|
|
GCE Advanced Level Maths Curriculum For 11th Grade And 12th Grade /
GCE A Level
Our Singapore Maths books for 11th Grade And
12th Grade / GCE A Level are written in English and based on GCE A
Level Maths curriculum for 11th Grade And 12th Grade / GCE A Level, which
covers the following topics.
If your child uses our Singapore Maths books for 11th Grade And 12th
Grade / GCE A Level, he will be able to:
Pure mathematics
-
Functions and graphs
- understand the terms function, domain, range and one-one
function
- find composite functions and inverses of functions, including
conditions for their existence
- understand and use the relation (fg)-1 = g-1f-1
where appropriate
- illustrate in graphical terms the relation between a one-one
function and its inverse
- understand the relationship between a graph and an associated
algebraic equation, and in particular show familiarity with the
forms of the graphs of:
- y = kxn where n is a positive or negative integer
or a simple rational number
- ax + by = c
 (knowledge
of geometrical properties of conics is not required)
- understand and use the relationships between the graphs of y =
f(x), y = af(x), y = f(x) + a, y = f(x+a), y = f(ax), where a is a
constant, and express the transformations involved in terms of
translations, reflections and scalings
- relate the equation of a graph to its symmetries
- understand, and use in simple cases, the expression of the
coordinates of a point on a curve in terms of a parameter
-
Partial fractions
- recall an appropriate form for expressing rational functions in
partial fractions, and carry out the decomposition, in cases where
the denominator is no more complicated than:
- (ax + b)(cx + d)(ex + f)
- (ax + b)(cx + d)2
- (ax + b)(x2 + c2)
- including cases where the degree of the numerator exceeds that
of the denominator
-
Inequalities; the modulus function
- use properties of inequalities, and in particular understand
that x > y and z > 0 imply that xz > yz while x > y
and z < 0 imply xz < yz
- find the solution set of inequalities that are reducible to the
form f(x) > 0, where f(x) can be factorised, and illustrate
such solutions graphically
- understand the meaning of |x| and sketch the graph of functions
of the form y = lax+ b|
- use relations such as lx - al < b
a - b < x < a + b and |al = |b|
a2 = b2 in the course of solving equations
and inequalities
-
Logarithmic and exponential functions
- recall and use the laws of logarithms (including change of base)
and sketch graphs of simple logarithmic and exponential functions
- recall and use the definition ax = ex ln a
- use logarithms to solve equations reducible to the form ax
= b, and similar inequalities
-
Sequences and series
- understand the idea of a sequence of terms, and use notations
such as un to denote the nth term of a sequence
- recognise arithmetic and geometric progressions
- use formulae for the nth term and for the sum of the first n
terms to solve problems involving arithmetic or geometric
progressions
- recall the condition for convergence of a geometric series, and
use the formula for the sum to infinity of a convergent geometric
series
- use
 notation
- use the binomial theorem to expand (a + b)n, where n
is a positive integer
- use the binomial theorem to expand (1 + x)n, where n
is rational, and recall the condition |x|< 1 for the validity
of this expansion
- recognise and use the notations n! (with 0! = 1) and
-
Permutations and combinations
- understand the terms 'permutation' and 'combination'
- solve problems involving arrangements (of objects in a line or
in a circle), including those involving
- repetition (e.g. the number of ways of arranging the letters
of the word NEEDLESS)
- restriction (e.g. the number of ways several people can
stand in a line if 2 particular people must - or must not -
stand next to each other)
-
Trigonometry
- use the sine and cosine formulae
- calculate the angle between a line and a plane, the angle
between two planes, and the angle between two skew lines in simple
cases
-
Trigonometrical functions
- understand the definition of the six trigonometrical functions
for angles of any magnitude
- recall and use the exact values of trigonometrical functions of
- use the notations sin-1x, cos-1x and
tan-1x to denote the principal values of the inverse
trigonometrical relations
- relate the periodicity and symmetries of the sine, cosine and
tangent functions to the form of their graphs, and use the
concepts of periodicity and / or symmetry in relation to these
functions and their inverses
- use trigonometrical identities for the simplification and exact
evaluation of expressions, and select an identity or identities
appropriate to the context, showing familiarity in particular with
the use of:
 
and equivalent statements- the expansion of sin(A ± B), cos(A ± B) and tan(A + B)
- the formulae for sin2A, cos2A and tan2A
- the formulae for sin A ± sin B and cos A ± cos B
- the expression of a
in the forms 
- find the general solution of simple trigonometrical equations,
including graphical interpretation
- use the small-angle approximations
-
Differentiation
- understand the idea of a limit and the derivative defined as a
limit, including geometrical interpretation in terms of the
gradient of a curve at a point as the limit of the gradient of a
suitable sequence of chords
- use the standard notations
for derived functions
- use the derivatives of xn (for any rational n), sin
x, cos x, tan x, ex, ax, ln x, sin-1x,
cos-1x and tan-1x; together with
constant multiples, sums, differences, products, quotients and
composites
- find and use the first derivative of a function which is defined
implicitly or parametrically
- locate stationary points, and distinguish between maxima, minima
and stationary points of inflexion (knowledge of conditions for
general points of inflexion is not required)
- find equations of tangents and normals to curves, and use
information about gradients for sketching graphs
- solve problems involving maxima and minima, connected rates of
change, small increments and approximations
- derive and use the first few terms of the Maclaurin series for a
function
-
Integration
- understand indefinite integration as the reverse process of
differentiation
- integrate xn (including the case where n = -1), ex,
sin x, cos x, sec2x, together with
- sums, differences and constant multiples of these
- expressions involving a linear substitution (e.g. e2x-1)
- applications involving the use of partial fractions
- applications involving the use of trigonometrical identities
(e.g.
 )
- recognise an integrand of the form
and integrate, e.g. 
or tan x
- integrate
- recognise when an integrand can usefully be regarded as a
product, and use integration by parts to integrate, e.g., x sin
2x, x2ex, ln x
- use the in method of integration by substitution to simplify and
evaluate either a definite or an indefinite integral (including
simple cases in which the candidates have to select the
substitution themselves, e.g.
 )
- evaluate definite integrals (including e.g.
- understand the idea of the area under a curve as the limit of a
sum of the areas of rectangles and use simple applications of this
idea
- use integration to find plane areas and volumes of revolution in
simple cases
- use the trapezium rule to estimate the values of definite
integrals, and identify the sign of the error in simple cases by
graphical considerations
-
Vectors
- use rectangular cartesian coordinates to locate points in three
dimensions, and use standard notations for vectors, i.e.
xi + yj + zk, a
- carry out addition and subtraction of vectors and multiplication
of a vector by a scalar, and interpret these operations in
geometrical terms
- use unit vectors, position vectors and displacement vectors
- recall the definition of and calculate the magnitude of a vector
and the scalar product of two vectors
- use the scalar product to determine the angle between two
directions and to solve problems concerning perpendicularity of
vectors
- understand the significance of all the symbols used when the
equation of a straight line is expressed in either of the forms r
= a + tb and
and convert equations of lines from vector to Cartesian form and
vice versa
- solve simple problems involving finding and using either form of
the equation of a line
- use equations of lines to solve problems concerning distances,
angles and intersections, and in particular:
- determine whether two lines are parallel, intersect or are
skew, and find the point of intersection of two lines when it
exists
- find the perpendicular distance from a point to a line
- find the angle between two lines
- use the ratio theorem in geometrical applications
-
Mathematical induction
- understand the steps needed to carry out a proof by the method
of induction
- use the method of mathematical induction to establish a given
result e.g. the sum of a finite series, or the form of an nth
derivative
-
Complex numbers
- understand the idea of a complex number, recall the meaning of
the terms 'real part', 'imaginary part', 'modulus', 'argument',
'conjugate', and use the fact that two complex numbers are equal
if and only if both real and imaginary parts are equal
- carry out operations of addition, subtraction, multiplication
and division of two complex numbers expressed in cartesian form (x
+ iy)
- recall and use the relation zz* = |z|2
- use the result that, for a polynomial equation with real
coefficients, any non-real roots occur in conjugate pairs
- represent complex numbers geometrically by means of an Argand
diagram
- carry out operations of multiplication and division of two
complex numbers expressed in polar form (
 )
- understand in simple terms the geometrical effects of
conjugating a complex number and of adding, subtracting,
multiplying, dividing two complex numbers
- illustrate simple equations and inequalities involving complex
numbers by means of loci in an Argand diagram (e.g.|z - a| < k,
|z - a| = |z - b|, arg(z - a) =
 ,
but excluding arg(z-a)-arg(z-b) = )
-
Curve sketching
- understand the relationships between the graphs of y = f(x), y2
= f(x) and y = |f(x)|
- determine, in simple cases, the equations of asymptotes parallel
to the axes
- use the equation of a curve, in simple cases, to make deductions
concerning symmetry or concerning any restrictions on the possible
values of x and / or y that there may be
- sketch curves of the form y = f(x), y2 = f(x) or y =
|f(x)| (detailed plotting of curves will not be required, but
sketches will generally be expected to show significant features,
such as turning points, asymptotes and intersections with the
axes)
-
First order differential fquations
- formulate a simple statement involving a rate of change as a
differential equation, including the introduction if necessary of
a constant of proportionality
- find by integration a general form of solution for a first order
differential equation in which the variables are separable
- find the general solution of a first order linear differential
equation by means of an integrating factor
- reduce a given first order differential equation to one in which
the variables are separable or to one which is linear by means of
a given simple substitution
- understand that the general solution of a differential equation
is represented in graphical terms by a family of curves, and
sketch typical members of a family in simple cases
- use an initial condition to find a particular solution to a
differential equation, and interpret a solution in terms of a
problem modelled by a differential equation
-
Numerical methods
- locate approximately a root of an equation by means of graphical
considerations and / or searching for a sign change
- use the method of linear interpolation to find an approximation
to a root of an equation
- understand the idea of, and use the notation for, a sequence of
approximations which converges to the root of an equation
- understand how a given simple iterative formula of the form
relates to the equation being solved, and use a given iteration to
determine a root to a prescribed degree of accuracy (conditions
for convergence are not included)
- understand, in geometrical terms, the working of the
Newton-Raphson method, and derive and use iterations based on this
method
- appreciate that an iterative method may fail to converge to the
required root
Particle mechanics
-
Forces and equilibrium
- identify the forces acting in a given situation
- understand the representation of forces by vectors, and find and
use resultants and components
- solve problems concerning the equilibrium of a particle under
the action of coplanar forces (using equations obtained by
resolving the forces, or by using a triangle or polygon of forces)
- recall that the contact force between two surfaces can be
represented by two components (the 'normal component' and the
'frictional component') and use this representation in solving
problems
- use the model of a 'smooth' contact and understand the
limitations of this model
- understand the concept of limiting friction and limiting
equilibrium, recall the definition of coefficient of friction; and
use the relationship
as appropriate (knowledge of angle of friction will not be
required)
- recall and use Newton's third law
-
Kinematics of motion in a straight line
- understand the concepts of distance and speed, as scalar
quantities, and of displacement, velocity and acceleration, as
vector quantities, and understand the relationships between them
- sketch and interpret x-t and v-t graphs, and in particular
understand and use the facts that:
- the area under a v-t graph represents displacement
- the gradient of an x-t graph represents velocity
- the gradient of a v-t graph represents acceleration
- use appropriate formulae for motion with constant acceleration
in a straight line
-
Newton's laws of motion
- recall and use Newton's first and second laws of motion
- apply Newton's laws to the linear motion of a particle of
constant mass moving under the action of constant forces
(including friction)
- solve problems on the motion of two particles, connected by a
light inextensible string which may pass over a fixed smooth light
pulley or peg
-
Energy, work and power
- understand the concept of the work done by a force, and
calculate the work done by a constant force when its point of
application undergoes a displacement not necessarily parallel to
the force (use of the scalar product is not required)
- understand the concepts of gravitational potential energy and
kinetic energy, and recall and use appropriate formulae
- understand and use the relationship between the change in energy
of a system and the work done by the external forces, and use
where appropriate the principle of conservation of energy
- recall and use the definition of power as the rate at which a
force does work, and use the relationship between power, force and
velocity for a force acting in the direction of motion
- solve problems involving, for example, the instantaneous
acceleration of a car moving on a hill with resistance
-
Linear motion under a variable force
- solve simple problems on the linear motion of a particle of
constant mass moving under the action of variable forces by
setting up and solving an appropriate differential equation (use
of
for velocity and  ,
as appropriate, for acceleration is expected, and any differential
equations to be solved will be first order with separable
variables)
-
Motion of a projectile
- model the motion of a projectile as a particle moving with
constant acceleration, and understand the limitations of this
model
- use horizontal and vertical equations of motion to solve
problems on the motion of projectiles (including finding the
magnitude and direction of the velocity at a given time or
position and finding the range on a horizontal plane)
- derive and use the cartesian equation of the trajectory of a
projectile, including cases where the initial speed and/or angle
of projection is unknown (knowledge of the range on an inclined
plane is not required)
-
Hooke's law
- recall and use Hooke's law as a model relating the force in an
elastic string or spring to the extension or compression, and
understand and use the term 'modulus of elasticity'
- understand the concept of elastic potential energy, and recall
and use the appropriate formula for its calculation
- use considerations of work and energy to solve problems
involving elastic strings and springs
-
Uniform circular motion
- understand the concept of angular speed for a particle moving in
a circle with constant speed, and recall and use the relation
(no proof required)
- understand that the acceleration of particle moving in a circle
with constant speed is directed towards the centre of the circle
and has magnitude
(no proof required)
- use Newton's second law to solve problems which can be modelled
as the motion of a particle moving in a circle with constant speed
Probability and statistics
-
Probability
- use addition and multiplication of probabilities, as
appropriate, in simple cases, and understand the representation of
events by means of tree diagrams
- understand the meaning of mutually exclusive and independent
events, and calculate and use conditional probabilities in simple
cases
- understand and use the notations P(A),
 P(A|B)
and the equations 
and  =
P(A) P(B|A) = P(B) P(A|B) (the general form of Bayes' theorem is
not required)
-
Discrete random variables
- understand the concept of a discrete random variable
- construct a probability distribution table relating to a given
situation, and calculate E(X) and Var(X)
- appreciate conditions under which a uniform distribution or a
binomial distribution B(n,p) may be a suitable probability model,
and recall and use formulae for the calculation of binomial
probabilities
- understand conditions under which a Poisson distribution Po(
 )
may be a suitable probability model, and recall and use the
formula for the calculation of Poisson probabilities
- recall and use the means and variances of binomial and Poisson
distributions
- use a Poisson distribution as an approximation to a binomial
distribution, where appropriate (candidates should know that the
conditions n > 50 and np < 5, approximately, can generally
be taken to be suitable)
-
The normal distribution
- recall the general shape of a normal curve, and understand how
the shape and location of the distribution
are affected by the values of 
(in general terms only; no knowledge of mathematical properties of
the normal density function is included)
- standardise a normal variable and use normal distribution tables
- use the normal distribution as a probability model, where
appropriate, and solve problems concerning a variable X, where
 ,
including:
- finding the value of P(X < x1) given the
values of
- use of the symmetry of the normal distribution
- finding a relationship between
given the value of P(X < x1)
- repeated application of the above
- recall conditions under which a normal distribution may be used
to approximate a binomial distribution (n sufficiently large to
ensure that np > 5 and nq > 5. approximately) or Poisson
distribution (
 ,
approximately), and calculate such approximations, including the
use of a continuity correction
-
Samples
- understand the distinction between a sample and a population,
and appreciate the necessity for randomness in choosing samples
- explain in simple terms why a given sampling method may be
unsatisfactory (a detailed knowledge of sampling and survey
methods is not required)
- recognise that the sample mean can be regarded as a random
variable, and use the facts that
- use the fact that X
is normally distributed if X is normally distributed
- use the Central Limit Theorem (without proof) to treat X
as being normally distributed when the sample size is sufficiently
large ('large' samples will usually be of size at least 50, but
candidates should know that using the approximation of normality
can sometimes be useful with samples that are smaller than this)
- calculate unbiased estimates of the population mean and
population variance from a sample (only a simple understanding of
the term 'unbiased' is required)
- determine, from a sample from a normal distribution of known
variance or from a large sample, a confidence interval for the
population mean
- determine, from a large sample, a confidence interval for a
population proportion
-
Linear combinations of random variables
- recall and use the results in the course of solving problems
that, for either discrete or continuous random variables,
- E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X)
- E(aX + bY) = aE(X) + bE(Y)
- Var(aX + bY) = a2 Var(X) + b2 for
independent X and Y
- recall and use the results that:
- if X has a normal distribution then so does aX + b
- if X and Y have independent normal distributions then aX +
bY has a normal distribution
- if X and Y have independent Poisson distributions then X + Y
has a Poisson distribution
-
Continuous random variables
- understand and use the concept of a probability density
function, and recall and use the properties of a density function
(which may be defined 'piecewise')
- use a given probability density function to calculate the mean,
mode and variance of a distribution, and in general use the result
in simple cases, where f(x) is the probability density function of
X and g(X) is a function of X
- understand and use the relationship between the probability
density function and the distribution function and use either to
evaluate the median, quartiles and other percentiles
- use a probability density function or a distribution function in
the context of a model, including in particular the continuous
uniform (rectangular) distribution
-
Hypothesis testing
- understand and use the concepts of hypothesis (null and
alternative), test statistic, significance level, and hypothesis
test (1-tail and 2-tail)
- formulate hypotheses and apply a hypothesis test concerning the
population mean using:
- a sample drawn from a normal distribution of known variance
- a large sample drawn from any distribution of unknown
variance
- formulate hypotheses concerning a population proportion, and
apply a hypothesis test using a normal approximation to a binomial
distribution
Please enter a number in the Qty box next to the products that you want to order, and then
click on the Add to your Shopping Cart button.
To learn more about a product, please scroll down this page or click on its name, where applicable.
|
|
Tell your friends about us! | |
|
|
