Set
language and notation

A
= {x
: x is a natural number}
B
=
C
=
D
= {a, b, c... }
Notation:
Union
of A and B: A ∪
B
Intersection
of A and B: A ∩
B
Number
of elements in set A: n(A)
“ .
. . is an element of . . . ”
: ∈
“ .
. . is not an element of . . . ”:
Complement
of set A
: A’
The
empty set: Ø
Universal
set
A is a subset of B
: A ⊆
B
A
is a proper subset of B
: A ⊂
B
A
is not a subset of B
: A ⊈
B
A
is not a proper subset of B: A
B

Functions

understand
the terms: function, domain, range (image set), oneone function,
inverse function and composition of functions
use
the notation f(x) = sin x, f: x lg x, (x > 0), f ^{−1}(x)
and f^{2}(x)
[= f(f(x))]
understand
the relationship between y = f(x) and y = f(x),
where f(x) may be linear, quadratic or trigonometric
explain
in words why a given function is a function or why it does not
have an inverse
find
the inverse of a oneone function and form composite functions
use
sketch graphs to show the relationship between a function and its
inverse

Equations
and inequalities

two
real roots, two equal roots, no real roots
intersect
a given curve , be a tangent to a given curve, not intersect a
given curve
Conditions
for ax^{2
}+
bx + c to be always positive (or always negative)
Solving
simultaneous equations in two variables with at least one linear
equation,
by substitution
Relationships
between the roots and coefficients of a quadratic equation
Solving
quadratic inequalities, and representing the solution on the
number line

Quadratic
functions

find
the maximum or minimum value of the quadratic function f : x ax^{2
}+
bx + c by any method
use
the maximum or minimum value of f(x) to sketch the graph or
determine the range for a given domain
know
the conditions for f(x) = 0 to have:
(i) two real roots,
(ii) two equal roots, (iii) no real roots
and the related
conditions for a given line to
(i) intersect a given
curve, (ii) be a tangent to a given curve, (iii) not intersect a
given curve
solve
quadratic equations for real roots and find the solution set for
quadratic inequalities

Indices
and surds


Factors
of polynomials

○ a^{3
}+
b^{3
}=
(a + b)(a^{2
}–
ab + b^{2})
○ a^{3
}–
b^{3
}=
(a – b)(a^{2
}+
ab + b^{2})

Simultaneous
equations


Logarithmic
and exponential functions

know
simple properties and graphs of the logarithmic and exponential
functions including lnx and e^{x
}(series
expansions are not required)
know
and use the laws of logarithms (including change of base of
logarithms)
solve
equations of the form a^{x
}=
b

Straight
line graphs

interpret
the equation of a straight line graph in the form
y = mx +
c
transform
given relationships, including y = ax ^{n
}and
y = Ab ^{x},
to straight line form and hence determine unknown constants by
calculating the gradient or intercept of the transformed graph
solve
questions involving midpoint and length of a line
know
and use the condition for two lines to be parallel or
perpendicular

Coordinate
geometry in two dimensions

Condition
for two lines to be parallel or perpendicular
Midpoint
of line segment
Area
of rectilinear figure
Graphs
of parabolas with equations in the form y^{2
}=
kx
Coordinate
geometry of circles in the form:
(x
– a)^{2
}+
(y – b)^{2
}=
r^{2}
x^{2
}+
y^{2
}+
2gx + 2fy + c = 0 (excluding problems involving 2 circles)

Circular
measure


Trigonometry

know
the six trigonometric functions of angles of any magnitude (sine,
cosine, tangent, secant, cosecant, cotangent)
Principal
values of sin^{–1}x,
cos^{–1}x,
tan^{–1}x
Exact
values of the trigonometric functions for special angles
(30°,45°,60°)

understand
amplitude and periodicity and the relationship between graphs of,
e.g. sin x and sin 2x
draw
and use the graphs of
where
a and b are positive integers and c is an integer
○ the
expansions of sin(AB), cos(A B) and tan(AB)
○ the
formulae for sin 2A, cos 2A and tan 2A
○ the
expression for in the form Rcos(α) or Rsin(α)
and
solve simple trigonometric equations involving the six
trigonometric functions and the above relationships (not
including general solution of trigonometric equations)
prove
simple trigonometric identities

Permutations
and combinations

recognise
and distinguish between a permutation case and a combination case
know
and use the notation n! (with 0! = 1), and the expressions for
permutations and combinations of n items taken r at a time
answer
simple problems on arrangement and selection (cases with
repetition of objects, or with objects arranged in a circle or
involving both permutations and combinations, are excluded)

Binomial
expansions


Vectors
in 2 dimensions

use
vectors in any form, e.g. , p,
ai
–
bj
know
and use position vectors and unit vectors
find
the magnitude of a vector, add and subtract vectors and multiply
vectors by scalars
compose
and resolve velocities
use
relative velocity, including solving problems on interception
(but not closest approach)

Matrices

display
information in the form of a matrix of any order and interpret
the data in a given matrix
solve
problems involving the calculation of the sum and product (where
appropriate) of two matrices and interpret the results
calculate
the product of a scalar quantity and a matrix
use
the algebra of 2 × 2 matrices (including the zero and
identity matrix)
calculate
the determinant and inverse of a nonsingular 2 × 2 matrix
and solve simultaneous linear equations

Differentiation
and integration

understand
the idea of a derived function
use
the notations
use
the derivatives of the standard functions
x ^{n
}(for
any rational n), sin x, cos x, tan x, e^{x},
ln x, together with constant multiples, sums and composite
functions of these
differentiate
products and quotients of functions
apply
differentiation to gradients, tangents and normals, stationary
points, connected rates of change, small increments and
approximations and practical maxima and minima problems
discriminate
between maxima and minima by any method
understand
integration as the reverse process of differentiation
integrate
sums of terms in powers of x, excluding
integrate
functions of the form (ax + b)^{n
}(excluding
n = –1),
e^{ax
+ b},
sin (ax + b), cos (ax + b)
evaluate
definite integrals and apply integration to the
evaluation
of plane areas
apply
differentiation and integration to kinematics problems that
involve displacement, velocity and acceleration of a particle
moving in a straight line with variable or constant acceleration,
and the use of xt and vt graphs
