Further Pure Mathematics Topic Questions from 2011 to 2013
Further Pure Mathematics Topic Questions from 2014 to 2016
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. |