matematik spm 2010

1

Projek Jawab Untuk Jaya 2010

MATHEMATICS SPM

This module aims to prepare the Form five students for the SPM examination and also for the Form four

students to reinforce as well as to enable them to master the selected topics. It also serves as a guidance for

effective acquisition of the various mathematical skills.

At the end of each topic, sample answers are given. Discussions on common mistakes that result in the students failure to obtain full mark are included as well. This module suggests specific strategies for each chosen topic

and strategies which can help the students in problem solving. It is hoped that this module can benefit all the

Pahang students as well as helping them towards achieving excellent results in SPM Mathematics.

TABLE OF CONTENTS

1.1 Authentic Examination Format 1.2 Topical Analysis of SPM Paper 1 1.3 Topical Analysis of SPM Paper 2 1.4 Selected topical

(BONUS) 1full set of Paper 1 and Paper 2 questions

2


FORMAT OF ASSESSMENT INSTRUMENT

MATHEMATICS 1449

NO TOPIC PAPER 1 (1449/1) PAPER 2 (1449/2)

1 Instrument type Objective questions Subjective questions

2 Item type Multiple Choice

Questions

Open ended structure and

limited response

3 Number of

Questions

40 questions

( answer all)

Section A

11 questions (answer all)

Section B

5 questions (choose 4)

4 Total Marks 40 Section A

Total marks : 52

Section B

Total marks : 48

(Each question 12 marks)

Overall Total :100

5 Test Duration 1 hour 15 minutes 2 hours 30 minutes

6 Constructs Weights Knowledge - 45%

Skills - 55%

Knowledge - 25%

Skills - 70%

Values - 5%

7 Context Scope Lower secondary school scope with

continuation in the

upper secondary.

All aspects of learning from Form

4 and Form 5.

Lower secondary school scope with continuation

in the upper secondary.

All aspects of learning from Form 4 and Form 5.

8 Difficulty level

Low - L

Medium - M

High - H

L : M : H = 5 : 3 : 2

L : M : H = 5 : 3 : 2

9 Additional Tools a. Scientific Calculator

b. Four-figure

Mathematical table

b. Geometry Set

a. Scientific Calculator

b. Four-figure Mathematical

table

b. Geometry Set

3


2.LIST OF MATHEMATICS TOPICS 1449

No. FORM 1 3 FORM 4 FORM 5

1 Circles I and II Standard Form Number Bases

2 Polygons I and II Quadratic Expressions

and Equations Graphs of Functions II

3 Solids and Volumes Sets Transformations III

4 Transformations I and II Mathematical

Reasoning Matrices

5 Trigonometry I The Straight Line Variations

6 Algebraic Expressions I

and II Statistics III

Gradient and Area under

a Graph

7 Algebraic fractions Probability I Probability II

8 Algebraic Formulae Circles III Bearing

9 Linear Equations(1 and

2 unknowns) Trigonometry II Earth as a Sphere

10 Indices Angles of Elevation

and Depression Plans and Elevations

11 Linear Inequalities Lines and Planes in 3

Dimensions

12 Statistics I and II

13

Arc Length & Area of

Sector

4


TOPICAL ANALYSIS OF SPM MATHEMATICS PAPER 1

Bil TOPIC FORM 2004 2005 2006 2007 2008 2009

1 Standard Form 4 1,2,3,4 1,2,3 1,2,3,4 1,2,3 1,2,3,4 1,2,3,4

2 Number Bases 5 5,6 4,5 5,6 4,5 5,6 5,6

3 Polygon II 3 7 6,7 7 6,7 7 7,8

4 Circles III 4 8 8 8 8 8 9

5 Transformation I 3 9,10. 9,10 9,10 9,10 9,10 10,11

6 Trigonometry I 3 13 - 11 - 12

7 Trigonometry II 4 11,12 11,12,13 12,13 11,12,13 11,12,

13 13

8 Lines and Planes in 3

Demension 4 14 14 14 14 14 14

9 Angles of Elevation

and Depression

4

15,16 15 15,16 15 15 15,16

10 Bearing 5 18 16 17 16 16,17 17

11 Earth as a Sphere 5 17 17,18 18 17,18 18 18

12 Algebraic Expression II 2 20 19 19.20 19 19 19

13 Algebraic Expression

III 3 19 20 - 21 20 20

14 Algebraic Formulae 3 21 21 21 20 21 21

15 Linear Equation I 2 22 22 22 22 22 22

16 Indices 3 23,24 23,24 23 23,24 23,24 23,24

17 Linear Inequalities 3 25,26 25 24 25 25,26

18 Statistic I 2 27 27 26,27 26 25 27

19 Statistic II 3 28 25 27 26,27 28

20 Statistic III 4 - 26 - - 29

21 Graph of Function II 5 29 28 28 28 30

22 Sets 4 30,31,32 29,30,31 29,30,31 29,30,31 29,30,3

1 30,31

23 The Straight Line 4 33,34 32,33 32,33 32,33 32,33 33,34

24 Probability I 4 35,36,37 34,35 34,35 34,35 34,35 35,36

25 Variations 5 38,39 36,37,38 36,37,38 36,37,38 36,37,3

8 37,38

26 Matrices 5 40 39,40 39,40 39,40 39,40 39,40

5


TOPICAL ANALYSIS OF SPM MATHEMATICS PAPER 2

No. Form Topic 04 05 06 07 08 09

Part A (Question 1 to Question 11)

1. 1-3 Simultaneous Linear Equations 1 1 1 1 1 1(Q3)

2 4 Quadratic Equation 1 1 1 1 1 1(Q2)

3

4 Sets (Shade Venn diagrams) 1 - 1 - 1

5 Region for Inequalities - 1 - 1(Q1)

4. 4 Mathematical Reasoning (Statement,

implications, argument, mathematical

induction, converse )

1 1 1 1 1

1Q(5)

5. 4 The Straight Line ( parallel, equation,y-

intercept)

1 1 1 1 1 1(Q6)

6. 5 Probability II 1 1 1 1 1 1(Q9)

7. 1-3 Arc Length & Area of Sector 1 1 1 1 1 1(Q10)

8. 1-3 Volume of Solids

a. Pyramids and half cylinders - - 1

b. Cones and Cylinders 1 1 -

c. Cones and cuboids

d. Pyramid and prism 1 1

e. Cones and hemisphere 1(Q8)

9. 5 Matrices ( Inverse, matrix equation ) 1 1 1 1 1 1(Q7)

10. 5 Gradient and Area Under a Graph

a. Speed-Time Graph - 1 1 1 1 1(Q11)

b. Distance-Time graph 1 - - 1

11. 4 Lines & Planes in 3 Dimensions (angle

between 2 planes)

1 1 1 1 1 1(Q4)

Part B (Question 12 to 16 - Choose any four)

12. 5 Graphs of Functions II

a. Quadratic - 1 1

b. Cubic - - - 1 1(Q12)

c. Reciprocal 1 - - 1

13. 5 Transformations III ( combined ) 1 1 1 1 1 1(Q13)

14. 5 Earth as a Sphere 1 1 1 1 1 1(Q16)

15. 5 Plans and elevations

a. Prism and cuboids 1 1(Q15)

b. Cuboids and half cylinder, prism - 1 - 1

c. Prism and prism - - 1 1

16. 4 Statistics III

a. Raw data, frequency table, mean,

frequency polygon, modal class

1

1

1

1 1 1(Q14a)

b. Ogive 1 - - 1

c. Histogram - 1 1 1(Q14b)

d. Frequency Polygon - - - 1

e. Communication 1 1 - 1 1(Q14c)

Total Questions 16 16 16 16 16

6


Question Allocation (Mathematics SPM 2007)

Form

Number of questions

Paper 1 % Paper 2 Marks % Total %

1-3 13 32.5 3 14 12.50 28.57

4 16 40.0 6 33 29.46 39.28

5 11 27.5 7 65 58.04 32.14

Total 40 100.00 16 112 100.00 100.00

Question Allocation (Mathematics SPM 2008)

Form

Number of questions


1-3 12 30.00 3 14 12.50 18.75

4 19 47.50 6 33 29.46 40.98

5 09 22.50 7 65 58.04 40.27

Total 40 100.00 16 112 100.00 100.00

Question Allocation (Mathematics SPM 2009 )

Form

Number of questions


1-3 15 37.50 3 14 12.50 18.75

4 16 40.00 6 33 29.46 40.98

5 09 22.50 7 65 58.04 40.27

Total 40 100.00 16 112 100.00 100.00

7


TOPICAL EXERCISE

1. SETS

PAPER 1

Given set F = {2, 3, 4} set G = {2,4,5,6} and set H = {4,6}.Which of the following is true?

Diberi set F = {2, 3, 4} set G = {2,4,5,6} dan H = {4,6}.Yang manakah benar?

A H G C H F B G F D F G

2

In the figure above, is an universal set. Set B is

Rajah di atas, ialah set semester. Set B ialah

A {2, 4, 7} C {3, 9, 10}

B {5, 6, 8} D {3, 5, 6, 8, 9, 10}

3

In the diagram above, is an universal set. n ( P Q ) =

A 2 C 7

B 4 D 8

4 Given S = { factors of 12} and T = {prime numbers that is less than 10}

the elements of set S T are

A {1} C { 3, 4, 6 }

B {2, 3 } D {2, 3, 4, 5, 6, 12}

.8

A B

.2

.7 .4

.5

.6

.9 .10

.3

P Q

.a

.f

.k

.b

.j

.c

.h .r

8


5 It is given that A = {a, b, c}, the number of subsets of A is

A 4 C 7

B 6 D 8

6 Given that = {x:x is an integer, 12 x 40}, A = { multiples of 8 } and

B = {integers that contain digit 4}, A B =

A {24, 34, 40} C {24, 40}

B {14, 24, 40} D {14, 40}

7 Given that = P Q R, P Q , Q R = and R P. The Venn diagram that represents

these relationship is

A

B

C

D

8

In the above diagram, n (P Q) = n (R). The value of x is

A 2 C 4

B 3 D 5

Q

P R

Q

R P

Q P R

Q

P R

P Q R

5 x + 7 3 2x

4

9


9 Diagram below is a Venn diagram showing the number of students in sets P, Q, and R. It is given that

set P = {English Society members}, set Q = {Science Society members}, set R = {Mathematics Society

members} and the universal set,

= P Q R

If the number of students who joined both the Science Society and the Mathematics Society is 11, find

the number of students who joined only two societies.

A 12 C 16

B 14 D 22

10 Given = P Q R. .Which of the following Venn diagrams represents ( PQ) R ?

A C

B D

R Q

5 x

6

4

3

2x

7

P Q

R

P Q

R

P Q

R

P Q

R

10


SET PAPER 2

1 The Venn diagram below shows sets R, S and T.Given the universal set = R S T. On

each of the diagrams, shade the region which represents

Gambarajah Venn di bawah menunjukkan set R, S dan T. Diberi set semesta = R S

T.Lorek pada setiap gambar rajah, kawasan yang mewakili

(a) R S (b) ( R S ) T

2 The Venn diagram below shows sets P, Q and R.Given the universal set = P Q R.

On the diagrams,

Gambar rajah Venn di bawah menunjukkan set P, Q dan R. . Diberi set semesta = P Q

R..Di atas rajah,

(a) shade set P Q (b) shade set PQ R Lorek set P Q lorek set PQ R

R

S T

R S

T

P Q R P Q R

11


3 The Venn diagram in the answer space show sets A, B, and C such that the universal set, =

A B C. On the Venn diagram in the answer space, shade Gambar rajah Venn pada ruang jawapan menunjukkan sets A,B,dan Cdi mana set semesta

= A B C.Di atas gambar rajah Venn di ruang jawapan, lorek

(a) A B (b) A (B C)

Answer:

(a) (b)

4 Given that the universal set, = {x : 20 x 30, x is an integer}, P = { x: x is a multiple

of 4},

Q = { x: x is a number such that the sum of its digits is less than 7} and

R = {x: x > 27}.

Diberi set semesta = {x : 20 x 30, x ialah integer},

P = { x: x ialah gandaan 4},Q = { x: x jumlah antara digit-digit kurang dari 7}

dan R = {x: x > 27}.

(a) List all the elements of the sets P and Q.

Senaraikan semua unsur bagi set P dan Q.

(b) Find /Cari the set P Q .

(c) Find/Cari n (P Q)

Answer:

(a) ..

(b) ..

( c) ..

A B

C

A B

C

12


5

The Venn diagram shows the universal set , set P, set Q and set R. List all the

elements for the following sets.

Gambar rajah Venn menunjukkan set semesta , set P, set Q dan set R.Senaraikan

semua unsur bagi set yang berikut.

(a) ( P R) (b) Q R P

Answer:

(a) .

(b) .

6

The Venn diagram shows the universal set , set P, set Q and set R.

Gambar rajah Venn menunjukkan set semesta set , set P, set Q and set R.

(a) Simplify/Ringkaskan the set PR. (b) State the set which represents the shaded region.

Nyatakan set yang mewakili kawasan yang berlorek

Answer:

(a) ..

(b) .

P Q R .2

.3 .7 .1

.6

.4

.5

P Q R

13


7 In the following Venn diagrams, = J K L .State the set which is represented by the

shaded region.

Dalam rajah berikut,gambar rajah Venn, = J K L. Nyatakan set yang mewakili

kawasan yang berlorek

(a) (b)

Answer:

(a) ...

(b) ...

8. Given that set E = {5, 6} , set F = {3, 4, 5, 6, 8, 9} , set G = {1, 2, 3, 6, 8} and

= E F G .

Diberi set E = {5, 6} , set F = {3, 4, 5, 6, 8, 9} , set G = {1, 2, 3, 6, 8} dan

= E F G .

(a) Draw a Venn diagram to show the relations between set E , F, and G . Lukis gambar rajah Venn menunjukkan hubungan antara set E , F, and G

(b) List the elements of set F G. Senaraikan unsur-unsur bagi set F G.

(c) Find/Cari n (E F).

Answer:

(a)

(b)

(c)

J

L K

J K

L

14


9. Given ={ x: 20 x 35, x is an integer, set P ={x:x is a multiple of 5}, set Q ={x: x is an

even number}

Diberi ={ x: 20 x 35, x ialah integer, set P ={x:x ialah gandaan 5}, set Q ={x: x

ialah nombor genap}

(a) List all the elements of set P and Q.

Senaraikan semua element bagi set P and Q.

(b) Find/Cari

(i) n(P Q)

(ii) n(P Q)

Answer:

(a) ..

(b)(i) ..

(ii) ..

10 The Venn diagram below shows the universal set , set X, set Y, set Z. In the

diagram, shade the region which represents the set.

Gambarajah Venn di bawah menunjukkan set semesta set , set X, set Y, set Z. Dalam rajah

lorek kawasan set yang diberi.

(a) X Z (b) (Y Z X)

X Y

Z X Y

Z

15


2. PROBABILITY

PAPER 1

1 A box contains 8 black balls, 6 yellow balls and some blue balls.A ball is chosen at random

from the box. The probability of choosing a yellow ball is 3

1. The probability of choosing

the blue ball is

Sebuah kotak mengandungi 8 biji bola hitam dan beberapa biji bola biru.Sebiji bola dipilih

secara rawak daripada kotak itu.Kebarangkalian untuk mendapatkan bola kuning ialah

3

1.Kebarangkalian untuk mendapatkan bola biru ialah

A 9

1 C

3

2

B 9

2 D

9

4

2 There are 45 packets of mango juice and orange juice in a refrigerator. A packet of drink is

chosen at random from the refrigerator. The probability that a packet of orange juice is

chosen is 5

2. How many packets of oranges need to be added to the refrigerator so that the

probability of choosing a packet of orange juice is 2

1?

Terdapat 45 kotak jus mangga dan jus oren di dalam sebuah peti sejuk.Satu kotak minuman

dipilih secara rawak daripada peti sejuk. Kebarangkalian sekotak jus oren dipilih ialah

5

2.Berapakah bilangan jus oren yang perlu ditambahkan ke dalam peti sejuk supaya

kebarangkalian sekotak jus oren dipilih ialah 2

1

A 5 C 8

B 6 D 9

3 There are 150 people in a theatre.Given that the probability of a chair being seated by a

person is 3

1, find the total numbers of chairs in the theatre.

Terdapat 150 orang di dewan teater.Diberi kebarangkalian kerusi yang diduduki ialah

3

1,cari bilangan kerusi di dewan teater itu.

A 50 C 300

B 150 D 450

16


4 The diagram shows six numbered cards.

If a card randomly selected, state the probability that the number is a multiple of 3.

Rajah menunjukkan enam keping kad nombor.Sekeping kad dipilih secara rawak,Nyatakan

kebarangkalian bahawa kad yang dipilih itu ialah nombor gandaan 3.

A 6

1 C

2

1

B 4

1 D

3

2

6 Among a class of 38 students, 21 of them are girls.If 4 new boys joined the class and a

student randomly selected from the class, state the probability of selecting a boy.

Diantara 38 orang pelajar dalam sebuah kelas, 21 adalah perempuan,4 pelajar lelaki baru

masuk dan seorang pelajar lelaki dipilih secara rawak,nyatakan kebarangkalian seorang

pelajar lelaki dipilih.

A 3

1 C

5

3

B 2

1 D

4

3

7 Paul has a collection of stamps from Perak, Sabah and Sarawak.12 of the stamps are from

Perak.He picks a stamp at random.The probability of picking a stamp from Sabah is 7

2 and

the probability of picking a stamp from Sarawak is 3

2.Calculate the total number of stamps

in his collection.

Paul mempunyai koleksi setem dari Perak,Sabah dan Sarawak.12 keping daripada setem itu

ialah dari Perak.Dia memilih sekeping setem secara rawak.Kebarangkalian sekeping setem

dariSabah dipilih ialah 7

2 dan kebarangkalian sekeping setem dari Sarawak dipilih ialah

3

2.Hitung jumlah bilangan setem dalam koleksinya.

A 252 C 278

B 260 D 350

12 14 18

20 21 51

17


8 The table shows three types of favourite drink of a group of students.

Jadual menunjukkan tiga jenis minuman kegemaran sekumpulan pelajar

Type of drink Number of students

Coffee x

Tea 14

Milo 26

A students is picked at random from the group.The probability of picking a students who likes

coffee is7

2.Calculate the value of x.

Seorang pelajar dipilih secara rawak daripada kumpulan tersebut.Kebarangkalian seorang

pelajar yang suka minuman kopi ialah 7

2.Hitung nilai x.

A 12 C 24

B 16 D 28

9 It is given that set M = {2, 3, 4, 5, 6, 7, 8, 9, 10}. A number is chosen at random from the

set. Find the probability that the number chosen is a factor of 32.

Diberi set M = {2, 3, 4, 5, 6, 7, 8, 9, 10}.Satu nombor dipilih secara rawak daripada set.

Cari kebarangkalian bahawa nombor yang dipilih itu ialah faktor bagi 32.

A 3

1 C

9

2

B 3

2 D

9

4

10

Subject Commerce Arts Science

Number

of

students

80 90 110

The table shows the subjects taken by a group of students.A student is chosen at random,

find the probability that he takes Commerce or Arts.

Jadual menunjukkan mata pelajaran yang diambil oleh sekumpulan pelajar.Seorang pelajar

dipilih secara rawak, cari kebarangkalian pelajar yang mengambil mata pelajaran

Perdagangan dan Pendidikan Seni.

A 28

11 C

28

17

B 28

16 D

98

9

18


PROBABILITY

PAPER 2

1 The table shows the number of buttons in a container.

Jadual menunjukkan bilangan butang di dalam sebuah bekas.

Colour Quantity

Red 12

Blue 15

Green x

Yellow 5

A button is picked at random and its colour is noted. The probability that a blue button is

picked is 103

Sebiji butang dipilih secara rawak dan warnanya dicatat.Kebarangkalian butang biru

dipilih ialah103

(a) Find the value of x

Cari nilai x

(b) Find the probability that either a red

or yellow button is picked.

Cari kebarangkalian sebiji butang merah atau sebiji butang kuning dipilih.

Answer:

(a)

(b).

2 A bag contains 5 yellow cards,4 red cards and 3 blue cards, all of the same size. A card is

drawn at random from the bag.

Sebuah beg mengandungi 5 kad kuning, 4 kad merah, dan 3 kad biru yang sama

saiz.Sekeping kad dipilih secara rawak daripada beg.

(i) Find the probability that the card is yellow. Cari kebarangkalian kad kuning dipilih.

(ii) If a card is drawn with replacement of 600 times, find the number of times the card is yellow.

Jika sekeping kad dipilih dan dikembalikan sebanyak 600 kali,cari berapa kali kad

kuning dipilih.

Answer:

(i)..

(ii) .

19


3 The diagram shows three numbered cards in

box P and two cards with letters in box Q.

Gambar menunjukkan tiga kad nombor dalam kotak P dan dua kad huruf dalam kotak Q.

A card is picked at random from box P and then a card is picked at random from box

Q.By listing the sample of all the possible outcomes of the event, find the probability that

Sekeping kad dipilih secara rawak dari kotak P dan sekeping kad dipilih dari kotak

Q.Dengan menyenaraikan sampel semua kesudahan perisriwa yang mungkin, cari

kebarangkalian bahawa

(a) a card with an even number and the

card labelled L are picked.

satu kad nombor genap dan kad berlabel L dipilih

(b) a card with number which is a multiple

of 2 or the card labelled A are picked.

satu kad nombor gandaan 2 atau kad

berlabel A dipilih.

Answer:

(a)

(b)

4 The table shows the masses of a group of students in a school.

Jadual menunjukkan berat badan sekumpulan pelajar sekolah.

Mass

(kg) 41- 45 46 -50 51 - 55 56 - 60

Number

of

students

12 32 20 16

If a student is randomly selected from the group,find the probability that the mass of the

selected student is

Jika seorang pelajar dipilih secara rawak daripada kumpulan pelajar itu,cari

kebarangkalian berat badan pelajar

(a) in the modal class (b) more than 50kg

(a) dalam kelas mod (b) lebih daripada 50kg

1 6 8 A L

P Q

20


5

Diagram above shows 11 letter cards which are put in a box.

Rajah di atas menunjukkan 11 kad huruf di dalam sebuah kotak.

Two cards are picked at random from the box, one by one, without returning the first card to

the box.Calculate the probability that

Dua kad dipilih secara rawak daripada kotak,satu demi satu, dengan kad pertama tidak

dikembalikan.Hitung kebarangkalian

(a) the first card picked is labelled T and the second card picked is not labelled T.

kad pertama dipilih bertanda T dan kad kedua

dipilih tidak bertanda T

(b) the first and the second card picked have the same letter Kad pertama dan kedua dipilih ialah sama nombor

Answer:

(a)

(b) .

6 A fair dice and a fair coin are tossed simultaneously.By listing the sample of all the possible

outcomes of the event,find the probability of obtaining

Sebiji dadu dan sekeping duit syiling dilambungkan secara serentak.Cari kebarangkalian

mendapat

a) the number 6 with the dice nombor 6 pada dadu itu

b) heads with the coin angka pada duit syiling itu

c) the number 4 with the dice and heads with the coin nombor 4 pada dadu itu dan angka pada duit syiling.

Answer:

(a) .

(b) ..

(c) .

M A T H E M

A T I C S

21


7

Fruit Durian Banana

Number

of

people

x 60

The table shows the number of people who likes durian or bananas. A person is chosen at

random from the group and the probability of choosing a person who likes banana is 4

1.

Jadual menunjukkan bilangan orang yang suka buah durian dan buah pisang.Seorang

dipilih secara rawak dari kumpulan itu dan kebarangkalian seorang yang suka pisang

dipilih ialah is 4

1

Find/Cari the value of x.

(a) 15 people who likes durians join the group.Find the probability of choosing a person who likes bananas.

15 orang yang suka pisang ditambah dalam kumpulan itu.Cari kebarangkalian

seorang yang dipilih suka pisang.

Answer:

(a)..

(b)..

8 The table above shows the probabilities that Krishnan and Halim will join the Science

stream or the Arts stream after completing form three.

Jadual di atas menunjukkan kebarangkalian Krishnan dan Halim masuk ke Aliran Sains

atau Aliran Sastera selepas tamat Tingkatan 3.

Calculate the probability that

Hitung kebarangkalian bahawa

Calculate the probability that

Student

The probability of

joining the

Science

stream

Arts

stream.

Krishnan 3

2

6

1

Halim 10

3

5

2

22


(a) Krishnan and Halim will join the Science Stream Krishnan dan Halim akan masuk Aliran Sains

(b) one of them joins the Science stream and the other joins the Arts stream. Salah seorang memilih Aliran Sains dan seorang

lagi memilih aliran Sastera

Answer:

(a) .

(b)

9 A survey conducted in a class shows that 20 students are not tongue rollers and y students

are tongue- rollers.A student is picked at random from the class.The probability of picking a

student who can roll his tongue is 7

3

Suatu tinjauan ke atas pelajar dalam sebuah kelas menunjukkan 20 orang pelajar ialah

bukan penggulung lidah dan y orang pelajar penggulung lidah.Kebarangkalian seorang

pelajar yang boleh menggulung lidah dipilih ialah 7

3.

(a ) If a student is picked at random from the class,find the probability that he cannot roll

his tongue.

Jika seorang pelajar dipilih daripada kelas itu, cari kebarangkalian dia tidak boleh

menggulung lidahnya.

(b) Calculate the value of y. Hitung nilai y.

Answer:

(a)

(b)

10 A box contains 1 red and 2 yellow identical balls.A ball is picked at random from the box

and its colour is noted and then returned to the box.Then the second ball is picked.

Sebuah kotak mengandungi 1 biji bola merah dan 2 biji bola kuning yang serupa. Sebiji

bola dipilih secara rawak daripada kotak itu dengan warnanya dicatat dan bola itu

dikembalikan semula ke dalam kotak itu.Kemudian,sebiji bola kedua dipilih.

23


(a) Draw a tree diagram to represent the possible outcomes using R to represent the red ball

and Y1 and Y2 to represent the yellow balls.

Lukis gambar rajah pokok untuk mewakili kesudahan yang mungkin dengan

menggunakan R untuk mewakili bola berwarna merah dan Y1 danY2 untuk mewakili bola

berwarna kuning.

(b) Find the probability that both balls picked at random are red or yellow

Cari kebarangkalian kedua-dua bola yang dipilih secara rawak ialah bola berwarna

merah atau kuning.

Answer:

(a)

(b)

3. THE STRAIGHT LINES

PAPER 1

1 The gradient of the straight line 5x + 7y = 14 is

Kecerunan bahi garis lurus 5x + 7y = 14 ialah

A 5

7 C

7

5

B 7

5 D

5

7

2 Diagram below shows a straight line MN on a Cartesian plane

Rajah di bawah menunjukkan garis lurus MN di atas satah Cartesan.

The gradient of MN is

Kecerunan MN ialah

A 3 C 3

1

B 3

1 D 3

N(0,3)

M(-9, 0)

0

y

x

24


3 The graph in the diagram below shows a straight line JK with gradient 5

2

Graf dalam rajah di bawah menunjukkan garis lurus JK dengan kecerunan 5

2

Calculate the x-intercept of JK

A 15 C 10 B 8 D 15

4 Find the y-intercept of the straight line 3x 5y = 15.

A 5

3 C 3

B 5

3 D 3

5 A straight line with a gradient of 3 passess through the point (2,0). The y-intercept of the line is

A 2

3 C 6

B 2

3 D 6

6

The gradient of QR is 4 and the distance of PQ is 15 units. Find the x-intercept of PQ

A -9 C 4

9

J(0,6)

K

0

y

x

R(1,8)

P

0

y

x

Q

25


B -12 D 4

15

7. The straight line RS has gradient of 2. The coordinates of R are (0,6).

The x-intercept of RS is

A 3 C 3

1

B 3

1 D 3

8 The coordinates of point T(-1,1) and the gradient of the straight line TU is 2. The coordinates of point U could be

A (3, 3) C (3,1) B (3, 9) D (3,7)

9 Given that the straight line y = mx + 6 is parallel to the straight line 4y 9x =16, find the value of m.

A 9 C 4

9

B 4 D 4

9

10 The straight lines 4x 3y = 9 and x 2y = 4 intersect at point A. Find the coordinates at point A.

A (6, 5) C (6, 5) B (6, 5) D (6, 5)

THE STRAIGHT LINES PAPER 2

1 Base on diagram above, find

(i) gradient of PQ

(ii) x-intercept for PQ

(iii) the equation of PR O

Q(3,5)

P(0,6)

x R

y

26


2 Base on diagram above, find

(i) the gradient of AB

(ii) the equation of ECF

(iii) x-intercept for ECF

3 The diagram above, 6OP units.

(i) find the gradient of PR

(ii) find the y-intercept of PQ

4 In the diagram , line DE is parallel to the line GF.

Given the equation of straight line DG is

x + 3y = 14 and the gradient of GF is -2, Find

(a) the value of h,

(b) the coordinate F

(c) the equation of straight line DE

5 In the diagram below, the gradient of the

straight line MN is 3

2

Find

(a) the value of p

(b) the equation of straight line LN

y

B(-3,3) C(5,4)

F

A(2,6

)

E

x

P

R

O

Q(7,8)

y

x

x

y

D(-4,6)

G(h,4)

E O F

3

y

-5

N(6, p)

M

L

27


6 Diagram below, JKLM is a trapezium and JK is

parallel to ML

Find

(i) the x-intercept for the straight line JK

(ii) the equation of the straight line ML

(iii) the coordinate M

7 In diagram below, the graph shows that PQ, QR, and RS are straight lines. P is on the y-axis.

OP is parallel to QR and PQ is parallel to RS.

The equation of PQ is y = 2x + 5

(a) State the equation of the straight line QR

(b) Find the equation of the straight line RS and hence , state its y-intercept.

8

P(0,6)

Q O

y

x

O Q

P

R

(8, -7)

x

y

s

y K (2,5)

J(-4,2)

L(4,3)

M O x

28


Diagram above, the gradient of PQ is2

3 . Line

RS is parallel to the line PQ and passes through a point (-5,3). Find

i) the coordinate of Q

ii) the equation of RS.

9 y = mx + c is the equation of a straight line AB, where m and c are constants.Find the value of m and of c

if the straight line AB is parallel to y = 3x + 5 and passes through the point (4, 3).

10 Given two points A and B with coordinates (0,1) and (1,3) respectively,

(a) calculate the gradient of the straight line AB

(b) find the equation of the straight line AB

(c) find the value of h if the point (10,h) lies on the straight line AB produced.

4. MATRICES

PAPER 1

1 Given that

83

622

4

3

w

ww

find the value of w.

Diberi bahawa

83

622

4

3

w

ww , cari nilai w.

A 1 C -1

B 2 D -2

2

6

4

03

12

A

018

68 C

7

14

29


B

518

68 D

12

14

3 Given that 2 ,46

01

03

21

M find the matrices of M.

Diberi bahawa 2 ,46

01

03

21

M Cari matriks M

A

40

41 C

412

213

B

40

21 D

412

43

4 Given that 71

22

1x 1511 , find the value of x

Diberi bahawa 71

22

1x 1511 , Cari nilai x

A 11 C -3

B 3 D -11

5 Given

e2

13

5

3

e

0, the value of e.

Diberi

e2

13

5

3

e

0, n ilai e ialah

A 1 C -4

B 0 D -9

6 Given 343

724

f . Find the value of f.

Diberi 343

724

f . Cari nilai f.

30


A -3 C 2

B -1 D 3

7

1

4

3

5

A.

4

9 C

3

20

B

3

9 D

4

1

8

74

52

10

43

36

12=

A

910

83 C

510

21

B

910

23 D

52

81

9 T - ,12

04

01

32

the matrix of T is

T - ,12

04

01

32

matriks bagi T ialah

A

13

36 C

11

32

B

11

32 D

11

16

10 Given that

3

4

2

3

30

12k , ind the value of k.

31


Diberi bahawa

3

4

2

3

30

12k , Cari nilal k

A 1 C 4

B 2 D 8

MATRIX PAPER 2

1 M is a matrix where M

45

23=

10

01

M ialah matriks di mana M

45

23=

10

01

a) Find the matrix M

Cari matriks M

b) Write the following simultaneous linear equations as a matrix equation.

Tulis persamaan linear serentak yang berikut dalam bentuk persamaan matriks:

3x 2y = 7 5x 4y = 9 Hence, calculate the values of x and y using matrices.

Seterusnya,menggunakan kaedah matriks,hitung nilai x dan nilai y.

2 a) Find the inverse matrix of

65

32.

Cari matriks songsang bagi

65

32.

b) Using matrices, calculate the value of x and of y thay satisfy the following

simultaneous linear equations:

Menggunakan kaedah matriks,hitung nilai x dan nilai y yang memuaskan persamaan linear

serentak berikut:

265

132

yx

yx

3 a) The inverse matrix of

65

43 is m

35

6 p, find the value of m and of p.

Matriks songsang bagi

65

43 ialah m

35

6 p, cari nilai m dan nilai p

32


b) Using matrices, calculate the value of x and of y that satisfy the

following simultaneous linear equations.

Menggunakan kaedah matriks,hitung nilai x dan nilai y yang

memuaskan persamaan linear berikut.

3x 4y = -1 5x 6y = 2

4 P is a 2 x 2 matrix such that

10

01

43

21P

P ialah matriks 2 x 2 dengan keadaan

10

01

43

21P

a) Find the matrix P

Cari matriks P

b) Write the following simultaneous linearequations as a matrix equation:

Tulis persamaan linear serentak yang berikut dalam bentuk persamaan matriks

643

82

yx

yx

Hence, using matrices, calculate the value of x and of y.

Seterusnya,menggunakan kaedah matriks,hitung

nilai x dan nilai y . Seterusnya,menggunakan kaedah matriks,hitung

5 The inverse matriks of

62

83is

k

1

3

86

m.

It is given that matrix P =

31

52 and

matrix Q = k

21

3 h such that PQ =

10

01


62

83ialah

k

1

3

86

m.

a) Find the value of m and of k

Cari nilai bagi m dan k.

b) Write the following simultaneous linear as a matrix equation :

Tulis persamaan linear serentak yang berikut dalam bentuk persamaan matriks.

2x 5y = -17

33


x + 3y = 8

6 It is given that matrix M =

21

43.

Diberi bahawa matriks M =

21

43.

(a) Find the inverse matrix of M Cari matriks songsang bagi M

(a) Write the following simultaneous linear equations as matrix equation: Tulis persamaan linear serentak yang berikut dalam bentuk persamaan matriks

62

1343

yx

yx

Hence, using matrices, calculate the value of x and of y.

Seterusnya,menggunakan kaedah matriks,hitung nilai x dan nilai y

7 (a) It is given that

n2

121

is inverse matrix of

21

43. Find the value of n.

Diberi bahawa

n2

121

adalah matriks songsang kepada

21

43.Cari nilai n.

(b) Write the following simultaneous linear equations as matrix equation:

Tulis persamaan linear serentak yang berikut

dalam bentuk persamaan matriks:

22

543

vu

vu

Hence, using matrices, calculate the value of u and of v

Seterusnya,menggunakan kaedah matriks,hitung nilai u dan nilai v.

8 a) The inverse matrix of

43

12 is m

2

14

p, find the value of m and p.


43

12 iala m

2

14

p, cari nilai m dan p.

34


b) Using matrices, calculate the value of x and y which satisfy the following

simultaneous linear equations.

Menggunakan kaedah matriks,hitung nilai x dan nilai y yang memuaskan

persamaan linear serentak berikut:

2x + y = 4

3x 4y = 17

9 Given that E =

23

45 and F =

10

84

k,

Diberi bahawa E =

23

45 dan F =

10

84

k,

a) Find the value of k where EF =

40

04

Cari nilai k di mana EF =

40

04

b) State the inverse matrix of E

Nyatakan matriks songsang bagi E

Hence, using matrices, find the value of x and y which satisfy

the following matrix equations.

Menggunakan kaedah matriks,hitung nilai x dan nilai y yang memuaskan persamaan

linear serentak berikut:

23

45

y

x=

4

6

10 a) Given that the inverse matrix of

23

54 is

7

47

5

k

h, state the value of h and k.

b) Using matrices, calculate the values of x and y which satisfy both of the following

equations.

4x + 5y = -1

3x + 2y = 8

5. STATISTICS

PAPER 1

35


1. The pictograph shows the sale of magazines in a particular week. The number of sports

magazines sold is not shown.

represents 12 copies of magazines

Entertainment magazines make up 25% of the total the sales that week. Calculate the number of

sports magazines sold.

A 26 C 84

B 39 D 94

2 The pie chart shows the amount spent on five types of groceries in a shop on a particular day.

flour

margarine

salt

sugar

rice

Given that the total spent on these groceries is

RM400. The amount spent on margarine is RM 80.Find the ratio of the amount spent on

margarine to the amount spent on sugar.

A 1:5 C 3: 10

B 3:4 D 4: 15

3 The table shows the heights of samples of seedlings.

Calculate the mean height, in cm, of the seedlings.

A 13.06 C 14.56

B 13.56 D 15.56

4 The bar chart shows the scores of a group of pupils in a test.

Number of pupils

Fashion Entertainment Sports

Heights (in cm) 9 11 12 14 15 17 18- 20

No. of seedlings 5 6 10 4

120

32

40

36


1 2 3 4 5 Score

State the modal score.

A 6 C 3

B 5 D 2

5 The table is a cumulative frequency table which shows the height of 40 siakap fish

caught by a fish breeder.

Which of the following is the correct calculation for the mean leight of the 40 siakap fish?

A 40

)10(41)16(36)8(31)6(26

B 40

))40(41)30(36)14(31)46(26

C 40

)10(43)16(38)8(33)6(28

D 40

)40(43)30(38)14(33)6(28

Leight

(cm)

2630

31 35

36 40

41 45

Cumulati

ve

Frequenc

y

4 16 34 44

37


6 The table below shows the distribution of the scores of a group of pupils in a quiz.

The median score of the distribution is

A 1 C 2

B 1.5 D 2.5

7 Given that the mean for 3, 4, p, 7, 5, 6, 4 and 3 is 5. Find the value of p.

A 5 C 7

B 6 D 8

8 The table shows the number of goals scored by a

soccer team in a certain number of matches.

Number of

goals 0 1 2 3 4

Number of

matches k 2 6 2 1

If the mean of the number of goals scored is 2, find the value of k.

A 1 C 3

B 2 D 4

9

Score Frequency

95 99 100 104 105 109

5

11

9

Which of the following statements

Score 0 1 2 3 4 5

Frequency 2 6 1 1 5 1

38


regarding the above frequency table is incorrect?

A The midpoint of the class 95 99 is 97. B The modal class is 100 104 . C The size of each class interval is 5.

D The upper boundary of the class 105 109 is 109

10 The frequency table shows the number of books in the bags of a group of students.

Given that the mode is 6 books, find the greatest possible value of a.

A 5 C 6

C 7 D 8

Number of books 2 3 4 5 6

Frequency 4 6 5 a 8

39


STATISTICS

ANSWERS

1. SET

Paper 1

1 A 2 D 3 C 4 B 5 D

6 C 7 B 8 C 9 C 10 D

Paper 2

1 (a) R S

(b) ( R S ) T

2 (a) shade set P Q

(b) shade set PQ R

3 (a) A B

(b) A (B C)

R

S T

S

R S

T

P Q R

P Q R

A B

C

A B

C

40


4 (a) set P = {20, 24, 28}

set Q = { 20, 21, 2, 23, 24, 30}

(b) P Q = {25, 26, 27, 28, 29 } (c ) n (P Q) = 1

5 (a) ( P R) = {4, 5, 7} (b) Q R P = {1, 2, 3}

6 (a) P R = { } (b) Q R

7 (a) J L (b) J K L

8 (a)

(b) {3, 6, 8}

(c ) 4

9 (a) set P = {20, 25, 30, 35}

set Q = {20, 22, 24, 26, 28, 30, 32, 34}

(b) (i) 2

(ii) 6

10 (a)

(b)

E F

G

6

5 9 3

8

1 2

X Y

Z

X Y

Z

PROBABILITY

PAPER 1

1. B 2. D 3. D 4. D 5. D 6. B

7. A 8. B 9. A 10. C

PAPER 2

1. Answer:

(a)1/8 (b).1/16

2. (i)..5/12 (ii) 250

3. Answer:

S = {(1,A), (1,L), (6,A), (6,L), (8,A),(8,L)}

(a) 2/6 or 1/3

(b) 5/6

4. Answer

(a) in the modal class (b) more than 50kg

dalam kelas mod lebih daripada 50kg

2/5 9/20

5. Answer:

(a) 18/110 (b) .9/55

6. Answer:

(a) S={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),

(4,T),(5,H),(5,T),(6,H),(6,T)}

(b).2/12 / 1/6 ..6/12 / (e).1/12

7. Answer:

(a)..20 (b).12/19

8.

(c) 1/5. (b) 19/60

9.

(c) 4/7 (d) 15

Answer:

(a)

(b) 9

5


1. B 2. C 3. C 4. C 5. C 6. A

7. A 8. 9. C 10. A


1 (i) gradient of PQ (-1/3)

(ii) x-intercept for PQ (18)

(iii) the equation of PR ( y = 5/3x+6)

2.

(iv) the gradient of AB (3/5)

(v) the equation of ECF (y=3/5x+1)

(vi) x-intercept for ECF (-5/3 )

R

Y1

Y2

R

Y1

Y2

R

Y1

Y2

R

Y1

Y2

RR

RY1

RY2

Y1R

Y1Y1

Y1Y2

Y2R

Y2Y1

Y2Y2

y

3 (ii) find the gradient of PR (4/3)

(ii) find the y-intercept of PQ (y = 8x 48 )

4 (a) the value of h, (h=2)

(b) the coordinate F (4,0)

(c) the equation of straight line DE

(y = -2x + 2)

5 (a) the value of p (p = 7)

(b) the equation of straight line LN(y = 2x 7 )

6

(iii) the x-intercept for the straight line JK (1/2)

(iv) the equation of the straight line ML(2y=x+2)

(iii) the coordinate M (-2,0)

7

(a) State the equation of the straight line QR (x = 5/2)

(b) Find the equation of the straight line RS and hence , state its y-intercept. (y = -2x + 9),

c = 9

8 i) the coordinate of Q.(4, 0)

ii) the equation of RS.(2y = -3x -9)

9

m = 3, c = -9

10 (a) calculate the gradient of the straight line

AB (m = 2)

(b) find the equation of the straight line AB(y= 2x+1)

(c) find the value of h if the point (10,h) lies

on the straight line AB produced.(21)

STATISTICS

PAPER 1

1. A 2. B 3. C 4. D 5. D 6. B

7. D 8. A 9. D 10. B

PAPER 2

1 (a) The table shows the speed, in km h-1

, of 56 cars on a road.

Speed ( km h-1

) Frequency Midpoint

51 55 2

56 60 6

61 65 18

66 70 15

71 75 9

76 80 5

81 85 1

(i) Find the midpoints of the class intervals.

(ii) Calculate the mean speed of the cars.

(b) Complete the table given in the answer space below.

(c) For this part of question, use the graph paper.

By using a scale of 2 cm to 5 km h-1

on the x axis and 2 cm to 5 cars on the y-axis,

draw an ogive for the data.

Hence, find

(i) the median

(ii) the interquartile range.

2. The data above shows the pocket money, in RM, per week of a group of 40 students.

10 13 18 7 19 5 21 21

18 16 18 10 15 19 20 8

4 14 12 6 9 20 20 15

8 10 16 11 13 17 9 11

19 15 17 12 14 11 16 17

(a) Based on the data, complete the table below.

Pocket money (RM) Frequency Midpoint

1 3 0

4 6

(b) Based on the table,

(i) state the modal class,

(ii) calculate the mean,

of the data.

(c) For this part of the question, use a graph paper.

By using a scale of 2 cm to RM 3 on the x-axis and 2 cm to 1 student on the y-axis,

draw a histogram and a frequency polygon on the same graph paper.

3. (a) The table above shows the scores of a group of players in the game.

Score 10 11 12 13 14 15

Frequency 4 7 3 X 5 8

(i) Given the median is 12.5, find the value of x.

(ii) Given the mode is 15, find the maximum value of x.

(b) The data shows the mass, in kg, of a group of students.

49 45 47 66 66 62 68 43 46

53 50 61 65 67 63 68 65 44

48 63 62 45 61 67 56 43 44

57 64 60 52 58 46 59 41 43

64 65 47 46 52 50 54 42 60

(i) Based on the data, complete the frequency table below.

Mass (kg) Frequency

41 44

65 68

(ii) For this part of question, use the graph paper.

(a) By using a scale of 2 cm to 4 kg on the x-axis and 2cm to 1 student on the

y-axis, draw a histogram for the data.

(b) State one information obtained from the histogram.

4. The histogram shows the donations from a group of 40 students to a school fund.

(a) Based on the histogram, state the modal class of the data.

(b) Complete the frequency table below.

Donation (RM) Frequency Cumulative

frequency

Upper boundary

0 0

1 4 2

(c) For this part of question, use a graph paper.

(i) By using a scale of 2 cm to RM4 on the x-axis and 2 cm to 5 students on the

y-axis, draw an ogive for the data.

(ii) From the ogive, find the inter quartile range.

0

2

4

6

8

10

12

14

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

Fre

qu

en

cy

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

32.5

5. The ogive shows the scores of a group of 48 students in a Mathematics test.

(a) Based on the ogive, calculate the percentage of students who scored more than 80.

(b) Complete the table below.

Score Cumulative

frequency

Frequency Midpoint

60 64 0

65 69

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.

By using a scale of 2 cm to 5 scores on the x-axis and 2 cm to 2 students on the y-

axis, draw a frequency polygon for the data

0

10

20

30

40

50

60

64.5 69.5 74.5 79.5 84.5 89.5 94.5 99.5

Score

Cu

mu

lati

ve f

req

uen

cy

6. The frequency table above shows the masses, in kg, of watermelons in a lorry.

Mass (kg)

Frequency

1.5 1.9 4

2.0 2.4 10

2.5 2.9 26

3.0 3.4 84

3.5 3.9 50

4.0 4.4 15

4.5 4.9 8

5.0 5.4 3

(a) From the table, state

(i) the size of the class interval,

(ii) the midpoint of the modal class.

(b) (i) Based on the information, complete the table below.

Upper

boundary 1.45 5.45

Cumulative

Frequency 0 200

(ii) For this part of question, use a graph paper.

By using a scale of 2 cm to 0.5 kg on the x-axis and 2cm to 20 watermelons

on the y-axis, draw an ogive for the data.

(iii) From the ogive,

a) state the median of the data,

b) find the number of watermelons with the weight less than 3.8 kg.

7. The cumulative frequency table shows the marks of a group of 40 students in a test.

Mark 60 64 65 69 70 74 75 79 80 84 85 89 90 94

Cumulative

frequency 7 11 17 26 34 37 40

(a) Based on the information, complete the table in the answer corner.

(b) Find

(i) the size of the class interval,

(ii) the modal class of the data,

(iii) the midpoint of the modal class.

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.

By using a scale of 2 cm to 5 marks on the x-axis and 2 cm to 1 student on the

y-axis, draw a histogram for the data.

8. The data shows the ages of a group of 40 workers in a factory.

38 41 32 39 32 50 31 38

36 51 31 27 44 30 49 28

26 31 22 36 32 33 34 43

31 27 46 41 35 35 34 31

40 35 25 27 50 37 33 45

(a) Find the range of the data.

(b) Based on the data and by using a class interval of 5 years, complete the table

below.

Ages (years) Frequency Midpoint

21 25

26 30

(c) From the table,


(ii) calculate the mean,of the data.

(d) For this part of question, use a graph paper.

By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 workers on the y-

axis, draw a frequency polygon for the data.

9. The data shows the ages of a group of 40 workers in a factory.

38 41 32 39 32 50 31 38

36 51 31 27 44 30 49 28

26 31 22 36 32 33 34 43

31 27 46 41 35 35 34 31

40 35 25 27 50 37 33 45

(a) Find the range of the data.

(b) Based on the data and by using a class interval of 5 years, complete the table below.


21 25

26 30

(c) From the table,


(ii) calculate the mean,

of the data.

(d) For this part of question, use a graph paper.

By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 workers on the y-axis,

draw a frequency polygon for the data.

10. The frequency polygon shows the daily incomes, in RM, of 28 workers.

0

2

4

6

8

10

12

18 23 28 33 38 43 48

Daily income

Fre

qu

en

cy

(a) Based on the frequency polygon, complete the table below.

Daily

income

(RM)

Midpoint Upper

boundary Frequency

Cumulative

frequency

16 20 18 20.5 0 0

(b) Based on the table,


(ii) calculate the mean, of the data.

(c) For this part of question, use a graph paper.

(i) By using a scale 2 cm to RM5 on the x-axis and 2 cm to 4 workers on the

y-axis,draw an ogive for the data.

(i) From the ogive, find the number of workers whose daily income is more than RM38.

STATISTICS

Answer/Jawapan :

1.

(a) (i) and (b)

Speed (km h-1

)

Frequency Midpoint Upper

Boundary

Cumulative

Frequency

46 50 0 48 50.5 0

51 55 2 53 55.5 2

56 60 6 58 60.5 8

61 65 18 63 65.5 26

66 70 15 68 70.5 41

71 75 9 73 75.5 50

76 80 5 78 80.5 55

81 85 1 83 85.5 56

(ii) Mean = 56

)183()578()973()1568()1863()658()253(

= 56

3738

= 66.75 km h-1

.

(c)

(a) Median = 66.5 km h-1

(b) Interquartile range = 71.0 62.5 = 8.5 km h

-1

2. (a)

Pocket money(RM) Frequency Midpoint

1 3 0 2

4 6 3 5

7 9 5 8

10 12 8 11

13 15 7 14

16 18 9 17

19 21 8 20

(b) (i) Modal class = 16 18

(ii) Mean = 8978530

)820()917()714()811()58()35(

= 40

554

= RM 13.85

0

5

10

15

20

25

30

35

40

45

50

55

60

50.5 55.5 60.5 65.5 70.5 75.5 80.5 85.5 90.5

Speed (km/h)

Cu

mu

lati

ve

fre

qu

en

cy

From the ogive,

First quartile = 62.5

Median = 66.5

Third Quartile = 71.0

(iii)

0

1

2

3

4

5

6

7

8

9

10

3.5 6.5 9.5 12.5 15.5 18.5 21.5

Pocket money

Fre

qu

en

cy

3.5 6.5 9.5 12.5 15.5

3. (a) (i) x = ( 4+ 7 +3 ) ( 5 + 8 ) = 14 13 = 1

(ii) Maximum value of x = 7

Mass (kg) Frequency

41 44 7

45 48 8

49 52 5

53 56 3

57 60 5

61 64 8

65 68 9

0

1

2

3

4

5

6

7

8

9

10

40.5

44.5

48.5

52.5

56.5

60.5

64.5

68.5

Mass(kg)

Fre

qu

en

cy

40.5 44.5 48.8 52.5 56.5 60.5

64.5 68.5

(iii) The modal class of the data is 65 68 kg.

4 (a) Modal class = 9 12

(b)

Donation (RM) Frequency Cumulative

frequency

Upper boundary

0 0 0 0.5

1 4 2 2 4.5

5 8 5 7 8.5

9 12 12 19 12.5

13 16 8 27 16.5

17 20 5 32 20.5

21 24 3 35 24.5

25 28 1 36 28.5

(c) (i)

0

5

10

15

20

25

30

35

40

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

Cu

mu

lati

ve f

req

uen

cy Third quartile

First

9. 16.5

(ii) Interquartile range = 16.5 9.3 = RM 7.20

5 (a) The percentage of students who scored more than 80

= 48

1948 x 100%

= 48

29 x 100%

= 60.42 %

(b)

Score Cumulative frequency Frequency Midpoint

60 64 0 0 62

65 69 2 2 67

70 74 7 5 72

75 79 17 10 77

80 84 34 17 82

85 89 43 9 87

90 94 47 4 92

95 99 48 1 97

(c) Mean = 1491710520

)197()492()987()1782()1077()572()267(

= 48

3906

= 81.375

(d)

0

2

4

6

8

10

12

14

16

18

62 67 72 77 82 87 92 97 102

Score

Fre

qu

en

cy

6 (a) (i) Size of class interval = 1.95 1.45 = 0.5 kg

(ii) Modal class = 3.0 3.4

Midpoint of the modal class = 2

4.30.3

= 3.2

(b) (i)

Upper boundary Cumulative

frequency

1.45 0

1.95 4

2.45 14

2.95 40

3.45 124

3.95 174

4.45 189

4.95 197

5.45 200

(iii) (a) Median = 3.40 kg

(b) The number of watermelons with the weight less than 3.8 kg = 200 164 = 36.

7. (a)

Mark Cumulative

frequency

Frequency Midpoint

55 59 0 0 57

60 64 7 7 62

65 69 11 4 67

70 74 17 6 72

75 79 26 9 77

80 84 34 8 82

85 89 37 3 87

90 - 94 40 3 92

(b) (i) Size of class interval = 59.5 54.5 = 5 marks

(ii) Modal class = 75 79

(iii) Midpoint of the modal class = 2

7975

= 77

(c) Mean = 33896470

)392()387()882()977()672()467()762(

= 40

3020

= 75.5 marks

(d)

0

1

2

3

4

5

6

7

8

9

10

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5

Mark

Fre

qu

en

cy

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5

8. (a) Range = 51 22 = 29 years.

(b)


21 25 2 23

26 30 6 28

31 35 15 33

36 40 7 38

41 45 5 43

46 50 4 48

51 55 1 53

(c) (i) Modal class = 31 35

(ii) Mean = 14571562

)153()448()543()738()1533()628()223(

= 40

1435

= 35.875 years

(d)

0

2

4

6

8

10

12

14

16

18 23 28 33 38 43 48 53 58

Age (years)

Fre

qu

en

cy

9 (a) Range = 51 22 = 29 years.

(b)


21 25 2 23

26 30 6 28

31 35 15 33

36 40 7 38

41 45 5 43

46 50 4 48

51 55 1 53

(c) (i) Modal class = 31 35

(ii) Mean = 14571562

)153()448()543()738()1533()628()223(

= 40

1435

= 35.875 years

(d)

0

2

4

6

8

10

12

14

16

18 23 28 33 38 43 48 53 58

Age (years)

Fre

qu

en

cy

10. (a)

(b) (i) Modal class = 26 30

(ii) Mean = 2481040

)243()438()833()1028()423(

= 28

874

= RM 31.21

Daily

income(RM)

Midpoint Upper

boundary

Frequency Cumulativ

e

frequency

16 20 18 20.5 0 0

21 25 23 25.5 4 4

26 30 28 30.5 10 14

31 35 33 35.5 8 22

36 40 38 40.5 4 26

41 45 43 45.5 2 28

(c ) (i)

0

4

8

12

16

20

24

28

20.5 25.5 30.5 35.5 40.5 45.5

Daily income (RM)

Cu

mu

lati

ve

fr

eq

ue

nc

y

(ii) The number of workers whose daily income is more than RM 38 = 28 24 = 4

SULIT

NO. KAD PENGENALAN

ANGKA GILIRAN

PROJEK JAWAB UNTUK JAYA (JUJ) 2010

PRAKTIS BESTARI

1449/2 MATHEMATICS

Kertas 2

22

1 jam Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1 Tulis nombor kad pengenalan dan

angka giliran anda pada ruangan

yang disediakan .

2 Kertas soalan ini adalah dalam

dwibahasa .

3 Soalan dalam bahasa Inggeris

mendahului soalan yang sepadan

dalam bahasa Melayu .

4 Calon dibenarkan menjawab

keseluruhan atau sebahagian soalan

sama ada dalam bahasa Inggeris

atau bahasa Melayu.

5 Calon dikehendaki membaca

maklumat di halaman belakang

kertas soalan ini .

Kertas soalan ini mengandungi 34 halaman bercetak

Untuk Kegunaan Pemeriksa

Kod Pemeriksa :

Bahagian Soalan Markah

Penuh

Markah

Diperoleh

A

Jumlah

1

2

4

3

5

6

7

8 9

10

11

12

13

14

15

16

12

12

12

12

12

3

4

4

7

3

5

4

6

5

6

5

B

MATHEMATICAL FORMULAE

RUMUS MATEMATIK

The following formulae may be helpful in answering the questions. The symbols given are

the ones commonly used.

Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang

diberi adalah yang biasa digunakan.

RELATIONS

PERKAITAN

1 nmnm aaa 10 Pythagoras Theorem

Teorem Pithagoras

2 nmnm aaa 222 bac

3 nmnm aa )( 11 )(

)()(

Sn

AnAP

4

ac

bd

bcadA

11 12 )(1)( ' APAP

5 Distance / Jarak 13 12

12

xx

yym

2122

12 )()( yyxx

14 m = 6 Midpoint / Titik tengah

2,

2),( 21

21 yyxxyx m =

7 Average speed =

Purata laju =

8 Mean =

Min =

9 Mean =

Min =

1449/2 2010 Hak Cipta JUJ Pahang

SULIT

distance travelled time taken

jarak yang dilalui masa yang diambil

sum of data

number of data

hasil tambah nilai data

bilangan data

sum of (classmark frequency )

sum of frequencies

hasil tambah (nilai titik tengah kelas kekerapan)

hasil tambah kekerapan

y - intercept

x - intercept

pintasan - y

pintasan - x

SULIT 3

1449/2

SHAPES AND SPACE

BENTUK DAN RUANG

1 Area of trapezium = 2

1 sum of parallel sides height

Luas trapezium = 2

1 hasil tambah dua sisi selari tinggi

2 Circumference of circle = d = 2 r

Lilitan bulatan = d = 2 j

3 Area of circle = r2

Luas bulatan = j2

4 Curved surface area of cylinder = 2 rh

Luas permukaan melengkung silinder = 2 jt

5 Surface area of sphere = 4 r2

Luas permukaan sfera = 4 j2

6 Volume of right prism = cross sectional area length

Isipadu prisma tegak = luas keratan rentas panjang

7 Volume of cylinder = r2h

Isipadu silinder = j2t

8 Volume of cone = 3

1 r2h

Isipadu kon = 3

1 j2t

9 Volume of sphere = 3

4 r3

Isipadu sfera = 3

4 j3

10 Volume of right pyramid = 3

1 base area height

Isipadu pyramid tegak = 3

1 luas tapak tinggi

11 Sum of interior angles of a polygon

Hasil tambah sudut pedalaman poligon

= ( n 2 ) 180o

1449/2 2010 Hak Cipta JUJ Pahang [Lihat halaman sebelah

SULIT

SULIT 4

1449/2

12 =

=

13 =

=

14 Scale factor, PA

PAk

'

Faktor skala, PA

PAk

'

15 Area of image = k2 area of object

Luas imej = k2 luas objek

1449/2 2010 Hak Cipta JUJ Pahang

SULIT

Section A

arc length

circumference of circle

angle subtended at centre

360o

panjang lengkuk

lilitan bulatan

sudut pusat

360o

area of sector

area of circle

angle subtended at centre

360o

luas sektor

luas bulatan

sudut pusat

360o

Bahagian A

[52 marks]

[52 markah]

Answer all questions from this section.

Jawab semua soalan dalam bahagian ini.

1. The Venn diagram in the answer space shows sets P, Q and R, such that the universal set,

= P Q R. On the diagrams in the answer space, shade the set

Gambar rajah Venn di ruang jawapan menunjukkan set P, set Q dan set R dengan

keadaan = P Q R. Pada rajah di ruang jawapan, lorek set

(a) 'QP

(b) )( RPQ [3 marks] / [3 markah]

Answer / Jawapan :

(a)

(b)

P

R

Q

R

Q P

2. Solve the quadratic equation :

Selesaikan persamaan kuadratik :

)53(2

12 pp [4 marks] / [4 markah]

Answer / Jawapan:

3. Calculate the value of x and of y that satisfy the following simultaneous linear equations :

Hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut:

932 yx

23

1 yx

[4 marks] / [4 markah]

Answer / Jawapan :

4. (a) Determine whether the following sentence is a statement or non-statement.

Tentukan sama ada ayat berikut adalah pernyataan atau bukan pernyataan.

(b) Write two implications based on the following statement:

Tulis dua implikasi berdasarkan pernyataan berikut:

(c) Make a general conclusion by induction for the sequence of numbers 5, 16, 37, 80, ...

which follows the following pattern:

Buat satu kesimpulan umum secara aruhan untuk urutan nombor 5, 16, 37, 80, ... yang

mengikut pola berikut:

5 = 4(1) + 13

16 = 4(2) + 23

37 = 4(3) + 33

80 = 4(4) + 43


Answer / Jawapan :

(a) .

(b) Implication 1 / Implikasi 1 :

..

Implication 2 / Implikasi 2 :

(c)

7 is a factor of 30 7 ialah faktor bagi 30

x is an even number if and only if x is divisible by 2 x ialah nombor genap jika dan hanya jika x boleh dibahagi tepat dengan 2

5. Diagram 5 shows a right prism LMNOPQ with horizontal base LMNO and POL as its

cross-section.

Rajah 5 menunjukkan sebuah prisma tegak LMNOPQ dengan tapak mengufuk dan POL

sebagai keratan rentasnya.

(a) Name the angle between the plane POM and the plane PONQ. Namakan sudut antara satah POM dengan satah PONQ

(b) Calculate the angle between the plane POM and the plane PONQ Hitung sudut antara satah POM dengan satah PONQ.


Answer / Jawapan :

(a)

(b)

P

Q

N

O

L

M

16 cm

Diagram 5

Rajah 5

12 cm

6. It is given that matrix A =

k2

34

Diberi bahawa matriks A =

k2

34

(a) Find the value of k if the inverse of matrix A does not exist.

Cari nilai k jika songsangan matriks A tidak wujud.

(b) Write the following simultaneous linear equations as matrix equation :

Tuliskan persamaan linear serentak berikut dalam bentuk persamaan matriks:

4m + 3n = 5

2m - n = 5

Hence, by using matrix method, calculate the value of m and of n.

Seterusnya dengan menggunakan kaedah matriks, hitungkan nilai m dan nilai n.


Answer / Jawapan :

(a)

(b)

7. Diagram 7 shows a circle PQR with centre O, and three semi circles. Given that POQ

= QOR = ROP and OP = OQ = OR = 28 cm. Rajah 7 menunjukkan sebuah bulatan PQR berpusat O dan tiga buah separuh bulatan.

Diberi bahawa POQ = QOR = ROP dan OP = OQ = OR = 28 cm.

Using 7

22 , calculate

Menggunakan 7

22 , hitung

a) the perimeter, in cm, of the shaded region

perimeter, dalam cm, kawasan berlorek

b) the area, in cm2, the shaded region

luas, dalam cm2, kawasan berlorek


Answer / Jawapan :

(a)

(b)

R

P

Q Diagram 7 /

Rajah 7

O

8. Diagram 8 shows a composite solid. PQRSTU is a right prism with PQ = QR. SUV is a

half circular cylinder. They are joined at the rectangular plane PRSU. STUV is its cross-

section.

Rajah 8 menunjukkan sebuah gabungan pepejal. PQRSTU ialah sebuah prisma tegak

dengan PQ = QR. SUV ialah sebuah separuh silinder. Kedua-dua pepejal ini dicantum

pada satah PRSU. STUV ialah keratan rentasnya.

Using 7

22 , calculate the volume, in cm3, of the composite solid.

Menggunakan 7

22 , hitung isipadu, dalam cm3 , gabungan pepejal itu.


Answer / Jawapan :

U

V

S

R

Q

T

P

W Diagram 8

Rajah 8

13 cm

10 cm

10 cm

9. Diagram 9 shows three cards labelled with letters in box A and four numbered cards in

box B.

Rajah 9 menunjukkan tiga kad huruf di dalam kotak A dan empat kad nombor di dalam

kotak B.

A B

A code is form by a letter selected from box A and a number from box B.

Satu kod dibina menggunakan huruf yang dipilih daripada kotak A dan satu nombor

yang dipilih daripada kotak B.

(a) Listing the sample space of the possible outcomes of the event.

Senaraikan ruang sample bagi kesudahan peristiwa.

(b) Hence, find the probability of

Seterusnya, cari kebarangkalian bahawa

(i) choosing a code that contain number 2 memilih kod yang mengandungi nombor 2

(ii) choosing a code that contain a vowel or number 3 memilih kod yang mengandungi huruf vokal atau nombor 3


Answer / Jawapan :

(a)

(b)(i)

(ii)

E R K 1 2 3 4

10. In diagram 10, PQ is parallel to SR and O is the origin. Dalam rajah 10, PQ adalah selari dengan SR dan O ialah titik asalan.

Find

Cari

(a) the gradient of PQ.

Kecerunan PQ.

(b) the equation of straight line SR.

persamaan garis lurus SR.

(c) the y-intercept of SR.

pintasan-y bagi garis lurus SR.


Answer / Jawapan :

(a)

(b)

(c)

y

x

Q(1, 7)

R(6, 2) P(-2, 1)

S

Diagram 10

Rajah 10

11. Diagram 11 shows a speed-time graph for the movement of a particle for a period of

18 seconds.

Rajah 11 menunjukkan graf laju-masa bagi pergerakan satu zarah dalam tempoh 18

saat.

Speed (ms-1

)

(a) How long does the particle move in uniform speed ? Berapa lamakah zarah itu bergerak dengan laju malar ?

(b) Calculate the rate of change of speed, in ms-1, in the first 8 seconds. Hitung kadar perubahan laju, dalam ms

-1, dalam tempoh 8 saat yang pertama.

(c) Calculate the value of u, if the distance travelled in the last 10 seconds is 172 m. Hitung nilai u, jika jumlah jarak yang dilalui dalam tempoh 10 saat terakhir ialah

172m.


Answer / Jawapan :

(a)

(b)

(c)

Diagram 11

Rajah 11

32

18 12

u

Time (s)

Masa(s)

16

8 0

Laju(ms-1

)

Section B

Bahagian B

[48 marks]

[48 markah]

Answer any four questions from this section.

Jawab mana-mana empat soalan dalam bahagian ini.

12.(a) Table 12 in the answer space shows the values of x and of y for the equation

432 2 xxy .

Find the value of m and of p. [2 marks]

Jadual 12 di ruang jawapan menunjukkan nilai-nilai x dan y bagi persamaan

432 2 xxy .

Cari nilai bagi m dan p. [2 markah]

(b) For this part of the question, use graph paper. You may use a flexible curve ruler. Untuk ceraian soalan ini, gunakan kertas graf. Anda boleh menggunakan pembaris

fleksibel.

By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw

the graph of 432 2 xxy for 52 x . [4 marks]

Dengan menggunakan skala 2 cm kepada 1 unit pada paksi-x dan 2 cm kepada 5 unit

pada paksi-y, lukis graf 432 2 xxy untuk 52 x . [4 markah]

(c) Using the graph drawn in 12(b), find Menggunakan graf yang dilukis di 12(b), cari

(i) the value of y when x = -1.5.

nilai y apabila x = -1.5

(ii) the values of x when y = 0. [3 marks]

Nilai-nilai x apabila y = 0 [3 markah]

(d) Draw a suitable straight line on your graph to find the value of x that satisfies the

equation 1822 2 xx for 52 x . State the value of x. [3 marks]

Lukis garis lurus yang sesuai pada graf anda untuk mencari nilai x yang memuaskan

persamaan 1822 2 xx untuk 52 x . Nyatakan nilai x itu. [3 markah]

Answer / Jawapan :

(a)

x -2 -1 0 1 2 3 4 4.5 5

y m 1 -4 -5 -2 p 16 23 31

Table 12 / Jadual 12

m = ..............................................

p = ..............................................

(b) Refer graph / rujuk graf .

(c) (i) y = ...........................................

(ii) x = ............................................ , ...............................................

(d) ...................................................................

13. (a) Transformation R is a reflection in the line x = 6 and transformation P is a clockwise

rotation of 90 about the centre (4, 2).

Penjelmaan R ialah pantulan pada garis x = 6 dan penjelmaan P ialah putaran 90 arah

jam pada pusat (4,2).

State the coordinates of the image of point (9, 4) under each of the following

transformations:

Nyatakan koordinat imej bagi titik (9, 4) di bawah setiap penjelmaan berikut:

(i) R

(ii) P

(iii) PR [4 marks] / [4 markah]

(b) Diagram 13 shows three pentagons, ABCDE, HJKLM and PQRST, drawn on a Cartesian

plane.

Rajah 13 menunjukkan tiga pentagon, ABCDE, HJKLM dan PQRST dilukis di atas satah

Cartesan.

Diagram 13

Rajah 13

6

4

2

0 10 8 6 4 2 x

y

10

8

T

P Q

S

M L

R

J

K

D

E A

B

C

H

(i) HJKLM is the image of ABCDE under the combined transformation TU. Describe in full the transformation :

HJKLM ialah imej bagi ABCDE di bawah gabungan penjelmaan TU. Huraikan

selengkapnya penjelmaan :

(a) U

(b) T [5 marks] / [5 markah]

(ii) It is given that pentagon ABCDE represents a region of area 12 m2. Calculate the

area , in m2 , of the region represented by the shaded region.

Di beri bahawa pentagon ABCDE mewakili suatu kawasan yang mempunyai luas

12 m2. Hitungkan luas, dalam m

2, kawasan yang diwakili oleh rantau berlorek.

[3 marks]

Answer / Jawapan :

(a) (i)

(ii)

(iii)

(b)(i) U :

.

T : .

..

(ii)

14. Table 14.1 shows the frequency distribution of the number of books read by 40 students in a

reading programme in a school.

Jadual 14.1 menunjukkan taburan kekerapan bagi bilangan buku yang dibaca oleh 40

orang pelajar dalam suatu program membaca di sebuah sekolah.

Class interval

Selang Kelas

Frequency

Kekerapan

5 7 3

8 10 5

11 13 9

14 16 10

17 19 8

20 22 4

23 25 1

(a) Based on the data in Table 14.1 complete Table 14.2 in the answer space. [4 marks]

Berdasarkan Jadual 14.1, lengkapkan Jadual 14.2 di ruang jawapan. [4 markah]

(b) Calculate the estimated mean of books read by a student. [3 marks]

Hitung min anggaran buku yang dibaca oleh seorang pelajar. [3 markah]

(c) For this part of the question, use the graph paper

Untuk ceraian soalan ini gunakan kertas graf

By using a scale of 2 cm to 3 books on the horizontal axis and 2 cm to 5 students on the

vertical axis, draw an ogive for the data. [4 marks]

Dengan menggunakan skaia 2 cm kepada 3 buah buku pada paksi mengufuk dan 2 cm

kepada 5 orang pelajar pada paksi mencancang , lukiskan satu ogif bagi data di atas.

[4 markah]

(d) Based on the ogive in 14(c), give one information about the marks obtained. [1 marks]

Berdasarkan ogif (b), beri satu maklumat tentang markah yang diperoleh. [1 markah]

Table 14.1

Jadual 14.1

Answer / Jawapan :

(a)

Class interval

matematik spm 2010

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