Further Pure Mathematics (Summary)
Chapter 1: SURDS AND LOGARITHMIC FUNCTIONS
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:
$\text{The graph of } y = e^x \text{ is an increasing exponential curve.}$
$\text{The graph of } y = \ln x \text{ is its inverse and only defined for } x \gt 0.$
Chapter 2: THE QUADRATIC FUNCTION
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:
- $\alpha + \beta = -\frac{b}{a}$
- $\alpha \beta = \frac{c}{a}$
Chapter 3: INEQUALITIES AND IDENTITIES
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.
Chapter 4: SKATCHING POLYNOMIALS
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. |
Chapter 5: SEQUENCES AND SERIES
Summary: SEQUENCES AND SERIES
- All arithmetic series can be written in the form: $$a + (a + d) + (a + 2d) + (a + 3d) + \dots + (a + (n - 1)d) + \dots$$ First term | Second term | Third term | Fourth term | $n$th term
- The $n$th term of an arithmetic sequence is $a + (n - 1)d$, where $a$ is the first term and $d$ is the common difference.
- The sum of an arithmetic series is: $$S_n = \frac{n}{2}[2a + (n - 1)d] \text{ or } S_n = \frac{n}{2}(a + L)$$ where $a$ is the first term, $d$ is the common difference, $n$ is the number of terms and $L$ is the last term in the sequence.
- You can use the symbol $\Sigma$ to indicate ‘sum of’. $\Sigma$ is used to write a series in a quick and concise way. For example, $$\sum_{r=1}^{500}(2r + 50) = 52 + 54 + 56 + \dots + 1050$$
- In a geometric sequence you can get from one term to another by multiplying by a constant called the common ratio.
- The formula for the $n$th term is $ar^{n-1}$ where $a$ is the first term and $r$ is the common ratio.
- The formula for the sum of the $n$ terms of a geometric series is $$S_n = \frac{a(1 - r^n)}{1 - r} \text{ or } S_n = \frac{a(r^n - 1)}{r - 1}$$
- The sum to infinity exists if $-1 \lt r \lt 1$ and $S_\infty = \frac{a}{1 - r}$.
Chapter 6: THE BINOMIAL SERIES
Summary: THE BINOMIAL SERIES
The binomial expansion
$$ (1+x)^n=1+n x+\frac{n(n-1)}{2!} x^2+\frac{n(n-1)(n-2)}{3!} x^3+\ldots $$can be used to give an exact expression if $n$ is a positive integer.
It can also be used to give an approximation for any rational number. The expansion of
$$ (1+x)^n=1+n x+\frac{n(n-1)}{2!} x^2+\frac{n(n-1)(n-2)}{3!} x^3+\ldots $$where $n$ is negative or a fraction, is only valid if $|x|\lt1$.
Chapter 7: SCALAR AND VECTOR QUANTITIES
Summary: SCALAR AND VECTOR QUANTITIES
- Vectors that are equal have both the same magnitude and the same direction.
- Two vectors can be added using the ‘triangle law’, $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$.
- The modulus of the vector is another way of saying its magnitude. The words are interchangeable.
- The modulus of vector $\mathbf{a}$ is written as $|\mathbf{a}|$.
- The modulus of vector $\overrightarrow{PQ}$ is written as $|\overrightarrow{PQ}|$.
- The modulus of $x\mathbf{i}+y\mathbf{j}$ is $\sqrt{x^2+y^2}$.
- The vector $-\vec a$ has the same magnitude as the vector $\vec a$ but in the opposite direction.
- If vector $\mathbf{a}$ is parallel to vector $\mathbf{b}$, $\mathbf{a}=\lambda \mathbf{b}$ where $\lambda$ is a scalar. $\mathbf a-\mathbf b$ is the same as $\mathbf{a}+(-\mathbf{b})$.
- A unit vector is a vector that has a modulus (or magnitude) of 1 unit.
- If $\lambda \mathbf{a}+\mu \mathbf{b}=\alpha \mathbf{a}+\beta \mathbf{b}$ and the non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are not parallel, then $\lambda=\alpha$ and $\mu=\beta$.
- The position vector of point $A$ is the vector $\overrightarrow{OA}$, where $O$ is the origin.
- $\overrightarrow{OA}$ is usually written as vector $\mathbf{a}$.
- $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$, where $\mathbf{a}$ and $\mathbf{b}$ are the position vectors of $A$ and $B$ respectively.
Chapter 8: RECTANGULAR CARTESIAN COORDINATES
Summary: RECTANGULAR CARTESIAN COORDINATES
-
The general form of the equation of the straight line is $y=mx+c$ where $m$ is the gradient and $c$ is the $y$-intercept or constant.
Another form of this equation is $ax+by+c=0$ where $a$, $b$, $c$ are integers. - The formula for calculating the gradient is $$m=\frac{y_2-y_1}{x_2-x_1}$$
- You can find the equation of the line with gradient $m$ that passes through the point with coordinates $(x_1,y_1)$ by using $$y-y_1=m(x-x_1)$$
- Two lines with the same gradient are parallel.
- If a line has a gradient of $m$, a line perpendicular to it has a gradient of $-\frac{1}{m}$.
- You can also say that if two lines are perpendicular, the product of their gradients is $-1$.
- You can find the distance $d$ between $(x_1,y_1)$ and $(x_2,y_2)$ using the formula: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
- The coordinates of the point $(x_p,y_p)$ dividing the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ are given by $$x_p=\frac{nx_1+mx_2}{m+n}\mbox{ and } y_p=\frac{ny_1+my_2}{m+n}$$
Chapter 9: DIFFERENTIATION
Summary: DIFFERENTIATION
- You should know, and be able to use, these standard formulae:
| Function | Derivative |
|---|---|
| $x^n$ | $nx^{n-1}$ |
| $\sin ax$ | $a\cos ax$ |
| $\cos ax$ | $-a\sin x$ |
| $\mathrm{e}^{ax}$ | $a\mathrm{e}^{ax}$ |
- You also need to know these rules.
- Chain rule: $$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} u}\times\frac{\mathrm{d} u}{\mathrm{d} x}$$
-
Product rule:
If $y=uv$ then $$\frac{\mathrm{d} y}{\mathrm{d} x}=u\frac{\mathrm{d} v}{\mathrm{d} x}+v\frac{\mathrm{d} u}{\mathrm{d} x}$$ - Quotient rule: $$\text{If } y=\frac{u}{v} \text{ then } \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{v\frac{\mathrm{d} u}{\mathrm{d} x}-u\frac{\mathrm{d} v}{\mathrm{d} x}}{v^2}$$
- A turning point is a point where $$\frac{\mathrm{d} y}{\mathrm{d} x}=0$$
- If $\frac{\mathrm{d} y}{\mathrm{d} x}=0$ and $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}\gt0$ then the point is a minimum.
- If $\frac{\mathrm{d} y}{\mathrm{d} x}=0$ and $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}\lt0$ then the point is a maximum.
Chapter 10: INTEGRATION
Summary: INTEGRATION
You should know, and be able to use, these standard integral formulae.
| FUNCTION | INTEGRAL |
|---|---|
| $x^n$ | $\frac{1}{n+1}x^{n-1},\ n\neq-1$ |
| $\sin ax$ | $-\frac{1}{a}\cos ax+c$ |
| $\cos ax$ | $\frac{1}{a}\sin ax+c$ |
| $\mathrm{e}^{ax}$ | $\frac{1}{a}\mathrm{e}^{ax}+c$ |
You should also know how to calculate areas and volumes:
- Area between a curve and $x$-axis $=\int_a^b y\mathrm{d}x,\ y\geqslant0$
- Area between a curve and $y$-axis $=\int_c^d x\mathrm{d}y,\ x\geqslant0$
- Volume of revolution about the $x$-axis $=\pi\int_a^b y^2\mathrm{d}x$
- Volume of revolution about the $y$-axis $=\pi\int_\pi^d x^2\mathrm{d}y$
Chapter 11: TRIGONOMETRY
Summary: TRIGONOMETRY
- $1$ radian $=\frac{180}{\pi}$ degrees.
- The length of an arc of a circle is $l=r\theta$
- The area of a sector is $A=\frac{1}{2}r^2\theta$
- The sine rule is $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ or $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
- The cosine rule is $$a^2=b^2+c^2-2bc\cos A$$ or $$b^2=a^2+c^2-2ac\cos B$$ or $$c^2=a^2+b^2-2ab\cos C$$
- You can find an unknown angle using rearranged form of the cosine rule. $$\cos A=\frac{b^2+c^2-a^2}{2bc}\text{ or }\cos B=\frac{a^2+c^2-b^2}{2ac}\text{ or }\cos C=\frac{a^2+b^2-c^2}{2ab}$$
- You can find the area of a triangle using the formula $$\text{Area}=\frac{1}{2}ab\sin C$$ if you know the length of two sides ($a$ and $b$) and the value of the angle $C$ between them.
- You need to know these identities.
- $\tan\theta=\frac{\sin\theta}{\cos\theta}$
- $\sin^2\theta+\cos^2\theta=1$
- $\sin(A+B)=\sin A\cos B+\cos A\sin B$
- $\cos(A+B)=\cos A\cos B-\sin A\sin B$
- $\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
- The table below will help you solve trigonometrical equations.
| EQUATION | (CALC) | 2nd SOL | 3rd SOL | 4th SOL |
|---|---|---|---|---|
| $\sin x=k$ | $a$ | $180-a$ | All cases are $\pm360$ | All cases are 2nd solution |
| $\cos x=k$ | $a$ | $360-a$ | $\pm360$ | |
| $\tan x=k$ | $a$ | $180+a$ | $\pm360$ |
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