gce n(t)-level syllabuses examined in 2010_mathematics syllabus t

8/8/2019 GCE N(T)-Level Syllabuses Examined in 2010_Mathematics Syllabus T

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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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4043 MATHEMATICS N LEVEL (2010)

1

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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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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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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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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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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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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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

12


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

gce n(t)-level syllabuses examined in 2010_mathematics syllabus t

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