A-Level · H1 · H2 · Further Maths · 2017–2024

MF26 Formula List, and what changed in MF27

Q: Is MF26 still used?

No. MF26 was used from 2017 to 2024. From 2025 every A-Level maths paper (H1, H2, H3 Mathematics and H2 Further Mathematics) comes with MF27 instead.

Q: What changed from MF26 to MF27?

MF27 dropped the four factor formulae, the two-sample pooled variance and the normal, t and chi-squared tables. It added arc length, surface area of revolution, the two-variable quadratic approximation, and a page of mathematical results (AM-GM, Cauchy-Schwarz, the triangle inequality, inclusion-exclusion). The binomial, Maclaurin, partial fractions, trigonometry, derivatives, integrals, vectors and statistics formulas are the same.

Q: Can I still practise with past papers that used MF26?

Yes. Papers from 2017 to 2024 are still good practice. Two things to watch: learn the factor formulae by heart if a question needs them, and use your graphing calculator wherever the paper expected the normal table.

Q: Do I need to memorise the factor formulae now?

If your syllabus or school uses them, yes — they are no longer printed on MF27. They are listed under Trigonometry on our MF27 page, under "Not on the list".

MF26 was the List of Formulae and Statistical Tables for Singapore-Cambridge A-Level maths from 2017 to 2024. Past papers from those years were set with it beside you.

Every formula on it is below. Each section says what happened to it in MF27, the list used from 2025.

Used: 2017–2024
Replaced by: MF27
Booklet: 12 pages
Tables: 4

Print the MF26 PDF

Sitting A-Levels in 2025 or later? You get MF27, not MF26. Go to the MF27 formula list →

Algebraic series

H2 Maths

Binomial expansions and Maclaurin series — Sequences & Series and the small-x approximations.

Binomial expansion, n a positive integer

\begin{aligned} (a+b)^n = a^n &+ \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 \\ &+ \binom{n}{3}a^{n-3}b^3 + \dots + b^n \end{aligned}

where

\binom{n}{r} = \frac{n!}{r!\,(n-r)!}

Maclaurin series

\begin{aligned} f(x) = f(0) &+ x f'(0) + \frac{x^2}{2!}f''(0) \\ &+ \dots + \frac{x^n}{n!}f^{(n)}(0) + \dots \end{aligned}

Binomial series, any n

\begin{aligned} (1+x)^n = 1 &+ nx + \frac{n(n-1)}{2!}x^2 + \dots \\ &+ \frac{n(n-1)\cdots(n-r+1)}{r!}x^r + \dots \end{aligned}

valid for $|x| < 1$

\begin{aligned} e^x = 1 &+ x + \frac{x^2}{2!} + \frac{x^3}{3!} \\ &+ \dots + \frac{x^r}{r!} + \dots \end{aligned}

valid for $\text{all } x$

\begin{aligned} \sin x = x &- \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \\ &+ \frac{(-1)^r x^{2r+1}}{(2r+1)!} + \dots \end{aligned}

valid for $\text{all } x$

\begin{aligned} \cos x = 1 &- \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \\ &+ \frac{(-1)^r x^{2r}}{(2r)!} + \dots \end{aligned}

valid for $\text{all } x$

\begin{aligned} \ln(1+x) = x &- \frac{x^2}{2} + \frac{x^3}{3} - \dots \\ &+ \frac{(-1)^{r+1}x^r}{r} + \dots \end{aligned}

valid for $-1 < x \le 1$

Not on MF26 — memorise

AP: nth term and sum

u_n = a + (n-1)d

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

GP: nth term and sum

u_n = ar^{n-1}

S_n = \frac{a(1-r^n)}{1-r}

GP: sum to infinity

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

valid for $|r| < 1$

Sum and nth term

u_n = S_n - S_{n-1}

Small-angle approximations (x in radians)

\sin x \approx x

\cos x \approx 1 - \frac{x^2}{2}

\tan x \approx x

Expanding (a + bx)ⁿ: take out aⁿ first

(a+bx)^n = a^n\left(1 + \frac{b}{a}x\right)^n

valid for $\left|\tfrac{b}{a}x\right| < 1$

Partial fractions

H2 Maths

Splitting a proper fraction before you integrate it or expand it as a series.

Distinct linear factors

\frac{px+q}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}

A repeated linear factor

\frac{px^2+qx+r}{(ax+b)(cx+d)^2} = \frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{(cx+d)^2}

A quadratic factor that does not factorise

\frac{px^2+qx+r}{(ax+b)(x^2+c^2)} = \frac{A}{ax+b} + \frac{Bx+C}{x^2+c^2}

Not on MF26 — memorise

Improper fraction (top degree ≥ bottom degree): divide first

\frac{x^2+1}{(x-1)(x+2)} = 1 + \frac{-x+3}{(x-1)(x+2)}

Trigonometry

H2 Maths

Compound and double angles, the factor formulae, and principal values.

MF27 keeps everything here except the four factor formulae.

\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 2A \equiv 2\sin A\cos A

\begin{aligned} \cos 2A &\equiv \cos^2 A - \sin^2 A \\ &\equiv 2\cos^2 A - 1 \\ &\equiv 1 - 2\sin^2 A \end{aligned}

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

Factor formulae

\sin P + \sin Q \equiv 2\sin\tfrac12(P+Q)\cos\tfrac12(P-Q)

\sin P - \sin Q \equiv 2\cos\tfrac12(P+Q)\sin\tfrac12(P-Q)

\cos P + \cos Q \equiv 2\cos\tfrac12(P+Q)\cos\tfrac12(P-Q)

\cos P - \cos Q \equiv -2\sin\tfrac12(P+Q)\sin\tfrac12(P-Q)

Principal values:

-\tfrac12\pi \le \sin^{-1}x \le \tfrac12\pi

valid for $|x| \le 1$

0 \le \cos^{-1}x \le \pi

valid for $|x| \le 1$

-\tfrac12\pi < \tan^{-1}x < \tfrac12\pi

Not on MF26 — memorise

Pythagorean identities

\sin^2 A + \cos^2 A \equiv 1

1 + \tan^2 A \equiv \sec^2 A

1 + \cot^2 A \equiv \operatorname{cosec}^2 A

R-formula

a\sin\theta + b\cos\theta \equiv R\sin(\theta + \alpha)

R = \sqrt{a^2+b^2}

\tan\alpha = \frac{b}{a}

cos 2A turned round (for integrating sin² and cos²)

\sin^2 A \equiv \tfrac12(1 - \cos 2A)

\cos^2 A \equiv \tfrac12(1 + \cos 2A)

Derivatives

H2 Maths

Only the five awkward derivatives are given. Everything else in differentiation is yours to remember.

Unchanged in MF27.

$f(x)$	$f'(x)$
$\sin^{-1}x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1}x$	$-\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1}x$	$\dfrac{1}{1+x^2}$
$\operatorname{cosec} x$	$-\operatorname{cosec} x\cot x$
$\sec x$	$\sec x\tan x$

Not on MF26 — memorise

Basic derivatives

\frac{d}{dx}\sin x = \cos x

\frac{d}{dx}\cos x = -\sin x

\frac{d}{dx}\tan x = \sec^2 x

\frac{d}{dx}\cot x = -\operatorname{cosec}^2 x

\frac{d}{dx}e^x = e^x

\frac{d}{dx}\ln x = \frac1x

\frac{d}{dx}a^x = a^x\ln a

Product and quotient rules

\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}

\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

Chain rule and parametric form

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

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Integrals

H2 Maths

The eight standard forms. Constants of integration are left out and a is a positive constant.

Unchanged in MF27.

$f(x)$	$\displaystyle\int f(x)\,dx$	Valid for
$\dfrac{1}{x^2+a^2}$	$\dfrac1a\tan^{-1}\!\left(\dfrac xa\right)$
$\dfrac{1}{\sqrt{a^2-x^2}}$	$\sin^{-1}\!\left(\dfrac xa\right)$	$\|x\|<a$
$\dfrac{1}{x^2-a^2}$	$\dfrac{1}{2a}\ln\!\left(\dfrac{x-a}{x+a}\right)$	$x>a$
$\dfrac{1}{a^2-x^2}$	$\dfrac{1}{2a}\ln\!\left(\dfrac{a+x}{a-x}\right)$	$\|x\|<a$
$\tan x$	$\ln(\sec x)$	$\|x\|<\tfrac12\pi$
$\cot x$	$\ln(\sin x)$	$0<x<\pi$
$\operatorname{cosec} x$	$-\ln(\operatorname{cosec} x+\cot x)$	$0<x<\pi$
$\sec x$	$\ln(\sec x+\tan x)$	$\|x\|<\tfrac12\pi$

Not on MF26 — memorise

Basic integrals

\int x^n\,dx = \frac{x^{n+1}}{n+1}\ (n \ne -1)

\int \frac1x\,dx = \ln|x|

\int e^{ax}\,dx = \frac1a e^{ax}

\int \sin ax\,dx = -\frac1a\cos ax

\int \cos ax\,dx = \frac1a\sin ax

\int \sec^2 ax\,dx = \frac1a\tan ax

Standard forms

\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|

\int [f(x)]^n f'(x)\,dx = \frac{[f(x)]^{n+1}}{n+1}

Integration by parts

\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx

Volume of revolution

V = \pi\int_a^b y^2\,dx \ \ (\text{about the } x\text{-axis})

V = \pi\int_c^d x^2\,dy \ \ (\text{about the } y\text{-axis})

Vectors

H2 Maths

The ratio theorem and the cross product. Lines, planes, angles and distances are not given.

Unchanged in MF27.

The point dividing AB in the ratio λ : μ

\frac{\mu\mathbf a + \lambda\mathbf b}{\lambda + \mu}

Vector (cross) product

\mathbf a \times \mathbf b = \begin{pmatrix} a_1\\a_2\\a_3 \end{pmatrix} \times \begin{pmatrix} b_1\\b_2\\b_3 \end{pmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2\\ a_3b_1 - a_1b_3\\ a_1b_2 - a_2b_1 \end{pmatrix}

Not on MF26 — memorise

Scalar product

\mathbf a\cdot\mathbf b = |\mathbf a||\mathbf b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3

Length of projection of a on b, and area of triangle

|\mathbf a\cdot\hat{\mathbf b}|

\text{Area} = \tfrac12|\mathbf a\times\mathbf b|

Line and plane

\mathbf r = \mathbf a + \lambda\mathbf d

\mathbf r\cdot\mathbf n = D

Distance from point P to the plane r·n = D

\frac{|\mathbf p\cdot\mathbf n - D|}{|\mathbf n|}

Angle between line and plane, and between two planes

\sin\theta = \frac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|}

\cos\theta = \frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}

Numerical methods

Further Maths

Further Maths: estimating an integral, a root, or a step of a differential equation. Not in H2 Maths.

Unchanged in MF27.

Trapezium rule, one strip

\int_a^b f(x)\,dx \approx \tfrac12(b-a)\big[f(a) + f(b)\big]

Simpson's rule, two strips

\int_a^b f(x)\,dx \approx \tfrac16(b-a)\left[f(a) + 4f\!\left(\frac{a+b}{2}\right) + f(b)\right]

Newton-Raphson, x₁ a first approximation to a root of f(x) = 0

x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}

Euler method, step size h

y_2 = y_1 + h f(x_1, y_1)

Improved Euler method, step size h

u_2 = y_1 + h f(x_1, y_1)

y_2 = y_1 + \frac h2\big[f(x_1, y_1) + f(x_2, u_2)\big]

Standard distributions

H1 & H2 Maths

H1 and H2 Maths use the binomial row only. Poisson, geometric and exponential are Further Maths.

Unchanged in MF27.

Discrete

Distribution of X	$P(X=x)$	Mean	Variance
Binomial B(n, p)	$\dbinom nx p^x(1-p)^{n-x}$	$np$	$np(1-p)$
Poisson Po(λ)	$e^{-\lambda}\dfrac{\lambda^x}{x!}$	$\lambda$	$\lambda$
Geometric Geo(p)	$(1-p)^{x-1}p$	$\dfrac1p$	$\dfrac{1-p}{p^2}$

Continuous

Distribution of X	p.d.f.	Mean	Variance
Exponential	$\lambda e^{-\lambda x}$	$\dfrac1\lambda$	$\dfrac1{\lambda^2}$

Not on MF26 — memorise

Expectation and variance

E(X) = \sum x\,P(X=x)

\operatorname{Var}(X) = E(X^2) - [E(X)]^2

Linear combinations (X, Y independent for the variance)

E(aX+b) = aE(X)+b

\operatorname{Var}(aX \pm bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y)

Sample mean (central limit theorem, n large)

\bar X \sim N\!\left(\mu, \frac{\sigma^2}{n}\right) \text{ approximately}

Probability

P(A\cup B) = P(A) + P(B) - P(A\cap B)

P(A\mid B) = \frac{P(A\cap B)}{P(B)}

Independent events, and counting

P(A\cap B) = P(A)P(B)

{}^nP_r = \frac{n!}{(n-r)!}

{}^nC_r = \frac{n!}{r!\,(n-r)!}

Sampling and testing

H1 & H2 Maths

Estimating the population variance, from one sample or from two samples pooled.

MF27 keeps the one-sample formula and drops the pooled one.

Unbiased estimate of population variance

\begin{aligned} s^2 &= \frac{n}{n-1}\left(\frac{\sum(x-\bar x)^2}{n}\right) \\ &= \frac{1}{n-1}\left(\sum x^2 - \frac{(\sum x)^2}{n}\right) \end{aligned}

Unbiased estimate of the common population variance from two samples

s^2 = \frac{\sum(x_1 - \bar x_1)^2 + \sum(x_2 - \bar x_2)^2}{n_1 + n_2 - 2}

Not on MF26 — memorise

Unbiased estimate of the mean

\bar x = \frac{\sum x}{n}

z-test statistic for a mean

Z = \frac{\bar X - \mu_0}{\sigma/\sqrt n}

Regression and correlation

H1 & H2 Maths

r and the y-on-x line. Your GC gives both; the formula is for questions that hand you summary totals.

Unchanged in MF27.

Product moment correlation coefficient

\begin{aligned} r &= \frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2\sum(y-\bar y)^2}} \\[4pt] &= \frac{\sum xy - \frac{\sum x\sum y}{n}}{\sqrt{\left(\sum x^2 - \frac{(\sum x)^2}{n}\right)\left(\sum y^2 - \frac{(\sum y)^2}{n}\right)}} \end{aligned}

Regression line of y on x

y - \bar y = b(x - \bar x)

b = \frac{\sum(x-\bar x)(y-\bar y)}{\sum(x-\bar x)^2}

Not on MF26 — memorise

Regression line of x on y (not given — swap the roles)

x - \bar x = d(y - \bar y)

d = \frac{\sum(x-\bar x)(y-\bar y)}{\sum(y-\bar y)^2}

Both lines pass through the mean point

(\bar x, \bar y)

Statistical tables

H1 & H2 Maths

MF26 printed four tables: the normal distribution function Φ(z), critical values of t, critical values of χ², and the Wilcoxon signed rank test.

MF27 keeps only the Wilcoxon table. Normal values now come from your graphing calculator.

Old papers that say "use the tables in MF26" can be done on the graphing calculator.

Φ(z): normalcdf. Its inverse: invNorm.

The t and χ² tables served Further Maths tests.

Wilcoxon signed rank test

Further Maths

Further Maths non-parametric testing. Reject the null hypothesis when T is at most the value in the table.

Unchanged in MF27.

P = sum of the ranks of the positive differences.

Q = sum of the ranks of the negative differences.

T = the smaller of P and Q.

Each entry is the LARGEST T that still rejects the null hypothesis at that level.

A dash: no T can reject at that level for that n.

Critical values of T

n	One-tail 0.05 · Two-tail 0.1	One-tail 0.025 · Two-tail 0.05	One-tail 0.01 · Two-tail 0.02	One-tail 0.005 · Two-tail 0.01
6	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37

Questions students ask

Is MF26 still used?+

No. MF26 was used from 2017 to 2024. From 2025 every A-Level maths paper (H1, H2, H3 Mathematics and H2 Further Mathematics) comes with MF27 instead.

What changed from MF26 to MF27?+

MF27 dropped the four factor formulae, the two-sample pooled variance and the normal, t and chi-squared tables. It added arc length, surface area of revolution, the two-variable quadratic approximation, and a page of mathematical results (AM-GM, Cauchy-Schwarz, the triangle inequality, inclusion-exclusion). The binomial, Maclaurin, partial fractions, trigonometry, derivatives, integrals, vectors and statistics formulas are the same.

Can I still practise with past papers that used MF26?+

Yes. Papers from 2017 to 2024 are still good practice. Two things to watch: learn the factor formulae by heart if a question needs them, and use your graphing calculator wherever the paper expected the normal table.

Do I need to memorise the factor formulae now?+

If your syllabus or school uses them, yes — they are no longer printed on MF27. They are listed under Trigonometry on our MF27 page, under "Not on the list".

The current list: MF27 (from 2025) · All formula pages · JC H2 Maths tuition

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n	One-tail 0.05 · Two-tail 0.1	One-tail 0.025 · Two-tail 0.05	One-tail 0.01 · Two-tail 0.02	One-tail 0.005 · Two-tail 0.01
6	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37

n	One-tail 0.05 · Two-tail 0.1	One-tail 0.025 · Two-tail 0.05	One-tail 0.01 · Two-tail 0.02	One-tail 0.005 · Two-tail 0.01
6	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37

n	One-tail 0.05 · Two-tail 0.1	One-tail 0.025 · Two-tail 0.05	One-tail 0.01 · Two-tail 0.02	One-tail 0.005 · Two-tail 0.01
6	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37