mixing sin, cos, tan.
Compound & Double Angle Formulae
Compound Angle — SineGiven on Exam Sheet
$$\sin(A \pm B) \;=\; \sin A \cos B \;\pm\; \cos A \sin B$$
Sign ruleThe signs match — straightforward.
Compound Angle — CosineGiven on Exam Sheet
$$\cos(A \pm B) \;=\; \cos A \cos B \;\mp\; \sin A \sin B$$
Sign ruleThe signs are opposite. Most common slip in the entire topic.
Compound Angle — TangentMemorize
$$\tan(A \pm B) \;=\; \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$
Sign ruleTop signs match, bottom signs flip.
Double-Angle Formulae (set $B = A$)
$\sin 2A$Memorize
$$\sin 2A \;=\; 2\sin A \cos A$$
$\tan 2A$Memorize
$$\tan 2A \;=\; \dfrac{2\tan A}{1 - \tan^2 A}$$
$\cos 2A$ — Three Equivalent Forms
When equation has both
$\cos 2A = \cos^2 A - \sin^2 A$
Use this form for identity proofs or when a mix of $\sin^2$ and $\cos^2$ is present.
When other $\sin^2$ terms appear
$\cos 2A = 1 - 2\sin^2 A$
Substitute to collapse the equation into a quadratic in $\sin A$.
When other $\cos^2$ terms appear
$\cos 2A = 2\cos^2 A - 1$
Substitute to collapse the equation into a quadratic in $\cos A$.
Triangle Area & Trig Graphs
Area of a TriangleMemorize
$$\text{Area} = \tfrac{1}{2}\, ab \sin C$$
UseWhen two sides $a, b$ and the included angle $C$ are known.
Graphs of $\sin kx$, $\cos kx$, $\tan kx$
Function
Amplitude
Period
Key Feature
$\sin kx$
$1$
$\dfrac{2\pi}{k}$
Starts at 0, rises first
$\cos kx$
$1$
$\dfrac{2\pi}{k}$
Starts at 1 (the maximum)
$\tan kx$
undefined
$\dfrac{\pi}{k}$
Vertical asymptotes at odd multiples of $\dfrac{\pi}{2k}$
The role of $k$
$k$ compresses the graph horizontally — doubling $k$ halves the period and doubles the number of cycles in $[0, 2\pi]$.
03
Section Three
Introductory Calculus
Calculus is the mathematics of change. Differentiation gives the gradient at a point; integration reverses that and recovers the original function — or the area beneath the curve.
Differentiation — The Basic Rules
Power Rule (Standard)Memorize
$$\dfrac{d}{dx}\big(x^n\big) = n\,x^{\,n-1}$$
UseFor any power of $x$. This is the rule you reach for every single day.
UseFor a function of a function. Let $u$ be the inner expression, differentiate the outer in terms of $u$, then multiply by the derivative of $u$ with respect to $x$.
Chain Rule — Linear Inner ShortcutMemorize
$$\dfrac{d}{dx}\big[f(ax + b)\big] \;=\; a \cdot f'(ax + b)$$
MeaningThe rate of change of the gradient itself — tells you how the curve is bending.
Stationary Points & Their Nature
A stationary point is where the gradient is momentarily zero. From there it is either a maximum, a minimum, or a point of inflexion (excluded from CSEC).
Stationary PointMemorize
$$\dfrac{dy}{dx} = 0$$
ProcessDifferentiate, set $f'(x) = 0$, solve for $x$, then back-substitute to find the $y$-coordinate.
IdeaThe limit of the gradient of a secant line as the two points squeeze together. The coordinate-geometry gradient taken at infinitesimally close points.
ExampleFor a circle of radius $r$, the area $A = \pi r^2$ changes with time via $\dfrac{dA}{dt} = \dfrac{dA}{dr} \cdot \dfrac{dr}{dt} = 2\pi r \cdot \dfrac{dr}{dt}$.
Integration — The Basic Rules
Integration reverses differentiation. It also accumulates change — giving the area under a curve.
Note$F(x)$ is any antiderivative. The constant of integration $c$ cancels when you subtract.
Area & Volume by Integration
Area Under a CurveMemorize
$$A = \int_a^b f(x)\, dx$$
ScopeCSEC restricts this to regions in the first quadrant, bounded by the curve, the $x$-axis, and the lines $x = a$ and $x = b$.
Volume of Revolution about the $x$-axisMemorize
$$V = \pi \int_a^b \big[f(x)\big]^2 dx$$
IdeaRotate the region under $y = f(x)$ a full turn about the $x$-axis. Each thin strip becomes a disc of area $\pi y^2$. CSEC restricts this to polynomials up to degree 2.
Equation of a Curve from its Gradient
Reverse the DifferentiationMemorize
$$y \;=\; \int \dfrac{dy}{dx}\, dx \;+\; c$$
UseWhen given the gradient function and a point on the curve. Integrate, then substitute the point to find $c$.
Kinematics
Kinematics is calculus applied to motion. Differentiate to move forward through the chain; integrate to move backward.
$\displaystyle s = \int_a^b v(t)\, dt$ can be negative when the particle moves backwards. For total distance use $|v(t)|$: $\displaystyle \text{Distance} = \int_a^b |v(t)|\, dt$. Find when $v(t) = 0$, split the integral there, and add the absolute values.
SUVAT — Equations of Motion Optional
SUVAT applies only when acceleration is constant. CSEC Add Math does not require these — every kinematics question can be solved using the calculus approach above. SUVAT can occasionally provide a faster route, so it is worth knowing.
Final Velocity
$$v = u + at$$
Displacement
$$s = ut + \tfrac{1}{2} at^2$$
Without Time
$$v^2 = u^2 + 2as$$
04
Section Four
Probability & Statistics
Statistics is half computation, half careful reading of data. Probability is half logic, half clean diagram work. Lock the formulas in, then practise reading questions slowly — that is where the marks slip.
NoteUse class midpoints for $x_i$, weighted by frequency.
Standard DeviationMemorize
$$S \;=\; \sqrt{S^2}$$
NoteSame units as the original data — easier to interpret than variance. Always find variance first.
Box-and-Whisker Plots & Stem-and-Leaf
A box-and-whisker plot displays the five-number summary: Minimum, $Q_1$, Median ($Q_2$), $Q_3$, Maximum. The box spans $Q_1$ to $Q_3$; the line inside marks the median; the whiskers extend to the extremes.
Skewness from the box plot: the position of the median line inside the box tells you which way the data is pulled.
Stem-and-Leaf
Split each value into a stem (leading digits) and a leaf (final digit). Preserves the original values, so median, quartiles, and mode can all be read directly. Always include a key such as "$1\,|\,2$ means $12$."
Probability Foundations
An experiment produces an outcome. The sample space $S$ is the set of all possible outcomes. An event $A$ is a subset of the sample space.
Classical ProbabilityMemorize
$$P(A) \;=\; \dfrac{\text{number of outcomes in } A}{\text{total number of outcomes in } S}$$
ConditionAll outcomes must be equally likely — coins, fair dice, balls drawn at random.
Range
$$0 \le P(A) \le 1$$
Sample Space
$$\sum_{\text{outcomes}} P = 1$$
Complement
$$P(A') = 1 - P(A)$$
Addition Rule & Mutually Exclusive Events
Addition RuleGiven on Exam Sheet
$$P(A \cup B) \;=\; P(A) + P(B) - P(A \cap B)$$
Why subtract?Adding $P(A)$ and $P(B)$ counts the overlap $P(A \cap B)$ twice. Subtracting it once removes the duplicate.
Mutually Exclusive EventsMemorize
$$P(A \cap B) = 0 \;\;\Longrightarrow\;\; P(A \cup B) = P(A) + P(B)$$
MeaningThe events cannot both happen — rolling a 2 and rolling a 5 on the same die. No overlap, no subtraction.
Conditional Probability
Conditional ProbabilityMemorize
$$P(A | B) \;=\; \dfrac{P(A \cap B)}{P(B)}$$
Read"The probability of $A$, given that $B$ has occurred." Knowing $B$ restricts the sample space to $B$, and we recompute within that restricted space.
Independent Events
Independence ConditionMemorize
$$P(A \cap B) = P(A) \times P(B) \qquad\text{or equivalently}\qquad P(A | B) = P(A)$$
MeaningOne event does not affect the other — like tossing two separate coins. Either condition can be used to prove independence.
Common Confusion
Mutually exclusive is not the same as independent. If two events are mutually exclusive (and both have non-zero probability), they cannot be independent — knowing one happened tells you the other did not.
Tree Diagrams, Venn Diagrams & Possibility Spaces
Tree Diagrams
Multiply along branches for "and" — joint probability of $A$ then $B$.
Add across branches for "or" — total probability of an outcome reached by more than one path.
CSEC limits trees to two initial branches.
Venn Diagrams (Two Sets)
$A \cap B$: inside both circles $A \cap B'$: inside $A$ only $A' \cap B$: inside $B$ only $(A \cup B)'$: outside both
Always fill in $P(A \cap B)$ first, then work outward.
Possibility Spaces
A grid of all outcome pairs from two experiments — e.g. rolling two dice gives a 6 × 6 grid.
Every CSEC Additional Mathematics formula, arranged in syllabus order, presented the way I teach it. Search anything, jump anywhere, share with your class — this book is yours. — Kerwin
