TOP ONE MATHS F4

Contents





Redod Pencapaian Pentaksiran Murid iii – vi 6.2 Systems of Linear Inequalities in Two Variables 71
Sistem Ketaksamaan Linear dalam Dua Pemboleh Ubah
Quadratic Functions and Equations in One SPM Practice 6 76
CHAPTER Variable 1 HOTS Challenge 78
1
Fungsi dan Persamaan Kuadratik dalam Satu Pemboleh
Ubah QQ 78
1.1 Quadratic Functions and Equations 1
Fungsi dan Persamaan Kuadratik CHAPTER Graphs of Motion 79
7
SPM Practice 1 11 Graf Gerakan
HOTS Challenge 12 7.1 Distance-Time Graphs 79
QQ 12 Graf Jarak-Masa
7.2 Speed-Time Graphs 82
CHAPTER Number Bases 13 SPM Practice 7 85
Graf Laju-Masa
2
Asas Nombor
HOTS Challenge 90
2.1 Number bases 13 QQ 90
Asas nombor
SPM Practice 2 29 Measures of Dispersion for Ungrouped
HOTS Challenge 30 CHAPTER Data 91
8
QQ 30 Sukatan Serakan Data Tak Terkumpul
CHAPTER Logical Reasoning 31 8.1 Dispersion 91
3
Serakan
Penaakulan Logik
8.2 Measures of Dispersion 94
Sukatan Serakan
3.1 Statements 31
Pernyataan SPM Practice 8 101
3.2 Argument 38 HOTS Challenge 104
Hujah QQ 104
SPM Practice 3 42
HOTS Challenge 45 CHAPTER Probability of Combine Events 105
9
QQ 45 Kebarangkalian Peristiwa Bergabung
CHAPTER Operations of Sets 46 9.1 Combined Events 105
4

Peristiwa Bergabung
Operasi Set
9.2 Dependent Events and Independent Events 108
4.1 Intersection of Sets 46 Peristiwa Bersandar dan Peristiwa Tidak Bersandar
Persilangan Set 9.3 Mutually Exclusive Events and Non-Mutually
4.2 Union of Sets 49 Exclusive Events 111
Kesatuan Set Peristiwa Saling Eksklusif dan Peristiwa
4.3 Combined Operations on Sets 51 Tidak Saling Eksklusif
Gabungan Operasi Set 9.4 Application of Probability of Combined Events 113
Aplikasi Kebarangkalian Peristiwa Bergabung
SPM Practice 4 54
HOTS Challenge 59 SPM Practice 9 116
QQ 59 HOTS Challenge 122
QQ 122
CHAPTER Network in Graph Theory 60 PAK-21 Corner 123
5
Rangkaian dalam Teori Graf
CHAPTER Consumer Mathematics: Financial
5.1 Network 60 10 Management 124
Rangkaian Matematik Pengguna: Pengurusan Kewangan
SPM Practice 5 65 10.1 Financial Planning and Management 124
HOTS Challenge 67 Perancangan dan Pengurusan Kewangan
QQ 67 SPM Practice 10 138
140
CHAPTER Linear Inequalities in Two Variables 68 HOTS Challenge QQ 141
6
Ketaksamaan Linear dalam Dua Pemboleh Ubah
142
PAK-21 Corner
6.1 Linear Inequalities in Two Variables 68
Ketaksamaan Linear dalam Dua Pemboleh Ubah SPM Year-End Assessment 143

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00 CONTENT TOP 1 MATH F4.indd 2 13/02/2020 5:21 PM

4 Operations on Sets

Operasi Set





4.1 Intersection of Sets pg. 94 – 105
Textbook
Persilangan Set


SMART Notes

1. Symbol  represent the same elements in all sets in a Example
universal set, . Shade the set / Lorekkan set (A  B)  (B  C).
Simbol  mewakili unsur-unsur yang sama dalam semua A B C
set dalam satu set semesta, .
2. The intersection of set A and set B, written as set A  B,
is the set of all elements which are common to both
sets A and B.
Persilangan set A dan set B, ditulis sebagai set A  B, ialah
set semua unsur yang serupa bagi kedua-dua set A dan B. Step 1: Numbered all parts involved.
Langkah 1: Nomborkan semua bahagian yang terlibat.
3. For any set A within the universal set, , the compliment
of set A, written as A, is a set of elements of  that are A B C
not the elements of set A.
Bagi mana-mana set A dalam set semesta, , pelengkap 1 2 3 4 5
bagi set A, ditulis sebagai A, adalah satu set unsur dalam
 yang bukan unsur bagi set A.
4. The most effective method to solve problems involving Step 2: Determine the elements in brackets first.
sets is by numbering the parts involved. Langkah 2: Kenal pasti unsur dalam tanda kurung
Kaedah yang paling berkesan untuk menyelesaikan masalah terlebih dahulu.
berkaitan set ialah dengan menomborkan bahagian yang (A  B)  (B  C)
terlibat. (1, 2  2 , 3, 4)  (2, 3, 4  4 , 5)
(a) List the elements of sets one by one based on the (2)  (4)
given information.
Senaraikan unsur-unsur set itu satu persatu Step 3: Hence, shade area 2 and 4 only in the
berdasarkan maklumat yang diberi. diagram.
(b) For intersection of sets, identify the same elements Langkah 3: Maka, lorekkan kawasan 2 dan 4 sahaja
in two or more sets involved. pada rajah.
Bagi persilangan set, kenal pasti unsur-unsur yang A B C
sama dalam dua atau lebih set yang terlibat.
(c) After doing the operations on intersection or union
of sets, the last elements listed will be the answer. 1 2 3 4 5
Shade only the answer in the Venn diagram.
Selepas melakukan operasi persilangan atau kesatuan
set, unsur-unsur terakhir yang tersenarai ialah
jawapannya. Lorekkan jawapan sahaja dalam gambar
rajah Venn. NOTES


1. Determine the intersection of the following sets. PL 3
Tentukan persilangan bagi set berikut.
Example (a) Set D = {G, A, R, D, E, N},
Set A = {2, 7, 8, 9}, set E = {D, O, M, A, I, N},
set B = {2, 3, 6, 7}, and/ dan set F = {B, A, C, K}.
and/ dan set C = {5, 6, 7, 8}.
D  E  F = {A}
A  B  C = {7}






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Mathematics Form 4 Chapter 4 Operation on Set

(b) Set  = {x : x is an integer and 1  x  15}, (c) Set  = {1, 2, 3, 4, 5, 6, 7},
Set  = {x : x ialah integer dan 1  x  15}, Set  = {1, 2, 3, 4, 5, 6, 7},
set G = {factors of 12}, set M = {3, 6, 8, 9},
set G = {faktor bagi 12}, set M = {3, 6, 8, 9},
set H = {multiple of 4}, set N = {factors of 6},
set H = {Gandaan 4}, set N = {faktor bagi 6},
and set J = {integers more than 10}. and set P = {prime numbers}.
dan set J = {integer lebih daripada 10}. dan set P = {nombor perdana}.

G = {1, 2, 3, 4, 6, 12 } M = { 3 , 6, 8, 9}
H = {4, 8, 12 } N = {1, 2, 3 , 6}
J = {11, 12 , 13, 14, 15} P = {2, 3 , 5, 7}
∴ G  H  J = {12} ∴ M  N  P = {3}






2. Determine the compliment of the following sets. PL 3
Tentukan pelengkap bagi set berikut.
Example
Given that/ Diberi bahawa
 = {x : x is an integer and 10  x  20},
 = {x : x ialah integer dan 10  x  20},
set S = {x : x is multiple of 4},
set S = {x : x adalah gandaan 4},
and set R ={x : x is a prime number}.
dan set R = {x : x ialah nombor perdana}.
List the elements of set S, S, R and R.
Senaraikan unsur-unsur set S, S, R dan R.

 = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
Set S = {12, 16, 20}
Set S= {10, 11, 13, 14, 15, 17, 18, 19}
Set R = {11, 13, 17, 19}
Set R = {10, 12, 14, 15, 16, 18, 20}

(a) Based on the Venn diagram below, list the (b) Draw a Venn diagram to illustrate the relationship
elements of set J, J, K, K and . between the following set.
Berdasarkan gambar rajah Venn di bawah, senaraikan unsur- Lukis satu gambar rajah Venn untuk menggambarkan
unsur bagi set J, J, K, K dan . hubungan antara set berikut.
 = {a, b, c, d, e, f, g, h, i}
J = {a, e, i}
J K
3 2 B = {a, b, c, d, e}
1
6 9
5 4 8 7 J B
b c f
g i
d
h
Set J = {1, 3, 4, 6, 8}
Set J’ = {2, 5, 7, 9} a e
Set K = {2, 3, 6, 7, 8, 9}
Set K = {1, 4, 5}
Set  = {1, 2, 3, 4, 5, 6, 7, 8, 9}








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Mathematics Form 4 Chapter 4 Operation on Set

3. Solve the following problems involving intersection of sets. PL 4
Selesaikan masalah yang melibatkan persilangan set berikut.

Example
In a hostel of 70 students, 56 are members of Briget Bestari and 32 are members of Taekwondo Club. How
many students are the members of both clubs if 4 students are members of neither?
Di sebuah asrama dengan 70 orang pelajar, 56 orang adalah ahli Briget Bestari dan 32 orang adalah ahli Kelab Taekwondo. Berapakah
bilangan pelajar yang menjadi ahli kedua-dua kelab jika 4 orang pelajar bukan ahli kedua-duanya?

Let x = Number of Students who join both club.
Biar x = bilangan pelajar yang menyertai kedua-dua kelab.
(56 – x) + (32 – x) + x + 4 = 70
92 – x = 70
x = 22
Thus, 22 students join both clubs.
Maka, 22 orang pelajar menyertai kedua-dua kelab.

(a) One quarter of the passenger in a bus are left- (b) A sport club providing facilities in badminton (B),
1 squash (S) and tennis (T) has 60 active members.
handed, are left-handed and short-sighted, 17
12 The table below shows the number of members
are short-sighted while 23 are neither left-handed who play various games.
and short-sighted. Sebuah kelab sukan yang menyediakan kemudahan untuk
Satu perempat daripada penumpang sebuah bas adalah permainan badminton (B), squash (S) dan tenis (T) mempunyai
1
kidal, adalah kidal dan rabun jauh, 17 orang adalah 60 orang ahli aktif. Jadual di bawah menunjukkan bilangan
12
rabun jauh manakala 23 orang lagi tidak kidal mahupun ahli yang bermain pelbagai permainan.
rabun jauh. B S T B B S
and/dan and/dan and/dan
(i) Find the total number of passengers of the S T T
bus. 30 31 36 13 16 15
Cari jumlah penumpang bas itu.
(ii) Find the number of passenger who are left- (i) Find the number of members who play all
handed and short-sighted. games.
Cari bilangan penumpang yang kidal dan rabun jauh. Cari bilangan ahli yang bermain semua permainan.
(ii) Find the number of members who play
(i) Let/ Biar tennis only.
x = Total number of passengers Cari bilangan ahli yang bermain tenis sahaja.
x = Jumlah penumpang
L = Passengers who is left-handed (i) Let/ Biar
L = Penumpang yang kidal x = Number of members who play all games
S = Passengers who is short-sighted x = Bilangan ahli yang bermain semua permainan
S = Penumpang yang rabun jauh n(B  S) = 13 – x
1 1 n(B  T ) = 16 – x
N(L) = x, n(L  S) = x,
4 12 n(S  T ) = 15 – x
n(S) = 17, n(L  S) = 23 n(B only / sahaja) = 30 – (13 – x) – (16 – x) – x
 
 1 x – 12 x + 17 – 12x  + 12 x + 23 = x n(S only / sahaja) = 31 – (13 – x) – (15 – x) – x
1
= 1 + x
1
1
4
1 x + 40 = x = 3 + x
6 n(T only / sahaja) = 36 – (16 – x) – (15 – x)
5 x = 40 = 5 + x
6 x + 1 + 13 – x + 16
x = 48 – x + 3 + x + 15 – x + 5 + x + x = 60
Thus, the total number of passengers in the 53 + x = 60
bus are 48 passengers. x = 7
Maka, jumlah penumpang di dalam bas tersebut ialah Thus, the total number of members who play
48 orang. all games are 7 members.
Maka, jumlah ahli yang bermain semua permainan
1
(ii) n(L S) = x ialah 7 orang.
12
1 (ii) n(T only / sahaja) = 5 + x
= (48)
12 = 5 + 7
= 4 = 12

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Mathematics Form 4 Chapter 4 Operation on Set

4.2 Union of Sets pg. 106 – 115
Textbook
Kesatuan Set

SMART Notes


1. Symbol  represent all emelents in all sets in a universal set, .
Simbol  mewakili semua unsur dalam semua set dalam satu set semesta, .
2. The union of sets A and B, written as set A  B, is the set of all elements in both sets.
Kesatuan set A dan B, ditulis sebagai set A  B, adalah set semua unsur dalam kedua-dua set.


4. Determine the union of the following sets. PL 3
Tentukan kesatuan set bagi set berikut.
Example (a) Set Q = {3, 6, 9},
Set J = {C, L, I, E, N, T} and/ dan set R = {3, 5, 7, 9},
set K = {C, U, S, T, O, M, E, R} and/ dan set S ={2, 5, 6, 8}.
Q  R  S = {2,3,5,6,7,8,9}
J  K = {C, L, I, E, N, T, U, S, O, M, R}





(b) Set J = {b, d, g, h}, (c) Shade the region A  B  C on the Venn
set K = {c, d, e, h}, diagram below.
and/ dan set L = {b, d, e, g}. Lorekkan kawasan A  B  C pada gambar rajah Venn
di bawah.
J  K  L = {b, c, d, e, g, h}

A
B
C





5. Determine the complement of the union of the following sets. PL 3
Tentukan pelengkap bagi kesatuan set berikut.
Example (a) Given the universal set,  = {integers from 1 to
Given the universal set,  = {integers from 1 to 15}, 10}, set J = {3, 4, 5, 6}, set K = {5, 6, 7, 8} and set
set P = {prime numbers} and set Q = {odd numbers}. L = {4, 6, 7, 10}. Determine set J  K  L and
List the elements of set P  Q. shade the region on the Venn diagram below.
Diberi set semesta,  = {integer dari 1 hingga 15}, Diberi set semesta,  = {integer dari 1 hingga 10},
set P = {nombor perdana} dan set Q = {nombor ganjil}. set J = {3, 4, 5, 6}, set K = {5, 6, 7, 8} dan set
Senaraikan unsur-unsur bagi set P  Q. L = {4, 6, 7, 10}. Tentukan set J  K  L dan lorekkan
pada gambar rajah Venn di bawah.
 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
P = {2, 3, 5, 7, 11, 13} J  K  L = {1, 2, 3, 5, 8, 9, 10}
Q = {1, 3, 5, 7, 9, 11, 13, 15}
P = {1, 4, 6, 8, 9, 10, 12, 14, 15} J K
Q = {2, 4, 6, 8, 10, 12, 14} 3 5 8
1
P  Q = {1, 2, 4, 6, 8, 9, 10, 12, 14, 15} 6
2 4 7
9 L 10






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Mathematics Form 4 Chapter 4 Operation on Set

(b) The Venn diagram below shows universal set, , (c) The Venn diagram below shows number of
set S, set T and set U. elements in the universal set, , set M and set N.
Gambar rajah Venn di bawah menunjukkan set semesta, , Gambar rajah Venn di bawah menunjukkan bilangan unsur
set S, set T dan set U. dalam set semesta, , set M dan set N.


S T
2 M N
1 3
7 3x x 2
6 4
5 8
U 5

Find set S’  T’  U’. If n(N) = n(M  N), find the value of x.
Cari set S  T  U. Jika n(N) = n(M  N), cari nilai x.

S’  T’  U’ = {1, 2, 3, 4, 5, 6, 8} n(N’) = n(M  N)
3x + 5 = 3x + x + 2
3 x + 5 = 4x + 2
x = 3









6. Solve the following problems involving union of sets. PL 4
Selesaikan masalah yang melibatkan kesatuan set berikut.

Example (a) In a class of 36 students, 24 students passed in
In a group of students, 26 students play volleyball, Mathematics, 16 students passed in Geography
20 students play hockey and 8 students play both and 8 students passed in both subjects. Find the
volleyball and hockey. If all students play at least number of students who failed in both subjects.
one of the two games, how many students are there Dalam sebuah kelas seramai 36 orang pelajar, 24 orang
in the group? lulus Matematik, 16 orang lulus Geografi dan 8 orang lulus
Dalam sekumpulan pelajar, 26 orang bermain bola tampar, kedua-dua subjek. Cari bilangan pelajar yang gagal dalam
20 orang bermain hoki dan 8 orang bermain kedua-duanya. kedua-dua subjek.
Sekiranya semua pelajar bermain sekurang-kurangnya satu
daripada dua permainan tersebut, berapakah bilangan pelajar Number of students who passed mathematics only
dalam kumpulan itu? Bilangan pelajar yang lulus matematik sahaja
= 24 – 8
Number of students who play volleyball only = 16
Bilangan pelajar yang bermain bola tampar sahaja
= 26 – 8 Number of students who passed Geography only
= 18 Bilangan pelajar yang lulus Geografi sahaja
= 16 – 8
Number of students who play hockey only = 8
Bilangan pelajar yang bermain hoki sahaja
= 20 – 8 Number of students who failed both subject
= 12 Bilangan pelajar yang gagal kedua-dua subjek
= 36 – 16 – 8 – 8
Total number of students = 4 students / orang
Jumlah pelajar
= 18 + 12 + 8
= 38 students / orang












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Mathematics Form 4 Chapter 4 Operation on Set

(b) In a survey, 60 men drink tea or coffee or both. (c) The Venn diagram below shows the number of
If 30 men drink coffee but not tea and 50 men teachers who like pineapples (P), rambutans (R)
drink coffee, how many men drink tea but not and mangoesteens (M).
coffee? Gambar rajah Venn di bawah menunjukkan bilangan guru
Dalam satu kaji selidik, 60 orang lelaki minum teh atau kopi yang menggemari nanas (P), rambutan (R) dan manggis (M).
atau kedua-duanya. Jika 30 orang minum kopi tetapi tidak P R M
teh dan 50 orang minum kopi, berapakah bilangan lelaki
yang minum teh tetapi tidak kopi?
10 6
Number of men who drink coffee and tea
Bilangan lelaki yang minum kopi dan teh
= 50 – 30 All teachers like at least one of the three type of
fruits, 24 like rambutans, 26 like mangoesteens
= 20
and 22 like pineapples. Find the total number of
Number of men who drink tea but not coffee teachers in the school.
Bilangan lelaki yang minum teh tetapi tidak kopi Semua guru menggemari sekurang-kurangnya satu daripada
= 60 – 30 – 20 tiga jenis buah-buahan tersebut, 24 orang menggemari
= 10 men / orang rambutan, 26 orang menggemari manggis dan 22 orang
menggemari nanas. Cari jumlah guru di sekolah itu.

Number of teachers who like pineapples and
rambutan
Bilangan guru yang menggemari nanas dan rambutan
= 22 – 10 = 12
Number of teachers who like rambutan only
Bilangan guru yang menggemari rambutan sahaja
= 24 – 12 – 6 = 6
Number of teachers who like mangoesteens only
Bilangan guru yang menggemari manggis sahaja
= 26 – 6 = 20
Total number of teachers
Jumlah guru
= 10 + 12 + 6 + 6 + 20
= 54 teachers / orang



4.3 Combined Operation on Sets pg. 116 – 127
Textbook
Gabungan Operasi Set


7. Determine the following combined operations on sets using. PL 2
Tentukan gabungan operasi set berikut.
Example (a) The Venn diagram below shows the elements in
Given the universal set,  = {1, 2, 3, 4, 5, 6, 7, 8}, sets , P, Q and R.
set J = {1, 2, 3}, set K = {2, 3, 5, 7} and set Gambar rajah Venn di bawah menunjukkan unsur-unsur dalam
L = {5, 6, 7}, find set J  K  L. set , P, Q dan R.
Diberi set semesta,  = {1, 2, 3, 4, 5, 6, 7, 8}, set J = {1, 2, 3},
set K = {2, 3, 5, 7} dan set L = {5, 6, 7}, cari set J  K  L. P Q
b
a d
J  K  L = {1, 2, 3, 5, 6, 7}
c
f e
h
g
R

Find set (P  Q)  R.
Cari set (P  Q)  R.
(P  Q) = {b, c}
R = {c, e, f, g}
(P  Q)  R = {b, c, e, f, g}

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Mathematics Form 4 Chapter 4 Operation on Set

(b) State the relationship represented by the shaded region in each of the following Venn diagrams using
the symbols  and .
Nyatakan hubungan yang diwakili oleh rantau berlorek dalam setiap rajah Venn berikut menggunakan simbol  dan .



(i) (ii)
P Q
R S


R
T



(P  Q)  R (S  T)  R





8. Determine the complement of the following combined operations on sets. PL 3
Tentukan pelengkap bagi gabungan operasi set berikut.
Example (a) The Venn diagram below shows the elements in
Given the universal set,  = {1, 2, 3, 4, 5, 6, 7, 8}, set sets , P, Q and R.
J = {1, 2, 3}, set K = {2, 3, 5, 7} and set L = {5, 6, 7}. Gambar rajah Venn di bawah menunjukkan unsur dalam
Find set J  (K  L). set-set , P, Q dan R
Diberi set semesta,  = {1, 2, 3, 4, 5, 6, 7, 8}, set J = {1, 2, 3},
set K = {2, 3, 5, 7} dan set L = {5, 6, 7}. P Q
Cari set J  (K  L). b
a d
c
(K  L) = {2, 3, 5, 6, 7} f e
J = {4, 5, 6, 7, 8} h
g
J  (K  L) = {5, 6, 7} R

Find set (P  Q)  R.
Cari set (P  Q)  R.

(P  Q) = {a, d, e, f, g, h}
(R) = {c, e, f, g}
(P  Q)  R = {e, f, g}



(c) State the relationship represented by the shaded region in each of the following Venn diagrams using
the symbols  and .
Nyatakan hubungan yang diwakili oleh rantau berlorek dalam setiap rajah Venn berikut menggunakan symbol  dan .


(i) (ii)
A B C E F




G




A  (B  C) (E  G)  F




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Mathematics Form 4 Chapter 4 Operation on Set

9. Solve the following problems involving combined operations on sets. PL 4 PL 5 PL 6
Selesaikan masalah melibatkan gabungan operasi set berikut.

Example
In a group of 25 athletes, 15 take part in track events, 14 take part in field events, 15 take part in relay race,
3 take part in track events only, 2 take part in relay race only, 4 take part in track and field events only
and 6 take part in tracks events and relay race only. Draw a Venn diagram to represent this information.
Dalam sekumpulan 25 orang atlet, 15 orang mengambil bahagian dalam acara trek, 14 orang dalam acara padang, 15 orang
dalam acara lari berganti-ganti, 3 orang dalam acara trek sahaja, 2 orang dalam acara lari berganti-ganti, 4 orang dalam acara
trek dan padang sahaja dan 6 orang mengambil bahagian dalam acara trek dan lari berganti-ganti sahaja. Lukis sebuah gambar
rajah Venn untuk mewakili maklumat tersebut.
Use the diagram to find the number of athletes
Gunakannya rajah tersebut untuk mencari bilangan atlet
(i) who take part in one type of event only,
yang mengambil bahagian dalam satu jenis acara sahaja,
(ii) who take part in field events and relay race but not track events.
yang mengambil bahagian dalam acara padang dan lari berganti-ganti tetapi tidak acara trek.

T = {athletes who take part in track events} Let y = Number of athletes who take part in field
T = {atlet yang mengambil bahagian dalam acara trek} events and relay race only
F = {athletes who take part in field events} Biar y = Bilangan atlet yang mengambil bahagian dalam acara
F = {atlet yang mengambil bahagian dalam acara padang} padang dan lumba lari berganti-ganti sahaja
R = {athletes who take part in relay race} 6 + 2 + 2 + y = n(R)
R = {atlet yang mengambil bahagian dalam acara lari berganti- 10 + y = 15
ganti} y = 5
From the information given, Let z = Number of athletes who take part in field
Daripada maklumat yang diberi, events only
n(T) = 15, n(F) = 14, n(R) = 15 Biar z = Bilangan atlet yang mengambil bahagian dalam
n(F  R) = 3, n (T  F) = 2 acara padang sahaja
n(T  F only / sahaja) = 4, n(T  R only / sahaja) = 6 4 + 2 + 5 + z = n(F)
11 + z = 14
Let x = Number of athletes who take part in all three z = 3
events
Biar x = Bilangan atlet yang mengambil bahagian dalam
kesemua acara T F
3 + 4 + 6 + x = n(T) 3 4 3
13 + x = 15
x = 2 2
6 5

2
R


(i) =3 + 3 + 2
= 8 athletes/orang atlet

(ii) 5 athletes/orang atlet























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Mathematics Form 4 Chapter 4 Operation on Set

(b) The Venn diagram below shows the number of (c) Draw a Venn diagram to represent universal set,
students in sets A, B and C where universal set, , set P and set Q. Given that
 = A  B  C.  = {x : x is an even number and 10  x  25},
Gambar rajah Venn di bwah menunjukkan bilangan pelajar set P = {x : x is a multiple of 4} and
dalam set A, B dan C yang mana set semesta,  = A  B  C. set Q = {x : x is a number greater than 20}.
Lukis gambar rajah Venn untuk mewakili set semesta, , set
A B P dan set Q. Diberi bahawa
5
6 9  = {x : x ialah nombor genap dan 10  x  25},
set P = {x : x ialah gandaan 4} dan
4
4 3 set Q = {x : x ialah nombor lebih besar daripada 20}.

8 15 17 19
P Q
C 12
16 24 22
A = {students who like lychee},
A = {pelajar yang suka laici}, 14 20 23
B = {students who like orange}, 18 11 13 14
B = {pelajar yang suka oren},
and C = {students who like banana}.
dan C = {pelajar yang suka pisang}.  = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24}
Find the number of students P = {12, 16, 20, 24}
Cari bilangan pelajar Q = {21, 22, 23, 24}
(i) who like one type of fruit only, P  Q = {24}
yang suka satu jenis buah sahaja,
(ii) who like two types of fruits only.
yang suka dua jenis buah sahaja.

(i) = 6 + 9 + 8
= 23 students/orang pelajar

(ii) = 5 + 4 + 3
= 12 students/orang pelajar





SPM Practice 4




Paper 1 Given that set N = {students who like nasi lemak},
set K = {students who like fried kuetiau},
set R = {students who like roti canai},
1. The Venn diagram below shows the number of students in
SPM sets N, K and R. and universal set,  = N  K  R.
2016 Diberi bahawa set N = {pelajar yang menggemari nasi lemak},
Gambar rajah Venn di bawah menunjukkan bilangan pelajar
bagi set N, K dan R. set K = {pelajar yang menggemari kuetiau goreng},
set R = {pelajar yang menggemari roti canai},
N
dan set semesta,  = N  K  R.
8 If the number of students who like all three types of food
is 21, find the number of students who like only two types
2 2x
3x of food.
Jika bilangan pelajar yang menggemari kesemua tiga jenis
3 7
1 makanan ialah 21 orang, cari bilangan pelajar yang menggemari
K R dua jenis makanan sahaja.
A 1
B 2
C 14
D 17



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Mathematics Form 4 Chapter 4 Operation on Set

2. The Venn diagram below shows a universal set, . 5. The diagram below shows a Venn diagram with the universal
SPM Gambar rajah Venn di bawah menunjukkan sebuah set semesta, set,  = P  Q  R.
2017
. Rajah di bawah menunjukkan gambar rajah Venn dengan set
semesta,  = P  Q  R.

a Q
P d P Q R
b c
e
A B C D
f


List all the elements of set Q. Which of the regions A, B, C, and D represents the set
Senaraikan semua unsur bagi set Q. (P  Q)  R?
A {a} Antara kawasan A, B, C, dan D, kawasan yang manakah
B {b, f} mewakili set (P  Q)  R?
C {a, b, f}
D {a, b, c, e, f} 6. The diagram below shows a Venn diagram with the
universal set,  = {form 5 students},
3. The Venn diagram below shows a universal set, . set L = {students who are good in Bahasa Melayu},
Gambar rajah Venn di bawah menunjukkan sebuah set semesta, and set M = {students who are good in English}.
.
Rajah di bawah menunjukkan gambar rajah Venn dengan set
semesta, ξ = {pelajar tingkatan 5},

X Y set L = {pelajar yang bagus dalam Bahasa Melayu},
dan set M = {pelajar yang bagus dalam Bahasa Inggeris}.


L M


Which region is equivalent to X  Y?
Kawasan yang manakah setara dengan X  Y?
A (X  Y)
B X  Y
C X  Y
D X  Y Given that n(L) = 125, n(M) = 92, n(L  M) = 40 and the
number of students who are not good in both languages are
4. Given that the universal set,  = E  F  G, where E  F ≠ φ 28, find the total number of form 5 students in the school.
and G  F. Which of the following Venn diagram represents Diberi bahawa n(L) = 125, n(M) = 92, n(L  M) = 40 dan
the relationships? bilangan pelajar yang tidak bagus dalam kedua-dua bahasa
Diberi bahawa set semesta,  = E  F  G, dengan keadaan ialah 28 orang. cari jumlah pelajar tingkatan 5 di sekolah
E  F ≠ φ dan G  F. Antara yang berikut, gambar rajah Venn tersebut.
yang manakah mewakili hubungan tersebut? A 285 C 245
A C B 257 D 205

E F E G 7. Given that the universal set,  = {x : 2  x  10; x is an
G F integer}, set H = {x : x is an even number},
and set N = {x : x  4}.
Diberi bahawa set semesta, ξ = {x : 2  x  10; x ialah integer},
set H = {x : x ialah nombor genap}, dan set N = {x : x  4}.
B D
What are the elements in set H  N?

E F F G Apakah unsur dalam set H  N?
G E
A {5, 7, 9} C {2, 4, 6, 8}
B {6, 8, 10} D {3, 5, 6, 7, 8, 9, 10}
8. Given that F = {a, b}, find all the subsets of F.

Diberi bahawa F = {a, b}, cari semua subset bagi F.
A {a}, {b}
B {a}, {b}, { }
C {a}, {b}, {a, b}, {b, a}
D {a}, {b}, {a, b}, { }




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Mathematics Form 4 Chapter 4 Operation on Set

9. The diagram below is a Venn diagram showing the elements A 7 C 9
SPM of the universal set, , set R and set S. B 8 D 10
2017
Rajah di bawah ialah gambar rajah Venn yang menunjukkan 13. The diagram below shows universal set,  = L  M  N.
unsur-unsur bagi set semesta, , set R dan set S.
Rajah di bawah menunjukkan set semesta,  = L  M  N.

M
m L
S
e
R
f w a N
g
k

List all the elements of set R. Which of the following represents set of the shaded region?
Senaraikan semua unsur bagi set R. Antara berikut, yang manakah mewakili set bagi rantau berlorek?
A {k, m} C {a, e, k, m} A L  M  N
B {k, m, w} D {a, e, k, m, w}
B L  (M  N)
10. The diagram below is a Venn diagram showing sets P, Q C M  (M  N)
SPM and R. D M  (L  N)
2016
Rajah di bawah ialah sebuah gambar rajah Venn yang 14. The diagram below shows a Venn diagram.
menunjukkan set-set P, Q dan R.
Rajah di bawah menunjukkan sebuah gambar rajah Venn.
Q
P Q
R P
R
A B C D



Given that the universal set,  = P  Q  R, find the set
that represents the shaded region.
Diberi bahawa set semesta,  = P  Q  R, cari set yang Given that the universal set,  = {integers}, set P =
mewakili rantau yang berlorek. {multiples of 2}, set Q = {multiples of 3} and set R =
A P  Q  R C (P  Q)  R {multiples of 12}. State the region where element 102 is
B P  Q  R D (P  Q)  R located.
Diberi bahawa set semesta set,  = {integer}, set P = {gandaan 2},
11. The diagram below is an incomplete Venn diagram showing set Q = {gandaan 3} dan set R = {gandaan 12}. Nyatakan
the number of elements in sets J, K and L. kawasan di mana unsur 102 terletak.
Rajah di bawah ialah gambar rajah Venn yang tidak lengkap
yang menunjukkan bilangan unsur dalam set J, K dan L. 15. The diagram below shows a Venn diagram with the
universal set,  = P  Q  R.
J L Rajah di bawah menunjukkan gambar rajah Venn dengan set
3
6 8 semesta,  = P  Q  R.
P Q
5 4 R
A B C D
7
K
Given that the universal set,  = J  K  L and n( J  K) = 7,
find n[ (J  K)  L]. Which of the region, A, B, C and D, represents P  Q  R?
Diberi bahawa set semesta, set  = J  K  L dan n(J  K) = 7, Antara kawasan, A, B, C dan D, yang manakah mewakili
cari n[(J  K)  L]. P  Q  R?
A 7 C 15 16. Given that set J = {1, 7, 9} and set Q = {3, 7, 1}, find the
B 13 D 17 elements of J  Q.
12. Given that universal set,  = J  K  L, set J = {G, E, M, Diberi bahawa set J = {1, 7, 9} dan set Q = {3, 7, 1}, cari unsur
SPM A, S}, set K = {P, A, R, O ,I} and set L = {T, A, M, P, I, N}, bagi J  Q.
2019
Diberi bahawa set semesta,  = J  K  L, set J = {G, E, M, A {7}
A, S}, set K = {P, A, R, O ,I} dan set L = {T, A, M, P, I, N}, B {1, 7}
C {1, 3, 7, 11}
find n( J  L). D {1, 3, 7, 9, 11}
cari n(J  L).


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Mathematics Form 4 Chapter 4 Operation on Set

17. The diagram below shows the number of elements in set 20. Given that set P = {1, 2, 3, 4, 5, 6, 7, 8}, set Q = {2, 4, 6, 8}
F, set G and set H. and set R = {4, 8, 12}. List all the elements of set
Rajah di bawah menunjukkan bilangan unsur dalam set F, P  (Q  R).
set G, dan set H. Diberi bahawa set P = {1, 2, 3, 4, 5, 6, 7, 8}, set Q = {2, 4, 6, 8}
dan set R = {4, 8, 12}. Senaraikan semua unsur bagi set
F G
H P  (Q  R).
x 4 7 A {2, 6} C {2, 4, 6, 8}
B {4, 8} D {2, 4, 6, 8, 12}
5
21. The diagram below is a Venn diagram showing the number
of members of Badminton Club, marked as set B, and Tennis
Given the universal set,  = F  G  H and n(G  H) = Club, marked as set T.
n(F). Find the value of x. Rajah di bawah ialah gambar rajah Venn yang menunjukkan
Diberi set semesta,  = F  G  H dan n(G  H) = n(F). bilangan ahli dalam Kelab Bandminton, ditandakan sebagai set
Cari nilai x. B, dan Kelab Tenis, ditandakan sebagai set T.
A 3 C 5
B 4 D 6 B T

18. The diagram below shows a Venn diagram with set R = x 2x 6 + 2x
{x : x is multiple of 5}, set S = {x : x is multiple of 7} and
set T = {x : x is multiple of 9}.
Rajah di bawah menunjukkan sebuah gambar rajah Venn
dengan set R = {x : x gandaan 5}, set S = {x : x gandaan 7} If the number of members who join only one club is 30,
dan set T = {x : x gandaan 9}. find the total number of members in both clubs.
Jika bilangan ahli yang menyertai hanya satu kelab sahaja ialah
R S 30 orang, cari jumlah ahli dalam kedua-dua kelab itu.
I
A 16 C 46
B 22 D 62
III
II IV
22. Given that the universal set,  = {x : 20  x  40, x is an
integer}, set P = {x : x is 2 digit number where the first
digit is 2} and set Q = {x : x is multiple of 5}.
T Diberi bahawa set semesta, ξ = {x : 20  x  40, x ialah integer},
set P = {x : x ialah nombor 2 digit dengan digit pertama ialah 2}
The number 63 should be written in which region? dan set Q = {x : x ialah gandaan 5}.
Nombor 63 sepatutnya ditulis dalam kawasan yang mana?
A I C III Find the value of n(P  Q).
B II D IV Cari nilai n(P  Q).
A 2
19. Given that the universal set,  = G  H  K, H  G and B 4
G  K ≠ φ. C 10
Diberi bahawa set semesta,  = G  H  K, H  G and D 19
G  K ≠ φ. 23. The diagram below is a Venn diagram showing the elements
of universal sets,  , set J and set K.
Which of the following Venn diagram represents the
relationship? Rajah di bawah ialah gambar rajah Venn yang menunjukkan
Antara gambar rajah Venn berikut yang manakah mewakili unsur-unsur bagi set semesta,  , set J dan set K.
hubungan tersebut?
A C
J K
G K G K q
t v
H H r
u w
s
p
B D
H K H K What are the elements of set J  K?
G G Apakah unsur-unsur bagi set J  K?
A {q, r, s}
B {p, q, r, s}
C {q, r, s, t ,u}
D {p, q, r, s, t ,u}



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Mathematics Form 4 Chapter 4 Operation on Set

24. The diagram below is a Venn diagram showing number of 3. The Venn diagram in the answer space shows set P, set Q
elements in universal set, , set X and set Y. and set R.
Rajah di bawah ialah gambar rajah Venn yang menunjukkan Gambar rajah Venn pada ruang jawapan menunjukkan set P,
bilangan unsur dalam set semesta, , set X dan set Y. set Q dan set R.
On the diagram, shade
X
Pada rajah tersebut, lorekkan
Y
5 (a) (P  Q)  R,
2 6 (b) P  (Q  R).
Answer / Jawapan:
(a)
P Q R

Given that n() = 21, what is the value of n(X  Y)?
Diberi bahawa n() = 21, apakah nilai bagi n(X  Y)?
A 8 C 11
B 10 D 13


Paper 2

1. The Venn diagram in the answer space shows set P, set Q (b) P R
SPM and set R.
2013
Gambar rajah Venn pada ruang jawapan menunjukkan set P,
set Q dan set R.
On the diagram, shade
Pada rajah tersebut, lorekkan
(a) P  Q,
(b) P  (Q  R). Q
Answer / Jawapan:
(a) (b)
P P
Q Q 4. The Venn diagram in the answer space shows set P, set Q
and set R.
Gambar rajah Venn pada ruang jawapan menunjukkan set P,
R R set Q dan set R.

On the diagram, shade
Pada rajah tersebut , lorekkan

(a) P  Q,
(b) R  (P  Q).
2. The Venn diagram in the answer space shows set Answer / Jawapan:
J, set K and set L.
Gambar rajah Venn pada ruang jawapan menunjukkan set J, (a) P Q R
set K dan set L.
On the diagram, shade
Pada rajah tersebut, lorekkan
(a) J  L,
(b) (K  L)  J.
Answer / Jawapan:
(a) (b) (b)
J P R
J K Q
K
L



L




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Mathematics Form 4 Chapter 4 Operation on Set

HOTS Challenge



1. The table below shows 3 groups, P, Q and R with 100 members respectively. Calculate the value of x, y and z then complete
the Venn diagram in the answer space.
Jadual di bawah menunjukkan 3 kumpulan, P, Q dan R dengan bilangan ahli masing-masing 100 orang. Hitung nilai x, y dan z
kemudian lengkapkan gambar rajah Venn pada ruang jawapan.

P 3 12 5 0 x 0 0
Q 7 0 4 7 0 y 0
R 4 8 0 6 0 0 z
P  Q  R = 14 P  R = P  Q = 9 Q  R = 13 P = x Q = y R = z

Answer / Jawapan:
9 + 14 + 20 + x = 100 20 + 14 + 13 + z = 100
P Q
43 + x = 100 47 + z = 100
9 x = 57 z = 53
57 64
9 + 14 + 13 + y = 100
14
20 13 36 + y = 100
y = 64
53

R






2. Given the universal set,  = L  M  (L  M), n() = 35, n(L  M) = 7, n(L) = 18 and n(M) = 16, find n(L  M).
Diberi set semesta  = L  M  (L  M), n() = 35, n(L  M) = 7, n(L) = 18 dan n(M) = 16, cari n(L  M).
Answer / Jawapan:
ξ
L M
18 – x x 16 – x

7

Let n(L  M) = x
35 – 7 = 18 – x + x + 16 – x
28 = 34 – x
x = 6
n(L  M) = (18 – 6) + (16 – 6) + 7
= 29




HOTS
















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