TECHNIQUE APPLICATION ADD MATH

8/3/2019 TECHNIQUE APPLICATION ADD MATH

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SECURE AND OUTSTANDING FOR TERTIARY

ADMISSION LOCALLY OR ABROAD

Erected by :

KHAIRIL ANUAR BIN MOHD RAZALI

Enhanced, refined and continued by :


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A SYLLABUS

God created the natural numbers; everything else is the work of man.LEOPOLD KRONECKER (18231891)

(in a reference to the development of mathematics)

CORE PACKAGE

Compulsory for all students and contains

5 components.

ELECTIVE PACKAGE

Students only have to choose 1

from 2 elective packages.

Geometry Component Trigonometry Component- Coordinate Geometry - Circular Measure

- Vector - Trigonometric Function

Algebraic Component Calculus Component

- Functions - Differentiation- Quadratic Equations - Integration- Quadratic Functions

- Simultaneous Equations Statistics Component

- Indices & Logarithms - Statistics

- Progressions - Probability Distributions- Linear Law - Probability

- Permutations & Combinations

Science & Technology

Application Package- Solutions of Triangle

- Motion Along A StraightLine

Social Science Application

Package- Index Numbers

- Linear Programming

1. In Elective Package, students who are keen to science and technology are encourage tochoose Science and Technology Application Package while students who are keen to

commerce, literature and economy are encourage to choose Social Science ApplicationPackage.

2. In Core Package, each teaching component contains topics that have connection with onebranch of mathematics. Topics in any component were arranged in a hierachy so that

any simple topic, was learned, before moving on to more complex topics.

3. Majority of topics listed above are continuation of Lower Secondary Mathematics, Form 4 orForm 5 Mathematics that have or still being taught. For instance, the basic of Linear

Programming topic is the drawing of Inequalities Region that was learnt from Graphs ofFunctions topic in Form 5 Mathematics.

4. Although there are a few new topics and concepts, candidates can still apprehend them witha high algebraic dan arithmetic skills and plenty of time going through the facts and technical

things in those topics.

5. Project Work is encourage to give opportunity for students to use knowledge and skill that

have been learned in a real and challenging situation. It gives benefit such as mind

stimulation, makes learning more meaningful, chance to apply mathematical skill and

upgrade communication skill.

Project Work Report must contain the following items:


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a. Topicb. Introductionc. Method / Strategy / Procedured. Findingse. Discussion / Method of solvingf. Conclusion / Pengitlakan

6. To make teaching and learning easier, two annual schemes are suggested; ComponentScheme and Topical Scheme.In Component Scheme all topics on Algebra are taught first before moving on to otherschemes. This scheme suggest contents from those that have been taught to the new ones.

Topical Scheme gives more room by introducing algebraic and geometrical topics before

introducing new branch such as calculus.

Example Of Teaching and Learning for Form 5 :

COMPONENT SCHEME TOPICAL SCHEME

Algebraic Component- Progressions- Linear Law

Progressions

Integration

Linear Law

Vector

Trigonometric Functions

Permutations and

Combinations

Probability

Probability Distributions

Motion Along A Straight Line

Or

Linear Programming

PROJECT WORK

Calculus Component

- Integration

Geometri Component

- Vector

Trigonometry Component

- Trigonometric Functions

Statistic Component- Permutations &

Combinations

- Probability- Probability Distributions

Science & Technology

Application Package

- Motion Along A Staight Line

Social Science

Application Package

- Linear Programming

PROJECT WORK PROJECT WORK


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B EXAMINATION FORMAT

Half the battle in mathematics is the invention of a good notation.PIERE SIMON LAPLACE (17491827)

(as paraphrased by E. T. Bell)

ITEM PAPER 1 (3472/1) PAPER 2 (3472/2)

Type Of

Instrument

Subjective Test(Short Question)

Subjective Test(Limited Response and structure)

Number Of

Question

25 questions (Answer All)

Part A [ 40 Marks ]6 questions (Answer all)

Part B [ 40 Marks ]5 questions (Choose 4)

Part C [ 20 Marks ]

4 questions (Choose 2)(2 questions from Science &

Technology Application Package ; 2

Questions from Social Science

Application Package)

Total Marks 80 marks 100 marks

Exam Duration 2 hours 2 hours 30 minutes

Contextual

Coverage

Covers all field of studies

From Form 4 to form 5.

Covers all field of studies from Form 4

to Form 5.

Constructual

Inclination

Knowledge : 20 %

Application Skill : 80 %

Application Skill : 60 %

Problem Solving Skill : 40 %

Difficulty Level

Easy : Moderate : Difficult

6 : 3 : 1

Easy : Moderate : Difficult

4 : 3 : 3

Overall

Easy : Moderate : Difficult = 5 : 3 : 2

Additional Tools1. Scientific Calculator

2. Mathematical Table Book

3. Geometrical Tools

1. Scientific Calculator

2. Mathematical Table Book

3. Geometrical Tools

1. Other than providing time as suggested above, candidates should also set aside the last 5 10minutes to recheck and arrange answers.

2. Short questions usually are on basic skill in a topic. It also doesnt involve other topics. Forexample, short question on Circular Measure only involves matters and facts about Circular

Measure only and doesnt involve other topics.


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3. The situation differs in long questions where usually skills from a few topics are gatheredtogether. A clear example is the long question from the Circular Measure topic whichincludes the Coordinate Geometry and Trigonometry topics.

4. Content of Paper 1 is short questions from Core Package of the syllabus. Have a good graspat least on the basic works of these topics.

5. To build self confidence in preparation for examination, there are a few strategies androutines that can be carried out. Among them are

5.1 Start with solving Paper 1s short questions because usually they are not outside the

topic asked, what the questions want are very clear and they do not need a lot of workingmethod. Then, go to long question in Paper 2.

5.2 Follow the topical flow as suggested in page 2.

5.1Sharpen your Algebraic and Lower Secondary Mathematics Skills.The key word to our strategy is begin from the easiest and move to the more difficult work.

C ANSWERS AND MARKS

The moving power of mathematical invention is not reasoning but imagination.AUGUSTUS DE MORGAN (18061871)

1. Do not cancel out non completed answers or answers you feel are not correct because correctideas and working method will be considered and given marks.

2. Long subjective questions usually are divided into a few parts such as part (a), (b) and (c),where answer in (a) will be used in part (b) and so on. Though mistake in part (a) will cause

mistake in part (b) and (c), marks will still be given to correct working methods. (If working

mark is 1 and answer is 0 in part (a), then chance of getting working marks in latter parts are

still there).

3. The important things to note are ANSWER IS NEAT AND TIDY,WORKING METHODSHOWN CLEARLY and FINAL ANSWER IS DENOTED.

4. Marks allocated for a question, predict level of difficulty and number of working method thathas to be written.

5. Answers, where possible, should be written in the simplest form.6. Give the precise answer base on what the question want.7. In sketching graphs, characteristics like shape of the graph, min/max points and x or y-

intercepts must be priortize. In drawing graphs, characteristics like uniform scale, a fewpoints correctly plotted, and smooth curve are the important things looked upon.

8. In general, types of marks given are INDEPENDENT marks,WORKING/METHODmarks and ANSWER marks. INDEPENDENT marks do not need working method,

candidates will loose ANSWER marks if WORKING marks are not obtained.

9. To demonstrate these important points, let us together study examples below. (Empty spacesare provided for student to jot down notes and other important matters).


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Example 1 (Paper 2)

A slice of cake has the surface OAB in the shape of a sector with radius 15 cm. Length of arc ABis 10 cm and the cake is 6 cm thick. Find

O(a)Angle of the sector in radian (1 mark)(b)Total surface area of the cake (4 marks) A

B

Example 2 (Paper 1)

Given f(x) = 4x(2x1)4. Find f (x). (2 marks)

Example 3 (Paper1)

Given the geometric progression 8, 24, 72, . . Find the smallest number of term that has tobe taken in order that its sum exceed 50,000. (4 marks)


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D LIST OF FORMULAE

A chess player may offer the sacrifice of a pawn or even a piece, but a mathematician the game.G. H. HARDY (18771947)

(commenting on the method of proof by contradiction)

1. List of formulae supplied in the examination is limited to certain formulas and facts. Thismeans there are still many formulas / facts / concepts that are not given.

2. Candidates are encourage to familiarise themselves working with this short and limited list offormulae and memorise other formulas / facts / concepts that are not listed.

3. Among the techniques that can be practised in order not to forget formula / fact / concepteasily are by :

3.1Doing as many as possible exercises repeatedly.3.2Doing all sub-topic exercises from your text book or other reference book.3.3From sub-topic exercises, move to sumative exercises that incorporates back these sub-

topics.

4. Following are a few problems that might be encountered on the list of formulae supplied :4.1List of formulae are long and plenty and might cause candidates to be in doubt which

formulae is the right one.

Example 4: Solve 23 sin AKos 2A = 0 for 90o A 270

o

Try to look at the suitableTrigonometric Identity (Basic

Identity @ Addition Identity @

Double Angle Identity.

4.2There are formulas that are not given or listed.Example 5 : Solve the equation log x 16log x 2 = 3.

Indices and logarithms law are

not supplied. So write down theselaws ini and then make a choice

which is suitable to be used

- Laws are not necessarily read

from left to right but can also

be done the other way round.)


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4.3Cannot use the precise formulae because the problem given cannot be interpretedcorrectly.

Example 6 : A square has a perimeter of 160 cm. The second square is form by joining

midpoints for the sides of the first square and so on as depicted in thediagram. Find :

(a) Perimeter of the eight square

(b) The sum of perimeter of 5 squares formed.

4.4Formulae is given,but candidates still cant use it properly.Example 7 : Find the median of the data below.

Age Number of resident

1 - 20

21 - 40

41 - 60

618081 - 100

50

79

47

1410

4.5Formulae / fact / concept that you thought are only used in Mathematics only (not inAdditional Mathematics) and didnt bother about them.

Example 8 : Area and volume of solids formulas (used in Differentiation topic).

Translation concepts (used in Coordinate Geometry topic ).

Tangen to circles law (used in Circular Measure topic)


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FOR PAPER 2 :

1. Choose long questions in Part B where the form of question and answering method does notchange a lot form year to year.

Example 9 : Long questions from Linear Law topic

Table shows values of x and y obtained from an experiment. It is known that y and x is related

by the equation y = axn

x 1.2 1.8 2.3 2.6 3

y 2.3 5.2 8.5 10.8 14.4

(a) Plot lg x against lg y (5 marks)(b) Find the value of a and n (4 marks)(c) Find the value of y when x = 2 (1 mark)An experiment involving adiabatic expansion is carried out. The pressure, P, for mercury and

volume, V, for air obtained are as follows:

V 100 125 150 175 200

P 58.6 42.4 32.8 27.0 22.3

Variables P and V is related by P = kVn

where k and n are constant.

(a) Change the equation that relates P and V to the linear form (1 mark)

(b) Draw the graph of lg P against lg V. (5 marks)

(c) From your graph, find the value of k and n. (4 marks)

2. Choose long questions in Part C where the form of question and answering method does not

change a lot from year to year.

Example 10: Long question from Linear Programming topic.

Kasmah wish to sew shirts and pants to be sold to the public. A pants needs 40 minutes preparation

and 1 hour of sewing time. A shirt needs 50 minutes preparation time and 40 minutes of sewing time.

Kasmah sews x pants and y shirts.

(a)

Given at least 10 hours are use for preparation and the maximum sewing time is 16 hours.Write 2 inequalities base on these informations.

(b) Given the total preparation time is less than or the same as the total sewing time. Show thaty 2x.

(c) Contruct and shade the region that satisfies the above inequalities.(d) The profit from the sale of a pants and shirt respectively is RM5 and RM8. Find from the graph

the maximum profit that Kasmah can obtain.

Housing developer Rejeki Halal wish to build type A and type B houses. To build type A houses needs

120 m2

of land and a cost of RM56,000. To build type B houses needs 300 m2

of land and a cost of

RM84,000. The developer wish to build x type A houses and y type B houses according to the

following constraints:I : Number of type A houses built must exceed number of type B houses.

II : Land area that can be used to build both type of houses is 2400 m2.

III : Maximum capital for building the houses is RM840,000.

(a) Write 3 inequalities that satisfy the above constraints.(b) With a scale of 1 cm to 1 unit on each axis, draw graphs of the three inequalities. Hence, shade

The region, R, that satisfies the above constraints.

(c) Base on your graph,(i) What is the maximum unit of type B house built if number of type A houses is 10.

(ii) By selling all the houses, the developer obtain profit of RM15,000 for each unit of type A

house and RM24,000 for each unit of type B house. How much are the maximum profit obtain by

the developer ?


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3. There are questions that incorporate topics. Among those that usually are combined together(2 IN 1) are Differentiation and Integration topic.

4. Also, choose long questions from solo topics (topic that does not involve other topic).

F GENERAL GUIDE ON PROBLEM SOLVING

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not

refer to reality.ALBERT EINSTEIN (18791955)

1. In our short seminar session, a detail guide for each question is not possible. Even though, abit of general guide are provided below.

2. Among existing methods are as follows:2.1Directly use formula/fact/concept/algorithm method

Example 11 : Given f(x) = 2x21 , find f (x)

x + 1

Example 12 : Find the straight line equation that passes through (2,1) and perpendicular

with the line 2x + y3 = 0

2.2 Forming equation (whether linear, quadratic or simultaneous) method.

(a)Forming equation from information given.Example 13: Given f : x ax + b and 3 - - 5

Find a and b.

-2 - - - 1

f

x ax + b


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Example 14: Distance of the point (k, 3) to (5, 7) is 5 units. Find the values of k.

Example 15: A point P(x, y) is moving such that its distance from the line y = - 2is equal its distance from the point (6, 6). Find the points loci equation.

(b)Forming equation after making comparison.Example 16 : Given the function f(x) = 3x + c and its inverse function given as

f-1(x) = mx + 4/3. Find the value of m and c.

2.3 Forming own equation method.Example 17: The sum of the first three terms of a geometric progression that has

a common ratio1/3 is 42. Calculate the sum of the third term until

the fifth term.

3 Other than the methods given above, skills listed below are also important :3.1Algebrasolving linear, simultaneous, quadratic equation and simplifying

expressions.

3.2Arithmeticadding, subtracting, multiplying, dividing fractions, decimals andnegative number.

3.3Reading Mathematical Tables.3.4Using Scientific calculator.Without these skills, plenty of questions are answered only half way..


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G TOPICAL STUDY (ANALYSIS)

All the measurements in the world are not the equivalent of a single theorem that produces a significant advance in

our greatest sciences.KARL FRIEDRICH GAUSS (17771855)

(generally ranked with Archimedes and Newton at the pinnacle of mathematical achievement)

1. It is very important for candidates to know inter topics relations in the syllabus. From thisinter topics relations, at least we can see

1.1Which topic that is always used in other topics.1.2Which topic that usually stands alone (does not involve other topics)1.3Important topics whereby without them, many other topics cannot be completed with ease

and flow.

Following are the relations between topics in Additional Mathematics

LINEAR PROGRAMMING

COORDINATE GEOMETRY

LINEAR LAW

QUADRATIC EQUATION

PROGRESSIONS

QUADRATIC FUNCTION

DIFFERENTIATION

FUNCTIONS

INTEGRATION

CIRCULAR MEASURE

MOTION ALONG A STRAIGHT LINE

INDICES & LOGARITHMS

SOLUTIONS OF TRIANGLE

STATISTICSPERMUTATIONS & COMBINATIONS

INDEX NUMBERS

VECTORSIMULTANEOUS EQN.

PROBABILITY

TRIGONOMETRIC FUNCTION PROBABILITY DISTRIBUTIONS

Question : What is the importance of (i) Trigonometry (ii) Graphs of Function

(iii) Statistics (iv) Gradient & Area Under Graphs (v) Volume & Area of

Solids topics (from Mathematics subject) to our Additional Mathematics ?


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2. After spending a lot of time giving attention and solving questions, try to identify what arethe important formulas, concepts and facts needed for each topic. Following are among the

important points that can be listed down for each topic.

CIRCULAR MEASURE

Short question : - Converting radians to degrees and vice versa.

- Using the formulae s = j and L = j2- Can apply the Tangen to a circle and Trigonometric Law

Long question : - Other than the above skills, long questions also involves Coordinate

Geometry, Solutions of Triangles and Trigonometry topics.

- Involves the Area of a segment = j2 - j

2sin formulae.

COORDINATE GEOMETRY

Short question : - Sketch the information given.- If sketches are given, add the information given in the diagram.

- Involves the Distance, Midpoints, Area, Gradient, Equation of aStraight Line, Paralllel and Perpendicular formulas..

- Quadratic Equations are involve too.

Long question : - Other than the above items, long questions usually involves

Dividor with the ratio of m:n and Locus Equation.

PROGRESSIONS

Short question : - Type of progression is usually informed and this makes it easier for usto choose formulas.

- Sometimes we have to use own formula first.

Long question : - Type of progression is not informed. We have to carry out a test

Whether it is an arithmetic ar geometric progressions first

- Algebra involves sometimes is quite difficult.

- Logarithms sometimes are needed for problems involving GeometricProgressions.

FUNCTIONS

Short question : - Using the concept if f(x) = 2x + 1 then f(u) = 2u + 1 @

f(t1) = 2(t1) + 1 @ f(#*) = 2(#*) + 1 and so on.- Finding the inverse and composite function.- Sketches of graphs of Absolute Value Function (this does not restricted

to linear graphs only but also quadratic and trigonometric graphs).

- Forming Equation by Comparison.

Long question : - Other the above items, question of finding f if given fg and g, or

finding g if given fg and f, usually is in this part.


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QUADRATIC EQUATIONS

Short and : - Finding roots

Long question - Using Condition of type of roots

- Forming equation from roots .- Relations with Quadratic Function

QUADRATIC FUNCTIONS

Short and : - Sketching graph with Finding x-intercept method.

Long question - Sketching graph with Completing the square method.- Relation with Quadratic Equation topic.

- Solving Quadratic Inequalities.

INDICES & LOGARITHMS

Short and : - Write down indices and logarithmic laws.

Long question -Not necessarily laws are read from left to right but can also be the

other way around.

Example 18: Solve 5logx3 + 2logx2logx324 = 4

- Dont create ownformulae

Example 19: Solve log3x + log93x = -1

- Sometimes we can be given linear, quadratic orsimultaneous equation in indeks and logarithm form.

Example 20 : Solve 22x2

x2 = 0

- Once in a while readings oflog table or calculator are needed.

Example 21 : Evaluate log 4 5

LINEAR LAW

Short question : - Linearise non-linear function (i.e change to the form of Y = mX + c)- Finding value of constant by comparing constant of non-linear function

with constant of linear function obtained from graph.


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Long question : - Other than the above skills, long question involves drawing of Best Fit

Line using transparent long ruler.

- This is quite easy to be done because we only plots graph by following

the instruction but be careful when shifting points to the graph paper.- A little algebra and knowledge on logarithm usually needed in

linearising function.

- Make sure you can find gradient and read y-intercept of the graph drawn

- Scale used is very important. Draw as big as possible but not until

y-intercept cant be read.

STATISTICS

Short and : - Which formulae wish to be used depends on data type (either Grouped

Long question Data or Ungrouped Data)

-You must be able to identify which is data and which is frequency.

Example 22 : Find mean of the number of student for the data below.

Number of classes Number of Students

54

3

3035

40

- Experience in drawing ogives and histograms from Mathematics subject

are needed.

SIMULTANEOUS EQUATIONS

Short and : - Make x or y subject of a formula from the linear equation and subtitute

Long question in the non-linear equation.

- Quadratic equation will be obtained and solve it using formulae orfactorisation.

- Dont forget to find the other variable value.

- Long question usually is in implicit form concealed in questions fromother topics.

Example 23 : Find the distance between two points of intersection of

the graph x + y = 10 with the graph x2y + y

2+ 10 = 0

DIFFERENTIATIONS

Short question : - Can differentiate functions using the various techniques.- Other terms for differentiationGradient Function, Derivative of x

with respect to y, Gradient of Curve and Tangent Gradient

- Differentiation usage in finding Tangent/Normal equations,Rate of Change, Small Changes/Approximation and Min/Max Problems.

- A rough visualisation of diagram is very useful.

Example 24 :Find equation of the tangent to the curve y = x32 at (-2, -10).


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Long question : - Application of Differentiation (tangent/normal equation, min/max valueRate of change, small changes and approximations) usually were

incorporated together.

INTEGRATIONS

Short question : - Can integrate functions using techniques available.

- Integration application in finding curve equation and area under graphs

are needed.

Long question : - Integration appllication usually were incorporated together.- Volume of revolution is involve in this section..

- Because graphs are involved, knowledge from Coordinate Geometry,

Function, Quadratic Function, Simultaneous Equation are very

useful.

MOTION ALONG A STRAIGHT LINE

Short and : - Sketch the problems visualisation if possible.

Long question - Can derive the function s, v and a by diffrentiating or integrating.- Dont forget + c when doing indefinite integrals .

- Interpretation certain terms as listed below is very important whensolving problems..

Displacement - position from O (point of reference)

Maximum displacement/Stationary/Momentarily at rest - ds/dt = 0- v = 0

Maximum Velocity/Uniform Velocity/Fix Velocity - dv/dt = 0

- a = 0

Beginning from a fix point O - when t = 0, s = 0.

Change direction - ds/dt = 0- v = 0

To the left of point O - s < 0

Return to the point O - s = 0

LINEAR PROGRAMMING

Short question : - Defining inequalities region.

- Finding solution certain function from region.

- Changing constraints to inequalities and vice versa.

Long question : - Changing constraints to inequalities.- Constraints are usually given, but there are also some that are implicitly

stated.

- Drawing inequalities region (use Mathematics experience).

- Draw as big as possible but not too big until part of the region cannot be

shown.- Find solution (min cost @ max profit problem) by subtituting corner

points in the function that is to maximise or minimise.


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TRIGONOMETRIC FUNCTIONS

Short and : - Identify suitable tools to be used .

Long question TRIGONOMETRIC RATIOS DEFINITION

TRIGONOMETRIC RATIOS IN QUADRANTS FORMULASCOTANGENT, COSECANT, SECANT

TRIGONOMETRIC IDENTITIESBasic, Double Angle, Addition

TABLE BOOK

Example 25: Given tan x = 1/3. Without using table, find kos (x 45o)

- Usually question are on solving equations and graph sketching or

drawing where knowledge from Functions and Graphs of Functions

topics are gravely required.

SOLUTIONS OF TRIANGLES

Short and : - The precise use of formulae that is the sine rule, cosine rule or area

Long question of a triangle formula.

- Knowledge ofLines and Planes in 3D topic sometimes are required.

PROBABILITY

Short question : - Usually involves Simple Probability problems.

Long question : - Usually involves Permutations & Combinations problems.

VECTOR

Short question : - Usually, you have to shift problem to the Cartesan Plane for a clearer

picture.

- Hence, knowledge from Coordinate Geometry can be applied.- Vector Unit Formula is sometime needed.

Long question : - Usually find other alternative route that is suitable to arrive

at destination and state this route in defined vector notation.

- Addition, subtraction and multiplication of vector with scalar are

required.

PROBABILITY DISTRIBUTION

Short and : - Finding probability of Binomial Distribution

Long question - Finding n, r, p q of Binomial Distribution

- Mean, Variance and Standard Deviation of Binomial Distribution.- Normal Distribution Table- Finding probability of Normal Distribution- Given probability, find Normal Distribution Score


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H COMMON MISTAKES

1. Non-suitable final answer is not deleted.Example 26: Solve the equation 2log 5 (x1) = 1 + log 5(1x)

2. Answer given is not complete.Example 27 : Solve the equation sin (2x + 32

o) = -sin 72o for 0o x 360o

3. Using formulas that are not on any Mathematical ListExample 28 : If log 9 y = 2 + log 3 x , express y in terms of x.

Example 29 : Given sin x = m , express kos (90o + x) in terms of m.

4. Working method shown is correct, but still no marks awarded because the work is halfway completed.

Example 30 : Solve sin (2x + 32o) = - sin 72

ofor 0

o x 360

o

5. Mistakes on graphs - not drawn big enough- drawn too big till y-intercept cant be shown

- best fit line not drawn using long transparent ruler.

6. Answer not written explicitly.


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7. Linear, quadratic and simultaneous equation that involves log, indices and

trigonometrycant be solve properly.

8. Adding information not given and thus changing the original question.

Example 31 : The weather is defined as cloudy, rainy, sunny and windy. Find theprobability that it will rain in two consecutive days.

9. Solving quadratic inequalities like quadratic equation. Quadratic inequality by

right should be solve by graphical or number line method.

10. No precision in answer even though the question is quite easy.

Example 32:

Given diameter = 36 cm.

PQ = 3OP and PQ is tangent to the cicle.

Oo R Find(a) the angle made by the arc PR at

centre of the circle.

(b) area of shaded region.

P Q

I CONCLUSION

[The Universe] cannot be read until we have learnt the language and become familiar with the characters in which

it is written. It is written in mathematical language ..GALILEO GALILEI (15641642)

1. From our long discussion, it is observed that Additional Mathematics is a subject that

requires time to grasp and be fluent on.

2. With strong and sound basic skills (Algebra, Arithmetic, Shape, Number) and a lot of

exercises, only then the important and main content of Additional Mathematics can be

grasped.

3. A sound grasp of Mathematics SPM is required to make a notable advance in

Additional Mathematics..

4. Begin with the easy subtopic exercises and move to topical summative exercisesof short questions (Paper 1). After a lot of questions are studied, move to long questions(Paper 2) incorporate topics.

5. Hopefully candidates use their time as good as possible and with patience, work hard for

their SPM.

ALL THE VERY BEST

SELAMAT MAJU JAYA

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TECHNIQUE APPLICATION ADD MATH