International GCSE Further Pure Mathematics Formula Sheet

International GCSE in Further Pure Mathematics Formulae sheet

Mensuration

Surface area of sphere = $4\pi r^2$

Curved surface area of cone = $\pi r \times \text{slant height}$

Volume of sphere = $\dfrac{4}{3}\pi r^3$

Series

Arithmetic series

Sum to $n$ terms, $S_n = \dfrac{n}{2}\left[2a + (n - 1)d\right]$

Geometric series

Sum to $n$ terms, $S_n = \dfrac{a(1-r^n)}{1-r}$

Sum to infinity, $S_\infty = \dfrac{a}{1-r} \qquad |r| < 1$

Binomial series

$(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \ldots + \dfrac{n(n-1)\ldots(n-r+1)}{r!}x^r + \ldots \quad \text{for } |x| < 1,\; n \in \mathbb{Q}$

Calculus

Quotient rule (differentiation)

$\dfrac{d}{dx} \left( \dfrac{f(x)}{g(x)} \right) = \dfrac{f'(x)g(x) - f(x)g'(x)} {[g(x)]^2}$

Trigonometry

Cosine rule

In triangle $ABC$: $a^2 = b^2 + c^2 - 2bc\cos A$

$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\tan(A + B) = \dfrac{\tan A + \tan B} {1 - \tan A \tan B}$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\tan(A - B) = \dfrac{\tan A - \tan B} {1 + \tan A \tan B}$

Logarithms

$\log_a x = \dfrac{\log_b x} {\log_b a}$

Questions by Topic (Further Pure Mathematics)