Free resource · A-Level H1, H2 and H3 Mathematics

MF27 List of Formulae for H2 Math: Free PDF

MF27 is the SEAB List of Formulae and Results given in all A-Level H1, H2 and H3 Mathematics and H2 Further Mathematics papers from 2025. It prints the binomial and Maclaurin series, partial fractions, the addition and double angle formulae, selected derivatives and integrals, the vector product, numerical methods, standard distributions, and the variance, correlation and regression formulae. It does not give the AP and GP formulae, the laws of logarithms, the basic derivatives and integrals, integration by parts, the scalar product or the rules for E(X) and Var(X), so these must be memorised.

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$First page of the MF27 List of Formulae (H2 Math) PDF$

Printed in the paper

Formulas MF27 gives you

In the order SEAB prints them. You do not need to memorise these, but you do need to know when to use each one.

Algebraic series

Binomial expansionwhere n is a positive integer: $(a + b)^{n} = a^{n} + (1 n) a^{n - 1} b + (2 n) a^{n - 2} b^{2} + (3 n) a^{n - 3} b^{3} + \dots + b^{n}$; $(r n) = \frac{n !}{r ! ( n - r )!}$

Maclaurin expansion: $f (x) = f (0) + x f^{'} (0) + \frac{x ^{2}}{2 !} f^{''} (0) + \dots + \frac{x ^{n}}{n !} f^{(n)} (0) + \dots$; $(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots + \frac{n ( n - 1 ) \dots ( n - r + 1 )}{r !} x^{r} + \dots$ |x| < 1; $e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots + \frac{x ^{r}}{r !} + \dots$ all x; $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots + \frac{( - 1 ) ^{r} x ^{2 r + 1}}{( 2 r + 1 )!} + \dots$ all x; $cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots + \frac{( - 1 ) ^{r} x ^{2 r}}{( 2 r )!} + \dots$ all x; $ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots + \frac{( - 1 ) ^{r + 1} x ^{r}}{r} + \dots$ −1 < x ≤ 1

Partial fractions decomposition

Non-repeated linear factors: $\frac{p x + q}{( a x + b ) ( c x + d )} = \frac{A}{( a x + b )} + \frac{B}{( c x + d )}$

Repeated linear factors: $\frac{p x ^{2} + q x + r}{( a x + b ) ( c x + d ) ^{2}} = \frac{A}{( a x + b )} + \frac{B}{( c x + d )} + \frac{C}{( c x + d ) ^{2}}$
Non-repeated quadratic factor: $\frac{p x ^{2} + q x + r}{( a x + b ) ( x ^{2} + c ^{2} )} = \frac{A}{( a x + b )} + \frac{B x + C}{( x ^{2} + c ^{2} )}$

Trigonometry

sin (A \pm B) \equiv sin A cos B \pm cos A sin B

cos (A \pm B) \equiv cos A cos B \mp sin A sin B

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

sin 2 A \equiv 2 sin A cos A

cos 2 A \equiv cos^{2} A - sin^{2} A \equiv 2 cos^{2} A - 1 \equiv 1 - 2 sin^{2} A

tan 2 A \equiv \frac{2 tan A}{1 - tan ^{2} A}

Principal values|x| ≤ 1: $- \frac{1}{2} π ⩽ sin^{- 1} x ⩽ \frac{1}{2} π$; $0 ⩽ cos^{- 1} x ⩽ π$ |x| ≤ 1; $- \frac{1}{2} π < tan^{- 1} x < \frac{1}{2} π$

Derivatives

\frac{d}{d x} (sin^{- 1} x) = \frac{1}{1 - x ^{2}}

\frac{d}{d x} (cos^{- 1} x) = - \frac{1}{1 - x ^{2}}

\frac{d}{d x} (tan^{- 1} x) = \frac{1}{1 + x ^{2}}

\frac{d}{d x} (cosec x) = - cosec x cot x

\frac{d}{d x} (sec x) = sec x tan x

Integrals

Arbitrary constants are omitted, and a denotes a positive constant, in every row of this table: $\int \frac{1}{x ^{2} + a ^{2}} d x = \frac{1}{a} tan^{- 1} (\frac{x}{a})$; $\int \frac{1}{a ^{2} - x ^{2}} d x = sin^{- 1} (\frac{x}{a})$ |x| < a; $\int \frac{1}{x ^{2} - a ^{2}} d x = \frac{1}{2 a} ln (\frac{x - a}{x + a})$ x > a; $\int \frac{1}{a ^{2} - x ^{2}} d x = \frac{1}{2 a} ln (\frac{a + x}{a - x})$ |x| < a; $\int tan x d x = ln (sec x)$ |x| < ½π; $\int cot x d x = ln (sin x)$ 0 < x < π; $\int cosec x d x = - ln (cosec x + cot x)$ 0 < x < π; $\int sec x d x = ln (sec x + tan x)$ |x| < ½π

Vectors

The point dividing AB in the ratio λ : μ has position vector: $\frac{μ a + λ b}{λ + μ}$

Vector product: $a \times b = a_{1} a_{2} a_{3} \times b_{1} b_{2} b_{3} = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}$

Applications of definite integrals (not in H2 Mathematics)

Arc length of a curve defined in cartesian form: $\int_{a}^{b} 1 + (\frac{d y}{d x})^{2} d x$

Surface area of revolution about the x-axis for a curve defined in cartesian form: $\int_{a}^{b} 2 π y 1 + (\frac{d y}{d x})^{2} d x$

Functions of two variables (not in H2 Mathematics)

Quadratic approximation of f at (a, b): $f (x, y) \approx f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) + \frac{1}{2} f_{xx} (a, b) (x - a)^{2} + f_{x y} (a, b) (x - a) (y - b) + \frac{1}{2} f_{y y} (a, b) (y - b)^{2}$

Numerical methods (not in H2 Mathematics)

Trapezium rule (for single strip): $\int_{a}^{b} f (x) d x \approx \frac{1}{2} (b - a) [f (a) + f (b)]$

Simpson’s rule (for two strips): $\int_{a}^{b} f (x) d x \approx \frac{1}{6} (b - a) [f (a) + 4 f (\frac{a + b}{2}) + f (b)]$
The Newton-Raphson iteration for approximating a root of f(x) = 0where x₁ is a first approximation: $x_{2} = x_{1} - \frac{f ( x _{1} )}{f ^{'} ( x _{1} )}$
Euler Method with step size h: $y_{2} = y_{1} + h f (x_{1}, y_{1})$
Improved Euler Method with step size h: $u_{2} y_{2} = y_{1} + h f (x_{1}, y_{1}) = y_{1} + \frac{h}{2} [f (x_{1}, y_{1}) + f (x_{2}, u_{2})]$

Standard discrete distributions

Binomial B(n, p): $P (X = x) = (x n) p^{x} (1 - p)^{n - x}, Mean = n p, Variance = n p (1 - p)$

Poisson Po(λ)not in H2 Mathematics 9758: $P (X = x) = e^{- λ} \frac{λ ^{x}}{x !}, Mean = λ, Variance = λ$
Geometric Geo(p)not in H2 Mathematics 9758: $P (X = x) = (1 - p)^{x - 1} p, Mean = \frac{1}{p}, Variance = \frac{1 - p}{p ^{2}}$

Standard continuous distribution (not in H2 Mathematics)

Exponential: $p.d.f. = λ e^{- λ x}, Mean = \frac{1}{λ}, Variance = \frac{1}{λ ^{2}}$

Sampling and testing

Unbiased estimate of population variance: $s^{2} = \frac{n}{n - 1} (\frac{Σ ( x - x ˉ ) ^{2}}{n}) = \frac{1}{n - 1} (Σ x^{2} - \frac{( Σ x ) ^{2}}{n})$

Regression and correlation

Estimated product moment correlation coefficient: $r = \frac{Σ ( x - x ˉ ) ( y - y ˉ )}{{ Σ ( x - x ˉ ) ^{2} } { Σ ( y - y ˉ ) ^{2} }} = \frac{Σ x y - \frac{Σ x Σ y}{n}}{( Σ x ^{2} - \frac{( Σ x ) ^{2}}{n} ) ( Σ y ^{2} - \frac{( Σ y ) ^{2}}{n} )}$

Estimated regression line of y on x: $y - \overset{y}{ˉ} = b (x - \overset{x}{ˉ}), where b = \frac{Σ ( x - x ˉ ) ( y - y ˉ )}{Σ ( x - x ˉ ) ^{2}}$

Wilcoxon signed rank test (not in H2 Mathematics)

P Q T = the sum of the ranks corresponding to the positive differences = the sum of the ranks corresponding to the negative differences = the smaller of P and Q

Critical values of TLevel of significance for a one-tail (One) or two-tail (Two) test. For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the level of significance indicated.: $One Two n = 6 7891011121314151617181920 0.05 0.1 23581013172125303541475360 0.025 0.05 0235810131721252934404652 0.01 0.02 0135791215192327323743 0.005 0.01 01357912151923273237$

Mathematical Results (not in H2 Mathematics)

AM-GM inequalityFor any nonnegative real numbers x₁, x₂, …, xₙ, where the equality holds if and only if x₁ = x₂ = … = xₙ: $\frac{x _{1} + x _{2} + \dots + x _{n}}{n} ⩾ n x_{1} x_{2} \dots x_{n}$

Cauchy-Schwarz inequalityFor any real numbers u₁, u₂, …, uₙ and v₁, v₂, …, vₙ, where the equality holds if and only if there exists a nonzero constant k such that uᵢ = kvᵢ for all i = 1, 2, …, n: $(i = 1 \sum n u_{i} v_{i})^{2} ⩽ (i = 1 \sum n u_{i}^{2}) (i = 1 \sum n v_{i}^{2})$
Triangle inequalityFor any real numbers x₁, x₂, …, xₙ, where the equality holds if x₁, x₂, …, xₙ are all nonnegative: $∣ x_{1} + x_{2} + \dots + x_{n} ∣ ⩽ ∣ x_{1} ∣ + ∣ x_{2} ∣ + \dots + ∣ x_{n} ∣$
Inclusion-Exclusion PrincipleFor any subsets A₁, A₂, …, Aₙ of a set: $∣ A_{1} \cup A_{2} \cup \dots \cup A_{n} ∣ = ∣ A_{1} ∣ + ∣ A_{2} ∣ + \dots + ∣ A_{n} ∣ - [∣ A_{1} \cap A_{2} ∣ + ∣ A_{1} \cap A_{3} ∣ + \dots + ∣ A_{n - 1} \cap A_{n} ∣] + [∣ A_{1} \cap A_{2} \cap A_{3} ∣ + ∣ A_{1} \cap A_{2} \cap A_{4} ∣ + \dots + ∣ A_{n - 2} \cap A_{n - 1} \cap A_{n} ∣] ⋮ + (- 1)^{n - 1} ∣ A_{1} \cap A_{2} \dots \cap A_{n - 1} \cap A_{n} ∣$

Not given: memorise

Formulas MF27 does not give you

The syllabus expects you to use these, but the paper does not print them.

Algebra

Quadratic formula and discriminantfor ax² + bx + c = 0. The three signs of the discriminant give two real roots, two equal roots and no real roots: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}, b^{2} - 4 a c > 0, = 0, < 0$

Modulus inequalitieswhere b > 0: $∣ x - a ∣ < b ∣ x - a ∣ > b ⟺ a - b < x < a + b ⟺ x < a - b or x > a + b$
Laws of logarithmsy = aˣ is equivalent to x = logₐ y: $lo g_{a} x y = lo g_{a} x + lo g_{a} y, lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y, lo g_{a} x^{r} = r lo g_{a} x, lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}$

Coordinate geometry

Equation of a circlecentre (a, b) and radius r: $(x - a)^{2} + (y - b)^{2} = r^{2}$

Sequences and series

Arithmetic serieswhere l is the last term: $u_{n} = a + (n - 1) d, S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$

Geometric seriesr ≠ 1. The sum to infinity exists only when |r| < 1: $u_{n} = a r^{n - 1}, S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}, S_{\infty} = \frac{a}{1 - r}$
nth term from the sum to n terms: $u_{n} = S_{n} - S_{n - 1}$

Trigonometry

Pythagorean identities: $sin^{2} A + cos^{2} A \equiv 1, 1 + tan^{2} A \equiv sec^{2} A, 1 + cot^{2} A \equiv cosec^{2} A$

R formulaefor a, b > 0, where R = √(a² + b²) and tan α = b/a: $a sin θ + b cos θ = R sin (θ + α), a cos θ + b sin θ = R cos (θ - α)$
Small angle approximationsx in radians: $sin x \approx x, cos x \approx 1 - \frac{1}{2} x^{2}, tan x \approx x$

Differentiation

Standard derivativesx in radians for the trigonometric functions: $\frac{d}{d x} x^{n} = n x^{n - 1}, \frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} ln x = \frac{1}{x}$; $\frac{d}{d x} sin x = cos x, \frac{d}{d x} cos x = - sin x, \frac{d}{d x} tan x = sec^{2} x$

Chain, product and quotient rules: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}, \frac{d}{d x} (uv) = u \frac{d v}{d x} + v \frac{d u}{d x}, \frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$

Integration

Standard integralsn ≠ −1, arbitrary constants omitted. For (ax + b) in place of x, divide by a: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}, \int e^{x} d x = e^{x}, \int sin x d x = - cos x, \int cos x d x = sin x, \int sec^{2} x d x = tan x$

Integral of 1/xarbitrary constants omitted. This is the case n = −1 that the xⁿ rule leaves out: $\int \frac{1}{x} d x = ln ∣ x ∣, \int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣$
Integrals of sin²x, cos²x and tan²xarbitrary constants omitted. Rewrite with cos 2A from MF27, or tan²x = sec²x − 1, first: $\int sin^{2} x d x = \frac{x}{2} - \frac{sin 2 x}{4}, \int cos^{2} x d x = \frac{x}{2} + \frac{sin 2 x}{4}, \int tan^{2} x d x = tan x - x$
Integrals involving f′(x) and f(x)n ≠ −1, arbitrary constants omitted: $\int f^{'} (x) [f (x)]^{n} d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1}, \int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣, \int f^{'} (x) e^{f (x)} d x = e^{f (x)}$
Integration by parts: $\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$
Volume of revolution: $V = π \int_{a}^{b} y^{2} d x (about the x -axis), V = π \int_{c}^{d} x^{2} d y (about the y -axis)$

Vectors

Magnitude and scalar productwhere θ is the angle between a and b: $∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}, a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣∣ b ∣ cos θ$

Projection and perpendicular distancewhere n̂ is a unit vector: $length of projection of a on \hat{n} = ∣ a \cdot \hat{n} ∣, perpendicular component = ∣ a \times \hat{n} ∣$
Equations of lines and planesa line through the point with position vector a in direction d, and a plane with normal n: $r = a + λ d, r \cdot n = D, n_{1} x + n_{2} y + n_{3} z = D$
Cartesian equation of a linethe line through (a₁, a₂, a₃) with direction vector d: $\frac{x - a _{1}}{d _{1}} = \frac{y - a _{2}}{d _{2}} = \frac{z - a _{3}}{d _{3}}$
Angle between a line and a planed is the direction of the line and n is the normal to the plane: $sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}$
Angle between two lines or two planesd₁ and d₂ are the directions of the lines, n₁ and n₂ the normals to the planes. The modulus gives the acute angle: $cos θ = \frac{∣ d _{1} \cdot d _{2} ∣}{∣ d _{1} ∣∣ d _{2} ∣}, cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$
Distance from a point to a plane or a lineb is the position vector of the point. You can also find the foot of the perpendicular, then the distance between the two points: $plane r \cdot n = D : \frac{∣ b \cdot n - D ∣}{∣ n ∣}, line r = a + λ d : \frac{∣ ( b - a ) \times d ∣}{∣ d ∣}$

Complex numbers

Modulus and conjugate: $z = x + i y, ∣ z ∣ = x^{2} + y^{2}, z^{*} = x - i y, z z^{*} = ∣ z ∣^{2}$

Argumentfind the acute angle α first, then place arg z by the quadrant of z on the Argand diagram: $tan α = \frac{y}{x}, - π < ar g z ⩽ π$

Probability and statistics

Probability rulesfor independent events, P(A ∩ B) = P(A) P(B): $P (A^{'}) = 1 - P (A), P (A \cup B) = P (A) + P (B) - P (A \cap B), P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

Permutationsthe second is the number of ways to arrange n distinct objects in a circle: $^{n} P_{r} = \frac{n !}{( n - r )!}, in a circle: (n - 1)!$
Expectation and variance of a discrete random variable: $E (X) = \sum x P (X = x), Var (X) = E (X^{2}) - [E (X)]^{2}$
Linear combinationsthe last result needs X and Y to be independent: $E (a X + b) E (a X + bY) = a E (X) + b, = a E (X) + b E (Y), Var (a X + b) Var (a X + bY) = a^{2} Var (X) = a^{2} Var (X) + b^{2} Var (Y)$
Sample meanx̄ is the unbiased estimate of the population mean: $E (\overset{ˉ}{X}) = μ, Var (\overset{ˉ}{X}) = \frac{σ ^{2}}{n}, \overset{x}{ˉ} = \frac{Σ x}{n} = \frac{Σ ( x - a )}{n} + a$
Distribution of the sample meanexactly for a normal population, approximately for large n (for example n ≥ 30) by the Central Limit Theorem: $\overset{ˉ}{X} \sim N (μ, \frac{σ ^{2}}{n})$
Unbiased variance from coded datause when the data are summarised as Σ(x − a) and Σ(x − a)²: $s^{2} = \frac{1}{n - 1} (Σ (x - a)^{2} - \frac{( Σ ( x - a ) ) ^{2}}{n})$
StandardisingIf X ~ N(μ, σ²) then Z ~ N(0, 1). The second is the test statistic for a population mean, using s in place of σ for a large sample when σ is unknown: $Z = \frac{X - μ}{σ}, Z = \frac{X ˉ - μ _{0}}{σ / n}$
Regression line of x on yMF27 prints only the line of y on x: $x - \overset{x}{ˉ} = b^{'} (y - \overset{y}{ˉ}), b^{'} = \frac{Σ ( x - x ˉ ) ( y - y ˉ )}{Σ ( y - y ˉ ) ^{2}}$

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Questions students ask

Is MF27 given in the H2 Math exam?

Yes. The 9758 syllabus states that candidates will be provided in the examination with a list of formulae and results. MF27 is that list, for use from 2025 in all papers for H1, H2 and H3 Mathematics and H2 Further Mathematics.

Is integration by parts in MF27?

No. MF27 gives a table of standard integrals such as 1/(x² + a²), tan x and sec x, but not integration by parts. Section 5.3 of the 9758 syllabus includes integration by parts, so you must know ∫u (dv/dx) dx = uv − ∫v (du/dx) dx by heart.

Are the AP and GP formulae in MF27?

No. Section 2.1 of the 9758 syllabus expects the formula for the nth term and the sum of arithmetic and geometric series, and the sum to infinity of a convergent geometric series, but MF27 does not print any of them.

Why does MF27 contain formulae that are not in H2 Math?

One list serves H1, H2 and H3 Mathematics and H2 Further Mathematics. So it includes results that the 9758 H2 Mathematics syllabus does not cover: arc length and surface area of revolution, functions of two variables, the numerical methods (trapezium and Simpson's rules, Newton-Raphson and the Euler methods), the Poisson, geometric and exponential distributions, the Wilcoxon signed rank test and the Mathematical Results section. They are tagged not in H2 Mathematics on this page.

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