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Structured Mathematics
FURTHER PURE MATHEMATICS 1
(September 2004 version, based on FPM)
Assessment format
Examination
1h 30 mins
Section A: 5-7 questions, each worth at most 8 marks; section total: 36 marks
Section B: 3 questions, each worth about 12 marks; total: 36 marks
Coursework
None
Topic
COMPLEX
NUMBERS
Quadratic equations
Addition, subtraction,
multiplication and
division of complex
numbers
Appplication of
complex numbers to
the solution of
polynomial equations
with real coefficients
Modulus-argument
form
Simple loci in the
Argand diagram
CURVE
SKETCHING
Treatment and
sketching of graphs
of rational functions
Competence
Book
Reference
Be able to solve any quadratic equation with
real coefficients.
Understand the language of complex
numbers.
p 5 ex 1A
nos 4,10
p 1 + dept
notes
Be able to add, subtract, multiply and divide
complex numbers given in the form x+yj,
where x and y are real.
Know that a complex number is zero if and
only if both the real and imaginary parts are
zero.
Know that the complex roots of real
polynomial equations with real coefficients
occur in conjugate pairs.
p 5 ex 1A
nos 5, 6, 7
Be able to solve equations of higher degree
with real coefficients in simple cases.
Know how to represent complex numbers and
their conjugates on an Argand diagram.
Be able to represent the sum and difference
of two complex numbers on an Argand
diagram.
Be able to represent a complex number in
modulus-argument form.
Be able to represent simple sets of complex
numbers as loci in the Argand diagram.
p9
Be able to sketch the graph of y = f(x)
obtaining information about symmetry,
asymptotes parallel to the axes, intercepts
with the co-ordinate axes, behaviour near x=0
and for numerically large x.
Be able to ascertain the direction from which a
curve approaches an asymptote.
Be able to use a curve to solve an inequality.
p 138 ex 7a
nos 1 – 7
FPM1 (Structured Mathematics 2004)
page 1/3
p 1 + dept
notes
p 157 ex 8c
nos 1 - 14
p 9 ex 1B no
1
Dept notes
p 9 ex 1b no
2
p 19 ex 1c
nos 1, 2, 3
p 131 + dept
notes
p 145 ex 7b
nos 1 - 10
Progress
PROOF
Meaning of the terms
if, only if, necessary
and sufficient
Identities
Proof by induction
ALGEBRA
Summation of simple
finite series
Be able to use the terms if, only if, necessary
and sufficient correctly in any appropriate
context.
Know the difference between an equation and
an identity.
Be able to find unknown constants in an
identity.
Be able to construct and present a correct
proof using mathematical induction.
IPM p 515 526 + dept
notes
Dept notes
Know the difference between a sequence and
a series.
IPM p 230 241 + dept
notes
p 170 ex 9b
IPM p 231 234, 239 240 + dept
notes
p 6 - 9, 36 37 + dept
notes
p 151 ex 8a
nos 1 – 8
Be able to sum a simple series.
Know the meaning of the word converge
when applied to either a sequence or a series.
The manipulation of
simple algebraic
inequalities
Relations between
the roots and
coefficients of
quadratic, cubic and
quartic equations.
Be able to manipulate simple algebraic
inequalities, to deduce the solution of such an
inequality.
Appreciate the relationship between the roots
and coefficients of quadratic, cubic and
quartic equations.
Be able to form a new equation whose roots
are related to the roots of a given equation by
a linear transformation.
FPM1 (Structured Mathematics 2004)
page 2/3
Dept notes
p 164 ex 9a
no 1
p 153 ex 8b
nos 1 - 4, 7,
8, 9, 11
MATRICES
Matrix addition and
multiplication
Be able to add, subtract and multiply
conformable matrices, and to multiply a matrix
by a scalar.
Know the zero and identity matrices, and what
is meant by equal matrices.
Know that matrix multiplication is associative
but not commutative.
Linear
transformations in a
plane and their
associated 2x2
matrices
Combined
transformations in a
plane
Invariance
Determinant of a
matrix
The meaning of the
inverse of a square
matrix
The product rule for
inverses
Solution of equations
Be able to find the matrix associated with a
linear transformation and vice-versa.
p 306
p 303 - 304
+ dept
notes
p 301 - 302
+ dept
notes
p 309 - 314,
ex 14b no 4
+ dept notes
Understand successive transformations and
the connection with matrix multiplication.
ex 14b no 1
+ dept notes
Understand the meaning of invariant points
and lines of invariant points in a plane and
how to find them.
Be able to find the determinant of a 2x2
matrix.
Know that the determinant gives the area
scale factor of the transformation, and
understand the significance of a zero
determinant.
Understand what is meant by an inverse
matrix.
p 312 - 313
+ dept notes
Be able to find the inverse of a non-singular
2x2 matrix.
Appreciate the product rule for inverse
matrices.
Know how to use matrices to solve linear
equations.
In the case of 2 linear equations in 2
unknowns, be able to give a geometrical
interpretation of a case where the matrix is
singular.
p 306 ex
14a no 2
Dept notes
p 306 ex
14a no 2
Dept notes
p 304 + dept
notes
Dept notes
Dept notes
Page references are to Gaulter, Further Pure Mathematics, 2nd edition,
Oxford 2001. [HR: 10/04]
FPM1 (Structured Mathematics 2004)
page 3/3