A-Level · H1 · H2 · Further Maths

MF27 Formula List: every formula, typeset

MF27 is the List of Formulae and Results handed out in every Singapore-Cambridge A-Level maths paper from 2025 — H1, H2 and H3 Mathematics and H2 Further Mathematics. It replaced MF26.

Below is every formula on it, grouped the same way, with one line on when each group is used. Under each group you will find what MF27 does NOT give you — the formulas you still have to memorise.

Used from
2025
Replaces
MF26
Booklet
8 pages
In the exam
Given
Print the MF27 PDFWhat is NOT on MF27

Algebraic series

H2 Maths

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

Binomial expansion, n a positive integer

(a+b)n=an+(n1)an−1b+(n2)an−2b2+(n3)an−3b3+⋯+bn\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

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

Maclaurin series

f(x)=f(0)+xf′(0)+x22!f′′(0)+⋯+xnn!f(n)(0)+…\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

(1+x)n=1+nx+n(n−1)2!x2+…+n(n−1)⋯(n−r+1)r!xr+…\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|x| < 1

ex=1+x+x22!+x33!+⋯+xrr!+…\begin{aligned} e^x = 1 &+ x + \frac{x^2}{2!} + \frac{x^3}{3!} \\ &+ \dots + \frac{x^r}{r!} + \dots \end{aligned}

valid for all x\text{all } x

sin⁡x=x−x33!+x55!−…+(−1)rx2r+1(2r+1)!+…\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 all x\text{all } x

cos⁡x=1−x22!+x44!−…+(−1)rx2r(2r)!+…\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 all x\text{all } x

ln⁡(1+x)=x−x22+x33−…+(−1)r+1xrr+…\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≤1-1 < x \le 1

Not on MF27 — memorise

AP: nth term and sum

un=a+(n−1)du_n = a + (n-1)d
Sn=n2[2a+(n−1)d]=n2(a+l)S_n = \frac{n}{2}\big[2a + (n-1)d\big] = \frac{n}{2}(a + l)

GP: nth term and sum

un=arn−1u_n = ar^{n-1}
Sn=a(1−rn)1−rS_n = \frac{a(1-r^n)}{1-r}

GP: sum to infinity

S∞=a1−rS_\infty = \frac{a}{1-r}

valid for ∣r∣<1|r| < 1

Sum and nth term

un=Sn−Sn−1u_n = S_n - S_{n-1}

Small-angle approximations (x in radians)

sin⁡x≈x\sin x \approx x
cos⁡x≈1−x22\cos x \approx 1 - \frac{x^2}{2}
tan⁡x≈x\tan x \approx x

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

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

valid for ∣bax∣<1\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

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

A repeated linear factor

px2+qx+r(ax+b)(cx+d)2=Aax+b+Bcx+d+C(cx+d)2\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

px2+qx+r(ax+b)(x2+c2)=Aax+b+Bx+Cx2+c2\frac{px^2+qx+r}{(ax+b)(x^2+c^2)} = \frac{A}{ax+b} + \frac{Bx+C}{x^2+c^2}

Not on MF27 — memorise

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

x2+1(x−1)(x+2)=1+−x+3(x−1)(x+2)\frac{x^2+1}{(x-1)(x+2)} = 1 + \frac{-x+3}{(x-1)(x+2)}

Trigonometry

H2 Maths

Compound and double angles, and the principal values of the inverse functions. The factor formulae are gone from MF27.

sin⁡(A±B)≡sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) \equiv \sin A\cos B \pm \cos A\sin B
cos⁡(A±B)≡cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) \equiv \cos A\cos B \mp \sin A\sin B
tan⁡(A±B)≡tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
sin⁡2A≡2sin⁡Acos⁡A\sin 2A \equiv 2\sin A\cos A
cos⁡2A≡cos⁡2A−sin⁡2A≡2cos⁡2A−1≡1−2sin⁡2A\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≡2tan⁡A1−tan⁡2A\tan 2A \equiv \frac{2\tan A}{1 - \tan^2 A}

Principal values:

−12π≤sin⁡−1x≤12π-\tfrac12\pi \le \sin^{-1}x \le \tfrac12\pi

valid for ∣x∣≤1|x| \le 1

0≤cos⁡−1x≤π0 \le \cos^{-1}x \le \pi

valid for ∣x∣≤1|x| \le 1

−12π<tan⁡−1x<12π-\tfrac12\pi < \tan^{-1}x < \tfrac12\pi

Not on MF27 — memorise

Pythagorean identities

sin⁡2A+cos⁡2A≡1\sin^2 A + \cos^2 A \equiv 1
1+tan⁡2A≡sec⁡2A1 + \tan^2 A \equiv \sec^2 A
1+cot⁡2A≡cosec⁡2A1 + \cot^2 A \equiv \operatorname{cosec}^2 A

R-formula

asin⁡θ+bcos⁡θ≡Rsin⁡(θ+α)a\sin\theta + b\cos\theta \equiv R\sin(\theta + \alpha)
R=a2+b2R = \sqrt{a^2+b^2}
tan⁡α=ba\tan\alpha = \frac{b}{a}

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

sin⁡2A≡12(1−cos⁡2A)\sin^2 A \equiv \tfrac12(1 - \cos 2A)
cos⁡2A≡12(1+cos⁡2A)\cos^2 A \equiv \tfrac12(1 + \cos 2A)

Factor formulae (were on MF26, not on MF27)

sin⁡P+sin⁡Q≡2sin⁡12(P+Q)cos⁡12(P−Q)\sin P + \sin Q \equiv 2\sin\tfrac12(P+Q)\cos\tfrac12(P-Q)
sin⁡P−sin⁡Q≡2cos⁡12(P+Q)sin⁡12(P−Q)\sin P - \sin Q \equiv 2\cos\tfrac12(P+Q)\sin\tfrac12(P-Q)
cos⁡P+cos⁡Q≡2cos⁡12(P+Q)cos⁡12(P−Q)\cos P + \cos Q \equiv 2\cos\tfrac12(P+Q)\cos\tfrac12(P-Q)
cos⁡P−cos⁡Q≡−2sin⁡12(P+Q)sin⁡12(P−Q)\cos P - \cos Q \equiv -2\sin\tfrac12(P+Q)\sin\tfrac12(P-Q)

Derivatives

H2 Maths

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

f(x)f(x)f′(x)f'(x)
sin⁡−1x\sin^{-1}x11−x2\dfrac{1}{\sqrt{1-x^2}}
cos⁡−1x\cos^{-1}x−11−x2-\dfrac{1}{\sqrt{1-x^2}}
tan⁡−1x\tan^{-1}x11+x2\dfrac{1}{1+x^2}
cosec⁡x\operatorname{cosec} x−cosec⁡xcot⁡x-\operatorname{cosec} x\cot x
sec⁡x\sec xsec⁡xtan⁡x\sec x\tan x

Not on MF27 — memorise

Basic derivatives

ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos x
ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x
ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x = \sec^2 x
ddxcot⁡x=−cosec⁡2x\frac{d}{dx}\cot x = -\operatorname{cosec}^2 x
ddxex=ex\frac{d}{dx}e^x = e^x
ddxln⁡x=1x\frac{d}{dx}\ln x = \frac1x
ddxax=axln⁡a\frac{d}{dx}a^x = a^x\ln a

Product and quotient rules

ddx(uv)=udvdx+vdudx\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}
ddx(uv)=vdudx−udvdxv2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

Chain rule and parametric form

dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}
dydx=dy/dtdx/dt\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.

f(x)f(x)∫f(x) dx\displaystyle\int f(x)\,dxValid for
1x2+a2\dfrac{1}{x^2+a^2}1atan⁡−1 ⁣(xa)\dfrac1a\tan^{-1}\!\left(\dfrac xa\right)
1a2−x2\dfrac{1}{\sqrt{a^2-x^2}}sin⁡−1 ⁣(xa)\sin^{-1}\!\left(\dfrac xa\right)∣x∣<a|x|<a
1x2−a2\dfrac{1}{x^2-a^2}12aln⁡ ⁣(x−ax+a)\dfrac{1}{2a}\ln\!\left(\dfrac{x-a}{x+a}\right)x>ax>a
1a2−x2\dfrac{1}{a^2-x^2}12aln⁡ ⁣(a+xa−x)\dfrac{1}{2a}\ln\!\left(\dfrac{a+x}{a-x}\right)∣x∣<a|x|<a
tan⁡x\tan xln⁡(sec⁡x)\ln(\sec x)∣x∣<12π|x|<\tfrac12\pi
cot⁡x\cot xln⁡(sin⁡x)\ln(\sin x)0<x<π0<x<\pi
cosec⁡x\operatorname{cosec} x−ln⁡(cosec⁡x+cot⁡x)-\ln(\operatorname{cosec} x+\cot x)0<x<π0<x<\pi
sec⁡x\sec xln⁡(sec⁡x+tan⁡x)\ln(\sec x+\tan x)∣x∣<12π|x|<\tfrac12\pi

Not on MF27 — memorise

Basic integrals

∫xn dx=xn+1n+1 (n≠−1)\int x^n\,dx = \frac{x^{n+1}}{n+1}\ (n \ne -1)
∫1x dx=ln⁡∣x∣\int \frac1x\,dx = \ln|x|
∫eax dx=1aeax\int e^{ax}\,dx = \frac1a e^{ax}
∫sin⁡ax dx=−1acos⁡ax\int \sin ax\,dx = -\frac1a\cos ax
∫cos⁡ax dx=1asin⁡ax\int \cos ax\,dx = \frac1a\sin ax
∫sec⁡2ax dx=1atan⁡ax\int \sec^2 ax\,dx = \frac1a\tan ax

Standard forms

∫f′(x)f(x) dx=ln⁡∣f(x)∣\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|
∫[f(x)]nf′(x) dx=[f(x)]n+1n+1\int [f(x)]^n f'(x)\,dx = \frac{[f(x)]^{n+1}}{n+1}

Integration by parts

∫udvdx dx=uv−∫vdudx dx\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx

Volume of revolution

V=π∫aby2 dx  (about the x-axis)V = \pi\int_a^b y^2\,dx \ \ (\text{about the } x\text{-axis})
V=π∫cdx2 dy  (about the y-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.

The point dividing AB in the ratio λ : μ

μa+λbλ+μ\frac{\mu\mathbf a + \lambda\mathbf b}{\lambda + \mu}

Vector (cross) product

a×b=(a1a2a3)×(b1b2b3)=(a2b3−a3b2a3b1−a1b3a1b2−a2b1)\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 MF27 — memorise

Scalar product

a⋅b=∣a∣∣b∣cos⁡θ=a1b1+a2b2+a3b3\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

∣a⋅b^∣|\mathbf a\cdot\hat{\mathbf b}|
Area=12∣a×b∣\text{Area} = \tfrac12|\mathbf a\times\mathbf b|

Line and plane

r=a+λd\mathbf r = \mathbf a + \lambda\mathbf d
r⋅n=D\mathbf r\cdot\mathbf n = D

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

∣p⋅n−D∣∣n∣\frac{|\mathbf p\cdot\mathbf n - D|}{|\mathbf n|}

Angle between line and plane, and between two planes

sin⁡θ=∣d⋅n∣∣d∣∣n∣\sin\theta = \frac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|}
cos⁡θ=∣n1⋅n2∣∣n1∣∣n2∣\cos\theta = \frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}

Applications of definite integrals

Further Maths

Further Maths: the length of a curve and the area of the surface it sweeps out. New in MF27.

Arc length, y as a function of x

s=∫ab1+(dydx)2 dxs = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx

Surface area of revolution about the x-axis

S=∫ab2πy1+(dydx)2 dxS = \int_a^b 2\pi y\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx

Functions of two variables

Further Maths

Further Maths: approximating f(x, y) near a point (a, b), as Maclaurin does for one variable. New in MF27.

Quadratic approximation of f at (a, b)

f(x,y)≈f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)+12fxx(a,b)(x−a)2+fxy(a,b)(x−a)(y−b)+12fyy(a,b)(y−b)2\begin{aligned} f(x,y) \approx{} & f(a,b) \\ & + f_x(a,b)(x-a) + f_y(a,b)(y-b) \\ & + \tfrac12 f_{xx}(a,b)(x-a)^2 \\ & + f_{xy}(a,b)(x-a)(y-b) \\ & + \tfrac12 f_{yy}(a,b)(y-b)^2 \end{aligned}

Numerical methods

Further Maths

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

Trapezium rule, one strip

∫abf(x) dx≈12(b−a)[f(a)+f(b)]\int_a^b f(x)\,dx \approx \tfrac12(b-a)\big[f(a) + f(b)\big]

Simpson's rule, two strips

∫abf(x) dx≈16(b−a)[f(a)+4f ⁣(a+b2)+f(b)]\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

x2=x1−f(x1)f′(x1)x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}

Euler method, step size h

y2=y1+hf(x1,y1)y_2 = y_1 + h f(x_1, y_1)

Improved Euler method, step size h

u2=y1+hf(x1,y1)u_2 = y_1 + h f(x_1, y_1)
y2=y1+h2[f(x1,y1)+f(x2,u2)]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.

Discrete

Distribution of XP(X=x)P(X=x)MeanVariance
Binomial B(n, p)(nx)px(1−p)n−x\dbinom nx p^x(1-p)^{n-x}npnpnp(1−p)np(1-p)
Poisson Po(λ)e−λλxx!e^{-\lambda}\dfrac{\lambda^x}{x!}λ\lambdaλ\lambda
Geometric Geo(p)(1−p)x−1p(1-p)^{x-1}p1p\dfrac1p1−pp2\dfrac{1-p}{p^2}

Continuous

Distribution of Xp.d.f.MeanVariance
Exponentialλe−λx\lambda e^{-\lambda x}1λ\dfrac1\lambda1λ2\dfrac1{\lambda^2}

Not on MF27 — memorise

Expectation and variance

E(X)=∑x P(X=x)E(X) = \sum x\,P(X=x)
Var⁡(X)=E(X2)−[E(X)]2\operatorname{Var}(X) = E(X^2) - [E(X)]^2

Linear combinations (X, Y independent for the variance)

E(aX+b)=aE(X)+bE(aX+b) = aE(X)+b
Var⁡(aX±bY)=a2Var⁡(X)+b2Var⁡(Y)\operatorname{Var}(aX \pm bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y)

Sample mean (central limit theorem, n large)

Xˉ∼N ⁣(μ,σ2n) approximately\bar X \sim N\!\left(\mu, \frac{\sigma^2}{n}\right) \text{ approximately}

Probability

P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B) = P(A) + P(B) - P(A\cap B)
P(A∣B)=P(A∩B)P(B)P(A\mid B) = \frac{P(A\cap B)}{P(B)}

Independent events, and counting

P(A∩B)=P(A)P(B)P(A\cap B) = P(A)P(B)
nPr=n!(n−r)!{}^nP_r = \frac{n!}{(n-r)!}
nCr=n!r! (n−r)!{}^nC_r = \frac{n!}{r!\,(n-r)!}

Sampling and testing

H1 & H2 Maths

The unbiased estimate of the population variance from a sample. MF27 no longer gives the two-sample pooled version.

Unbiased estimate of population variance

s2=nn−1(∑(x−xˉ)2n)=1n−1(∑x2−(∑x)2n)\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}

Not on MF27 — memorise

Unbiased estimate of the mean

xˉ=∑xn\bar x = \frac{\sum x}{n}

z-test statistic for a mean

Z=Xˉ−μ0σ/nZ = \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.

Product moment correlation coefficient

r=∑(x−xˉ)(y−yˉ)∑(x−xˉ)2∑(y−yˉ)2=∑xy−∑x∑yn(∑x2−(∑x)2n)(∑y2−(∑y)2n)\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−yˉ=b(x−xˉ)y - \bar y = b(x - \bar x)
b=∑(x−xˉ)(y−yˉ)∑(x−xˉ)2b = \frac{\sum(x-\bar x)(y-\bar y)}{\sum(x-\bar x)^2}

Not on MF27 — memorise

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

x−xˉ=d(y−yˉ)x - \bar x = d(y - \bar y)
d=∑(x−xˉ)(y−yˉ)∑(y−yˉ)2d = \frac{\sum(x-\bar x)(y-\bar y)}{\sum(y-\bar y)^2}

Both lines pass through the mean point

(xˉ,yˉ)(\bar x, \bar y)

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.

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

nOne-tail 0.05 · Two-tail 0.1One-tail 0.025 · Two-tail 0.05One-tail 0.01 · Two-tail 0.02One-tail 0.005 · Two-tail 0.01
620——
7320—
85310
98531
1010853
11131075
12171397
132117129
1425211512
1530251915
1635292319
1741342723
1847403227
1953463732
2060524337

Mathematical results

Further Maths & H3

Inequalities and counting for proofs in Further Maths and H3. A new page in MF27.

AM-GM inequality — x₁, …, xₙ ≥ 0; equal only when all the xᵢ are equal

x1+x2+⋯+xnn≥x1x2⋯xnn\frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1x_2\cdots x_n}

Cauchy-Schwarz inequality — equal only when uᵢ = kvᵢ for every i, for some k ≠ 0

(∑i=1nuivi)2≤(∑i=1nui2)(∑i=1nvi2)\left(\sum_{i=1}^n u_iv_i\right)^2 \le \left(\sum_{i=1}^n u_i^2\right)\left(\sum_{i=1}^n v_i^2\right)

Triangle inequality — equal when the xᵢ all have the same sign (or are zero)

∣x1+x2+⋯+xn∣≤∣x1∣+∣x2∣+⋯+∣xn∣|x_1 + x_2 + \dots + x_n| \le |x_1| + |x_2| + \dots + |x_n|

Inclusion-exclusion principle

∣A1∪⋯∪An∣=∑i∣Ai∣−∑i<j∣Ai∩Aj∣+∑i<j<k∣Ai∩Aj∩Ak∣−…+(−1)n−1∣A1∩⋯∩An∣\begin{aligned} |A_1\cup\dots\cup A_n| ={} & \sum_i |A_i| - \sum_{i<j}|A_i\cap A_j| \\ & + \sum_{i<j<k}|A_i\cap A_j\cap A_k| \\ & - \dots \\ & + (-1)^{n-1}|A_1\cap\dots\cap A_n| \end{aligned}

Complex numbers (not on the list at all)

H2 Maths

A whole H2 topic with nothing given. Since 2025 H2 Maths uses Cartesian form only — no polar or exponential form.

Modulus and conjugate of z = x + iy

∣z∣=x2+y2|z| = \sqrt{x^2 + y^2}
z∗=x−iyz^* = x - iy
zz∗=∣z∣2zz^* = |z|^2

Real and imaginary parts

z+z∗=2Re⁡(z)z + z^* = 2\operatorname{Re}(z)
z−z∗=2iIm⁡(z)z - z^* = 2i\operatorname{Im}(z)

Dividing: multiply top and bottom by the conjugate

1z=z∗∣z∣2\frac{1}{z} = \frac{z^*}{|z|^2}

Argument: tan of the angle, then check the quadrant on the Argand diagram

tan⁡(arg⁡z)=yx\tan(\arg z) = \frac{y}{x}
−π<arg⁡z≤π-\pi < \arg z \le \pi

A polynomial with real coefficients

p(z)=0  ⟹  p(z∗)=0p(z) = 0 \implies p(z^*) = 0

What is not on MF27 (H2 Maths)

MF27 gives you the hard-to-remember results only. For H2 Maths you still need, by heart:

There is no normal table on MF27: normal probabilities come from your graphing calculator.

Questions students ask

What is MF27?+

MF27 is the List of Formulae and Results that SEAB gives you in Singapore-Cambridge A-Level maths exams. It is used from 2025 in every paper for H1 Mathematics, H2 Mathematics, H3 Mathematics and H2 Further Mathematics.

Is MF27 given in the exam?+

Yes. A clean copy is handed out in every A-Level maths paper. You may not bring your own copy or write on one beforehand, so practise with a clean copy beside you.

What is the difference between MF26 and MF27?+

MF27 replaced MF26 from 2025. It dropped the four factor formulae, the two-sample pooled variance, and the normal, t and chi-squared tables (you use your graphing calculator instead). 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 and inclusion-exclusion.

Is MF27 the same for H2 Maths and H2 Further Maths?+

Yes, it is one list for H1, H2 and H3 Mathematics and H2 Further Mathematics. H2 Maths students use the algebra, trigonometry, calculus, vectors, binomial, sampling and regression sections. Numerical methods, Poisson, geometric, exponential, Wilcoxon and the mathematical results are for Further Maths (and H3).

Which formulas must I memorise for H2 Maths?+

Everything not on the list: AP and GP formulas, the small-angle approximations, the R-formula and Pythagorean identities, basic derivatives and integrals, integration by parts, volume of revolution, the scalar product and every line-and-plane result in vectors, all of complex numbers, the E(X) and Var(X) rules, standardising a normal variable, the sample-mean distribution and the z-test statistic. Each section on this page lists them under "Not on the list".

Does MF27 have the normal distribution table?+

No. MF27 has no normal, t or chi-squared tables; you find normal probabilities and inverse-normal values on your graphing calculator. The only table left is the Wilcoxon signed rank test, which is for Further Maths.

Can I download MF27 as a PDF?+

Yes. Our typeset PDF of every formula on the list, with the must-memorise formulas added under each section, is free to print from this page. The official booklet is published by SEAB with the A-Level syllabus documents.

The old list: MF26 (2017–2024) · All formula pages · JC H2 Maths tuition

Typeset by Adrian's Math Tuition from the list SEAB publishes for the Singapore-Cambridge A-Level. The official booklet is the one handed out in the exam; if anything here ever differs from it, the booklet wins — tell us and we will fix it.

Knowing the formula is half of it.

JC H2 Maths in groups of up to 3, in Kovan. Send a question you are stuck on over WhatsApp.

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