AS Mathematics - Pure
Quadratic Equations


The general quadratic equation is
The roots of a quadratic equation can be found in three ways:
1. Factorising e.g.
2. Completing the Square e.g.
OR
3. Quadratic Formula =
e.g.
Discriminant (Δ) =



. The discriminant reveals what kind of roots the equation has:
When Δ > 0, the roots are real and different
When Δ = 0, the roots are real and equal
When Δ < 0, the roots are not real
Parabola – the graph of a quadratic equation
Vertex – the minimum or maximum point of a parabola. It can be found by completing the square:
Functions
Function (f) – a function, f, is a rule that maps a number, x, to another single number, f(x).
Domain of a Function – the set of input numbers i.e. the set of values for x. When x ∈ℝ, x can have
any real value. If f(x) is a fraction, e.g.
, then x cannot equal a number that will cause the
denominator to equal 0, so in that example x cannot equal 2.
Range of a Function – the set of output values i.e. the set of value for f(x)/ y.
Inverse Function (f-1) – the function that maps the output of f to its input. The range of f is the
domain of f-1 and the domain of f is the range of f-1. When the mapping is not one-to-one (i.e. when
one y value has two x values), the function does not have an inverse. To find the inverse of the
function you rearrange the function to make x the subject:
Composite function (gf) – when a function, g, is applied to a function, f. gf(x) is not always the same
as fg(x).
Intersections


A linear and a quadratic equation represent a line and a curve.
To solve a linear and quadratic equation simultaneously
o Eliminate one unknown to give a quadratic equation in the second unknown
o Substitute into the linear equation to find the values for the 1st unknown
o Solve for the 2nd unknown
Coordinate Geometry
With straight line, AB, passing through (x1, y1) and (x2, y2), the following are true:
The length of line AB =
The midpoint, M, of line AB = (
)
The gradient, m, of line AB =
Equation of line AB =




The standard equation of a straight line is
, where m is its gradient and c its
intercept on the y-axis.
Parallel lines have equal gradients
Any equation of the form
gives a straight line.
Given a line with
, then any perpendicular line has equation

When two lines are perpendicular the product of their gradient is -1
Differentiation
Differentiation – the process of finding a general expression for the gradient of a curve at any point
on the curve. This general expression is called the gradient function, or the derived function or the
derivative. The derivative is denoted by
or f’(x) or y’.
Second Derivative (y’’) – the second derivative is the derivative of the first derivative. When turning
point value of x is put into the second derivative:


y’’ > 0, means the curve is concave up
y’’ < 0, means the curve is concave down
The Chain Rule
For composite functions with large powers:
Calculus and Curve Properties
Stationary Points – where the gradient and
=0
Turning Points – points on a graph where the gradient function (the derivative) is 0 and it is changing
the sign where it is passing through this point. There are two types of turning point:


Maximum Point – the gradient changes from positive to negative. This occurs when the
second derivative is less than 0 (y” < 0)
Minimum Point – the gradient changes from negative to positive. This occurs when the
second derivative is greater than 0 (y” > 0)
Increasing Functions – when y becomes larger as x becomes larger. When y’ > 0, the function
increases.
Decreasing Functions – when y becomes smaller as x becomes larger. When y’ < 0, the function
decreases.
Connected Rates of Change
e.g. The equation of a curve is y = x2 – 5x. A point P is moving along the curve so that the x-coordinate
is increasing at the constant rate of 0.2 units per second. Find the rate at which the y-coordinate is
increasing when x = 4.
Integration
Integration – the process of finding a function given from its derivative.
Inverse Chain Rule
For derivatives with large powers and the x has a power of one:
Definite Integrals
Definite Integral – the integral from point a (upper boundary) to b (lower boundary) of y with
respect to x. In doing this, the constant of integration disappears.
e.g.
Finding the Constant of Integration


The constant of integration can be found when a point on a curve is provided
The points can be substituted into the integral so find the value of the constant.
e.g. Find the equation of a curve, such that
passes through point (1,5) on the curve.
Finding the Area
Where a is the upper boundary and b is the lower boundary:
The area between the y-axis and a curve can be found from subtracting the area between the x-axis
and curve from the area of a rectangle.
It can also be found though making x the subject of the equation, and finding the integral from the
upper and lower boundary of the y-axis.
e.g.
y
Area between curve and y-axis:
9
1.
OR
1
2.
0
2x
For compound areas, the area can be found by subtracting one area from the other:
Finding the Volume
Like with the area, the volume when rotated about the y-axis can be found by making x the subject
of the equation:
Circular Measure
One Radian – the size of the angle formed at the centre of a circle by two radii which join the ends of
an arc equal in length to the radians
For a sector of angle
radians of a circle of radius r:
Trigonometry
Ratios of 30°, 45°, 60°
Sin
Cos
Tan
1
45°
1
2
1
30°
2
60°
1
Sine Curve
The graph of
:
0

Graph of
The graph of
has a period of
and a frequency k times that of
:
0


Graph of
is the same shape as the graph
along the x-axis in the positive direction.
Graph of
is the same shape as the graph
along the x-axis in the negative direction
but moved by
but moved by
The graph of
:
0

For any graph of
, where k is a constant, has the same shape as
but moved by k units along the y-axis.
The graph of
or
:
0
The graph of
0
NOTE: These facts about the graph
also apply to
and
Cosine Curve
0
Tan Curve
0

Unlike the cosine and sine curves, the tan curve is not continuous, being undefined when


The range of values of
is unlimited
It is periodic with a period of
Trigonometric Identities
Vectors
Magnitude of a Vector – the length of the vector. It is presented as
.
Scalar Product/Dot Product – used to find the angle between two vectors
Parallel Vectors – vectors that are multiples of one another
Perpendicular Vectors – vectors that have a scalar product equal to 0
Unit Vectors – a vector of magnitude 1. It is present as
e.g. Find the vector
given that
.
is of length 5 units and is in the direction of the vector
Sequences and Series
Arithmetic Progression
Arithmetic Sequence – a sequence whose terms go up or down by constant steps/common
difference
For the nth term of a sequence; where a = the first term, d = the common difference:
For the sum of n terms in an arithmetic sequence; where a = the first term, l = the last (nth) term
These two formula can be combined to give:
Geometric Progression
Geometric Sequence – a sequence whose terms go up or down by a common ratio
For the nth term of a sequence; where a = the first term, r = the common ratio
For the sum of the first n terms:
For the sum to infinity of a geometric progression:
Binomial Expansion
The general term in the binomial expansion is:
AS Mathematics - Statistics
Data Representation
Qualitative Data – consists of descriptions, using names, e.g. colours of cars, types of vehicles etc.
Quantitative Data – takes numerical values. There are two types of quantitative data: discrete and
continuous.
Discrete Data – can only take exact values e.g. the number of hits on a website
Continuous Data – cannot take exact values but can be given only within a specified range or to a
specified degree of accuracy
Mean – the most commonly used average that is calculated by dividing the sum of all the
observations by the number of observations.
Median – an average that is not influenced by extreme values. It is the ½(n+1)th term. For a set of n
numbers arranged in ascending order:


When n is odd, the median is the middle value
When n is even, the median is the mean of the two middle values
Mode – the value that occurs most often.
Bar Chart


There must be gaps between the bars
The mode is given by the highest bar
Advantages
Different sets of data can
be compared using
comparative bar charts
Shows the mode clearly
Disadvantages
Only useful for qualitative
data
6
4
2
0
Pie Chart
Advantages
Shows the proportions of
each quantity
Bar Chart
Disadvantages
Has limited use with
quantitative data
Does not show frequencies
To determine the angles for each section of the pie chart:
Stem
Leaf
Pieand
Chart
Vertical Line Diagram


The height of the line gives the frequency
The mode is given by the highest line
Advantages
Gives an idea of the shape
of the distribution
Shows the mode clearly
Disadvantages
Only useful for illustrating a
small number of values
Vertical Line Diagram
14
12
10
8
6
4
2
0
Stem and Leaf Diagrams
Advantages
It shows the shape of the
distribution
It shows all the original data
The mode, median and
quartiles can be found from
the diagram
It is useful for comparing
two sets of data
Disadvantages
It is not suitable for large
amounts of data
Histograms



There are no gaps between the bars
Total area = total frequency
Area of bar = frequency in that interval

Frequency density =

Interval width = upper class boundary – lower class
boundary
The modal class is represented by the highest bar

Stem and Leaf
A key is essential
Equal intervals must be chosen
Advantages
The mean and standard
deviation can be estimated
from the histogram
It shows whether the
distribution is symmetrical
or skew
It can represent groups of
different widths
21
99653
33
1
6
1345
256
9
9
Histogram
5
Disadvantages
The visual impact can be
altered by choosing
different groups
Two distributions cannot be
shown on the same
diagram
5
4
3
2
1
0
Key: 1 | 2| 9 represents
21 for Column 1 and 29 for Column 2
Freuqncy Density


4
3
2
1
0
Lower Quartile (Q1) – is the median of all the values before the median
Upper Quartile (Q3) – is the median of all the values after the median
Interquartile Range – upper quartile – lower quartile
Range – highest value – lowest value
Cumulative Frequency Graph
Cumulative Frequency – total frequency up to a particular
value

Plot cumulative frequency against upper class
boundaries
Join the points with a smooth curve or straight lines
Advantages
The median and quartiles
can be estimated from the
graph
Sets of data can be
compared by drawing
graphs on the same
diagram
Disadvantages
The visual impact can be
altered by using different
scales
10
9
Cumulative Frequency

Cumulative Frequency
Graph
8
7
6
5
4
3
2
1
0
Box and Whisker Plot



The ends of the whiskers are at the maximum and
minimum values
The end of the box are drawn at the lower quartile and
upper quartile
The line in the box is drawn at the median
Advantages
It is easy to see whether the
distribution is symmetrical
or whether there is a larger
tail in one direction
It can be used to investigate
extreme values
It is easy to see the range
and interquartile range
You can compare two or
more sets of data by
drawing plots on the same
diagram
Disadvantages
It does not show
frequencies
Box and Whisker Plot
Max Value
Upper Q
Median
Lower Q
Min Value
Mean and Standard Deviation
For Raw Data:
For data in a Frequency Table:
x




x
When data are grouped, use the mid-interval value to represent the interval,
where mid-interval = ½ (1.c.b. + u.c.b.)
The variance is the standard deviation squared
When given
and
:
o To find the mean, you find the mean of (x – a) and then add a.
o To find the standard deviation, you find the standard deviation of (x – a); this is the
same as the standard deviation of x
When given two sets of data:
o
o
Arrangements in a Line

The number of different arrangements of n distinct objects is:

The number of different arrangements of n items of which p are alike is:

The number of different arrangements of n items of which p of one type are alike, q of
another type are alike, r of another type are alike and so on is:

By definition, 0! = 1
Permutations – the number of different arrangements available in a group. In permutations, order
matters.
The number of permutations of r items taken from n distinct items is: nPr =
Combinations – a selection of some items where the order of the selected items does not matter.
The number of combinations of r items taken from n distinct items is: nCr =
=
Probability




The probability of an event if the likelihood that it will happen
A probability of 0 indicates that the event is impossible
A probability of 1 indicates that the event is certain to happen
All other events have a probability between 0 and 1
For events A and B:
For mutually exclusive events A and B:
Mutually Exclusive – events are mutually exclusive when they cannot occur at the same time
If
, then A and B are independent
Discrete Random Variables
Random Variable – a quantity whose value depends on chance. The probability distribution of a
discrete random variable is a listing of the possible values of the variable and the corresponding
probabilities.
Statistical Model – uses probabilities to describe a situation and make predictions
Binomial Distribution
Binomial Distribution – used to model a situation if the following conditions are met:




A trial has two possible outcomes; a success (p) and a failure (q)
The trial is repeated n times
The trials are independent
The probability of success (p) is constant for each trial
Normal Distribution



Symmetrical with the mean, median and mode all at its centre
For Standard Normal Variable,
, the mean is 0 and the standard deviation 1 and
the results can be read off of the table
For General Normal Variable,
, the mean and standard deviation must be
converted back to Standard Normal Variable to record the results from the table
e.g. For  = 100,  = 5, and P(X < 110)
Normal Approximation to the Binomial Distribution
If X is a binomial random variable with np>5 and nq>5, then we can use the area under the normal
curve to approximate the probability of a binomial random variable.

Use  = np, and  =

This will give you the General Normal Variable which must be converted back into Standard
Normal Variable
To approximate a binomial (discrete random variable) with a normal (continuous random
variable) we add or subtract 0.5 from our discrete value to convert to a continuous value.

e.g. For
If it were to find
find
as the parameters