SPM2011

ADDITIONAL MATHEMATICS

BY : KHAIRIL ANUAR MOHD RAZALI

A. SYLLABUS

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 Statistics Component

- Simultaneous Equations - Statistics

- Indices & Logarithms - Probability Distributions

- Progressions - Probability

- Linear Law - Permutations & Combinations

Science & Technology

Application Package

- Solutions of Triangle

- Motion Along A Straight Line

Social Science Application

Package

- Index Numbers

- Linear Programming

SYLLABUS EXAMPLE OF TEACHING AND LEARNING IN FORM 5

ALGEBRAIC COMPONENT PROGRESSIONS

- Progressions

- Linear Law INTEGRATIONS

CALCULUS COMPONENT

- Integration LINEAR LAW

GEOMETRY COMPONENT

- Vektor VECTOR

TRIGONOMETRY COMPONENT

- Trigonometric Functions TRIGONOMETRIC FUNCTIONS

STATISTICS COMPONENT

- Permutation & Combination PERMUTATIONS & COMBINATION

- Probability

- Probability Distribution PROBABILITY

APPLICATION PACKAGE PROBABILITY DISTRIBUTION

PROJECT WORK

APPLICATION PACKAGE

PROJECT WORK

COMPONENT SCHEME TOPICAL SCHEME

B. EXAMINATION FORMAT

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

Test Duration 2 Hours 2 Hours 30 Minutes

Constructual Inclination Knowledge : 20 % Application Skill : 80 %

Application Skill : 60 %

Problem Solving Skill : 40 %

Contextual Coverage Covers all field of studies From Form 4 to form 5.

Covers all field of studies from Form 4 to

Form 5.

Level of Difficulty

Easy Moderate Difficult

E : M : D = 6 : 3 : 1 E : M : D = 4 : 3 : 3

Additional Tools 1. Scientific Calculator

2. Mathematical Table Book

3. Geometrical Tools

1. Scientific Calculator

2. Mathematical Table Book

3. Geometrical Tools

B. EXAMINATION FORMAT

1. Set aside 5 – 10 minutes to recheck and arrange answers.2. Short questions are on basic skill in a topic.3. Long questions sometimes incorporates a few topics together. 4. Content of Paper 1 is short questions from Core Package.5. To build self confidence, a few strategies and routines can be

practised : - Start with solving Paper 1’s short questions. Then, go to long question in Paper 2. - Follow the topical flow as suggested in page 2.- Sharpen your Algebraic and Lower Secondary Mathematics skills.

6. Begin from the easiest and move to the more difficult work.

C. ANSWERS AND MARKS

1. Don’t cancel answers you feel are not correct or unfinished.2. In long questions, though mistake in part (a) will cause mistake in

other parts, marks will still be given to correct working methods. 3. Answers should be NEAT AND TIDY, WORKING METHOD

SHOWN CLEARLY and FINAL ANSWER IS DENOTED.4. Marks allocated for a question, predict level of difficulty. 5. Answers, should be written in the simplest form.6. Give the precise answer base on what the question want.7. Sketching graphs - shape of graph, min/max points, x or y-

intercepts. 8. Drawing graphs - uniform scale, a few points correctly plotted,

smooth curve.9. Marks given are INDEPENDENT marks, WORKING/METHOD

marks and ANSWER marks.

C. ANSWERS AND MARKS

Example 1 (Paper 2)A slice of cake has the surface OAB in the shape of a sector withradius 15 cm. Length of arc AB is 10 cm and the cake is 6 cm thick. Find

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

Example 2 (Paper 1)Given f(x) = 4x(2x – 1) 4. Find f ’(x). (2 marks)

Example 3 (Paper1)Given the geometric progression 8, 24, 72, ……. . Find the smallest number ofterm that has to be taken in order that its sum exceed 50,000. (4

marks)

D. LIST OF FORMULAE

4.1 List of formulae are long and plenty and might cause candidates to

be in doubt which formulae is the right one.

Example 4: Solve 2 – 3 sin A – Kos 2A = 0 for 90o ≤ A ≤ 270o

Try to look at the suitable Trigonometric Identity - Basic Identity - Addition Identity- Double Angle Identity.

D. LIST OF FORMULAE

4.2 There are formulas that are not given or listed.

Example 5 : Solve the equation log x 16 – log x 2 = 3.

Indices and logarithms law are not supplied .So write down these lawsand 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.)

D. LIST OF FORMULAE

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

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 the diagram. Find :

(a) Perimeter of the eight square

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

D. LIST OF FORMULAE

4.4 Formulae is given,but candidates still can’t use

it properly.Example 7 : Find the median of the data below.

Age Number of resident

1 - 20

21 - 40

41 - 60

61 – 80

81 - 100

50

79

47

14

10

D. LIST OF FORMULAE

4.5 Formulae / fact / concept that you thought are only used in Mathematics only (not in Additional Mathematics) and didn’t 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)

E. OVERALL ANALYSIS – Paper 1TOPICS

2003 2004 2005 2006 2007 2008 2009 2010 2011

MARKS MARKS MARKS MARKS MARKS MARKS MARKS MARKS MARKS

2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

1. FUNCTION 1 1 1 1 1 1 2 2 2 1 2 1 1 1 1

2. QUADRATIC EQUATIONS 2 1 1 1 1 1 1

3. QUADRATIC FUNCTIONS 1 2 2 1 1 2 2 1 1

4. SIMULTANEOUS EQUATIONS

5. INDICES AND LOGARITHMS 2 1 1 2 1 3 1 1 2 2

6. COORDINATE GEOMETRY 2 2 1 1 2 1 1 1

7.STATISTCS 1 1 1 1 1

8. CIRCULAR MEASURE 1 1 1 1 1 1 1

9. DIFFERENTIATION 1 2 1 1 1 2 1 1 1 1 1

10. SOLUTIONS OF TRIANGLES

11. INDEX NUMBERS

12. PROGRESSIONS 1 1 2 1 1 1 2 1 1 1 2 2 1 1 1 1

13. LINEAR LAW 1 1 1 1 1 1

14. INTEGRATIONS 2 1 1 1 1 1 1 1 1 1

15. VECTORS 1 1 1 2 1 1 1 1 1 1 1 1 2

16. TRIGONOMETRIC FUNCTIONS 1 1 1 1 1 1 1 2

17. PERM. AND COMBINATIONS 1 1 1 1 1 1 1 1

18. PROBABILITY 1 1 1 1

19. PROBABILITY DISTRIBUTIONS 1 1 1 1 1 2 1 2

20. MOTION ALONG A STRAIGHT LINE

21. LINEAR PROGRAMMING

TOTAL QUESTIONS 3 14 8 4 12 9 4 12 9 5 1

19 4 1

29 4 12 9 4 12 9

E. OVERALL ANALYSIS – Paper 2TOPICS

2003 2004 2005 2006 2007 2008 2009 2010 2011

PART PART PART PART PART PART PART PART PART

A B C A B C A B C A B C A B C A B C A B C A B C A B C

1. FUNCTION 1

2. QUADRATIC EQUATIONS 1 ¾

3. QUADRATIC FUNCTIONS 1 1 ¼

4. SIMULTANEOUS EQUATIONS 1 1 1 ⅕ 1 1 1

5. INDICES AND LOGARITHMS

6. COORDINATE GEOMETRY 1 1 1 1 1 1 1

7.STATISTCS 1 1 1 1 1 1

8. CIRCULAR MEASURE 1 1 1 1 1 1 1

9. DIFFERENTIATION 1 ½ ½ ½ ½ ⅓ ⅝ ⅓ ⅓ ½

10. SOLUTION OF TRIANGLES 1 1 1 1 1 1 1

11 INDEX NUMBERS 1 1 1 1 1 1 1

12. PROGRESSIONS 1 1 1 1 1 1

13. LINEAR LAW 1 1 1 1 1 1 1

14. INTEGRATIONS ½ ½ ½ ½ ⅔ ⅘ ⅜ ⅔ ⅔ ½ 1

15. VECTORS 1 1 1 1 1 1 1

16. TRIGONOMETRIC FUNCTIONS 1 1 1 1 1 1 1

17. PERMUTATIONS AND COMBINATIONS

18. PROBABILITY

19. PROBABILITY DISTRIBUTIONS 1 1 1 1 1 1 1

20. MOTION ALONG A STRAIGHT LINE 1 1 1 1 1 1 1

21. LINEAR PROGRAMMING 1 1 1 1 1 1 1

TOTAL QUESTIONS 6 5 4 6 5 4 6 5 4 6 5 4 6 5 4 6 5 4 6 5 4

E. OVERALL ANALYSIS – Detail Analysis According To Subtopic

CHAPTER 9 : DIFFERENTIATION2003 2004 2005 2006 2007 2008 2009 2010

P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2

9.1 Idea of Limit

9.2 Differentiation By First Principle

9.3 Differentiation by Formulae ✓

4M 2M 3M 5M

9.4 Tangent & Normal Equations ✓

3M 3M 4M 3M

9.5 Minimum & Maximum Problems ✓ 3M 5M

9.6 Rate of Change ✓

✓ 3M 3M

9.7 Small Changes & Approximations ✓

4M 3M

9.8 Second derivatives 4M

9.9 Differential Equations ✓

E. OVERALL ANALYSIS – Detail Analysis According To Subtopic

CHAPTER 14 : INTEGRATION 2003 2004 2005 2006 2007 2008 2009 2010

P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2

14.1 Integration -Inverse of Differentiation √ √

14.2 Integration Formulae axn 3M 3M 3M

14.3 Integration by Substitution √

14.4 Definite Integrals √ 4M 4M 4M 3M

14.5 Area Under A Curve √ √ 4M 2M 5M 4M 4M

14.6 Volume of Revolution √ √ 3M 3M 3M 3M

FOR PAPER 2 :

1. Choose long questions in Part B that the form of question and

answering method does not change a lot.

Example 9 : Long question 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)

FOR PAPER 2 :

2. Choose long question in Part C where the form of question and

answering method does not change a lot.

EXAMPLE 10 : 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 that y ≤ 2x.

(c) Contruct and shade the region that satisfies the

above inequalities.

(d) 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.

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 scale of 1 cm to 1 unit on each axis, draw graphs of the 3

inequalities. Shade the region, R, that satisfies the 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 ?

FOR PAPER 2 :

There are questions that incorporates topics (2 in 1)

Also, choose questions from “SOLO” topics.

F. GENERAL GUIDE ON PROBLEM SOLVING

2.1 “Directly use formula / fact / concept / algorithm” method

Example 11 : Given f(x) = 2x2 – 1 , find f ’(x)

x + 1

Example 12 : Find the straight line equation that

passes through (2,1) and perpendicular

with the line 2x + y –3 = 0

F. GENERAL GUIDE ON PROBLEM SOLVING

2.2 “ Forming equation” (whether inear, quadratic @ 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

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 = - 2 is equal its distance from the point (6, 6). Find the point’s 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.

F. GENERAL GUIDE ON PROBLEM SOLVING

2.3 “ Forming own equation” method.

Example 17: The sum of the first three terms of a geometric progression that has

a common ratio –1/3 is 42. Calculate the sum of the third term until the fifth term.

G. TOPICAL STUDY (ANALYSIS)

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

STATISTICS PERMUTATIONS & COMBINATIONS

INDEX NUMBERSVECTOR

SIMULTANEOUS EQN. PROBABILITY

TRIGONOMETRIC FUNCTION PROBABILITY DISTRIBUTIONS

G. TOPICAL STUDY (ANALYSIS)

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 + 2logx2 – logx324 = 4

- Don’t create “own formulae” Example 19: Solve log3x + log93x = -1

- Sometimes we can be given linear, quadratic or simultaneous equation in indeks and logarithm form. Example 20 : Solve 22x – 2x – 2 = 0

- Once in a while readings of log table or calculator are needed. Example 21 : Evaluate log 4 5

INDICES AND LOGARITHMS

G. TOPICAL STUDY (ANALYSIS)

Short and : - Which formulae wish to be used depends on data Long Question type (either Grouped 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 5 30 4 35 3 40

- Experience in drawing ogives and histograms from Mathematics subject are needed.

STATISTICS

G. TOPICAL STUDY (ANALYSIS)

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

Long question equation and subtitute in the non-linear equation. - Quadratic equation will be obtained and solve it using formulae or factorisation. - Don’t forget to find the other variable value. - Long question usually is in implicit form concealed in questions from other topics. Example 23 : Find the distance between two points of intersection of the graph x + y = 10 with the graph x2 – y + y2 + 10 = 0

SIMULTANEOUS EQUATIONS

G. TOPICAL STUDY (ANALYSIS)

Short and : - Identify suitable “tools” to be used .Long question TRIGONOMETRIC RATIOS DEFINITION

TRIGONOMETRIC RATIOS IN QUADRANTS COTANGENT, COSECANT, SECANT TRIGONOMETRIC IDENTITIES TABLE BOOK / CALCULATOR

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.

TRIGONOMETRIC FUNCTIONS

H. COMMON MISTAKES

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

2. Answer given is not complete. Example 27 : Solve sin (2x + 32o) = - sin 72o for 0o ≤ x ≤ 360o

3. Using formulas that are not on any Mathematical List Example 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 half way completed. Example 30 : Solve sin (2x + 32o) = - sin 72o for 0o ≤ x ≤ 360o

H. COMMON MISTAKES

5. Mistakes on graphs - not drawn big enough - drawn too big till y-intercept can’t be shown - best fit line not drawn using long

transparent ruler.

6. Answer not written explicitly.

7. Linear, quadratic and simultaneous equation that involves log, indices and trigonometry can’t be solve properly.

8. Adding information not given, thus changing the original question. Example 31 : The weather is defined as cloudy, rainy, sunny and

windy. Find the probability that it will rain in two consecutive days.

H. COMMON MISTAKES

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.

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

centre of the circle. (b) area of shaded region.

P Q

I. CONCLUSION

1. Additional Mathematics requires time to grasp and be fluent on.

2. With sound basic skills (Algebra, Arithmetic, Shape, Number) and a lot of exercises, important 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 subtopic exercises and move to topical summative exercises. From short questions (Paper 1) move to long questions (Paper 2).

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

ALL THE BESTSELAMAT MAJU JAYA