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Mathematics(NORMAL TECHNICAL)
GCE Normal Level(Syllabus T 4043)
CONTENTS
Page
GCE NORMAL LEVEL SYLLABUS T MATHEMATICS 4043 1
MATHEMATICAL FORMULAE 11
MATHEMATICAL NOTATION 12
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AIMS
The syllabus is intended to provide students with the fundamental mathematical knowledgeand skills to prepare them for technical- or service-oriented education.
The general aims of the syllabus are to enable students to:
1. acquire the necessary mathematical concepts and skills for continuous learning inmathematics and related disciplines, and for applications to the real world;
2. develop the necessary process skills for the acquisition and application ofmathematical concepts and skills;
3. develop the mathematical thinking and problem solving skills and apply these skills toformulate and solve problems;
4. recognise and use connections among mathematical ideas, and betweenmathematics and other disciplines;
5. develop positive attitudes towards mathematics;
6. make effective use of a variety of mathematical tools (including information andcommunication technology tools) in the learning and application of mathematics;
7. produce imaginative and creative work arising from mathematical ideas;
8. develop the abilities to reason logically, to communicate mathematically, and to learncooperatively and independently.
ASSESSMENT OBJECTIVES
The assessment will test candidates abilities to:
AO1 understand and use mathematical concepts and skills in a variety of contexts;
AO2 organise and analyse data and information; formulate problems into mathematicalterms and select and apply appropriate techniques of solution;
AO3 apply mathematics in practical situations; interpret mathematical results and make
inferences.
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SCHEME OF ASSESSMENT
Paper Duration Description Marks Weighting
Paper 1 1 h
There will be 810 short questionscarrying 25 marks followed by 3 longquestions carrying 68 marks.
Candidates are required to answer allquestions which will cover topics from thestrands:
Numbers and Algebra Geometry and Measurement Integrative Contexts (see 4.1 of
Content Outline) related to Numbersand Algebra and Geometry andMeasurement
50 50%
Paper 2 1 h
There will be 810 short questionscarrying 25 marks followed by 3 longquestions carrying 68 marks.
Candidates are required to answer allquestions which will cover topics from thestrands:
Numbers and Algebra Statistics and Probability Integrative Contexts (see 4.1 of
Content Outline) related to Numbersand Algebra and Statistics andProbability
50 50%
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NOTES
1. Omission of essential working will result in loss of marks.
2. Relevant mathematical formulae will be provided for candidates.
3. Scientific calculators are allowed in both Paper 1 and Paper 2.
4. Candidates should have geometrical instruments with them for Paper 1.
5. Unless stated otherwise within a question, three-figure accuracy will be required foranswers. This means that four-figure accuracy should be shown throughout theworking, including cases where answers are used in subsequent parts of thequestion. Premature approximation will be penalised, where appropriate. Angles indegrees should be given to 1 decimal place.
6. SI units will be used in questions involving mass and measures.
Both the 12-hour and 24-hour clock may be used for quoting times of the day. In the24-hour clock, for example, 3.15 a.m. will be denoted by 03 15; 3.15 p.m. by 15 15,noon by 12 00 and midnight by 24 00.
7. Candidates are expected to be familiar with the solidus notation for the expression ofcompound units, e.g. 5 cm/s for 5 centimetres per second, 13.6 g/cm
3for 13.6 grams
per cubic centimetre.
8. Unless the question requires the answer in terms of , the calculator value for or
= 3.142 should be used.
9. Spaces will be provided in each question paper for working and answers.
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CONTENT OUTLINE
The syllabus consists of three content strands, namely, Numbers and Algebra,Geometry and Measurement, and Statistics and Probability, and a context strandcalled Integrative Contexts.
Application of mathematics is an important emphasis of the content strands. Theapproach to teaching should involve meaningful contexts so that students can see andappreciate the relevance and applications of mathematics in their daily life and the worldaround them.
Integrative Contexts are realistic contexts that naturally have practical applications ofMathematics, and the Mathematics can come from any part of the Content Outline.
Topic/Sub-topics Content
1 NUMBERS ANDALGEBRA1.1 Numbers and the
four operationsInclude:
negative numbers, integers, and their four operations
four operations on fractions and decimals (including negative fractionsand decimals)
calculations with the use of a calculator, including squares, cubes,square roots and cube roots
representation and ordering of numbers on the number line
use of the symbols , Y,
rounding off numbers to a required number of decimal places orsignificant figures
estimating the results of computation
use of index notation for integer powers examples of very large and very small numbers such as
mega/million (106), giga/billion (10
9), tera/trillion (10
12),
micro (10-6
), nano (10-9
) and pico (10-12
)
use of standard form 10nA , where n is an integer, and 1 YA < 10
Exclude:
use of the terms rational numbers, irrational numbers and realnumbers
primes and prime factorisation
fractional indices and surds
1.2 Ratio, rate andproportion
Include: comparison between two or more quantities by ratio
dividing a quantity in a given ratio
ratios involving fractions and decimals
equivalent ratios
writing a ratio in its simplest form
rates and average rates (including the concepts of speed and averagespeed)
conversion of units
map scales (distance and area)
direct and inverse proportion
problems involving ratio, rate and proportion
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Topic/Sub-topics Content
1.3 Percentage Include:
expressing percentage as a fraction or decimal
finding the whole given a percentage part
expressing one quantity as a percentage of another
comparing two quantities by percentage
percentages greater than 100%
finding one quantity given the percentage and the other quantity
increasing/decreasing a quantity by a given percentage
finding percentage increase/decrease
problems involving percentages
1.4 Algebraicrepresentationand formulae
Include:
using letters to represent numbers
interpreting notations:
ab as a
b
a
bas ab
a2as aa, a
3as a aa, a
2b as a ab,
3yas y+ y+ y or 3 y
3
5
yas (3 y) 5 or ( )
1 3
5y
evaluation of algebraic expressions and formulae
translation of simple real-world situations into algebraic expressions
recognising and representing number patterns (including finding analgebraic expression for the nth term)
1.5 Algebraic
manipulation
Include:
addition and subtraction of linear algebraic expressions simplification of linear algebraic expressions, e.g.
_2(3 5)+4_x x,( )2
53
3
2
xx
expansion of the product of two linear algebraic expressions
multiplication and division of simple algebraic fractions, e.g.
2
3 5
34
a ab
b
,
23 9
4 10
a a
changing the subject of a simple formula
finding the value of an unknown quantity in a given formula
factorisation of linear algebraic expressions of the form
ax + ay(where a is a constant)
ax + bx + kay + kby(where a, b and kare constants)
factorisation of quadratic expressions of the form 2x + px + q
Exclude:
use of special products:2 2 2( ) = 2 +a b a ab b
a2 b
2= (a + b)(ab)
factorisation of algebraic expressions of the form
a2x
2 b
2y2
2 2 2 +a ab b
2 + +ax bx c , where 1a
addition and subtraction of algebraic fractions
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Topic/Sub-topics Content
1.6 Functions and
graphs
Include:
cartesian coordinates in two dimensions graph of a set of ordered pairs
linear relationships between two variables (linear functions)
the gradient of a linear graph as the ratio of the vertical change to thehorizontal change (positive and negative gradients)
graphs of linear equations in two unknowns
graphs of quadratic functions and their properties
positive or negative coefficient of 2x
maximum and minimum points
symmetry
Exclude sketching of graphs of quadratic functions.
1.7 Solutions ofequations
Include:
solving linear equations in one unknown (including fractionalcoefficients)
formulating a linear equation in one unknown to solve problems
solving simple fractional equations that can be reduced to linearequations, e.g.
_2+ = 3
3 4x x
3= 6
2_
x
solving simultaneous linear equations in two unknowns by
substitution and elimination methods graphical method
solving quadratic equations in one unknown by use of formula
formulating a quadratic equation in one unknown or a pair of linearequations in two unknowns to solve problems
Exclude solving quadratic equations by:
method of completing the square
graphical methods
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Topic/Sub-topics Content
2 GEOMETRY ANDMEASUREMENT
2.1 Angles, trianglesand quadrilaterals
Include:
right, acute, obtuse and reflex angles, complementary andsupplementary angles, vertically opposite angles, adjacent angles on astraight line, adjacent angles at a point, interior and exterior angles
angles formed by two parallel lines and a transversal: correspondingangles, alternate angles, interior angles
properties of triangles and special quadrilaterals
classifying special quadrilaterals on the basis of their properties
properties of perpendicular bisectors of line segments and anglebisectors
construction of simple geometrical figures from given data (including
perpendicular bisectors and angle bisectors) using compasses, ruler,set squares and protractor where appropriate
Exclude properties of polygons.
2.2 Congruence,similarity andtransformations
Include:
congruent figures and similar figures
properties of similar polygons:
corresponding angles are equal
corresponding sides are proportional
drawing on square grids the following transformations of simple planefigures
reflection about a given horizontal or vertical line rotation about a given point through multiples of 90
clockwise/anticlockwise
translation represented by a given translation arrow
enlargement by a simple scale factor such as1
2, 2 and 3, given
the centre of enlargement
scale drawings
Exclude:
use of coordinates
negative scale factors
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Topic/Sub-topics Content
2.3 Symmetry,
tessellations andprojections
Include:
line and rotational symmetry of plane figures order of rotational symmetry
identifying the unit figure(s) of a tessellation and continuing atessellation
orthographic projection drawings, including plan (top view), front, leftand right views
Exclude symmetry of solids
2.4 Pythagorastheorem andtrigonometry
Include:
use of Pythagoras theorem
determining whether a triangle is right-angled given the lengths of
three sides use of trigonometric ratios (sine, cosine and tangent) of acute angles
to calculate unknown sides and angles in right-angled triangles(including problems involving angles of elevation and depression)
use of the formula2
1ab sinCfor the area of a triangle (extending sine
to obtuse angles)
Exclude:
sine rule and cosine rule
bearings
2.5 Mensuration Include:
area of triangle as2
1 base height
area and circumference of circle
area of parallelogram and trapezium
problems involving perimeter and area of composite plane figures
visualising and sketching cube, cuboid, prism, cylinder, pyramid, coneand sphere (including use of nets to visualise the surface area ofthese solids, where applicable)
volume and surface area of cube, cuboid, prism, cylinder, pyramid,cone and sphere
conversion between cm2
and m2, and between cm
3and m
3
problems involving volume and surface area of composite solids arc length and sector area as fractions of the circumference and area
of a circle
Exclude the radian measure of angle.
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Topic/Sub-topics Content
3 STATISTICS ANDPROBABILITY
3.1 Data handling Include:
data collection methods such as:
taking measurements
conducting surveys
classifying data
reading results of observations/outcomes of events
construction and interpretation of:
tables
bar graphs
pictograms
line graphs
pie charts
histograms
purposes and use, advantages and disadvantages of the differentforms of statistical representations
drawing simple inference from statistical diagrams
Exclude histograms with unequal intervals.
3.2 Data analysis Include:
interpretation and analysis of dot diagrams
purposes and use of averages: mean, mode and median
calculations of mean, mode and median for a set of ungrouped data percentiles, quartiles, range and interquartile range
interpretation and analysis of cumulative frequency diagrams
3.3 Probability Include:
probability as a measure of chance
probability of single events (including listing all the possible outcomesin a simple chance situation to calculate the probability)
Exclude probability of combined events: P(A and B), P(A orB)
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Topic/Sub-topics Content
4 INTEGRATIVECONTEXTS
4.1 Problems derivedfrom practicalreal-life situations
Include:
practical situations such as
profit and loss
simple interest and compound interest
household finance (earnings, expenditures, budgeting, etc.)
payment/subscription rates (hire-purchase, utilities bills, etc.)
money exchange
time schedules (including 24-hour clock) and time zone variation
designs (tiling patterns, models/structures, maps and plans,packagings, etc.)
everyday statistics (sport/game statistics, household and market
surveys, etc.) tasks involving:
use of data from tables and charts
interpretation and use of graphs in practical situations
drawing graphs from given data
creating geometrical patterns and designs
interpretation and use of quantitative information
Exclude use of the terms percentage profit and percentage loss
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MATHEMATICAL FORMULAE
Note:Below is the list of formulae for Paper 1. For Paper 2, only the section on Numbersand Algebra will be given.
Numbers and Algebra
Compound interest
Total amount = 1+100
nr
P
Quadratic equation ax2
+ bx + c = 0
x =2__ 4
2
b b ac
a
Geometry and Measurement
Curved surface area of a cone = rl
Surface area of a sphere 2= 4r
Volume of a cone 21=3
r h
Volume of a pyramid = 1 base area height3
Volume of a sphere = 343r
Area of triangleABC=2
1ab sinC
C
B
A
c
b
a
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MATHEMATICAL NOTATION
The list which follows summarises the notation used in the Syndicates Mathematicsexaminations. Although primarily directed towards A Level, the list also applies, whererelevant, to examinations at all other levels.
1. Set Notation
is an element of
is not an element of
{x1,x2, } the set with elementsx1,x2,
{x: } the set of allxsuch that
n(A) the number of elements in setA
the empty set
universal set
A the complement of the setA
the set of integers, {0, 1, 2, 3, }
+
the set of positive integers, {1, 2, 3, }
the set of rational numbers
+
the set of positive rational numbers, {x :x> 0}
+
0 the set of positive rational numbers and zero, {x :x[0}
the set of real numbers
+
the set of positive real numbers, {x :x> 0}
+
0the set of positive real numbers and zero, {x :x[0}
n
the real n tuples
`= the set of complex numbers
is a subset of
is a proper subset of
is not a subset of
is not a proper subset of union
intersection
[a, b] the closed interval {x: aYxYb}
[a, b) the interval {x: aYx< b}
(a, b] the interval {x: a
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2. Miscellaneous Symbols
= is equal to
is not equal to
is identical to or is congruent to
is approximately equal to
is proportional to
< is less than
Y; is less than or equal to; is not greater than
> is greater than
[; is greater than or equal to; is not less than
infinity
3. Operations
a + b aplusba b aminusba b, ab, a.b amultiplied byba b,
b
a, a/b adivided byb
a : b the ratio ofato b=
n
i
ia
1
a1 + a2 + ... + an
a the positive square root of the real numbera
a the modulus of the real numbera
n! nfactorial forn +
U {0}, (0! = 1)
r
n
the binomial coefficient )!(!
!
rnr
n
, forn, r + U {0}, 0 YrYn
!
1)(1)(
r
rn...nn +
, forn , r +U {0}
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4. Functions
f function f
f(x) the value of the function fatx
f:A B fis a function under which each element of setA has an image in setBf:xy the function fmaps the elementxto the elementy
f1 the inverse of the function f
g o f, gf the composite function offand g which is defined by(g o f)(x)orgf(x) = g(f(x))
limax
f(x) the limit off(x) asxtends to a
x ; x an increment ofx
x
y
d
d the derivative ofywith respect to x
n
n
x
y
d
d the nth derivative ofywith respect to x
f'(x), f(x), ,f(n)(x) the first, second, nth derivatives off(x) with respect tox
xyd indefinite integral ofywith respect tox
b
a
xyd the definite integral ofy with respect toxfor values ofxbetween a and b
x& , x&& , the first, second, derivatives ofxwith respect to time
5. Exponential and Logarithmic Functions
e base of natural logarithms
ex, expx exponential function ofxlog
ax logarithm to the base aofx
lnx natural logarithm ofx
lgx logarithm ofxto base 10
6. Circular Functions and Relations
sin, cos, tan,cosec, sec, cot
the circular functions
sin1, cos1, tan1cosec1, sec1, cot1
the inverse circular functions
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7. Complex Numbers
i square root of1
z a complex number, z =x +iy= r(cos + i sin ), r +
0
= rei, r +0
Rez the real part ofz, Re (x + iy) =x
Imz the imaginary part ofz, Im (x + iy) =yz the modulus ofz, yx i+ = (x2 +y2), rr =)sini+(cos
argz the argument ofz, arg(r(cos + i sin )) = , < Y
z* the complex conjugate ofz, (x + iy)* =x iy
8. Matrices
M a matrix M
M1 the inverse of the square matrix M
MT the transpose of the matrix M
detM the determinant of the square matrix M
9. Vectorsa the vectora
AB the vector represented in magnitude and direction by the directed line segmentAB
a unit vector in the direction of the vectora
i, j, k unit vectors in the directions of the cartesian coordinate axes
a the magnitude ofa
AB the magnitude of AB
a.b the scalar product ofaand baPb the vector product ofa and b
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10. Probability and Statistics
A,B, C,etc. events
A B union of eventsAandB
A B intersection of the eventsAandB
P(A) probability of the eventA
A' complement of the eventA, the event notAP(A |B) probability of the eventAgiven the eventB
X, Y,R,etc. random variables
x,y, r,etc. value of the random variablesX, Y,R, etc.
1x ,
2x , observations
1f ,
2f , frequencies with which the observations, x1,x2 occur
p(x) the value of the probability function P(X=x) of the discrete random variableX
1p ,
2p probabilities of the values
1x ,
2x , of the discrete random variableX
f(x), g(x) the value of the probability density function of the continuous random variableX
F(x), G(x) the value of the (cumulative) distribution function P(XYx) of the random variableX
E(X) expectation of the random variableX
E[g(X)] expectation ofg(X)
Var(X) variance of the random variableX
B(n,p) binominal distribution, parameters nandp
Po() Poisson distribution, mean
N(, 2) normal distribution, meanand variance 2
population mean
2 population variance
population standard deviation
x sample mean
s2
unbiased estimate of population variance from a sample,
( )221
1xx
n
s
=
probability density function of the standardised normal variable with distributionN (0, 1)
corresponding cumulative distribution function linear product-moment correlation coefficient for a populationr linear product-moment correlation coefficient for a sample