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Structured Mathematics
FURTHER PURE
MATHEMATICS FP2
(September 2004 version, based on FPM)
Assessment format
Examination
1h 30 mins (72 marks)
Section A: 3 compulsory questions, each worth about 18 marks, total: 54
marks.
Section B: One question to be chosen from two, both worth 18 marks, total: 18
marks.
Coursework
None
Assumed Knowledge
Candidates are expected to know the content for C1, C2, C3, C4 and FP1.
Topic
Competence
Book
Reference
Understand the meaning of
polar co-ordinates (r, θ) and
to be able to convert from
polar to Cartesian coordinates and vice versa.
Be able to sketch curves with
simple polar equations.
Be able to find the area
enclosed by a polar curve.
Ex 3B, q1-2,
p44-45
The inverse
functions of sine,
cosine and tangent.
Differentiation of
arcsin x , arccos x
and arctan x
Understand the definitions of
inverse trigonometric
functions.
Be able to differentiate
inverse trigonometric
functions.
p31-32
Use of
trigonometrical
substitutions in
integration.
Recognise integrals of
functions of the form
1
(a 2  x 2 )  2 and (a 2  x 2 ) 1 and
be able to integrate
associated functions by using
trigonometrical substitutions.
Ex 2D, q2-6,
p40
POLAR
COORDINATES
Polar co-ordinates
in two dimensions.
Ex 3C, p48
Ex 3D, p51
CALCULUS
Ex 2D, q1,
p40
Progres
s
SERIES
Maclaurin series.
Approximate
evaluation of a
function.
Be able to find the Maclaurin
series of a function, including
the general term in simple
cases.
Ex 9D, p179
Appreciate that the series
may converge only for a
restricted set of values of x.
Departmental
notes
Identify and be able to use the Ex 9D, p179
Maclaurin series of standard
functions.
COMPLEX
NUMBERS
Modulus-argument
form
Understand the polar
(modulus-argument) form of a
complex number, and the
definition of modulus,
argument.
Definition, p6
Ex 1B, q2,
5,6,7, p9
Be able to multiply and divide
complex numbers in polar
form
Departmental
notes
De Moivre’s
Understand de Moivre’s
theorem and simple theorem
applications.
Expression of
complex numbers
in the form
Ex 15A, p334
Be able to apply de Moivre’s
theorem to finding multiple
angle formulae and to
summing suitable series.
Ex 15C, p352
Understand the definition
e j  cos   j sin  and hence
the form z  re j .
Departmental
notes
The n nth roots of a
complex number
Know that every non-zero
complex number has n nth
roots, and that in the Argand
diagram these are the
vertices of a regular n-gon.
Departmental
notes
Know that the distinct nth
Ex 15B, q1ii,
j
4, 5, 12, 14,
roots of re are:
    2k   j sin    2k  p342
r cos 


  n 
 n 
for k= 0, 1, …, n-1.
1
n
Be able to explain why the
sum of all the nth roots is
zero.
Departmental
notes
Appreciate the effect in the
Argand diagram of
multiplication by a complex
number.
Ex 15D, q1, 3,
p359
Be able to represent complex
roots of unity on an Argand
diagram
Departmental
notes
Be able to apply complex
numbers to geometrical
problems
Ex 15D, p359
Determinant and
inverse of a 3x3
matrix
Be able to find the
determinant of any 3x3 matrix
and the inverse of a nonsingular 3x3 matrix.
Ex 14A, q2c,
d, e; q3, 6, 9,
10, 11, 12,
p306
Eigenvalues and
eigenvectors of 2x2
and 3x3 matrices
Understand the meaning of
eigenvalue and eigenvector,
and be able to find these for
2x2 and 3x3 matrices
whenever this is possible.
Ex 14B, q514, p324
Applications of
complex numbers
in Geometry.
MATRICES
Diagonalisation and Be able to form the matrix of
powers of 2x2 and
eigenvectors and use this to
3x3 matrices
reduce a matrix to diagonal
form.
Ex 14B, q7-9,
p325
Solutions of
equations
The use of the
Cayley-Hamiton
Theorem
Be able to find powers of a
2x2 or 3x3 matrix.
Departmental
notes
Be able to solve a matrix
equation or the equivalent
simultaneous equations, and
to interpret the solution
geometrically.
Understand the term
characteristic equation of a
2x2 or 3x3 matrix
Departmental
notes
Understand that every 2x2 or
3x3 matrix satisfies its own
characteristic equation, and
be able to use this.
.
Departmental
notes
Understand the definitions of
hyperbolic functions and be
able to sketch these graphs.
P189-191
Be able to differentiate and
integrate hyperbolic functions.
Ex 10A, q3, 4,
7b, 10i, 12,
13, 16a
Departmental
notes
Departmental
notes
SECTION B
Option 1
HYPERBOLIC
FUNCTIONS
Hyperbolic
functions:
definitions, graphs,
differentiation and
integration.
.
Inverse hyperbolic
functions, including
the logarithmic
forms. Use in
integration.
Understand and be able to
use the definitions of the
inverse hyperbolic functions.
Be able to use the logarithmic
forms of the inverse
hyperbolic functions.
Departmental
notes
Be able to integrate
Ex 10B, q2-4,
p206
x
2
a

1
2 2
and  x 2  a
related functions.
Option 2
INVESTIGATION
OF CURVES

1
2 2
and
Curves
Know vocabulary associated
with curves
Departmental
notes
Graphical
Calculator
Be able to use a suitable
graphical calculator to draw
curves.
Departmental
notes
Properties of
curves.
Be able to find, describe and
generalise properties of
curves
Departmental
notes
Be able to determine
asymptotes
Departmental
notes
Be able to identify cusps and
loops
Departmental
notes
Be able to find and work with
equations of chords, tangents
and normals
Know the names and shapes
of conics
Departmental
notes
Know the standard Cartesian
and parametric equations of
conics.
p218-229
Ex 11A-D
Conics
p218-229
Textbook: “Further Pure Mathematics” Brian and Mark Gaulter (FPM)
[HR: 06/05, 08/08]