0.5 Course of study.pdf

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PHILOSOPHY OF NATIONAL EDUCATION IN MALAYSIA

MISI dan VISI KEMENTERIAN PENDIDIKAN MALAYSIA( KPM)

VISI Pendidikan Berkualiti Insan Terdidik Negara Sejahtera

MISI Melestarikan Sistem Pendidikan yang berkualiti Untuk Membangunakan Potensi Individu Bagi Memenuhi Aspirasi Negara

MISI dan VISI BAHAGIAN PENGURUSAN SEKOLAH BERASRAMA PENUH DAN SEKOLAH KECEMERLANGAN(BPSBPSK)

VISI

Peneraju Pendidikan Bertaraf Dunia

MISI

Menggerak dan Memastikan Perkhidmatan Profesionalisme dan Pengurusan Pendidikan Bertaraf Dunia

Bagi Memenuhi Aspirasi Negara

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INTRODUCTION

The aspirations of the nation will be achieved with the emergence of a well informed and knowledgeable society well versed in the

use of Mathematics in daily life. To achieve this goal, efforts need to be taken to ensure that the society assimilates Mathematics into

their daily lives. Hence, the ability to solve problems and communicate mathematically must be nurtured from an early age so that

pupils acquire the skills to make decisions effectively.

Mathematics is the driving force behind the various developments in the fields of science and technology. The mastery of

Mathematics is essential towards preparing a workforce capable of meeting the demands of a progressive nation. In line with the

nations objective of becoming a knowledge-based economy, the skills of Research & Development in the field of mathematics must be nurtured and developed while still in school.

Additional Mathematics is an elective subject in the secondary schools. This subject focuses on the needs of students who are inclined

towards Science and Technology as well as Social Science. Thus, the content of the curriculum has been organised to achieve this

objective.

The Additional Mathematics syllabus has been designed bearing in mind the contents of the Mathematics subject. New branches of

mathematics are also introduced in this curriculum, coinciding with the new development in the teaching of Mathematics. Besides

that, the emphasis is placed on the heuristics of problem solving in the process of teaching and learning. Through this emphasis,

students can gain ability and confidence in using Mathematics under new circumstances.

The learning of a topic emphasises understanding of concepts and mastery of related skills. Problem solving is the main focus in the

teaching and learning process. Besides that, the skills of communication through mathematics are also stressed in the process of

learning Mathematics. When students explain concepts and their work, they should be guided to use the correct and precise

mathematical terms and sentences. Emphasis on Mathematics communications will develop students ability in interpreting certain matters into mathematics models or vice versa.

The use of technology especially Information and Communication Technology (ICT) is encouraged in the teaching and learning

process. It will benefit students by increasing their understanding of certain concepts, providing visual pictures and making complex

calculation easier. Calculators should be used as a tool in the teaching and learning process for related topics. The usage of software

that can help to visualize mathematical concepts are needed to help students to understand abstract mathematical concepts more

effectively. Besides that, the usage of software can help students to model problems, which they can explore more effectively.

Evaluation is part and parcel of the teaching and learning and is on-going to determine the strengths and weaknesses of students.

Continuous evaluations need to be conducted to provide feedback to students on their progress and to enable schools to design

internal programs to assist students. Evaluation in Additional Mathematics needs to include aspects such as concept understanding,

mastery in skills and non routine questions which demand the application of problem solving strategies.

Project work is encouraged in Additional Mathematics to provide opportunities for students to apply the knowledge and skills learnt

in the classroom into real-life and challenging situations. Project work carried out by students includes the exploration of

mathematical problems. With the introduction of this project, students will gain a lot of benefits. For instance, it activates their minds,

makes the learning of Mathematics more meaningful, and enables students to apply Mathematics concept and skills learnt and to

further develops their communication skills. Project work is limited to cases using mathematical knowledge from the compulsory and

elective package taken by the students.

Besides playing a role in developing students mathematical abilities, cultivating intrinsic and moral values of Malaysian community is also needed in the teaching of this curriculum.

AIM

The Additional Mathematics curriculum is designed to increase the knowledge, ability and interest of students in Mathematics. It is

hoped that students are able to use Mathematics responsibly and effectively to communicate and to solve problems, so as to provide

them with adequate preparation for future studies and be productive in their future careers.

OBJECTIVES

The Additional Mathematics curriculum enables students to:

1. widen their ability in the field of numbers, shapes and relationships as well as to gain knowledge in calculus, vector and linear

programming;

2. enhance problem-solving skills;

3. develop the ability to think critically, creatively and to reason out logically;

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4. make inference and reasonable generalization from given information;

5. relate the learning of Mathematics to daily activities and careers;

6. use the knowledge and skills of Mathematics to interpret and solve real-life problems;

7. debate solutions in accurate language of Mathematics;

8. relate Mathematical ideas to the needs and activities of human beings;

9. use hardware and software to explore Mathematics;

10. practice intrinsic mathematical values.

CURRICULUM ORGANISATION

The contents of the Additional Mathematics curriculum are arranged into two learning packages. They are the Core Package and the

Elective Package.

The Core Package is compulsory for all students and consists of five components, that is:

Geometry;

Algebra;

Calculus;

Trigonometry; and

Statistics.

The Elective Package consists of two packages, that is:

Science and Technology Application; and

Social Science Application.

Students need to choose only one Elective Package according to their inclination in their future field. Those who are inclined towards

science and technology are encouraged to choose Science and Technology Application Package while those who are inclined towards

commerce, literature and economy are encouraged to choose Social Science Application Package.

In the Core Package, each teaching component consists of topics related to one branch of mathematic. Topics in a teaching component

are organized hierarchically so that easier topics are learnt first before going on to more complex topics.

CONTENT

This part lists the topics contained in each learning package.

Core Package

The Core Package consists of five learning components.

Geometric Component

G1. Coordinate Geometry

1. Distance between two points. 2. Division of line segment. 3. Area of polygon. 4. Equation of straight line. 5. Parallel and perpendicular lines. 6. Equation of locus involving distance between two points.

G2. Vector

1. Introduction to vector and its properties. 2. Addition and subtraction of vectors. 3. Expressing a vector as a combination of other linear vectors. 4. Vectors in the Cartesian plane.

Algebraic Component

A1. Functions

1. Relation. 2. Functions. 3. Composite functions. 4. Inverse functions.

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A2. Quadratic Equations

1. Quadratic equation and its roots. 2. Solving quadratic equations. 3. Conditions for quadratic equations to have

Two different roots

Two equal roots

No roots.

A3. Quadratic Functions

1. Quadratic function and its graph. 2. Maximum and minimum values of quadratic functions. 3. Sketch graphs of quadratic functions. 4. Quadratic inequalities.

A4. Simultaneous Equations

1. Simultaneous equations in two unknowns; one linear equation and one non-linear equation.

A5. Indices and Logarithms

1. Indices and laws of indices. 2. Logarithms and laws of logarithms. 3. Change the base of logarithms. 4. Equations involving indices and logarithms.

A6. Progressions

1. Arithmetic progressions. 2. Geometric progressions.

A7. Linear Law

1. Line of best fit. 2. Application to non-linear functions.

Calculus Component

C1. Differentiation

1. Gradients of curves and differentiation. 2. Differentiation of ax

n; (n is an integer), differentiation of the sum of algebraic functions; tangents and normals to curves.

3. Differentiation of the products and quotients of algebraic functions; differentiation of composite functions. 4. Application to minimum and maximum values, rates of change, small changes and approximations. 5. Second derivative.

C2. Integration 1. Integration as an inverse of differentiation.

2. Integration of axn (n is an integer, but n 1). 3. Integration by substitution. 4. Definite integrals. 5. Integration as a sum; area and volume.

Trigonometric Component

T1. Circular Measure

1. Radian. 2. Length of arc of a circle. 3. Area of sectors.

T2. Trigonometric Functions

1. Positive and negative angles in degrees and radians. 2. Six trigonometric functions of any angle.

3. Graphs of sine, cosine and tangent functions.

4. Basic Identities:

122 AA cossin , AA 22 tan1sec , .cotcos AAec 22 1

5. Addition formulae and double angle formulae:

),sin( BA ),cos( BA )tan( BA , A2sin , A2cos , A2tan

Statistic Component

S1. Statistics

1. Measures of central tendency: mean, mode and median.

2. Measures of dispersion: range, interquartile range, variance and

standard deviation.

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S2. Permutations and Combinations

1. Permutations. 2. Combinations.

S3. Probability

1. Probability of an event. 2. Probability of mutually exclusive events. 3. Probability of independent events.

S4. Probability Distributions

1. Discrete probability distribution and binomial distribution. 2. Continuous probability distribution and normal distribution,

Elective Package

The Elective Package consists of two application packages. Students choose only one application package.

Application Package for Science and Technology.

AST1. Solutions of triangles.

1. Sine rule. 2. Cosine rule. 3. Area of triangles.

AST2. Motion along a straight line.

1. Displacement. 2. Velocity. 3. Acceleration.

Application Package for Social Science.

ASS1. Index Number

1. Index number 2. Composite index.

ASS2. Linear Programming

1. Graphs of linear inequalities. 2. Solving linear programming problems.

FIRST TERM (20 weeks)

TOPIC LEARNING OUTCOMES

OBTAIN

INFORMATION

PROCESSING INFORMATION PRESENTING INFORMATION

Learning Area:

Chapter 1

Progressions

1. Understand and

use the concept of

arithmetic

progressions

The examples from

real life situations,

scientific or

graphing calculators

and computer

software to explore

arithmetic

progressions.

1.1 Identify characteristic of arithmetic

progression.

1.2 Determine whether a given sequence is

an arithmetic progression.

1.3 Determine by using formula:

a) specific terms in arithmetic progression;

b) the number of term in arithmetic

progression.

1.4 Find:

a) the sum of the first terms of arithmetic

progressions.

b) The sum of a specific number of

consecutive terms of arithmetic

progressions.

c) the value of , given the sum of the first

terms of arithmetic progression.

1.5 Solve problem involving arithmetic

progression.

Begin with sequences to introduce

arithmetic and geometric

progressions.

Include examples in algebraic form.

Include the use of the formula

1 nnn SST

Include problems involving real-life

situations.

2. Understand the

use of concept of

geometric

progressions.

The examples from

real life situations,

scientific or


and computer

software to explore

geometric

progressions.

2.1 Identify characteristics of geometric

progressions.

2.2 Determine whether a given sequence is

a geometric progressions.

2.3 Determine by using formula:

a) specific terms in a geometric progression,

b) the number of terms in geometric

progression.

Include examples in algebraic form.

Discuss:

As , , then , read as sum to

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INFORMATION


2.4 Find:

a) the sum of the first terms of geometric

progression;

b) the sum of a specific number of

consecutive terms of geometric

progressions;

c) the value of given the sum of the first

term of geometric progressions.

2.5 Find:

a) the sum to infinity of geometric

progression.

b) the first term or common ratio, given the

sum to infinity of geometric progression.

2.6 Solve problems involving geometric

progressions.

infinity.

Include recurring decimals. Limit to

2 recurring digits such as 0.3, 0.15,

Exclude:

a) combinations of arithmetic

progressions and geometric

progressions.

b) cumulative sequences such

as, (1), (2,3), (4,5,6), (7,8,9,10),

Learning Area:

Chapter 6

Permutation &

Combination

1 Understand and

use the concept of

permutation.

Use pictures and role play to

introduce the

concept of relations.

Use manipulative materials to explore

multiplication rule.

Use real-life situations and

computer software

such as spreadsheet

to explore

permutations.

Introduce notation.

1.1 Determine the total number of ways

to perform successive events using

multiplication rule.

1.2 Determine the number of permutations

of n different objects.


of n different objects taken r at a time.

For this topic :

a) Introduce the concept by

using numerical examples.

b) Calculators should only be used

after students have

understood the concept.

Limit to 3 events.

Exclude cases involving

identical objects.

Explain the concept of

permutations by listing all

possible arrangements.

Include notations :

a) n! =

n (n 1) (n 2) (3).(2).(1) b) 0! = 1

n! read as n factorial.

Exclude cases involving

arrangement of objects in a circle.

Learning Area:

Chapter 3

Integration

1. Understand and

use the concept of

Slide presentation.

Works in groups to determine the

number of

permutations of n

different objects

taken r at a time for

given conditions.

Slide presentation. Works in groups to determine the

number of

combinations of r

objects chosen

from n different

objects for given

conditions.

Use computer

software such as

Geometers Sketchpad to explore

the concept of


of n different objects for given

conditions.


of n different objects taken r at a time for

given conditions.

2.1 Determine the number of

combinations of r objects chosen

from n different objects.


combinations of r objects chosen

from n different objects for given

conditions.

1.1 Determine integrals by reversing

differentiation.

1.2 Determine integrals of axn , where a is

a constant and n is an integer, n -1.

Explain the concept of

combinations by listing all

possible selections.

Use examples to illustrate

!r

Pc r

n

r

n

Emphasise constant of intergration.

ydx read as intergration of y with respect to x

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OBTAIN

INFORMATION


identified integral intergration.

1.3 Determine integrals of algebraic

expression.

1.4 Find constants of integration, c, in

indefinite integrals.

1.5 Determine equations of curves from

functions of gradients.

1.6 Determine by substitution the integrals

of expression of the form ( ax + b)n ,

where a and b are constants, n is an

integer and

n -1

Limit intergration of

where ,dxun

u = ax + b

2. Understand and

use the concepts

of definite

integral

Use scientific or graphing calculators

to explore the

concept of definite

integrals.

Use computer software and

graphing calculator

to explore areas

under curves and the

significance of

positive and

negative values of

areas

Use dynamic computer software

to explore volumes

of revolutions.

2.1 Find definite integrals of algebraic

expressions.

2.2 Find areas under curves as the limit of a

sum of areas.

2.3 Determine areas under curves using

formula.

2.4 Find volumes of revolutions when

region bounded by a curve is rotated

completely about the

a) x-axis

b) y-axis

as the limit of a sum of volumes.

2.5 Determine volumes of revolutions using

formula.

Include

b

a

b

adxxfkdxxkf )()(

b

a

b

adxxfdxxf )()(

Derivation of formulae not required

Limit to one curve

Derivation of formulae not required.

Limit volumes of revolution about the

x-axis or y-axis.

Learning Area :

Chapter 4 Vectors

1. Understand and

use the concept of

vector.

Use examples from

real-life situations

and dynamic

computer software

such as Geometers Sketchpad to

explore vectors

1.1 Differentiate between vector and scalar

quantities.

1.2 Draw and label directed line segments

to represent vectors

1.3 Determine the magnitude and direction

of vectors represented by directed line

segments

1.4 Determine whether two vectors are

equal

1.5 Multiply vectors by scalars

1.6 Determine whether two vectors are

parallel

Use notation: Vector : a, AB, a, AB

Magnitude :

a, AB, a, AB

Zero vector : 0

Emphasise that a zero vector has a

magnitude of zero

Emphasise negative vector :

-AB = BA

Include negative scalar

Include :

a) collinear points

b) non-parallel non-zero vectors

Emphasise :

If a and b are not parallel and ha = kb,

then h = k = 0

2. Understand and

use the concept of

addition and

subtraction of

vectors.

Use real-life

situations and

manipulative

materials to explore

addition and

subtraction of

vectors

2.1 Determine the resultant vector of two

parallel vectors

2.2 Determine the resultant vector of two

non- parallel vectors using :

a) triangle law

b) parallelogram law

2.3 Determine the resultant vector of three

or more vectors using the polygon law

2.4 Subtract two vectors which are :

a) parallel

Emphasise :

a b = a + (-b)

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OBTAIN

INFORMATION


b) non-parallel

2.5 Represent vectors as a

combination of other vectors

2.6 Solve problems involving addition and

subtraction of vectors

3. Understand and

use vectors in the

Cartesian plane.

Use computer

software to explore

vectors in the

Cartesian plane

3.1 Express vectors in the form : a) xi + yj b) column vector

3.2 Determine magnitudes of vectors 3.3 Determine unit vectors in given

directions

3.4 Add two or more vectors 3.5 Subtract two vectors 3.6 Multiply vectors by scalars 3.7 Perform combined operations on

vectors

3.8 Solve problems involving vectors

Relate unit vector i and j to Cartesian

coordinates.

Emphasise :

Vector i = (10) and vector j = (

01)

For learning outcomes 3.2 to 3. 7, all

vectors are given in the form xi + yj

or ()

Limit combined operations to

addition, subtraction and

multiplication of vectors by scalars

Learning Area:

Chapter 5

Trigonometry

1. Understand the

concept of

positive and

negative angles

measured in

degrees and

radians.

2. Understand and

use the six

trigonometric

functions of any

angle.

Use dynamic computer software

such as GSP to

explore angles in

Cartesian Plane

Use dynamic software to explore

trigonometric

functions in degrees

and radians.

Use scientific or graphic calculators

to explore

trigonometric

functions of any

angle

1.1 Represent in a Cartesian plane, angles

greater than 360 or radians for positive and

negative angles.

2.1 Define sine, cosine and tangent of any

angle in a Cartesian Plane.

2.2 Define cotangent, secant and cosecant

of any angle in a Cartesian Plane.

2.3 Find values of the six trigonometric

functions of any angle.

2.4 Solve trigonometric equations.

Use unit circle to determine the sign

of trigonometric ratios.

Emphasise:

)cos(sin 90

)sin(cos 90

)cot(tan 90

)sec(cos 90ec

)(cossec 90ec

)tan(cot 90

Emphasise the use of triangle to find

trigonometric ratios for special angles

30 , 45 and 60

3. Understand and

use graphs of

sine, cosine and

tangent functions

Use examples from real life situations to

introduce graphs of

trigonometric

functions and

emphasise the

characteristics of

sine, cosine and

tangent graphs by

discussion.

Use graphing calculators

and dynamic

computer software

such as Geometry

Sketchpad to explore

graphs of

trigonometric

functions.

Group work activities for solving

trigonometric

equations.

3.1 Draw and sketch graphs of

trigonometric functions:

a. y = c + a sin bx

b. y = c + a cos bx

c. y = c + a tan bx

( a,b,c are constants, b> 0)

3.2 Determine the number of solutions to a

trigonometric equation using sketched

graphs.

3.3 Solve trigonometric equations using

drawn graphs.

4.1 Prove basic identities:

a) sin2A + cos2A = 1

b) 1 + tan2A = sec2 A

c) 1 + cot2 A = cosec2A

Angles in degree or radians in terms

Include trignometric function

involving modulus.

Exclude combinations of

trigonometric functions.

Derivation of addition formulae not

required.

Discuss half-angle formulae

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OBTAIN

INFORMATION


Use scientific or graphing calculators

and dynamic

computer software

such as GSP to

explore basic

identities.

5.1 Prove trigonometric identities using

addition formulae for sin ( A B),

cos ( A B) and tan ( A B).

5.2 Derive double-angle formulae for

sin 2A, cos 2A and tan 2A.

5.3 Prove trigonometric identities using

addition formulae and/ or double-angle

formulae.

5.4 solve trigonometric equation.

Exclude

A cos x + B sin x=c,

Where c 0.

Learning Area:

Chapter 7

Probability

1.Understand and

use the concept of

probability.

2.Understand and

use the concept of

probability of

mutually

exclusive events.

3. Understand and

use the concept of

probability of

independents

events.

Use real-life

situations to

introduce probability

Use manipulative

materials, computer

software , and

scientific or

graphing calculator

to explore the

concepts of

probability.

Use manipulative

materials and


to explore the

concept of

probability of

mutually exclusive

events.

Use computer

software to simulate

experiments

involving probability

of mutually

exclusive events.

Use manipulative

materials and


to explore the

concept of

probability of

independent events.

Use computer

software to simulate

experiments

involving probability

of independent

events.

1.1 Describe the sample space of an

experiment.


outcomes of event.

1.3 Determine the probability of an

event.

2.1 Determine whether two events are

mutually exclusive.

2.2 Determine the probability of two or

more events that are mutually exclusive.

3.1 Determine whether two events are

independent.

3.2 Determine the probability of two

independent events.

3.3 Determine the probability of three

independent events.

Use set notations.

Discuss:

a) classical probability (theoretical probability)

b) subjective probability c) relative frequency

probability (experimental

probability)

Emphasise:

Only classical probability is used to

solve problem.

Emphasise

)()()()( BAPBPAPBAP

using Venn diagrams.

Include events that are mutually

exclusive and exhaustive.

Limit to three mutually exclusive

events.

Include tree diagrams.

SECOND TERM (20 WEEKS)

Learning Area:

Chapter 8 :

Probability

Use real-life situations to

1.1 List all possible values of a discrete

random variable.

Include the characteristics of

Bernoulli trials.

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OBTAIN

INFORMATION


Distributions

1. Understand and

use the concept of

binomial

distribution.

introduce the

concept of binomial

distribution.

Use graphing calculators and

computer software

to explore binomial

distribution.

1.2 Determine the probability of an event in

a binomial distribution.

1.3 Plot binomial distribution graphs.

1.4 Determine mean , variance and standard

deviation of a binomial distribution.

1.5 Solve problems involving binomial

distribution.

For learning outcomes 1.2 and 1.4,

derivation of formulae not required.

2. Understand

and use the

concept of normal

distribution.

Use real-life situations and

computer software

such as statistical

packages to explore

the concept normal

distributions.

2.1 Describe continuous random

variables using set notations.

2.2 Find probability of z-values for

standard normal distribution.

2.3 Convert random variable of

normal distributions, X, to standardized

variable, Z.

2.4 Represent probability of an event

using set notation.

2.5 Determine probability of an event.

2.6 Solve problems involving normal

distributions.

Discuss characteristics of :

a) normal distribution graphs

b) standard normal distribution

graphs.

Z is called standardized variable.

Integration of normal distribution

function to determine probability is

not required.

Learning Area:

Chapter 9

Motion along a

straight line

1. Understand and

use the concept of

displacement.

Use examples from

real-life situations,

scientific or


and computer

software such as

Geometers Sketchpad to explore

displacement.

1.1 Identify direction of displacement

of a particle from a fixed point.

1.2 Determine displacement of a

particle from a fixed point.

1.3 Determine the total distance

traveled by a particle over a time interval

using graphical method.

Emphasize the use of following

symbols:

s = displacement

v = velocity

a = acceleration

t = time

Where s, v, and a are functions of

time.

Emphasize the difference between

displacement and distance.

Discuss positive, negative and zero

displacements.

Include the use of number line.

2. Understand and

use the concept of

velocity

3. Understand and

use the concept of

acceleration.

Use real-life

examples, graphing

calculators and

computer software

such as Geometers Sketchpad to explore

the concept of

velocity.

Use real-life

examples, graphing

calculators and

computer software

such as Geometers Sketchpad to explore

the concept of

acceleration.

2.1 Determine velocity function of a

particle by differentiation.

2.2 Determine instantaneous velocity of a

particle.

2.3 Determine displacement of a particle

from velocity function by integration.

3.1 Determine acceleration function of a

particle by differentiation.

3.2 Determine instantaneous acceleration of

a particle.

3.3 Determine instantaneous velocity of a

particle from acceleration function by

integration.

3.4 Determine displacement of a particle

from acceleration function by integration.

3.5 Solve problems involving motion along

a straight line.

Emphasize velocity as the rate of

change of displacement.

Include graphs of velocity functions.

Discuss:

a) uniform velocity

b) zero instantaneous velocity

c) positive velocity

d) negative velocity

Emphasize acceleration as the rate of

change of velocity.

Discuss:

a) uniform acceleration

b) zero acceleration

c) positive acceleration

negative acceleration

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Takwim Pembelajaran Matematik Tambahan

Tingkatan 5 Tahun 2016 MINGGU TARIKH Tajuk Catatan

1 Progression

2 Arithmatics Progeression

3 Geometrics Progression

4 SPM Question - Progression

5 Integration

6 Definite Integral

7 SPM Question - Integration

8 Test 1

9 Vector

10 Vector

Cuti Pertengahan Penggal 1

11 Trigonometric Function

12 Graph of trigonometric Function

13 Basic Identities

14 Double Angle and Half Angle

15 SPM Question Trigonometric Function

16 Permutations And Combinations

17 Probability

18 Probability Distributions

19 Perperiksaan SPM Fasa 1

20 Peperiksaan SPM Fasa 1

Cuti Pertengahan Tahun 2015

21 Binomial Distributions

22 Normal Distributions

23 Motions Along A straight Lines *

24 Motions Along A straight Lines*

25 Linear Programming*

26 Linear Programming*

27 Ulangkaji

28 Ulangkaji

29 Ulangkaji

30 Ulangkaji

31 Ulangkaji

32 Trial SBP

33 Trail SPM SBP

34 Trail SPM SBP

Cuti Pertengahan Penggal 2

35 Ulangkaji

36 Ulangkaji

37 Ulangkaji

38 Ulangkaji

39 Ulangkaji

40 Ulangkaji

41 SPM BERMULA

42 SPM BERMULA

Cuti Akhir Tahun

* Linear Programming ( Boleh dilaksanakan secara bengkel ( 8 jam) * Motions Along A straight Lines ( Boleh dilaksanakan secara bengkel 8 jam)

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