SPM Additional Mathematics Exam Format

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Paper 2

Paper 1 Overview of SPM

2004 — 2011

questions trend

am Format SPM Additional Mathematics Ex stions Paper

Number of Que

Time

Paper 1 (80 marks) Subjective evaluation

25 questions (Answer all questions)

2 hours

Paper 2 (100 marks) Subjective evaluation

Section A: 6 questions (Answer all questions) Section B: 5 questions ) (Choose and answer 4 questions tions ques 4 C: ion Sect ) (Choose and answer 2 questions

2 hours 30 minutes

vi

FORM

CHAPTER

1

4

Functions SPM Topical Analysis

Year

2004

2005

2006

2007

2008

2009

2010

2011

Paper

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

Number of questions

3

0

3

0

2

1

3

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3

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3

0

3

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3

0

ONCEPT MAP Domain Domain

RELATIONS HUBUNGAN Types of Relations Jenis Hubungan • One-to-one • Many-to-one • One-to-many • Many-to-many

Codomain Kodomain FUNCTIONS FUNGSI

Objects Objek Images Imej Range Julat

Absolute Value Functions Fungsi Nilai Mutlak f : x→|g(x)|

Graphs of Absolute Value Functions Graf Fungsi Nilai Mutlak The graph of a linear absolute value function has a V shape

Composite Functions Fungsi Gubahan The function of f followed by g is gf.

Inverse Functions Fungsi Songsang Given that y = f(x), then f – 1(y) = x.

Problems that involve Composite Functions and Inverse Functions Masalah melibatkan Fungsi Gubahan dan Fungsi Songsang

COMPANION WEBSITE

Learning Objectives

SAMaths_01(01_08) 1

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1.1

Relations

1.1a Representation of a Relation

4

CHAPTER

F O R M

1

1 A relation from set A to set B is the linking (or pairing) of the elements of set A to the elements of set B. 2 A relation between two sets can be represented by (a) an arrow diagram, (b) ordered pairs, (c) a graph.

1 A relation from set A = {12, 14, 23, 25, 43} to set B = {3, 5, 7} is defined by ‘sum of digits of’. Represent the relation by (a) an arrow diagram, (b) ordered pairs, (c) a graph.

Facts A set is a well-defined collection of objects. For example: Universal set, ␰ = {x : 10 ⭐ x ⭐ 30, x is an integer} Set A = {Factors of 36} Set B = {Prime numbers} Set C = {Numbers where the sum of the digits is 3} The list of elements of each of the sets ␰, A, B and C is as follows: ␰ = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} A = {12, 18} B = {11, 13, 17, 19, 23, 29} C = {12, 21, 30}

Solution (a) Arrow diagram A

‘sum of digits of’ B

12 14

3

23

5

25

7

43

(b) Ordered pairs {(12, 3), (14, 5), (23, 5), (25, 7), (43, 7)} (c) Graph 7 Set B 5 3 12 14 23 25 43 Set A Try: Question 1, Self Assess 1.1.

1.1b Domain, Codomain, Objects, Images and Range In a relation between set (A) and another set (B), • the first set (A) is known as domain, • the second set (B) is known as codomain, • the elements in the domain are known as objects, • the elements in the codomain that are linked to the objects are known as images, • the set of images is known as range. Functions

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PAPER 1 Instructions: 1 2 3 4 5 6 7

Time: 2 hours This question paper consists of 25 questions. Answer all questions. Write your answers clearly in the spaces provided in the question paper. Show your working. It may help you to get marks. The diagrams provided in the questions are not drawn to scale unless stated otherwise. The marks allocated for each question are shown in brackets. You may use a scientific calculator. Answer all questions. [80 marks]

1 A function f maps the elements from set P = {k – 3, 4, 9, 16} to set Q = {0, 3, 8, 15} as shown in the following ordered pairs. {(k – 3, 0), (4, 3), (9, 8), (16, 15)} (a) Write down the function notation for f. (b) State the value of k. (c) State the codomain. [3 marks] Answer : (a)

(b)

(c)

2 Given that g(x) =

ax – 4 bx + 4 , x ≠ 4 and g–1(x) = , x ≠ –3, find the value of a and of b. 4–x x+3

[3 marks]

Answer:

1 3 The equation px2 + 3x – 4q = 0, where p and q are constants, has the roots 2 and –4. Find the value of p and of q. 2 [3 marks] Answer:

489

SAMaths5_MP1(489-494).indd 489

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Answers 1

Functions

Set Y

Self Assess 1.1 1 (i) (a)

4

A

'remainder when divided by 5'

3

B

1

6

1

12

2

18

3

24

4

14 21 23 34 48

2 (a)

(b) {(6, 1), (12, 2), (18, 3), (24, 4)} (c)

P

'product of digits of'

Set X Q

26

12

34

20

45

24

62

48

Set B 4 3 2 1

Set A

6 12 18 24

(ii) (a) P

'factors of'

Q

2

2

3

3

4

4

8

8

(b) (i) {26, 34, 45, 62} (ii) {12, 20, 24, 48} (iii) 20 (iv) 26, 34, 62 (v) {12, 20} 3 (a) One-to-one relation (b) Many-to-many relation 4 (a) One-to-many relation (b) Many-to-one relation 5 (a) One-to-one relation (b) {1, 3, 5} 1 (c) f(x) = x 2

10 (a) p = 2, q = –3 1 (b) x = 0 or 3 11 (a) (i) 1.4142 (ii) 1.3660 (b) 56.78° 12 (a) {–2, –1, 0, 1, 2} (b) {2, 3, 6} (c) 2 (d) (i) 1, –1 (ii) 2, –2 Self Assess 1.3 1 (a) 18 (b) 3 (c) 7 2 (a) (i) 8 (ii) 12 4 (b) x = –1 or 1 5 3 (a)

(3,5) 2 (-3,1) -2

(b) {(2, 2), (3, 3), (4, 2), (4, 4), (8, 2), (8, 4), (8, 8)} (c)

2 (a) (i)

Set Q 8

3 2 2

3

4

8

Set P

(iii) (a)

5 'difference of digits of'

Y

6

14 21

1

23

3

34

4

48

7

8 9

x

y

(7,9)

(-3,9)

1 2 2 0
f (x)
9

x

y (4,5)

(ii) 1

3

1 5

(i) 9 (ii) 6 a = 10, b = 5 x=5 a = 3, b = 2 1 (b) 0 (c) 3 m = –20 (a) –2 (b) 4 (c) 20 (a) p = 3, q = –5 1 (b) x = 3 or 1 4 (a) a = –1, b = –2 1 (b) n = or –3 2 2 1 (b) (a) 1 5 2 (a) a = 15, b = 8 (b) x = 8

4

5

(c)

(b) 3 (a) (b) 4 (a)

4

X

–2

O 0
f (x)
5

(b)

O

Self Assess 1.2 1 (a) Function (b) Non-function (c) Non-function (d) Function

F O R M

y

CHAPTER 1

(b) {(14, 3), (21, 1), (23, 1), (34, 1), (48, 4)} (c)

Form 4

O

(d)

(-2,11)

1 1 2 0
f (x)
9

x

y (4,7) 5

O

2 1 3 0
f (x)
11

x

Self Assess 1.4 1 (a) 6 (b) 2 (c) 16 (d) 1.732 2 (a) fg : x → 9x2 + 6x gf : x → 3x2 – 2

499

SAMaths Ans 4(499-514).indd 499

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