PURE
MATHEMATICS

Functions,
inverse functions and composite functions

Include:
concepts
of function, domain and range
use
of notations such as f( x ) = x ^{2
}+
5 , f : x x ^{2
}+
5 , f^{−1}(x),
fg(x)and f^{2}(x)
finding
inverse functions and composite functions
conditions
for the existence of inverse functions and
composite
functions
domain
restriction to obtain an inverse function
relationship
between a function and its inverse as
reflection
in the line y = x
Exclude
the use of the relation (fg)^{−1
}=
g^{−1}f
^{−1
}

Graphing
techniques

Include:
characteristics
of graphs such as symmetry, intersections with the axes, turning
points and asymptotes
determining
the equations of asymptotes, axes of symmetry, and restrictions
on the possible values of x and/or y
effect
of transformations on the graph of y = f(x) as represented by
and
combinations of these transformations

Equations
and inequalities

Include:
solving
inequalities of the form where
and
are quadratic expressions that are either
factorable
or always positive
solving
inequalities by graphical methods
formulating
an equation or a system of linear equations
from
a problem situation
finding
the numerical solution of equations (including
system
of linear equations) using a graphing calculator

Summation
of series

Include:
concepts
of sequence and series
relationship
between u_{n
}(the
n th term) and S_{n
}(the
sum
to n terms)
sequence
given by a formula for the n th term
sequence
generated by a simple recurrence relation of
the
form x_{n+1
}=
f(x_{n
})
use
of ∑ notation
summation
of series by the method of differences
convergence
of a series and the sum to infinity
binomial
expansion of (1+ x)^{n
}for
any rational n
condition
for convergence of a binomial series
proof
by the method of mathematical induction

Arithmetic
and geometric series

Include:
formula
for the n th term and the sum of a finite
arithmetic
series
formula
for the n th term and the sum of a finite
geometric
series
condition
for convergence of an infinite geometric
series
formula
for the sum to infinity of a convergent
geometric
series
solving
practical problems involving arithmetic and
geometric
series

Vectors
in two and three dimensions

Include:
addition
and subtraction of vectors, multiplication of a
vector
by a scalar, and their geometrical interpretations
use
of notations such as
a
position
vectors and displacement vectors
magnitude
of a vector
unit
vectors
distance
between two points
angle
between a vector and the x, y or
zaxis
use
of the ratio theorem in geometrical applications

The
scalar and vector products of vectors

Include:
concepts
of scalar product and vector product of
vectors
calculation
of the magnitude of a vector and the angle
between
two directions
calculation
of the area of triangle or parallelogram
geometrical
meanings of a.b
and a
× b
, where b is
a
unit vector
Exclude
triple products a.b
× c
and a
× b × c

Threedimensional
geometry

Include:
vector
and cartesian equations of lines and planes
finding
the distance from a point to a line or to a plane
finding
the angle between two lines, between a line and
a
plane, or between two planes
relationships
between
–
two lines (coplanar or skew)
–
a line and a plane
–
two planes
–
three planes
finding
the intersections of lines and planes
Exclude:
finding
the shortest distance between two skew lines
finding
an equation for the common perpendicular to
two
skew lines

Complex
numbers expressed in cartesian form

Include:
extension
of the number system from real numbers to
complex
numbers
complex
roots of quadratic equations
four
operations of complex numbers expressed in the
form
(x+iy)
equating
real parts and imaginary parts
conjugate
roots of a polynomial equation with real
coefficients

Complex
numbers expressed in polar form

Include:
complex
numbers expressed in the form
where r >0 and
calculation
of modulus ( r ) and argument ( θ ) of a
complex
number
multiplication
and division of two complex numbers
expressed
in polar form
representation
of complex numbers in the Argand
diagram
geometrical
effects of conjugating a complex number
and
of adding, subtracting, multiplying, dividing two
complex
numbers
loci
such as
z
− c
r, z – a = z – b and
arg(z
− a) = α
use
of de Moivre’s theorem to find the powers and n th
roots
of a complex number
Exclude:
loci
such as z – a =k z – b , where k 1 and arg(z −
a) − arg(z − b) = α
properties
and geometrical representation of the n th roots of unity
use
of de Moivre’s theorem to derive trigonometric identities

Differentiation

Include:
–
f^{′}(x)>0,
f^{′}(x)=0
and f^{′}(x)<0
–
f^{′′}(x)>0
and f^{′′}(x)<0
relating
the graph of y = f^{′}(x)
to the graph of y = f(x)
differentiation
of simple functions defined implicitly or parametrically
finding
the numerical value of a derivative at a given point using a
graphing calculator
finding
equations of tangents and normals to curves
solving
practical problems involving differentiation
Exclude:

Maclaurin’s
series

Include:
derivation
of the first few terms of the series expansion
of
(1+x)^{n
},
e^{x
},
sin x , ln(1+ x), and other simple
functions
finding
the first few terms of the series expansions of
sums
and products of functions, e.g. e^{x}cos2x,
using
standard
series
summation
of infinite series in terms of standard series
sinx
≈ x, cosx ≈ 1−, tanx ≈ x
concepts
of ‘convergence’ and ‘approximation’

Integration
techniques

Include:

Definite
integrals

Include:
concept
of definite integral as a limit of sum
definite
integral as the area under a curve
evaluation
of definite integrals
finding
the area of a region bounded by a curve and
lines
parallel to the coordinate axes, between a curve
and
a line, or between two curves
area
below the xaxis
finding
the area under a curve defined parametrically
finding
the volume of revolution about the x or yaxis
finding
the numerical value of a definite integral using a
graphing
calculator

Differential
equations

Include:
formulating
a differential equation from a problem situation
use
of a family of solution curves to represent the general solution
of a differential equation
use
of an initial condition to find a particular solution
interpretation
of a solution in terms of the problem
situation

STATISTICS

Permutations
and combinations

Include:
addition
and multiplication principles for counting
concepts
of permutation ( n ! or ^{n
}P_{r
})
and combination (^{n}C_{r
})
arrangements
of objects in a line or in a circle
cases
involving repetition and restriction

Probability

Include:
addition
and multiplication of probabilities
mutually
exclusive events and independent events
use
of tables of outcomes, Venn diagrams, and tree
diagrams
to calculate probabilities
calculation
of conditional probabilities in simple cases
use
of:
P(A^{′})
= 1−P(A)
P(A
∪
B) = P(A) + P(B) − P(A ∩
B)
P(A
 B) =

Binomial
and Poisson distributions

Include:
concepts
of binomial distribution B(n,p) and Poisson
distribution
Po(μ); use of B(n,p) and Po(μ) as
probability
models
use
of mean and variance of binomial and Poisson
distributions
(without proof)
solving
problems involving binomial and Poisson
variables
additive
property of the Poisson distribution
Poisson
approximation to binomial

Normal
distribution

Include:
concept
of a normal distribution and its mean and
variance;
use of N(μ,σ ^{2
})
as a probability model
standard
normal distribution
finding
the value of P( X < x_{1
})
given the values of μ,σ
use
of the symmetry of the normal distribution
finding
a relationship between x_{1},
μ , σ given the
value
of P(X < x_{1})
solving
problems involving normal variables
x_{1
},
solving
problems involving the use of E(aX + b) and Var (aX + b)
solving
problems involving the use of E(aX + bY ) and Var (aX + bY ) ,
where X and Y are independent
normal
approximation to binomial
normal
approximation to Poisson

Sampling

Include:
concepts
of population and sample
random,
stratified, systematic and quota samples
advantages
and disadvantages of the various sampling
methods
distribution
of sample means from a normal population
use
of the Central Limit Theorem to treat sample
means
as having normal distribution when the sample
size
is sufficiently large
calculation
of unbiased estimates of the population
mean
and variance from a sample
solving
problems involving the sampling distribution

Hypothesis
testing

Include:
concepts
of null and alternative hypotheses, test
statistic,
level of significance and pvalue
tests
for a population mean based on:
–
a sample from a normal population of known variance
–
a sample from a normal population of unknown variance
–
a large sample from any population

Correlation
coefficient and linear regression

Include:
concepts
of scatter diagram, correlation coefficient and
linear
regression
calculation
and interpretation of the product moment
correlation
coefficient and of the equation of the least
squares
regression line
concepts
of interpolation and extrapolation
use
of a square, reciprocal or logarithmic
transformation
to achieve linearity
