STPM/S(E)956 PEPERIKSAAN SIJIL TINGGI PERSEKOLAHAN MALAYSIA (MALAYSIA HIGHER SCHOOL CERTIFICATE EXAMINATION) FURTHER MATHEMATICS Syllabus and Specimen Papers This syllabus applies for the 2012/2013 session and thereafter until further notice. MAJLIS PEPERIKSAAN MALAYSIA (MALAYSIAN EXAMINATIONS COUNCIL)
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STPM/S(E)956
PEPERIKSAAN
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE EXAMINATION)
FURTHER MATHEMATICS
Syllabus and Specimen Papers
This syllabus applies for the 2012/2013 session and thereafter until further notice.
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
NATIONAL EDUCATION PHILOSOPHY
“Education in Malaysia is an on-going effort towards further
developing the potential of individuals in a holistic and
integrated manner, so as to produce individuals who are
intellectually, spiritually, emotionally and physically
balanced and harmonious, based on a belief in and devotion
to God. Such effort is designed to produce Malaysian
citizens who are knowledgeable and competent, who possess
high moral standards, and who are responsible and capable
of achieving a high level of personal well-being as well as
being able to contribute to the betterment of the family, the
society and the nation at large.”
FOREWORD
This revised Further Mathematics syllabus is designed to replace the existing syllabus which has been
in use since the 2002 STPM examination. This new syllabus will be enforced in 2012 and the first
examination will also be held the same year. The revision of the syllabus takes into account the
changes made by the Malaysian Examinations Council (MEC) to the existing STPM examination.
Through the new system, the form six study will be divided into three terms, and candidates will sit
for an examination at the end of each term. The new syllabus fulfils the requirements of this new
system. The main objective of introducing the new examination system is to enhance the teaching
and learning orientation of form six so as to be in line with the orientation of teaching and learning in
colleges and universities.
The Further Mathematics syllabus is designed to cater for candidates who are competence and have
intense interest in mathematics and wish to further develop their understanding of mathematical
concepts and mathematical thinking and acquire skills in problem solving and the applications of
mathematics.
The syllabus contains topics, teaching periods, learning outcomes, examination format, grade
description and specimen papers.
The design of this syllabus was undertaken by a committee chaired by Professor Dr. Abu Osman bin
Md Tap from International Islamic University Malaysia. Other committee members consist of
university lecturers, representatives from the Curriculum Development Division, Ministry of
Education Malaysia, and experienced teachers who are teaching Mathematics. On behalf of MEC, I
would like to thank the committee for their commitment and invaluable contribution. It is hoped that
this syllabus will be a guide for teachers and candidates in the teaching and learning process.
Chief Executive
Malaysian Examinations Council
CONTENTS
Syllabus 956 Further Mathematics
Page
Aims 1
Objectives 1
Content
First Term: Discrete Mathematics 2 – 4
Second Term: Algebra and Geometry 5 – 7
Third Term: Calculus 8 – 11
Scheme of Assessment 12
Performance Descriptions 13
Mathematical Notation 14 – 18
Electronic Calculators 19
Reference Books 19
Specimen Paper 1 21 – 28
Specimen Paper 2 29 – 34
Specimen Paper 3 35 – 40
1
SYLLABUS
956 FURTHER MATHEMATICS
[May only be taken with 954 Mathematics (T)]
Aims
The Further Mathematics syllabus caters for candidates who have high competence and intense
interest in mathematics and wish to further develop the understanding of mathematical concepts and
mathematical thinking and acquire skills in problem solving and the applications of mathematics.
Objectives
The objectives of the syllabus are to enable candidates to:
(a) use mathematical concepts, terminology and notation;
(b) display and interpret mathematical information in tabular, diagrammatic and graphical forms;
(c) identify mathematical patterns and structures in a variety of situations;
(d) use appropriate mathematical models in different contexts;
(e) apply mathematical principles and techniques in solving problems;
(f) carry out calculations and approximations to an appropriate degree of accuracy;
(g) interpret the significance and reasonableness of results;
(h) present mathematical explanations, arguments and conclusions.
2
FIRST TERM: DISCRETE MATHEMATICS
Topic Teaching
Period Learning Outcome
1 Logic and Proofs 20 Candidates should be able to:
1.1 Logic 10 (a) use connectives and quantifiers to form
compound statements;
(b) construct a truth table for a compound
statement, and determine whether the
statement is a tautology or contradiction or
neither;
(c) use the converse, inverse and contrapositive of
a conditional statement;
(d) determine the validity of an argument;
(e) use the rules of inference;
1.2 Proofs 10 (f) suggest a counter-example to negate a
statement;
(g) use direct proof to prove a statement, including
a biconditional statement;
(h) prove a conditional statement by
contraposition;
(i) prove a statement by contradiction;
(j) apply the principle of mathematical induction.
2 Sets and Boolean Algebras 14 Candidates should be able to:
2.1 Sets 8 (a) perform operations on sets, including the
symmetric difference of sets;
(b) find the power set and the partitions of a set;
(c) find the cartesian product of two sets;
(d) use the algebraic laws of sets;
2.2 Boolean algebras 6 (e) identify a Boolean algebra;
(f) use the properties of Boolean algebras;
(g) prove that two Boolean expressions are
logically equivalent.
3
Topic Teaching
Period Learning Outcome
3 Number Theory 26 Candidates should be able to:
3.1 Divisibility 12 (a) use the divisibility properties of integers;
(b) find greatest common divisors and least
common multiples;
(c) use the properties of greatest common divisors
and least common multiples;
(d) apply Euclidean algorithm;
(e) use the properties of prime and composite
numbers;
(f) use the fundamental theorem of arithmetic;
3.2 Congruences 14 (g) use the properties of congruences;
(h) use congruences to determine the divisibility
of integers;
(i) perform addition, subtraction and
multiplication of integers modulo n;
(j) use the Chinese remainder theorem;
(k) use Fermat’s little theorem;
(l) solve linear congruence equations;
(m) solve simultaneous linear congruence
equations.
4 Counting 20 Candidates should be able to:
(a) use combinations and permutations to solve
counting problems;
(b) prove combinatorial identities;
(c) expand 1 2
n
kx x x , where n, k
and k 2;
(d) use the multinomial coefficients to solve
counting problems;
(e) apply the principle of inclusion and exclusion;
(f) apply the pigeonhole principle;
(g) apply the generalised pigeonhole principle.
>
4
Topic Teaching
Period Learning Outcome
5 Recurrence Relations 14 Candidates should be able to:
(a) find the general solution of a first order linear
homogeneous recurrence relation with constant
coefficients;
(b) find the general solution of a first order linear
non-homogeneous recurrence relation with
constant coefficients;
(c) find the general solution of a second order
linear homogeneous recurrence relation with
constant coefficients;
(d) find the general solution of a second order
linear non-homogeneous recurrence relation
with constant coefficients;
(e) use boundary conditions to find a particular
solution;
(f) solve problems that can be modelled by
recurrence relations.
6 Graphs 26 Candidates should be able to:
6.1 Graphs 10 (a) relate the sum of the degrees of vertices and
the number of edges of a graph;
(b) use the properties of simple graphs, regular
graphs, complete graphs, bipartite graphs and
planar graphs;
(c) represent a graph by its adjacency matrix and
incidence matrix;
(d) determine the subgraphs of a graph;
6.2 Circuits and cycles 10 (e) identify walks, trails, paths, circuits and cycles
of a graph;
(f) use properties associated with connected
graphs;
(g) determine whether a graph is eulerian, and find
eulerian trails and circuits;
(h) determine whether a graph is hamiltonian, and
find hamiltonian paths and cycles;
(i) solve problems that can be modelled by
graphs;
6.3 Isomorphism 6 (j) determine whether two graphs are isomorphic;
(k) use the properties of isomorphic graphs;
(l) apply adjacency matrices to isomorphism.
5
SECOND TERM: ALGEBRA AND GEOMETRY
Topic Teaching
Period Learning Outcome
7 Relations 20 Candidates should be able to:
7.1 Relations 12 (a) identify a binary relation on a set;
(b) determine the reflexivity, symmetry and
transitivity of a relation;
(c) determine whether a relation is an equivalence
relation;
(d) find the equivalence class of an element;
(e) find the partitions induced by an equivalence
relation;
(f) use the properties of equivalence relations;
7.2 Binary operations 8 (g) identify a binary operation on a set;
(h) use an operation table;
(i) determine the commutativity and associativity
of a binary operation, and determine whether a
binary operation is distributive over another
binary opration;
(j) find the identity element and the inverse of an
element.
8 Groups 24 Candidates should be able to:
8.1 Groups 6 (a) determine whether a set with a binary
operation is a group;
(b) identify an abelian group;
(c) determine the subgroups of a group;
8.2 Cyclic groups 6 (d) find the order of an element and of a group;
(e) determine the generators of a cyclic group;
(f) use the properties of a cyclic group;
8.3 Permutation groups 6 (g) determine the cycles and transpositions in a
permutation;
(h) determine whether a permutation is odd or
even;
(i) use the properties of a permutation group;
8.4 Isomorphism 6 (j) determine whether two groups are isomorphic;
(k) prove the isomorphism properties for identities
and inverses;
(l) use the properties of isomorphic groups.
6
Topic Teaching
Period Learning Outcome
9 Eigenvalues and
Eigenvectors 14 Candidates should be able to:
9.1 Eigenvalues and
eigenvectors
6 (a) find the eigenvalues and eigenvectors of a
matrix;
(b) use the properties of eigenvalues and
eigenvectors of a matrix;
(c) use the Cayley-Hamilton theorem;
9.2 Diagonalisation 8 (d) determine whether a matrix is diagonalisable,
and diagonalise a matrix where appropriate;
(e) find the powers of a matrix;
(f) use the properties of orthogonal matrices;
(g) determine whether a matrix is orthogonally
diagonalisable, and orthogonally diagonalise a
matrix where appropriate.
10 Vector Spaces 24 Candidates should be able to:
10.1 Vector spaces 8 (a) determine whether a set, with addition and
scalar multiplication defined on the set, is a
vector space;
(b) determine whether a subset of a vector space is
a subspace;
(c) determine whether a vector is a linear
combination of other vectors;
(d) find the spanning set for a vector space;
10.2 Bases and dimensions 8 (e) determine whether a set of vectors is linearly
dependent or independent;
(f) find a basis for and the dimension of a vector
space;
(g) use the properties of bases and dimensions;
(h) change the basis for a vector space;
10.3 Linear transformations 8 (i) determine whether a given transformation is
linear;
(j) use the properties of linear transformations;
(k) determine the null space and the range of a
linear transformation, and find a basis for and
the dimension of the null space and the range;
(l) determine whether a linear transformation is
one-to-one.
7
Topic Teaching
Period Learning Outcome
11 Plane Geometry 24 Candidates should be able to:
11.1 Triangles 8 (a) use the properties of triangles: medians,
altitudes, angle bisectors and perpendicular
bisectors of sides;
(b) use the properties of the orthocentre, incentre
and circumcentre;
(c) apply Apollonius’ theorem;
(d) apply the angle bisector theorem and its
converse;
11.2 Circles 10 (e) use the properties of angles in a circle and
tangency;
(f) apply the intersecting chords theorem;
(g) apply the tangent-secant and secant-secant
theorems;
(h) use the properties of cyclic quadrilaterals;
(i) apply Ptolemy’s theorem;
11.3 Collinear points and
concurrent lines
6 (j) apply Menelaus’ theorem and its converse;
(k) apply Ceva’s theorem and its converse.
12 Transformation Geometry 14 Candidates should be able to:
(a) use 2 2 and 3 3 matrices to represent linear
transformations;
(b) determine the standard matrices for
transformations;
(c) find the image and inverse image under a
transformation;
(d) find the invariant points and lines of
transformations;
(e) relate the area or volume scale-factor of a
transformation to the determinant of the
corresponding matrix;
(f) determine the compositions of transformations.
8
THIRD TERM: CALCULUS
Topic Teaching
Period Learning Outcome
13 Hyperbolic and Inverse
Hyperbolic Functions 16
Candidates should be able to:
13.1 Hyperbolic and
inverse hyperbolic
functions
8 (a) use hyperbolic and inverse hyperbolic
functions and their graphs;
(b) use basic hyperbolic identities and the
formulae for sinh (x ± y), cosh (x ± y) and
tanh (x ± y), including sinh 2x, cosh 2x and
tanh 2x;
(c) derive and use the logarithmic forms for
sinh1x, cosh
1x and tanh
1x;
(d) solve equations involving hyperbolic and
inverse hyperbolic expressions;
13.2 Derivatives and
integrals
8 (e) derive the derivatives of sinh x, cosh x, tanh x,
sinh1x, cosh
1x and tanh
1x;
(f) differentiate functions involving hyperbolic
and inverse hyperbolic functions;
(g) integrate functions involving hyperbolic and
inverse hyperbolic functions;
(h) use hyperbolic substitutions in integration.
14 Techniques and
Applications of Integration
20 Candidates should be able to:
14.1 Reduction formulae 4 (a) obtain reduction formulae for integrals;
(b) use reduction formulae for the evaluation of
definite integrals;
14.2 Improper integrals 4 (c) evaluate integrals with infinite limits of
integration;
(d) evaluate integrals with discontinuous
integrands;
9
Topic Teaching
Period Learning Outcome
14.3 Applications of
integration
12 (e) calculate arc lengths for curves with equations
in cartesian coordinates (including the use of a
parameter);
(f) calculate areas of surfaces of revolution about
one of the coordinate axes for curves with
equations in cartesian coordinates (including
the use of a parameter);
(g) sketch curves defined by polar equations;
(h) calculate the areas of regions bounded by
curves with equations in polar coordinates;
(i) calculate arc lengths for curves with equations
in polar coordinates.
15 Infinite Sequences and
Series
24 Candidates should be able to:
15.1 Sequences 4 (a) determine the monotonicity and boundedness
of a sequence;
(b) determine the convergence or divergence of a
sequence;
15.2 Series 10 (c) use the properties of a p-series and harmonic
series;
(d) use the properties of an alternating series;
(e) use the nth-term test for divergence of a series;
(f) use the comparison, ratio, root and integral
tests to determine the convergence or
divergence of series;
15.3 Taylor series 10 (g) find the Taylor series for a function and the
interval of convergence;
(h) use a Taylor polynomial to approximate a
function;
(i) use the remainder term, in terms of the
(n + 1)th derivative at an intermediate point
and in terms of an integral of the (n + 1)th
derivative;
(j) use l’Hospital’s rule to find limits in
indeterminate forms.
10
Topic Teaching
Period Learning Outcome
16 Differential Equations 20 Candidates should be able to:
16.1 Linear differential
equations
14 (a) find the general solution of a second order
linear homogeneous differential equation with
constant coefficients;
(b) find the general solution of a second order
linear non- homogeneous differential equation
with constant coefficients;
(c) transform, by a given substitution, a
differential equation into a second order linear
differential equation with constant coefficients;
(d) use boundary conditions to find a particular
solution;
(e) solve problems that can be modelled by
differential equations;
16.2 Numerical solution of
differential equations
6 (f) use a Taylor series to find a polynomial
approximation for the solution of a first order
differential equation;
(g) use Euler’s method to find an approximate
solution for a first order differential equation,
and determine the effect of step length on the
error;
(h) find the series solutions for second order
differential equations.
17 Vector-valued Functions 16 Candidates should be able to:
17.1 Vector-valued
functions
6 (a) find the domain and sketch the graph of a
vector-valued function;
(b) determine the existence and values of the
limits of a vector-valued function;
(c) determine the continuity of a vector-valued
function;
17.2 Derivatives and
integrals
2
(d) find the derivatives of vector-valued functions;
(e) find the integrals of vector-valued functions;
17.3 Curvature 4 (f) find unit tangent, unit normal and binormal
vectors;
(g) calculate curvatures and radii of curvature;
17.4 Motion in space 4 (h) find the position, velocity and acceleration of a
particle moving along a curve;
(i) determine the tangential and normal
components of acceleration.
11
Topic Teaching
Period Learning Outcome
18 Partial Derivatives 24 Candidates should be able to:
18.1 Functions of two
variables
6 (a) find the domain and sketch the graph of a
function of two variables;
(b) determine the existence and values of the
limits of a function of two variables;
(c) determine the continuity of a function of two
variables;
18.2 Partial derivatives 8 (d) find the first and second order partial
derivatives of a function of two variables;
(e) use the chain rule to obtain the first derivative;
(f) find total differentials;
(g) determine linear approximations and errors;
18.3 Directional derivatives 4 (h) find the directional derivatives and gradient of
a function of two variables;
(i) determine the minimum and maximum values
of directional derivatives and the directions in
which they occur;
18.4 Extrema of functions 6 (j) use the second derivatives test to determine the
extremum values of a function of two
variables;
(k) use the method of Lagrange multipliers to
solve constrained optimisation problems.
12
Scheme of Assessment
Term of
Study
Paper Code
and Name Type of Test
Mark
(Weighting) Duration Administration
First
Term 956/1
Further
Mathematics
Paper 1
Written test
Section A Answer all 6 questions of variable
marks.
Section B Answer 1 out of 2 questions.
All questions are based on topics 1
to 6.
60
(33.33%)
45
15
1½ hours Central
assessment
Second
Term 956/2
Further
Mathematics
Paper 2
Written test
Section A Answer all 6 questions of variable
marks.
Section B Answer 1 out of 2 questions.
All questions are based on topics 7
to 12.
60
(33.33%)
45
15
1½ hours Central
assessment
Third
Term 956/3
Further
Mathematics
Paper 3
Written test
Section A Answer all 6 questions of variable
marks.
Section B Answer 1 out of 2 questions.
All questions are based on topics 13
to 18.
60
(33.33%)
45
15
1½ hours Central
assessment
13
Performance Descriptions
A grade A candidate is likely able to:
(a) use correctly mathematical concepts, terminology and notation;
(b) display and interpret mathematical information in tabular, diagrammatic and graphical
forms;
(c) identify mathematical patterns and structures in a variety of situations;
(d) use appropriate mathematical models in different contexts;
(e) apply correctly mathematical principles and techniques in solving problems;
(f) carry out calculations and approximations to an appropriate degree of accuracy;
(g) interpret the significance and reasonableness of results, making sensible predictions where
appropriate;
(h) present mathematical explanations, arguments and conclusions, usually in a logical and
systematic manner.
A grade C candidate is likely able to:
(a) use correctly some mathematical concepts, terminology and notation;
(b) display and interpret some mathematical information in tabular, diagrammatic and graphical
forms;
(c) identify mathematical patterns and structures in certain situations;
(d) use appropriate mathematical models in certain contexts;
(e) apply correctly some mathematical principles and techniques in solving problems;
(f) carry out some calculations and approximations to an appropriate degree of accuracy;
(g) interpret the significance and reasonableness of some results;
(h) present some mathematical explanations, arguments and conclusions.
14
Mathematical Notation
Miscellaneous symbols
= is equal to
≠ is not equal to
≡ is identical to or is congruent to
≈ is approximately equal to
< is less than
is less than or equal to
> is greater than
is greater than or equal to
∞ infinity
therefore
there exists
for all
Operations
a + b a plus b
a − b a minus b
a × b, ab a multiplied by b
a b, a
b a divided by b
a : b ratio of a to b
an nth power of a
12a , a positive square root of a
1na , n a positive nth root of a
|a| absolute value of a real number a
1
n
i
i
u u1 + u2 + ∙ ∙ ∙ + un
n! n factorial for n
n
r binomial coefficient
!
!( )!
n
r n r for n, r , 0 r n
1 2, ,. . ., k
n
r r r multinomial coefficient
1 2
!
! !... !k
n
r r r, where r1 + r2 + . . . + rk = n
Logic
p a statement p
p not p
p q p or q
p q p and q
p q p or q but not both p and q
<
>
< <
15
p q if p then q
p q p if and only if q
p q p is logically equivalent to q
p q p is not logically equivalent to q
Set notation
is an element of
is not an element of
empty set
{x | . . .} set of x such that . . .
set of natural numbers, {0, 1, 2, 3, . . .}
set of integers
set of positive integers
set of rational numbers
set of real numbers
[a, b] closed interval {x | x , a x b}
(a, b) open interval { x | x , a x b}
[a, b) interval { x | x , a x < b}
(a, b] interval { x | x , a < x b}
union
intersection
U universal set
A' complement of a set A
is a subset of
is a proper subset
is not a subset of
is not a proper subset
n(A) number of elements in a set A
P(A) power set of A
A B cartesian product of sets A and B, i.e. A B = {( a, b ) | a A, b B}
A B complement of set B in set A
A B symmetry difference of sets A and B, (A B) (B A)
Number theory
a b a divides b
a | b a does not divide b
gcd (a, b) greatest common divisor of integers a and b
lcm (a, b) least common multiple of integers a and b
< <
<
<
16
m ≡ n (mod d) m is congruent to n modulo d
n set of integers modulo n, {0, 1, 2, . . ., n 1}
x floor of x
x ceiling of x
Graphs
G a graph G
V(G) set of vertices of a graph G
E(G) set of edges of a graph G
deg (v) degree of vertex v
{v, w} edge joining v and w in a simple graph
n a complete graph on n vertices
,m n a complete bipartite graph with one set of m vertices and another set of
n vertices
Relations
y R x y is related to x by a relation R
y ~ x y is equivalent to x, in the context of some equivalence relation
[ a ] equivalence class of an element a
A / R a partition of set A induced by the equivalence relation R on A
Groups
(G, *) a set G together with a binary operation *
e identity element
a −1
inverse of an element a
is isomorphic to
Matrices
A a matrix A
0 null matrix
I identity matrix
AT transpose of a matrix A
A−1
inverse of a non-singular square matrix A
det A determinant of a square matrix A
Vector spaces
V a vector space V
2 set of real ordered pairs
3 set of real ordered triples
n set of real ordered n-tuples
T a linear transformation T
17
Geometry
AB length of the line segment with end points A and B
BAC angle between line segments AB and AC
ABC triangle whose vertices are A, B and C
// is parallel to
is perpendicular to
Vectors
a a vector a
| a | magnitude of a vector a
i, j, k unit vectors in the directions of the cartesian coordinates axes
AB
vector represented in magnitude and direction by the directed line
segment from point A to point B
| |AB
magnitude of AB
a b scalar product of vectors a and b
a × b vector product of vectors a and b
Functions
f a function f
f(x) value of a function f at x
f : A B f is a function under which each element of set A has an image in set B
f : x y f is a function which maps the element x to the element y
1f inverse function of f
f g composite function of f and g which is defined by f g( ) = f[g( )] x x
ex exponential function of x
loga x logarithm to base a of x
ln x natural logarithm of x, loge x
sin, cos, tan,
csc, sec, cot
sin1, cos
1, tan
1,
csc1, sec
1, cot
1
sinh, cosh, tanh,
csch, sech, coth
sinh1, cosh
1, tanh
1,
csch1, sech
1, coth
1
trigonometric functions
inverse trigonometric functions
inverse hyperbolic functions
hyperbolic functions
18
Derivatives and integrals
limf ( )x a
x limit of f(x) as x tends to a
d
d
y
x first derivative of y with respect to x
f '( )x first derivative of f(x) with respect to x
2
2
d
d
y
x second derivative of y with respect to x
f ''( )x second derivative of f(x) with respect to x
d
d
n
n
y
x nth derivative of y with respect to x
( )f ( )n x nth derivative of f(x) with respect to x
dy x indefinite integral of y with respect to x
db
ay x definite integral of y with respect to x for values of x between a and b
Vector-valued functions
curvature
T unit tangent vector
N unit normal vector
Partial derivatives
y
x partial derivative of y with respect to x
del operator, x y z
i j k
19
Electronic Calculators
During the written paper examination, candidates are advised to have standard scientific calculators
which must be silent. Programmable and graphic display calculators are prohibited.
Reference Books
Discrete Mathematics
1. Dolan, S., Neill, H. and Quadling, D., 2009. Further Mathematics for the IB Diploma:
Standard Level. United Kingdom: Cambridge University Press.
2. Gaulter, B. and Gaulter, M., 2001. Further Pure Mathematics. United Kingdom: Oxford