A-Level Mathematics formula sheet

This is Mini Mathematics’ own reference for H1, H2 and H3 Mathematics and H2 Further Mathematics. It covers the results the exam provides in SEAB’s MF27. It is not an official SEAB publication or an endorsement. Your syllabus determines which results you need.

Use the page outline to find a result. You can also print this page on A4. Angles in calculus and power series are in radians. Inverse trigonometric notation means an inverse function, not a reciprocal.

Summarise paired data and samples

Estimate population variance

For a random sample x₁,…,xₙ, with n > 1 and sample mean x bar, the unbiased variance estimate is

Measure linear association

For paired observations (xᵢ,yᵢ) and their means x bar,y bar, the sample product moment correlation coefficient has two equivalent forms:

Both variables must have nonzero sample variation. The least-squares line predicting y from x is

The slope needs nonzero variation in x.

Use signed ranks

For the Wilcoxon signed rank test, let R₊ total the ranks of positive differences and R₋ total the ranks of negative differences. Then

Here n counts the nonzero differences being ranked. The critical values below use distinct absolute differences and equally likely independent signs under the null hypothesis. Reject when T is at most the critical value. For a one-sided alternative, the small rank sum must also be in the direction of that alternative. A dash means that even T = 0 cannot reach that significance level.

Each row pairs a one-sided significance level with its corresponding two-sided level. The entries are critical values, not probabilities.

Choose a probability model

Count successes in a fixed number of trials

For X∼B(n,p), where n is a nonnegative integer and 0 ≤ p ≤ 1,

At p = 0 or p = 1, read the probabilities by their limiting values.

Count events with a Poisson model

For X∼Po(λ), where λ > 0,

Count trials up to the first success

For X∼Geo(p), where 0 < p ≤ 1, count the successful trial as well:

Model an exponential waiting time

With rate λ > 0, the exponential density is zero for x < 0 and, for x ≥ 0,

Rewrite and expand expressions

Separate rational terms

The constants on the right are chosen to make each identity hold. Denominators must be nonzero; the linear factors shown separately are distinct, with nonzero leading coefficients. For the last form, c > 0.

Expand a positive integer power

For a positive integer m,

Expand about zero

A Maclaurin representation is

Use it where the function equals its convergent Taylor series; differentiability alone does not guarantee that equality.

For real α and |t| < 1,

For every real t,

For -1 < t ≤ 1,

Work with angles and inverse functions

Combine or double angles

Choose the upper signs together, or the lower signs together. Tangent formulas apply only where every term is defined.

Select the principal inverse value

For -1 ≤ t ≤ 1,

For real t,

Differentiate the listed trigonometric functions

For |t| < 1 in the first two formulas, and real t in the third,

On intervals where the functions are defined,

Find antiderivatives

Take a > 0 throughout this section. Each expression includes an arbitrary integration constant C.

Rational and square-root forms

For real t,

For |t| < a,

For t > a,

Trigonometric forms

For |t| < π/2,

For 0 < t < π,

Approximate values and solutions

Estimate an integral from function values

For one interval with endpoints a < b, the trapezium estimate is

Using the midpoint as well gives Simpson’s two-strip estimate:

Take a root-finding step

For g(t) = 0, Newton–Raphson updates an estimate tₖ by

The derivative in the denominator must be nonzero.

Step along a differential equation

For dy/dx = F(x,y) and xₖ₊₁ = xₖ + h, Euler’s step is

Improved Euler first predicts y tilde ₖ₊₁, then averages two slopes:

Approximate a function near a point in the plane

For a sufficiently smooth g near (a,b), write h = x-a and k = y-b. The quadratic approximation is

Subscripts denote partial derivatives; every derivative here is evaluated at (a,b).

Calculate with points, vectors and curves

Divide a line segment

If A and B have position vectors p and q, the point with AP:PB = r:s has position vector

For internal division, r,s > 0.

Form a cross product

With components in a right-handed Cartesian frame,

Measure a curve or a rotated surface

For a smooth curve y = g(x) on a ≤ x ≤ b, its arc length is

For g(x) ≥ 0, rotating that curve about the x-axis gives surface area

Bound quantities and count unions

Compare arithmetic and geometric means

For m nonnegative real numbers z₁,…,zₘ, with positive integer m,

Equality occurs exactly when all the zᵢ are equal.

Bound a sum of products

For real lists (u₁,…,uₘ) and (v₁,…,vₘ), Cauchy–Schwarz gives

Equality occurs when the two vectors are linearly dependent, including when either vector is zero.

Bound the magnitude of a sum

For real z₁,…,zₘ,

In particular, equality holds when every zᵢ is nonnegative.

Count overlapping finite sets

For finite sets A₁,…,Aₘ, inclusion–exclusion is

The sum takes every nonempty choice J of set indices: add single-set sizes, subtract pair intersections, then continue with alternating signs.