ANSWERING ANSWERING TECHNIQUESTECHNIQUES

ADDITIONAL MATHEMATICSADDITIONAL MATHEMATICS

SPM SPM

PAPER 1PAPER 1

• CONTENTS OF THE SYLLABUS

• FORMAT OF THE EXAMINATION PAPERS

• EFFECTIVE TECHNIQUES OF LEARNING

• TECHNIQUES OF ANSWERING

CONTENTS OF THE SYLLABUS

CONTENTS

CORE PACKAGEELECTIVE PACKAGE

STATISTICS

TRIGONOMETRY CALCULUS

ALGEBRA

GEOMETRY

CORE PACKAGE

Component Topics

Algebra

(i) Functions

(ii) Quadratic Equations

(iii) Quadratic Functions

(iv) Simultaneous Equations

(v) Indices and Logarithms

(vi) Progressions

(vii) Linear Law

Geometry(i) Coordinate Geometry

(ii) Vector

Calculus(i) Differentiation

(ii) Integration

Component Topics

Trigonometry(i) Circular Measures

(ii) Trigonometric Function

Statistics

(i) Statistics

(ii) Permutation and Combination

(iii) Simple Probability

(iv) Probability Distribution

ELECTIVEPACKAGE

Applied ScienceAnd

TechnologySocial Science

Application Of Index NumberLinear Programming

Solutions of TriangleMotion In A Straight Line

FORMAT OF PAPER 1

ADDITIONAL

MATHEMATICS

SPM

Type of instrument: Objective Type of item: Graded objective

(Item requires candidates to give responses with their own answers)

Number of question : 25 ( Answer all ) Total marks : 80 Duration : 2 jam Construct : Knowledge and Understanding: 20 % Application skills: 80 % Level of Difficulty : R : S : T = 6 : 3 : 1

Item of Knowledge and Understanding1. Does not involve complicated calculations

2. Measures the ability of candidates to Recall definitions, concepts, formulas and laws Translate idea from one form to another. Reason out a basic idea

3. Task Word : State…, Name…, Write….

Item of Application Skills

1. Measures the ability of candidates to carry out the calculation using the

definitions, concepts, formulas and laws sketch, draw and interpret graphs. generate formulas or relation

2. Word task:

Find…, Calculate…,Solve…, Differentiate…, Integrate…, Evaluate…, Express…

EXAMPLES OF ITEMS

3472/1

The first term of a geometric progression is 3 and the common ratio of the geometricprogression is −2.List down the first four terms of the geometricprogression.

[2 marks]

Answer:……………

Knowledge and Understanding

Diagram1 shows the graph of the quadratic function y = (x + b) 2 + c.

y

(2,3)

x0

DIAGRAM 1

State the values of b and c. [2 marks]

Answer: b =…………… c=…...………..

Application Skills

Given , find the value of n.

[3 marks]

Answer:………………….

n n3 15 25 5

On the axes provided in the answer space,

sketch the graph of y = sin2x for 0 x Answer :

y

0x

RANGE OF TOPICSIN PAPER 1

• ALL the topics in the syllabus except

- topics from the AST and SS Packages

- Simultaneous Equations • Questions involving proving will not be asked in

Paper 1

Marking Scheme• Full marks are given to the correct

answers.

• However, if the answer is wrong, marks will be given to the correct stage of the candidates’ working.

ExampleThe quadratic equation x(x + 1) = px – 4 has two distinct roots.Find the range of values of p .

[3 marks]Marking Scheme:

p < 3 , p >5

B2: (p + 3)(p 5) > 0

B1: (1 p)2 4(1)(4) > 0

TECHNIQUES OF ANSWERING

Start by answering the easy questions first.

Answer according to the requirements of the question.

(This determines how the answer must be written.)

Example: Find , in radians... Give your answer correct to two decimal

places...

1. Given and , find in the form

Answer: or

OA i j@@@@@@@@@@@@@@

3 OB i j@@@@@@@@@@@@@@

5

AB@@@@@@@@@@@@@@

x

y

AB

@@@@@@@@@@@@@@ 2

6

AB i j@@@@@@@@@@@@@@

2 6

2. The quadratic px² + qx +1= 0 has two equal roots. Express p in terms of q.

• q = 2√pq = 2√p• q² = 4pq² = 4p• p = p = q²q² 44

Answer:Answer:

Answer according to the instructions of the question.

(This determines the method that must be used when solving.)

Example:• Using , calculate… (Circular

Measures)• Sketch the graph…(Quadratic Functions/

Trigonometric)

3.142

• Understand the key words

Key Words

Action Example

State

Name

Write

Answers can be obtained without calculation

Find the values of m and n

Find Determine

Calculate

Evaluate

Involves calculation and usually formulas are used.

Given

,

find f´´(x)

10.090909

p

k m n

53 1f x x

PRESENTATIONOF

ANSWERS

READABLE AND

NEAT

HANDWRITING

• Workings must be shown clearly

• Common mistake: Answers without workings.

Answer : 7 [0 mark]

2 + 7 + 9 + 15 + x = 12 5 33 + x = 60 x = 27

[1 mark]

Answer : 7

Given the mean of the numbers 2,7,9,15 and x is 12, find the value of x. [2 marks]

• Final answer must be in simplified form.

• Common mistakes:

; 2x2 – 4x + 6 =0, 6

225

• Solutions involving ,answers can be given in terms of , unless stated

“Using = 3.142”

Hence, this value of must be used to obtain the answers.

• Precision

Answers involving decimal numbers must be rounded off to 4 significant numbers.

Example:

tan = 0.33 , = 180° 16’ [not precise]

tan = 0.333, = 180° 25’ [not precise]

tan = 0.3333, = 180° 26’ [precise ]

Solve the equation 3cos 2x = 8sinx – 5 for 0 x 360 [3 marks] B1: 3(1 - 2sin2 x ) = 8sinx – 5 B2 :

sin x = 0.67 x = 42.06

sin x = 0.6667 x = 41.81

3sin 2 sin 2 0

2sin , sin 2

3

x x

x x

not precise

Solve the equation 42x 1 = 7x

[4 marks]

(2x - 1 )lg 4 = x lg 7

2x(0.60) – x(0.85)= 0.60

x = 1.714

0.60

2 0.60 0.85x

Solve the equation 42x 1 = 7x

[4 marks]

1.677

B3 :

B2 : 2xlg 4 – xlg7=lg 4

B1: (2x - 1 )lg 4 = x lg 7

lg 4

2lg 4 lg 7x

More Common Mistakes

Graph of Trigonometric Function

On the axes provided in the answer space, sketch the graph of y = |3sin2x| for 0 x

Answer : y

x

y3

x

0

y3

x0

Progression

Find the ninth term of the arithmetic progression

7,4,1,…

Tn=a+(n-1)d

T9=7+(9-1)-3

=12

Tn=a+(n-1)d

T9=7+(9-1)(-3)

=-17

Make sure brackets is

written to show multiplication

Functions

Defining functions.

23

1

x xf x

x

23

, 11

x xf x x

x

Condition for the function to exist must be

written

Quadratic functions

Quadratic inequalities

1 5

1 , 5

x or x

x x

1 5x and x

Quadratic Equations

Solving quadratic equations using the formula.

2

2

3 5 1 0

5 5 4 3 1

2 3

x x

x

3 5 1x x Make sure ‘= 0’ is written

Show how the values of a, b and c are

substituted into the formula

IN THE IN THE EXAMINATION HALLEXAMINATION HALL

The use of non-programmable The use of non-programmable scientific calculator is allowed.scientific calculator is allowed.

Any valid method can be used Any valid method can be used to solve a problem. (paper 1)to solve a problem. (paper 1)

Do not waste time Do not waste time sketchingsketching a a graph.graph.

Give only one answer in the Give only one answer in the answer space.answer space.

Do not cross out the solution Do not cross out the solution that has been done; probably that has been done; probably the first attempt is better than the first attempt is better than the second.the second.

ExampleExample

A badminton team consists of 7 players. TheA badminton team consists of 7 players. The

team will be chosen from a group of 8 boys team will be chosen from a group of 8 boys

and 5 girls.and 5 girls.

Find the number of teams that can be formedFind the number of teams that can be formed

such that the team consists of 4 boys. such that the team consists of 4 boys.

[2 marks][2 marks]

Answer: 100800Answer: 100800

8 5 8 54 3 4 3C C P P