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Further mathematics

GCE AS and A level subject content

July 2014

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Contents
Introduction

3

Purpose

3

Aims and objectives

3

Subject content

5

Structure

5

Background knowledge

5

Overarching themes

5

Use of technology

7

Detailed content statements

7

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The content for further mathematics AS and A levels
Introduction
1
...


Purpose
2
...

3
...
It
is normally taken as an extra subject, typically as a fourth A level
...
As well as building on
algebra and calculus introduced in A level mathematics, the A level further mathematics
core content introduces complex numbers and matrices, fundamental mathematical ideas
with wide applications in mathematics, engineering, physical sciences and computing
...
A level further mathematics prepares students for further study and
employment in highly mathematical disciplines that require knowledge and understanding
of sophisticated mathematical ideas and techniques
...

AS further mathematics, which can be co-taught with the A level as a separate
qualification, is a very useful qualification in its own right, broadening and reinforcing the
content of A level mathematics, introducing complex numbers and matrices, and giving
students the opportunity to extend their knowledge in applied mathematics
...


Aims and objectives
5
...


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Subject content
Structure
6
...
The core content is set out below in sections A
to H
...
The content of
these options is not prescribed and will be defined within the different awarding
organisations’ specifications
...

7
...
It must not overlap with, or
depend upon, other A level mathematics content
...
Core content that must be included in AS further mathematics is
indicated in sections A to C in bold text within square brackets, and this must represent
20% of the overall assessment of AS further mathematics
...


Background knowledge
8
...
Problem solving, proof and mathematical
modelling will be assessed in further mathematics in the context of the wider knowledge
which students taking AS/A level further mathematics will have
...

A level specifications in further mathematics must require students to demonstrate
the following knowledge and skills
...
The knowledge and skills are similar to those specified for A level mathematics,
but they will be examined against further mathematics content and contexts
...

The minimum knowledge and skills required for AS further mathematics are shown
in bold text within square brackets
...
1

[Construct and present mathematical arguments through
appropriate use of diagrams; sketching graphs; logical deduction;
precise statements involving correct use of symbols and connecting
language, including: constant, coefficient, expression, equation,
function, identity, index, term, variable]

OT1
...
3
OT1
...
5

Understand and use the definition of a function; domain and range of
functions

OT1
...
1 [Recognise the underlying mathematical structure in a situation and
simplify and abstract appropriately to enable problems to be solved]
OT2
...
3 [Interpret and communicate solutions in the context of the original
problem]
OT2
...
5 [Evaluate, including by making reasoned estimates, the accuracy or
limitations of solutions], including those obtained using numerical
methods
OT2
...
]
OT2
...
1

[Translate a situation in context into a mathematical model, making
simplifying assumptions]

OT3
...
3

[Interpret the outputs of a mathematical model in the context of the
original situation (for a given model or a model constructed or
selected by the student)]

OT3
...
5

[Understand and use modelling assumptions]

Use of technology
11
...

12
...


Detailed content statements
13
...


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14
...

Assessment of the bold content must represent 20% of the overall assessment of AS
further mathematics
...


A11

Use complex roots of unity to solve algebraic and geometric problems

B

Matrices
Content
B1

[Add, subtract and multiply conformable matrices; multiply a
matrix by a scalar]

B2

[Understand and use zero and identity matrices]

B3

[Use matrices to represent linear transformations in 2-D;
successive transformations; single transformations in 3-D (3-D
transformations confined to reflection in one of x = 0, y = 0, z = 0 or
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rotation about one of the coordinate axes)]
B4

[Find invariant points and lines for a linear transformation]

B5

[Calculate determinants of 2 x 2] and 3 x 3 matrices and interpret as
scale factors, including the effect on orientation

B6

[Understand and use singular and non-singular matrices;
properties of inverse matrices]
[Calculate and use the inverse of a non-singular 2 x 2] and use the
inverse of a 3 x 3 matrix

B7

C

Solve three linear simultaneous equations in three variables by use of the
inverse matrix
Further algebra and functions
Content

C1

[Understand and use the relationship between roots and coefficients
of polynomial equations up to quartic equations]

C2

[Form a polynomial equation whose roots are a linear transformation
of the roots of a given polynomial equation (of at least cubic degree)]

C3

Understand and use formulae for the sums of integers, squares and cubes
and use these to sum other series

C4

Understand and use the method of differences for summation of series
including use of partial fractions

C5

Find the Maclaurin series of a function including the general term

C6

Recognise and use the Maclaurin series for e x , ln(1 + x) , sin x , cos x and
n
(1 + x ) and know the ranges of values for which they are valid

C7

Calculate errors in sums, differences, products and quotients

C8

Estimate the error in f(x) when there is an error in x using
f( x + h ) ≈ f( x ) + h f'( x )

C9

D

Estimate a derivative using the forward and central difference methods and
have an empirical understanding of the relative level of accuracy of the two
methods
Further calculus
Content

D1

Evaluate improper integrals where either the integrand is undefined at a
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value in the range of integration or the range of integration extends to
infinity
D2

Derive formulae for and calculate volumes of revolution

D3

Understand and evaluate the mean value of a function

D4

Integrate using partial fractions (extend to quadratic factors ax2 + c in the
denominator)

D5

Differentiate inverse trigonometric functions

D6

Integrate functions of the form ( a 2 − x 2 )

−

1
2

and ( a 2 + x 2 ) and be able to
−1

choose trigonometric substitutions to integrate associated functions
E

Further vectors
Content
E1

Understand and use the vector and cartesian forms of an equation of a
straight line in 3D

E2

Understand and use the vector and cartesian forms of the equation of a
plane

E3

Calculate the scalar product and use it in the equations of planes, for
calculating the angle between two lines, the angle between two planes and
the angle between a line and a plane

E4

Check whether vectors are perpendicular by using the scalar product

E5

Find the intersection of a line and a plane
Calculate the perpendicular distance between two lines

E6

Interpret the solution and failure of solution of three simultaneous linear
equations geometrically

E7

Calculate the vector product of two vectors including link to 3x3
determinant and understand the properties of the vector product

F

Polar coordinates
Content
F1

Understand and use polar coordinates and be able to convert between
polar and cartesian coordinates

F2

Sketch curves with r given as a function of θ ; including use of trigonometric
functions from A level Mathematics

F3

Find the area enclosed by a polar curve
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G

Hyperbolic functions
Content
G1

Understand the definitions of hyperbolic functions sinh x, cosh x and tanh
x and be able to sketch their graphs

G2

Differentiate and integrate hyperbolic functions

G3

Understand and be able to use the definitions of the inverse hyperbolic
functions and their domains and ranges

G4

Derive and use the logarithmic forms of the inverse hyperbolic functions

G5

Integrate functions of the form ( x 2 + a 2 )

−

1
2

and ( x 2 − a 2 )

−

1
2

and be able to

choose substitutions to integrate associated functions
H

Differential equations
Content
H1

Find and use an integrating factor to solve differential equations of form
dy
+ P( x) y =
Q( x) and recognise when it is appropriate to do so
dx

H2

Find both general and particular solutions to differential equations

H3

Use differential equations in modelling in kinematics and in other
contexts

H4

0 where a and b are
Solve differential equations of form y ''+ ay '+ by =

constants by using the auxiliary equation
H5

f( x) where a and b are
Solve differential equations of form y ''+ ay '+ by =

constants by solving the homogeneous case and adding a particular
integral to the complementary function (in cases where f(x) is a
polynomial, exponential or trigonometric function)
H6

Understand and use the relationship between the cases when the
discriminant of the auxiliary equation is positive, zero and negative and
the form of solution of the differential equation

H7

Solve the equation for simple harmonic motion x = −ω 2 x and relate the
solution to the motion

H8

Model damped oscillations using 2nd order differential equations and
interpret their solutions

H9

Analyse and interpret models of situations with one independent variable
and two dependent variables as a pair of coupled 1st order simultaneous
equations and be able to solve them, for example predator-prey models
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