Is this the official A-Math formula list?

The first half has every formula on SEAB's list for G3 Additional Mathematics (K341), set out in our own layout with our own examples. The paper prints the list at the front, so the second half, what it leaves out, is the part to learn. Scholar Tests is not connected with MOE, SEAB or Cambridge.

Did the formula list change with the SEC?

No. From 2027 the O-Level becomes the Secondary Education Certificate, and the subject is called G3. SEAB's 2027 syllabus has the same topics and the same formula list as the 2026 O-Level one, so this sheet serves both.

How should answers be given?

To 3 significant figures unless the question says otherwise, and angles in degrees to 1 decimal place. Use your calculator's π or 3.142, unless the answer is wanted in terms of π. Show the working: essential working left out loses marks.

Yes. The print button gives both halves on plain paper, without this page's buttons or questions. Keep it beside you while you do past-year and practice papers until you no longer need to look.

G3 Additional Mathematics (K341), was O-Level A-Math (4049)

O-Level A-Math formula sheet

The A-Math paper prints a list of formulas: the quadratic formula, the binomial expansion, the trigonometric identities and the triangle formulas. Below is every formula on that list, with a worked example for each, and then the longer list the paper does not give you: discriminants, logs, surds, partial fractions, the R-formula, circles and all of calculus.

What to learn by heart E-Math sheet

Given in the exam

SEAB prints these at the front of every paper. Know what each letter means and when to use each one, so you are not reading the list for the first time on the day.

Algebra

Quadratic formula
$x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
For ax² + bx + c = 0.
Example: 2x² − 5x + 1 = 0: x = (5 ± √17) ÷ 4, so x ≈ 2.28 or 0.219.
Binomial expansion
$(a + b)^{n} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + \dots$
$\qquad + \binom{n}{r}a^{n-r}b^{r} + \dots + b^{n}$
n a positive integer.
Example: (1 + 2x)⁵ = 1 + 10x + 40x² + ...
Binomial coefficient
$\binom{n}{r} = \dfrac{n!}{r!\,(n - r)!} = \dfrac{n(n - 1)\cdots(n - r + 1)}{r!}$
Example: 6C2 = (6 × 5) ÷ 2 = 15.

Trigonometric identities

Pythagorean identities
$\sin^{2}A + \cos^{2}A = 1$
$\sec^{2}A = 1 + \tan^{2}A$
$\csc^{2}A = 1 + \cot^{2}A$
Example: sin A = 3/5 with A acute: cos A = √(1 − 9/25) = 4/5.
Addition formulas
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
Example: sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√6 + √2) ÷ 4.
Double angle formulas
$\sin 2A = 2\sin A\cos A$
$\cos 2A = \cos^{2}A - \sin^{2}A$
$\cos 2A = 2\cos^{2}A - 1 = 1 - 2\sin^{2}A$
$\tan 2A = \dfrac{2\tan A}{1 - \tan^{2}A}$
Example: tan A = ½: tan 2A = 1 ÷ (1 − ¼) = 4/3.

Formulas for a triangle ABC

Sine rule
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
Example: a = 10 cm, A = 40°, B = 65°: b = 10 sin 65° ÷ sin 40° ≈ 14.1 cm.
Cosine rule
$a^{2} = b^{2} + c^{2} - 2bc\cos A$
Example: b = 5 cm, c = 7 cm, A = 60°: a = √39 ≈ 6.24 cm.
Area
$\Delta = \dfrac{1}{2}bc\sin A$
Example: b = 8 cm, c = 5 cm, A = 30°: Δ = 10 cm².

Not given: learn these by heart

The paper does not print any of these, and the questions expect you to know them.

Quadratics

Discriminant
$b^{2} - 4ac > 0: \text{ two real roots}$
$b^{2} - 4ac = 0: \text{ two equal roots}$
$b^{2} - 4ac < 0: \text{ no real roots}$
For a line and a curve, substitute the line in first: equal roots means the line is a tangent.
Example: x² + kx + 9 = 0 has equal roots when k² = 36, so k = ±6.
Always positive
$ax^{2} + bx + c > 0 \text{ for all } x \iff a > 0 \text{ and } b^{2} - 4ac < 0$
Always negative: a < 0 and b² − 4ac < 0.
Example: x² + 2x + 5: a = 1 > 0 and 4 − 20 < 0, so it is always positive.
Completing the square
$ax^{2} + bx + c = a\left(x + \dfrac{b}{2a}\right)^{2} + c - \dfrac{b^{2}}{4a}$
Example: 2x² − 8x + 3 = 2(x − 2)² − 5, so the minimum value is −5, at x = 2.

Surds and polynomials

Rationalising the denominator
$\dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b}$
Example: 1 ÷ (√3 + 1) = (√3 − 1) ÷ 2.
Remainder and factor theorems
$f(x) \div (x - a) \text{ leaves remainder } f(a)$
$f(a) = 0 \iff (x - a) \text{ is a factor}$
For (ax − b), use f(b/a).
Example: x³ − 2x + 4 divided by (x − 2): f(2) = 8 − 4 + 4 = 8.
Sum and difference of cubes
$a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})$
$a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})$
Example: x³ − 8 = (x − 2)(x² + 2x + 4).
Partial fractions
$\dfrac{px + q}{(ax + b)(cx + d)} = \dfrac{A}{ax + b} + \dfrac{B}{cx + d}$
$\dfrac{px^{2} + qx + r}{(ax + b)(cx + d)^{2}} = \dfrac{A}{ax + b} + \dfrac{B}{cx + d} + \dfrac{C}{(cx + d)^{2}}$
$\dfrac{px^{2} + qx + r}{(ax + b)(x^{2} + c^{2})} = \dfrac{A}{ax + b} + \dfrac{Bx + C}{x^{2} + c^{2}}$
If the top's degree is not less than the bottom's, divide first.
Example: 3 ÷ ((x − 1)(x + 2)) = 1/(x − 1) − 1/(x + 2).
General term of a binomial
$T_{r+1} = \binom{n}{r}a^{n-r}b^{r}$
Example: The x³ term of (2 + x)⁵: 5C3 × 2² × x³ = 40x³.

Exponentials and logarithms

Laws of logarithms
$\log_{a}xy = \log_{a}x + \log_{a}y, \quad \log_{a}\dfrac{x}{y} = \log_{a}x - \log_{a}y$
$\log_{a}x^{n} = n\log_{a}x, \quad \log_{a}a = 1, \quad \log_{a}1 = 0$
Example: log₂ 8 + log₂ 4 = log₂ 32 = 5.
Exponential and log form
$y = a^{x} \iff x = \log_{a}y$
$e^{\ln x} = x, \quad \ln e^{x} = x$
Example: eˣ = 5, so x = ln 5 ≈ 1.61.
Change of base
$\log_{a}b = \dfrac{\log_{c}b}{\log_{c}a}$
Example: log₄ 8 = log₂ 8 ÷ log₂ 4 = 3/2.

Trigonometry

The six functions
$\tan A = \dfrac{\sin A}{\cos A}, \quad \cot A = \dfrac{\cos A}{\sin A}$
$\sec A = \dfrac{1}{\cos A}, \quad \csc A = \dfrac{1}{\sin A}$
Example: cos A = 0.8: sec A = 1.25.
Exact values
$\sin 30^{\circ} = \tfrac{1}{2}, \quad \sin 45^{\circ} = \tfrac{\sqrt{2}}{2}, \quad \sin 60^{\circ} = \tfrac{\sqrt{3}}{2}$
$\cos 30^{\circ} = \tfrac{\sqrt{3}}{2}, \quad \cos 45^{\circ} = \tfrac{\sqrt{2}}{2}, \quad \cos 60^{\circ} = \tfrac{1}{2}$
$\tan 30^{\circ} = \tfrac{\sqrt{3}}{3}, \quad \tan 45^{\circ} = 1, \quad \tan 60^{\circ} = \sqrt{3}$
Example: In radians: 30° = π/6, 45° = π/4, 60° = π/3.
R-formula
$a\cos\theta + b\sin\theta = R\cos(\theta - \alpha)$
$R = \sqrt{a^{2} + b^{2}}, \quad \tan\alpha = \dfrac{b}{a}$
The R sin(θ ± α) form works the same way.
Example: 3 cos θ + 4 sin θ = 5 cos(θ − 53.1°), so its greatest value is 5.
Amplitude and period
$y = a\sin(bx) + c: \text{ amplitude } |a|, \text{ period } \dfrac{360^{\circ}}{b}$
$y = a\tan(bx): \text{ period } \dfrac{180^{\circ}}{b}$
Example: y = 3 sin 2x + 1: amplitude 3, period 180°, between −2 and 4.
Principal values
$-90^{\circ} \le \sin^{-1}x \le 90^{\circ}, \quad 0^{\circ} \le \cos^{-1}x \le 180^{\circ}$
$-90^{\circ} < \tan^{-1}x < 90^{\circ}$
sin is positive in quadrants 1 and 2, cos in 1 and 4, tan in 1 and 3.
Example: sin x = −0.5 for 0° ≤ x ≤ 360°: the basic angle is 30°, so x = 210° or 330°.

Coordinate geometry

Midpoint
$\left(\dfrac{x_{1} + x_{2}}{2}, \dfrac{y_{1} + y_{2}}{2}\right)$
Example: (2, 3) and (8, −1): midpoint (5, 1).
Parallel and perpendicular lines
$\text{parallel: } m_{1} = m_{2}, \quad \text{perpendicular: } m_{1}m_{2} = -1$
Example: A line perpendicular to y = 2x + 1 has gradient −½.
Equation of a circle
$(x - a)^{2} + (y - b)^{2} = r^{2}$
$x^{2} + y^{2} + 2gx + 2fy + c = 0: \text{ centre } (-g, -f), \; r = \sqrt{g^{2} + f^{2} - c}$
Example: x² + y² − 4x + 6y − 3 = 0: centre (2, −3), radius √(4 + 9 + 3) = 4.
Area of a rectilinear figure
$\dfrac{1}{2}\left|\sum (x_{i}y_{i+1} - x_{i+1}y_{i})\right|$
Take the vertices in order round the figure, and come back to the first at the end.
Example: (0, 0), (4, 0), (4, 3): ½ × |12 − 0| = 6.
Linear law
$y = ax^{n} \Rightarrow \lg y = n\lg x + \lg a$
$y = kb^{x} \Rightarrow \lg y = (\lg b)x + \lg k$
Example: Plot lg y against lg x: the gradient is n and the intercept is lg a.

Differentiation

Standard derivatives
$\dfrac{d}{dx}x^{n} = nx^{n-1}, \quad \dfrac{d}{dx}e^{x} = e^{x}, \quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}$
$\dfrac{d}{dx}\sin x = \cos x, \quad \dfrac{d}{dx}\cos x = -\sin x, \quad \dfrac{d}{dx}\tan x = \sec^{2}x$
x in radians for the trigonometric ones.
Example: y = 4x³ − √x: dy/dx = 12x² − 1/(2√x).
Chain rule
$\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}$
Example: y = (3x + 1)⁴: dy/dx = 4(3x + 1)³ × 3 = 12(3x + 1)³.
Product and quotient rules
$\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$
$\dfrac{d}{dx}\left(\dfrac{u}{v}\right) = \dfrac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^{2}}$
Example: y = x sin x: dy/dx = x cos x + sin x.
Stationary points
$\dfrac{dy}{dx} = 0$
$\dfrac{d^{2}y}{dx^{2}} > 0: \text{ minimum}, \quad \dfrac{d^{2}y}{dx^{2}} < 0: \text{ maximum}$
Example: y = x² − 6x: dy/dx = 2x − 6 = 0 at x = 3, and d²y/dx² = 2 > 0, so (3, −9) is a minimum.
Tangents and normals
$\text{tangent gradient } m = \dfrac{dy}{dx}, \quad \text{normal gradient} = -\dfrac{1}{m}$
Example: y = x² at x = 1: tangent gradient 2, normal gradient −½.
Connected rates of change
$\dfrac{dy}{dt} = \dfrac{dy}{dx} \times \dfrac{dx}{dt}$
Example: A = πr² and dr/dt = 0.5 cm/s: when r = 4, dA/dt = 8π × 0.5 = 4π cm²/s.

Integration

Standard integrals
$\int x^{n}\,dx = \dfrac{x^{n+1}}{n + 1} + c \quad (n \ne -1)$
$\int (ax + b)^{n}\,dx = \dfrac{(ax + b)^{n+1}}{a(n + 1)} + c \quad (n \ne -1)$
$\int \dfrac{1}{ax + b}\,dx = \dfrac{1}{a}\ln(ax + b) + c$
Example: ∫(2x + 1)³ dx = (2x + 1)⁴ ÷ 8 + c.
Trigonometric and exponential
$\int \sin(ax + b)\,dx = -\dfrac{1}{a}\cos(ax + b) + c$
$\int \cos(ax + b)\,dx = \dfrac{1}{a}\sin(ax + b) + c$
$\int \sec^{2}x\,dx = \tan x + c, \quad \int e^{ax + b}\,dx = \dfrac{1}{a}e^{ax + b} + c$
Example: ∫cos 3x dx = ⅓ sin 3x + c.
Area under a curve
$\text{area} = \int_{a}^{b} y\,dx$
A region below the x-axis gives a negative integral: take its size, and work it out apart from any part above.
Example: ∫ from 0 to 2 of x² dx = 8/3.

Kinematics

Displacement, velocity, acceleration
$v = \dfrac{ds}{dt}, \quad a = \dfrac{dv}{dt}$
$s = \int v\,dt, \quad v = \int a\,dt$
At rest means v = 0. Distance travelled: split the time where v changes sign.
Example: s = t³ − 3t: v = 3t² − 3, so the particle is at rest when t = 1.

Use the formulas on a paper

A free A-Math Paper 1 in the O-Level's shape for each year, with every answer, to print or sit online, and a whole written Paper 2 to do on paper and mark yourself.

Sec 3 A-Math paper Sec 4 A-Math paper

Questions parents ask

Is this the official A-Math formula list?: The first half has every formula on SEAB's list for G3 Additional Mathematics (K341), set out in our own layout with our own examples. The paper prints the list at the front, so the second half, what it leaves out, is the part to learn. Scholar Tests is not connected with MOE, SEAB or Cambridge.
Did the formula list change with the SEC?: No. From 2027 the O-Level becomes the Secondary Education Certificate, and the subject is called G3. SEAB's 2027 syllabus has the same topics and the same formula list as the 2026 O-Level one, so this sheet serves both.
How should answers be given?: To 3 significant figures unless the question says otherwise, and angles in degrees to 1 decimal place. Use your calculator's π or 3.142, unless the answer is wanted in terms of π. Show the working: essential working left out loses marks.
Can I print it?: Yes. The print button gives both halves on plain paper, without this page's buttons or questions. Keep it beside you while you do past-year and practice papers until you no longer need to look.