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Complex Numbers
COPlC
I)
Complex Numbers
XNTENT
- Introduction to complex numbers.
qo particular’ Style of introduction is
Real and imaginary parts of a complex number.
- Complex conjugates.
mposed. ConceDts should be put forward,
jut a derivation kom first principals is not
desirable.
- Operations on complex numbers:
sum, product, quotient of two complex numbers.
Reciprocal of a non-zero complex number.
- Square roots of a complex number.
Solution over ‘9? of quadratic equations with complex coefficients.
- Geometric representation of a complex number. (Argand diagram)
L.
- Trigonometric form.
- Modulus of a complex number. Modulus of a product, of a quotient,
of the reciprocal.
Argument of a non-zero complex number. Argument of a product, of
a quotient. Argument of the reciprocal of a non-zero complex
number.
Powers, nth roots.
de Moivre’s Theorem.
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Coefficients oft 1 wsc equations sl~ould be
free of parametc t-s
Definition of a real function of a real variable.
variable
Zeros of a tinction. Sign of a function.
- Even and odd fknctions Periodic functions.
Composition of two functions
- Inverse of a bijection
- Increasing and decreasing functions, constant, monotonic, over an
interval Local and global extrema
- Graph of a function.
2) Continuity and limits.
- Notion of continuity of a tknction at a point; k
examples and counter-examples.
- Continuity of a function from the right [left] of a point.
- Continuity of a tinction over an open [closed] interval.
- Statement (without proof) of theorems concerning continuity:
- of the absolute value of a continuous tinction.
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The order in which the two notions of
continuity and limit will be studied is 1ef-I
to the teacher. The study should largely
Continuity and limits
- of the product of a continuous function with a real number.
of the sum, product, quotient, composition of two continuous
- Continuity over Vof polynomial functions.
Continuity of rational functions over their domain.
- Notion of the limit of a function at a point;
examples and counter-examples
removable discontinuity.
Right-hand [left-hand] limit of a function at a point.
- Extension of the notion of limit: infinite limit, limit as the
variable tends to +CO or -00 .
- Statement (without proof) of theorems concerning limits:
- of the absolute value of a function.
- of the product of a function with a real number.
- of the sum, product, quotient, composition of two functions.
- Indeterminant forms.
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I) Differentiation.
- Value of the derivative of a function at a point. Geometrical
- Equation of the tangent at a point on the graph of a tin&on.
- If a function is differentiable at a point, it is also continuous at that
- Value of the right-hand [left-hand) derivative at a point.
- Derivative of a function Successive derivatives.
- Derivative
of a product of a differentiable function with a real
Inclusion of the proof of these results is ai
the teacher’s discretion.
Derkative of the sum. product, quotient and compostition of two
ditkrentiable fknctions
Derivative of the inverse of a function.
The result is to be stated and used.
Application of the notions of limit and derivative? to the analysis of a
The fImctions studied should be limited to
those for which the use of the derivative i!
ncrease and decrease of a function and identification of any extremum,
asymptotes on the graph of a fkrction.
Concave/convex nature of the graph of a function, points of inflection;
tangents at such points.
- Application of these ideas, and those of previous paragraphs,
o the study of polynomial, rational, circular ( sine, cosine, tangent )
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GEOh-IETRY 1N 3-DIhlESIONAL SPACE
ZQNTENT
Points, lines, planes, spheres.
,) Common objects in 3-D
Ipace.
Relative positions of these.
2) Vectors in 3-D space.
Definition.
Sum of vectors
Product of a vector with a scalar Collinear vectors.
Study of this section could be based on
revision of vectors in 2 dimensional space
and the obvious extension into 3-D space.
Vector equation of a line
Linear combination of two vectors. Coplanar vectors.
Vector equation of a plane.
Scalar product of two vectors
Magnitude of a vector, distance between two points.
b.
Orthogonal vectors.
Orthogonal, normalised, orthonormal basis.
Application of these concepts to problems in analytical geometry
3) Analytical geometry of
the point, the plane, the line.
Parametric and Cartesian equations of a plane.
-
Parametric and Cartesian equations of a line.
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In this section, the basis will always be
orthonormal.
PROBABILITY
raw
:ONTENT
I ) The counting of
u-rangement s and
. Permutations and combinations of a finite set with or without
?epetition
ielections
Introduction
the
theory of prohahilit)
- random outcomes. possibilities. possibility space.
- events. simple events
- certainty, impossibility
- the negation of an event.
- P(A n H), P(A u H), P(A n noti) .
- mutually exclusive events.
- The relation between probability and relative frequency.
- Probability defined on a finite possibility space.
- Elementary properties.
- General idea of a probability distribution.
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dote-that different languages differ in the
nterpretation to be given to the terms
lere
Analysis
) Analysis of real functions
ofa real variable
- domain of a fkction.
- zeros of a firnction Sign of a function.
- even and odd functions, Periodic functions.
- limit of a function at a point
- continuity at a point. over an interval
- dilrerentiability of a function at a point, over an interval.
The majority of this paragraph is studied
in year 6. It is not necessary to repeat the
entire content, but simply to revise and
where necessary to extend understanding.
increase and decrease of a function. Extremum.
- graph of a function, tangent at a point on the graph of a tinction,
‘.
asymptotes, concavity, points of inflection.
The study of these functions puts into
practice the concepts studied in the
previous section and those introduced in
To the following. functions:
- absolute value.
- rational functions.
The fbnctions studied should be limited to
those for which the use of the derivative iI
- fimctions of the type x + ,/m,
where P(x) is a 1’ or 2”d degree polynomial.
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ON-I’EN’I
circular functions. sine, cosine, tangent.
I liaison with the teaching of other
natural logarithm function, exponential function with base e.
ubjects, mention base 10 logarithms
functions obtained by addition, multiplication, division or
composition of the preceding functions.
!) Integration
Integral of a function defined on a closed and bounded interval
Graphical interpretation of such integrals as area.
Properties of integrals
I
I
b
a
a
f(x) dx = 0 ;
3
bqxjdx
b
C
f(x) dx =
a
f(x) dx
= -
a
C
f(x) dx +
bf(x) dx
a
Linearity:
b
s”i f(x) + g(x) ] dx = Ib f(x) dx +
a
a
J g(x) dx
a
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[OPlC
ntegration continued
CONTENT
b
b
I a [ h fix) ] dx = h I a f(x) dx
Lower and upper rectangle sums, enclosure thereby
- given a 5 b and !Jx) 2 0 on (a.b] , then 1: f(x) dx 2 0
- given a 5 b and flu) < g(u) on [a,b] ,
rhen ,:fix) du *. I:g(x)dx
- given a s b and m 5 9x) I hl on [a, b) ,
t h e n m(b-a)<J:qx) dx<M(b-a)
- Mean value of a function f on an interval [a,q :
p
&
_
I
b
f(x) dx
a
- Primitives (indefinite integral) of a fimction continuous over an
interval
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Practical examples should be used to
illustrate this point.
Integration
- If/is continuous over an interval 1, and a E 1, then the function F
defined on / by F(x) =
f(t) dt is the primitive off over I which
is zero when x = a.
- indefinite integral
- Evaluation of integrals by the following methods :
- Integration by inspection
- Integration by parts.
- Integration by substitution.
- Application of these methods to the functions studied in
- Application of the theory of integration to finding plane areas and
volumes of revolution generated by rotation about the x-axis.
- First order differential equations with variables separable leading to
the form y’.f(x) = g(x).
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The integration of rational functions may
require their transformation to the sum of
The form of the
decomposition should be given in all cases
May be applied to problems emanating
from physics, biology, economics, etc.
GEOMETRY OF THREE DIMENSIONAL SPACE
row
:ONTENT
) Vectors in 3-D Space
‘hrormphout lhr chwter w Ckonw!r~ the basis will alwaw he choserl
!.Y
orrhorrormal
-_-A
Collinear vectors Vector equation of a line.
Toplanar vectors Vector equation of a plane.
Scalar product of two vectors in 3-D.
The majority of this paragraph is studied
n year 6. It is not necessary to repeat the
:ntire content, but simply to revise and
where necessary to extend understanding.
rna_unitude of a vector. distance between two points.
Orthogonal Lectors Orthonornlal basis
. l’ector product of two vectors
. Triple scalar product
- Application of these concepts to problems in analytical geometry.
- The use of the preceding concepts :
_.
- in the calculation of areas of common plane figures:
triangle, trapezium, parallelogram.
- in the calculation of volumes of common solids:
prism, parallelepiped, cylinder, pyramid.
2) Analytical geometry of
the point, the plane, the line.
- Parametric and Cartesian equations of a plane.
- Parametric and Cartesian equations of a line.
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The use of matrices and determinants of
order 3 is left to the discretion of the
teacher.
2) Analytical geometry of
.he point, the plane, the line
lcontinued)
Relative position of two planes, of a line and a plane, of two
Orthogonal projection of a point onto a plane.
Distance between a point and a plane.
Distance between two parallel planes
Distance between a plane and a parallel line.
Orthogonal projection of a point on a line.
Distance of a point from a line
Distance between two lines
Angle between two vectors in 3-D.
Angle between two lines.
Angie between two planes.
_.
Angle between a line and a plane.
3 j Analytic geometry of the
- Cartesian equation of a sphere.
- Relative positions of a point and sphere; of a plane and sphere ; of a
- Volume and surface area of the sphere.
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Permutations and combinations of a finite set with or without
u-rangement s and
elections
The concepts of paragraphs 1) and 2) are
introduced in years 6. What is required
here is revision.
Note that different languages differ in the
interpretation to be given to the terms
!) Probability
- Probability defined on a finite possibility space.
Elementary properties
Probability distribution.
Probability conditional upon an event with non-zero probability;
notation P(BI A).
P(A n B) = P(A) x P(B 1 A)
Use of this result within tree diagrams for random trials with several
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CONTEN
) Conditional probability
continued)
_ Independent events:
I’( A n I{) = I’( A) x I’(H)
- total probability:
I’(H) = 2 /$4,)X l’(N) /I,).
1 I
- Bayes Theorem
I) Discrete random variables
In application, avoid situations in which the
relevant contingency table cannot be
partitioned as 2x2.
General prooerties
- discrete random variables.
- sample space.
- Probability fknction of a discrete random variable.
- Cumulative distribution fkction of a discrete random variable.
- Expected value, variance and standard deviation of a discrete random
variable.
Adopting the convention:
F(x) = P(X 5 x).
5) Continuous random
variables. (continued)
Normal (or Gaussian) distribution
I
- Definition.
- Expected value, variance and standard deviation of a Normal
distribution.
- Normal curve and cumulative Normal curve.
_ Standardised Normal distribution, use of tables.
- Normal approximation to the binomial distribution given 11py>9.
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