FPM Topic Questions from 2017 to 2019

Further Pure Mathematics Topic Questions from 2017 to 2019

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

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

Summary: Surds and logarithmic functions

You can simplify expressions by using the power (indices) laws.

$c^x \times c^y = c^{x+y}$

$c^x \div c^y = c^{x-y}$

$(c^p)^q = c^{p \times q}$

$\frac{1}{c} = c^{-1}$

$c^1 = c$

$c^0 = 1$

You can manipulate surds using these rules:

• $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

The rules for rationalising surds are:

• $\text{If you have } \frac{1}{\sqrt{a}}, \text{ multiply top and bottom by } \sqrt{a}$
• $\text{If you have } \frac{1}{1+\sqrt{a}}, \text{ multiply top and bottom by } (1-\sqrt{a})$
• $\text{If you have } \frac{1}{1-\sqrt{a}}, \text{ multiply top and bottom by } (1+\sqrt{a})$

Logarithms:

$\log_a n = x \;\;\text{can be rewritten as}\;\; a^x = n$

The laws of logarithms are:

$\log_a(xy) = \log_a x + \log_a y$

$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$

$\log_a(x^q) = q \log_a x$

$\log_a\left(\frac{1}{x}\right) = -\log_a x$

$\log_a(a) = 1$

$\log_a(1) = 0$

Change of base rule:

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

$\log_a b = \frac{1}{\log_b a}$

Natural logarithm:

$\log_e x \equiv \ln x$

Graphs:

$\text{The graph of } y = e^x \text{ is an increasing exponential curve.}$

$\text{The graph of } y = \ln x \text{ is its inverse and only defined for } x \gt 0.$

Summary: The quadratic function

$x^2 - y^2 = (x - y)(x + y)$ is known as the difference of two squares.

Quadratic equations can be solved by:

• factorisation
• completing the square: $x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$
• using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant of a quadratic expression is:

$b^2 - 4ac$

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$ then:

• $\alpha + \beta = -\frac{b}{a}$
• $\alpha \beta = \frac{c}{a}$

Summary: Inequalities and identities

When you multiply or divide an inequality by a negative number you need to reverse the inequality sign.

The steps for solving a quadratic inequality are:

• Solve the corresponding quadratic equation.
• Sketch the graph of the quadratic function.
• Use the sketch to find the required set of values.

$\text{If } f(x) \text{ is a polynomial and } f(a) = 0, \text{ then } (x - a) \text{ is a factor of } f(x).$

This is known as the factor theorem.

$\text{If } f(x) \text{ is a polynomial and } f\left(\frac{b}{a}\right) = 0, \text{ then } (ax - b) \text{ is a factor of } f(x).$

This is also known as the factor theorem.

$\text{If a polynomial } f(x) \text{ is divided by } (ax - b) \text{ then the remainder is } f\left(\frac{b}{a}\right).$

This is known as the remainder theorem.

Summary: Sketching Polynomials

You need to know the shapes of these basic curves.

You also need to know the basic rules of transformations.

NAME	$f(x)$	DESCRIPTION
Horizontal translation of $-a$	$f(x+a)$	The value of $a$ is subtracted from all the $x$-coordinates, but the $y$-coordinates stay unchanged. In other words the curve moves $a$ units to the left.
Vertical translation of $+a$	$f(x)+a$	The value of $a$ is added to all the $y$-coordinates, but the $x$-coordinates stay unchanged. In other words the curve moves $a$ units up.
Horizontal stretch of scale factor $\frac{1}{a}$	$f(ax)$	All the $x$-coordinates are multiplied by $\frac{1}{a}$, but the $y$-coordinates stay unchanged. In other words the curve is squashed in a horizontal direction.
Vertical stretch of scale factor $a$	$a f(x)$	All the $y$-coordinates are multiplied by $a$, but the $x$-coordinates stay unchanged. In other words the curve is stretched in a vertical direction.