Top Class F5 - Mathematics (Chapter 1)

CLASS
Form 5

KSSMKSSMTeacher’s Edition

Mathematics Samantha Neo
Tan Soon Chen
Matematik Tee Hock Tian

Topical Practices

SPM Practices

PAK-21 PdPRPakej

Pengajaran di Rumah
dan Pembelajaran

HOTS 2021NEW SPM ASSESSMENT Rekod Pencapaian PAK-21
FORMAT Online Quick Quiz
nerbit Rasm SPM Model Paper

Buku Teks

KSSM

ematik Tingka
Pe

tan 5
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Mat

Contents

CHA PTER 5.2 Enlargement ........................................................................................................... xx
Pembesaran
1 Variation 1
Ubahan 5.3 Combined Transformation.......................................................................... xx
Gabungan Transformasi
1.1 Direct Variation .................................................................................................... xx
Ubahan Langsung 5.4 Tessellation.............................................................................................................. xx
1.2 Inverse Variation ................................................................................................ xx Teselasi
Ubahan Songsang
1.3 Combined Variation ......................................................................................... xx SPM Practice 5 .................................................................................................................... xx
Ubahan Bergabung
SPM Practice 1 ..................................................................................................................... xx HOTS Challenge .................................................................................................................. xx
HOTS Challenge .................................................................................................................. xx Online Quick Quiz QR code .......................................................................................... xx
Online Quick Quiz QR code .......................................................................................... xx PAK-21 Corner QR code ................................................................................................... xx
PAK-21 Corner QR code ................................................................................................... xx
CHA CHA CHA CHA PTER Ratios and Graphs of Trigonometric
PTER
6 Functions xx
2 Matrices xx Nisbah dan Graf Fungsi Trigonometri
Matriks
6.1 The Value of Sine, Cosine and Tangent for
2.1 Matrices....................................................................................................................... xx Angle θ, 0°  θ  360°................................................................................. xx
Matriks Nilai Sinus, Kosinus dan Tangen bagi Sudut θ,
0°  θ  360°
2.2 Basic Operation on Matrices .................................................................. xx 6.2 The Graphs of Sine, Cosine and Tangent
Operasi Asas Matriks Functions.................................................................................................................... xx
Graf Fungsi Sinus, Kosinus dan Tangen
SPM Practice 2 .................................................................................................................... xx SPM Practice 6 .................................................................................................................... xx
HOTS Challenge .................................................................................................................. xx
HOTS Challenge .................................................................................................................. xx Online Quick Quiz QR code .......................................................................................... xx
Online Quick Quiz QR code .......................................................................................... xx PAK-21 Corner QR code ................................................................................................... xx
PAK-21 Corner QR code ................................................................................................... xx

CHA CHA PTER xx PTER Measures of Dispersion for Grouped

3 Consumer Mathematics: Insurance 7 Data xx
Matematik Pengguna: Insurans Sukatan Serakan Data Terkumpul

3.1 Risk and Insurance Coverage.................................................................. xx 7.1 Dispersion.................................................................................................................. xx
Risiko dan Perlindungan Insurans Serakan
7.2 Measures of Dispersion............................................................................... xx
SPM Practice 3..................................................................................................................... xx Sukatan Serakan
SPM Practice 7 .................................................................................................................... xx
HOTS Challenge .................................................................................................................. xx HOTS Challenge .................................................................................................................. xx
Online Quick Quiz QR code .......................................................................................... xx Online Quick Quiz QR code .......................................................................................... xx
PAK-21 Corner QR code ................................................................................................... xx PAK-21 Corner QR code ................................................................................................... xx

PTER xx

4 Consumer Mathematics: Taxation PTER
Matematik Pengguna: Percukaian
8 Mathematical Modeling
4.1 Taxation........................................................................................................................ xx Pemodelan Matematik xx
Percukaian
8.1 Mathematical Modeling................................................................................ xx
SPM Practice 4 .................................................................................................................... xx Pemodelan Matematik
SPM Practice 8 .................................................................................................................... xx
HOTS Challenge .................................................................................................................. xx HOTS Challenge .................................................................................................................. xx
Online Quick Quiz QR code .......................................................................................... xx Online Quick Quiz QR code .......................................................................................... xx
PAK-21 Corner QR code ................................................................................................... xx PAK-21 Corner QR code ................................................................................................... xx

CHA Congruency, Enlargement and xx SPM Model Paper.. ............................................................................................................ xx
PTER Combined Transformations

5 Kekongruenan, Pembesaran dan
Gabungan Transformasi

5.1 Congruency............................................................................................................... xx Answers
Kekongruenan

© Penerbitan Pelangi Sdn. Bhd. ii

Rekod Pencapaian Pentaksiran Murid

Form 5 Mathematics / Matematik Tingkatan 5

Student’s name: ..............................................................................……… Class: ..........................................…………..
Nama murid: Kelas:

Performance Achievement
level Penguasaan
Tahap
Chapter Descriptor (✓) (✗)
Bab penguasaan Deskriptor Achieve Not yet
1 1 Menguasai achieve
2 Demonstrate basic knowledge of variation. Belum
VARIATION 3 Mempamerkan pengetahuan asas tentang ubahan. menguasai
UBAHAN 4 Demonstrate understanding of variation.
Mempamerkan kefahaman tentang ubahan.
2 5 Apply the understanding of variation to carry out simple
MATRICES tasks.
MATRIKS 6 Mengaplikasikan kefahaman tentang ubahan untuk
melaksanakan tugasan mudah.
1 Apply appropriate knowledge and skills of variations in
2 the context of simple routine problem solving.
3 Mengaplikasikan pengetahuan dan kemahiran yang sesuai
4 tentang ubahan dalam konteks penyelesaian masalah rutin
yang mudah.
5 Apply appropriate knowledge and skills of variation in the
context of complex routine problem solving.
6 Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang ubahan dalam konteks penyelesaian masalah rutin
yang kompleks.
Apply appropriate knowledge and skills of variation in
the context of non-routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang ubahan dalam konteks penyelesaian masalah bukan
rutin secara kreatif.
Demonstrate basic knowledge about matrices.
Mempamerkan pengetahuan asas tentang matriks.
Demonstrate understanding of matrices.
Mempamerkan kefahaman tentang matriks.
Apply understanding of matrices to carry out simple
tasks.
Mengaplikasikan kefahaman tentang matriks untuk
melaksanakan tugasan mudah.
Apply appropriate knowledge and skills of matrices in the
context of simple routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang matriks dalam konteks penyelesaian masalah rutin
yang mudah.
Apply appropriate knowledge and skills of matrices in the
context of complex routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang matriks dalam konteks penyelesaian
masalah rutin yang kompleks.
Apply appropriate knowledge and skills of matrices in
the context of non-routine problem solving in a creative
manner.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang matriks dalam konteks penyelesaian masalah bukan
rutin secara kreatif.

iii © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Rekod Pencapaian Pentaksiran Murid

3 1 Demonstrate basic knowledge of insurance.
CONSUMER MATHEMATICS: Mempamerkan pengetahuan asas tentang insurans.

INSURANCE 2 Demonstrate understanding of insurance.
MATEMATIK PENGGUNA: Mempamerkan kefahaman tentang insurans.

INSURANS 3 Apply the understanding of insurance to carry out simple
tasks.
Mengaplikasikan kefahaman tentang insurans untuk
melaksanakan tugasan mudah.

4 Apply appropriate knowledge and skills of insurance in
the context of simple routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang insurans dalam konteks penyelesaian masalah rutin
yang mudah.

5 Apply appropriate knowledge and skills of insurance in
the context of complex routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang insurans dalam konteks penyelesaian masalah rutin
yang kompleks.

6 Apply appropriate knowledge and skills of insurance in
the context of complex non-routine problem solving in
a creative manner.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang insurans dalam konteks penyelesaian masalah bukan
rutin secara kreatif.

4 1 Demostrate basic knowledge of taxation.
CONSUMER MATHEMATICS: Mempamerkan pengetahuan asas tentang percukaian.

TAXATION 2 Demostrate understanding of taxation.
MATEMATIK PENGGUNA: Mempamerkan kefahaman tentang percukaian.

PERCUKAIAN 3 Apply the understanding of taxation to perform simple
tasks.
Mengaplikasikan kefahaman tentang percukaian untuk
melaksanakan tugasan mudah.

4 Apply appropriate knowledge and skills on taxation in
the context of simple routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang percukaian dalam konteks penyelesaian masalah rutin
yang mudah.

5 Apply appropriate knowledge and skills on taxation in the
context of complex routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang percukaian dalam konteks penyelesaian masalah rutin
yang kompleks.

6 Apply appropriate knowledge and skills on taxation in
the context of non-routine problem solving in a creative
manner.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang percukaian dalam konteks penyelesaian masalah
bukan rutin secara kreatif.

5 1 Demonstrate the basic knowledge of congruency,
CONGRUENCY, ENLARGEMENT enlargement and combined transformation.
AND COMBINED Mempamerkan pengetahuan asas tentang kekongruenan,
pembesaran dan gabungan transformasi.
TRANSFORMATIONS
KEKONGRUENAN,
PEMBESARAN DAN GABUNGAN 2 Demostrate the understanding of congruency,
enlargement and combined transformation.
TRANSFORMASI Mempamerkan kefahaman tentang kekongruenan,

pembesaran dan gabungan transformasi.

3 Apply the understanding of congruency, enlargement
and combined transformation to perform simple tasks.
Mengaplikasikan kefahaman tentang kekongruenan,
pembesaran dan gabungan transformasi untuk melaksanakan
tugasan mudah.

© Penerbitan Pelangi Sdn. Bhd. iv

6 Mathematics  Form 5  Rekod Pencapaian Pentaksiran Murid    
RATIOS AND GRAPHS OF
TRIGONOMETRIC FUNCTIONS 4 Apply appropriate knowledge and skills on congruency,
NISBAH DAN GRAF FUNGSI enlargement and combined transformation in the context
of simple routine problem solving.
TRIGONOMETRI Mengaplikasikan pengetahuan dan kemahiran yang
7 sesuai tentang kekongruenan, pembesaran dan gabungan
transformasi dalam konteks penyelesaian masalah rutin yang
MEASURES OF DISPERSION mudah.
FOR GROUPED DATA
5 Apply appropriate knowledge and skills on congruency,
SUKATAN SERAKAN DATA enlargement and combined transformation in the context
TERKUMPUL of complex routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang
sesuai tentang kekongruenan, pembesaran dan gabungan
transformasi dalam konteks penyelesaian masalah rutin yang
kompleks.

6 Apply appropriate knowledge and skills on congruency,
enlargement and combined transformation in the context
of non-routine problem solving in a creative manner.
Mengaplikasikan pengetahuan dan kemahiran yang
sesuai tentang kekongruenan, pembesaran dan gabungan
transformasi dalam konteks penyelesaian masalah bukan rutin
secara kreatif.

1 Demonstrate basic knowledge of trigonometric ratios
and graphs.
Mempamerkan pengetahuan asas tentang nisbah dan graf
fungsi trigonometri.

2 Demonstrate the understanding of trigonometric ratios
and graphs.
Mempamerkan kefahaman tentang nisbah dan graf fungsi
trigonometri.

3 Apply the understanding of ratios and graphs of
trigonometric functions to perform simple tasks.
Mengaplikasikan kefahaman tentang nisbah dan graf fungsi
trigonometri untuk melaksanakan tugasan mudah.

4 Apply appropriate knowledge and skills about ratios
and graphs of trigonometric functions in the context of
simple routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang nisbah dan graf fungsi trigonometri dalam konteks
penyelesaian masalah rutin yang mudah.

5 Apply appropriate knowledge and skills of ratios and
graphs of trigonometric functions in the context of
complex routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang nisbah dan graf fungsi trigonometri dalam konteks
penyelesaian masalah rutin yang kompleks.

6 Apply appropriate knowledge and skills about ratios and
graphs of trigonometric functions in the context of non-
routine problem solving in a creative manner.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang nisbah dan graf fungsi trigonometri dalam konteks
penyelesaian masalah bukan rutin secara kreatif.

1 Demonstrate basic knowledge of dispersion and
measures of dispersion of grouped data.
Mempamerkan pengetahuan asas tentang serakan dan
sukatan serakan data terkumpul.

2 Demonstrate the understanding of dispersion and
measures of dispersion of grouped data.
Mempamerkan kefahaman tentang serakan dan sukatan
serakan data terkumpul.

3 Apply the understanding of dispersion and measures
of dispersion of grouped data to perform simple tasks.
Mengaplikasikan kefahaman tentang serakan dan sukatan
serakan data terkumpul untuk melaksanakan tugasan mudah.

v © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Rekod Pencapaian Pentaksiran Murid

8 4 Apply appropriate knowledge and skills about measures
MATHEMATICAL MODELING of dispersion of grouped data in the context of simple
routine problem solving.
PEMODELAN MATEMATIK Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang sukatan serakan data terkumpul dalam konteks
penyelesaian masalah rutin yang mudah.

5 Apply appropriate knowledge and skills to the measures
of dispersion of grouped data in the context of complex
routine problem solving.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang sukatan serakan data terkumpul dalam konteks
penyelesaian masalah rutin yang kompleks.

6 Apply appropriate knowledge and skills to the measures
of dispersion of grouped data in the context of non-
routine problem solving in a creative manner.
Mengaplikasikan pengetahuan dan kemahiran yang sesuai
tentang sukatan serakan data terkumpul dalam konteks
penyelesaian masalah bukan rutin secara kreatif.

1 Demonstrate basic knowledge of mathematical modeling.
Mempamerkan pengetahuan asas tentang pemodelan
matematik.

2 Demonstrate the understanding of mathematical
modeling.
Mempamerkan kefahaman tentang pemodelan matematik.

3 Apply the understanding of mathematical modeling to
perform simple tasks.
Mengaplikasikan kefahaman tentang pemodelan matematik
untuk melaksanakan tugasan mudah.

4 Apply the knowledge and skills of mathematical
modeling in the context of solving real-life problems
involving linear functions.
Mengaplikasikan pengetahuan dan kemahiran tentang
pemodelan matematik dalam konteks penyelesaian masalah
kehidupan sebenar yang melibatkan fungsi linear.

5 Apply the knowledge and skills of mathematical
modeling in the context of solving real-life problems
involving quadratic and exponential functions.
Mengaplikasikan pengetahuan dan kemahiran tentang
pemodelan matematik dalam konteks penyelesaian masalah
kehidupan sebenar yang melibatkan fungsi kuadratik dan
eksponen.

6 Apply the knowledge and skills of mathematical modeling
in the context of solving real-life problems involving
quadratic and exponential functions in a creative manner.
Mengaplikasikan pengetahuan dan kemahiran tentang
pemodelan matematik dalam konteks penyelesaian masalah
kehidupan sebenar yang melibatkan fungsi kuadratik dan
eksponen secara kreatif.

© Penerbitan Pelangi Sdn. Bhd. vi

CHA PTER Variation

1 Ubahan

1.1 Direct Variation Textbook
Ubahan Langsung pg. 2 – 16

SMART Notes

1. In a direct variation, the value of a variable y increases 4. When y ∝ xn, the graph of y against xn is a straight line
when the value of variable x increases with the same that passes through the origin.
rate, and vice versa.
Apabila y ∝ xn, graf y melawan xn ialah satu garis lurus yang melalui
Bagi ubahan langsung, nilai suatu pemboleh ubah y bertambah asalan.
apabila nilai pemboleh ubah x bertambah dengan kadar yang sama y
dan sebaliknya.

2. The relation between the variables can be written as y
varies directly as xn.
Hubungan antara pemboleh ubah ini boleh ditulis sebagai y berubah
secara langsung dengan xn. O xn
y ∝ xn (variation relation / hubungan ubahan)

y = kxn (equation form / bentuk persamaan) 5. A joint variation is a direct variation where a variable
1 1 varies as the product of two or more variables. For
where / dengan keadaan n = 1, 2, 3, 2 , 3 , example, y varies directly as x and z.

and k is a constant / dan k ialah pemalar Ubahan tercantum ialah ubahan langsung dengan keadaan satu
3. kkWAepiahasdbeacinlaaanylylekbvdedarikrtuhiebeenasahcldisosierneecbsacatrtaaglynalaitanposgefsmxupannr,lgoatrphdpeoeenrrtgvkiaoaannlduaaxernla,itonny.i.flaxiynxyn== pemboleh ubah berubah sebagai hasil darab dua atau lebih
k, where pemboleh ubah yang lain. Misalnya, y berubah secara langsung
dengan x dan z.
k, dengan

1. For each of the following situations, state the change on the given quantity. PL 1
Bagi setiap situasi berikut, nyatakan perubahan pada kuantiti yang diberikan.

Example (a) The weight of an object on the Moon varies

The total sales of papaya, RMy, varies directly as the directly as its weight on the Earth. State the
weight of papaya, x kg, sold. State the change on change on
Jumlah jualan betik, RMy, berubah secara langsung dengan berat, Berat suatu objek di Bulan berubah secara langsung dengan
x kg, betik yang dijual. Nyatakan perubahan pada beratnya di Bumi. Nyatakan perubahan pada
(i) the total sales of papaya if the weight of papaya (i) the weight of an object on the Moon if its

sold increases 3 times. weight on the Earth increases 10%.
jumlah jualan betik jika berat betik yang dijual bertambah berat objek di Bulan jika beratnya di Bumi bertambah
3 kali ganda. 10%.
(ii) the weight of papaya sold if the total sales is (ii) the weight of an object on the Earth if its
weight on the Moon decreases 2 times.
halved. berat objek di Bumi jika beratnya di Bulan berkurang
berat betik yang dijual jika jumlah jualan betik adalah 2 kali ganda.
separuh daripada jualan asal.

(i) the total sales of papaya increases 3 times (i) the weight of the object on the Moon
jumlah jualan betik bertambah 3 kali ganda increases 10%
berat objek itu di Bulan bertambah 10%
(ii) the weight of papaya sold is halved
berat betik yang dijual adalah separuh daripada berat asal (ii) the weight of the object on the Earth
betik decreases 2 times
berat objek itu di Bumi berkurang 2 kali ganda

1 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(b) Yusnita earns RM18 for 2 hours of working in a (c) The table below shows the relation between the

restaurant. State the change on distance and the time taken by a car at a speed

Yusnita memperoleh RM18 dengan bekerja 2 jam di sebuah of 80 km/h.

restoran. Nyatakan perubahan pada Jadual di bawah menunjukkan hubungan antara jarak dan

(i) the income if the number of working hours masa yang diambil oleh sebuah kereta dengan laju 80 km/j.
1
decreases 5 times. 1 Time (hours)
5 Masa (jam)
gaji jika bilangan jam bekerja berkurang daripada 0.5 1 2 3

bilangan jam bekerja yang asal. Distance (km)
Jarak (km)
(ii) the working hours if the income is doubled. 40 80 160 240

bilangan jam bekerja jika gaji didarabkan dengan dua.

(i) the binerckoumranegd51ecdraeraipseasda15gatjiimaseasl State the change on
gaji Nyatakan perubahan pada
(i) the distance if the time taken is tripled.
(ii) the working hours is doubled jarak jika masa yang diambil didarabkan dengan tiga.
(ii) the time taken if the distance decreases 25%.
bilangan jam bekerja didarabkan dengan dua masa yang diambil jika jarak berkurang 25%.

(i) the distance is tripled
jarak didarabkan dengan tiga

(ii) the time taken decreases 25%
masa yang diambil berkurang 25%

2. Determine whether the relation is a direct variation. If yes, write the relation in the form of variation. PL 2

Tentukan sama ada hubungan yang diberikan ialah ubahan langsung atau bukan. Jika ya, tuliskan hubungan tersebut dalam bentuk
ubahan.

Example (a) The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y.
The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y. x 2.25 4 9 16

x1 234 y 31.5 42 63 84
y4 16 36 64
Determine whether y varies directly as x.
Determine whether y varies directly as x2. Tentukan sama ada y berubah secara langsung dengan x.
Tentukan sama ada y berubah secara langsung dengan x2. y
x 21 21 21 21
y
x2 4 4 4 4

y is a constant, therefore y varies directly as x2 and y is a constant, therefore y varies directly as x
x2 x
y ∝ x2. and y ∝ x  .
y y ialah satu pemalar, maka y berubah secara langsung
x2 ialah satu pemalar, maka y berubah secara langsung dengan x
dengan x dan y ∝ x .
x2 dan y ∝ x2.

© Penerbitan Pelangi Sdn. Bhd. 2

Mathematics  Form 5  Chapter 1  Variation    

(b) The table below shows the values of x and y. (c) The table below shows the values of current, I,
Jadual di bawah menunjukkan nilai-nilai x dan y. flowing through a 12-volt circuit with various

x2 3 4 5 resistance, R.
y 16 54 96 200 Jadual di bawah menunjukkan nilai-nilai arus, I, yang
mengalir melalui suatu litar 12 voltan dengan pelbagai
Determine whether y varies directly as x3. rintangan, R.
Tentukan sama ada y berubah secara langsung dengan x3.
R (Ω) 12 6 4 3

y I (A) 1 2 3 4
x3 2
2 1.5 1.6 Determine whether I varies directly as R.

y is not a constant, therefore y does not vary Tentukan sama ada I berubah secara langsung dengan R.
x3
directly as x3. I 0.083 0.333 0.750 1.333
R
y
x3 bukan satu pemalar, maka y tidak berubah secara I is not a constant, therefore I does not vary
R
langsung dengan x3. directly as R.

I bukan satu pemalar, maka I tidak berubah secara
R
langsung dengan R.

3. Determine whether each of the following variations is a direct variation using graph method. PL 3
Tentukan sama ada setiap ubahan berikut ialah ubahan langsung atau bukan dengan menggunakan kaedah graf.

Example (a) The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y.
The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y. x4 9 16 25
y4 6 8 10
x4 9 16 25
y4 6 8 10 Determine whether y varies directly as x.
Tentukan sama ada y berubah secara langsung dengan x.

Determine whether y varies directly as x. y
Tentukan sama ada y berubah secara langsung dengan x.

x 2 3 4 5 10
y 4 6 8 10
8

6

y 4

10 2
8
6 O x
4 5 10 15 20 25
2

O 12345 x The graph of y against x does not show a straight
line that passes through the origin. Therefore, y
does not vary directly as x.
Graf y melawan x bukan satu garis lurus yang melalui
asalan. Maka, y tidak berubah secara langsung dengan x.

The graph of y against x shows a straight line that
passes through the origin. Therefore, y varies directly

as x.
Graf y melawan x menunjukkan satu garis lurus yang melalui
asalan. Maka, y berubah secara langsung dengan x.

3 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

4. For each of the following, express m in terms of n. PL 4
Bagi setiap yang berikut, ungkapkan m dalam sebutan n.

Example (a) m varies directly as 3n and m = 12 when n = 125.
m berubah secara langsung dengan 3n dan m = 12 apabila
m varies directly as the square root of n and m = 45 n = 125.
when n = 9.

m berubah secara langsung dengan punca kuasa dua n dan m ∝ 3n

m = 45 apabila n = 9. m = k3n

m ∝ n m = 12 when / apabila n = 125

m = kn 12 = k3125
k = 12
m = 45 when / apabila n = 9 3125
45 = k9
k = 45
9 = 2.4
Therefore / Maka, m = 2.43n
= 15

Therefore / Maka, m = 15n

(b) It is given that m ∝ n3 and m = 108 when n = 6. (c) The surface area, m, of a sphere varies directly

Diberi m ∝ n3 dan m = 108 apabila n = 6. as the square of its radius, n. It is given that

m = 55.44 when n = 2.1.

m ∝ n3 Luas permukaan, m, sebuah sfera berubah secara langsung

m = kn3 dengan kuasa dua jejarinya n. Diberi bahawa m = 55.44

m = 108 when / apabila n = 6 apabila n = 2.1.

108 = k(6)3 m ∝ n2
m = kn2
k = 108
63

= 21 m = 55.44 when / apabila n = 2.1
55.44 = k(2.1)2

Therefore / Maka, m = n3 k = 55.44
2 2.12

= 878

Therefore / Maka, m = 88n2
7

5. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example

It is given that y ∝ x2 and y = 18 when x = 3. Find the value of x when y = 4.5 and x is a positive value.
Diberi y ∝ x2 dan y = 18 apabila x = 3. Cari nilai x apabila y = 4.5 dan x ialah nilai positif.

y ∝ x2 Alternative Method

y = kx2 Concept of proportion:

y = 18 when / apabila x = 3 Konsep kadaran:

1 8 = k(3)2 y1 y2
18 x12 x22
k = 32 = when y1 = 18, x1 = 3 and y2 = 4.5

= 2 y1 = y2 apabila y1 = 18, x1 = 3 dan y2 = 4.5
x12 x22
Therefore / Maka, y = 2x2

When / Apabila y = 4.5, 1382 = 4.5
x22
4.5 = 2x2
4.5 (4.5)(32)
x2 = 2 x22 = 18

x = 2.25 x2 = 2.25

= 1.5 = 1.5

© Penerbitan Pelangi Sdn. Bhd. 4

Mathematics  Form 5  Chapter 1  Variation    

(a) It is given that y varies directly as the cube of x (b) It is given that y ∝ x and y = 168 when x = 196.

and y = 540 when x = 6. Find the value of x when Find the value of y when x = 3.24.

y = 20. Diberi y ∝ x  dan y = 168 apabila x = 196. Cari nilai y
Diberi y berubah secara langsung dengan kuasa tiga x dan apabila x = 3.24.
y = 540 apabila x = 6. Cari nilai x apabila y = 20.

y ∝ x3 y ∝ x
y = kx3 y = kx

y = 540 when / apabila x = 6 y = 168 when / apabila x = 196

540 = k(6)3 168 = k196
168
k = 540 k =  196
63
= 12
= 2.5
Therefore / Maka, y = 12x
Therefore / Maka, y = 2.5x3

When / Apabila y = 20, When / Apabila x = 3.24,

20 = 2.5x3 y = 123.24
20 = 21.6
x3 = 2.5

x = 38

= 2

6. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example (a) p varies directly as q2 and p = 40.5 when q = 1.5.

p varies directly as the cube root of q and p = 2.4 p berubah secara langsung dengan q2 dan p = 40.5 apabila
when q = 216. q = 1.5.

p berubah secara langsung dengan punca kuasa tiga q dan (i) Write the equation of p in terms of q.

p = 2.4 apabila q = 216. Tuliskan persamaan bagi p dalam sebutan q.

(i) Write the equation of p in terms of q. (ii) Find the value of p when q = 0.6.
Tuliskan persamaan bagi p dalam sebutan q. Cari nilai p apabila q = 0.6.

(ii) Find the value of p when q = 0.027. (i) p ∝ q2
Cari nilai p apabila q = 0.027. p = kq2

(i) p ∝ 3q p = 40.5 when / apabila q = 1.5
p = k3q
40.5 = k(1.5)2

p = 2.4 when / apabila q = 216 k = 40.5
1.52
2.4 = k3216
= 18
2.4
k =  3216 Therefore / Maka, p = 18q2

= 0.4 (ii) When / Apabila q = 0.6,
p = 18(0.62)
Therefore / Maka, p = 0.43q = 6.48

(ii) When / Apabila q = 0.027,

p = 0.430.027
= 0.12

5 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(b) The table below shows the values of p and q. (c) It is given that p ∝ q and p = 3 when q = 18.
Jadual di bawah menunjukkan nilai-nilai p dan q. Diberi bahawa p ∝ q dan p = 3 apabila q = 18.

q 1.69 x (i) Write the equation of p in terms of q.
p 9.1 84 Tuliskan persamaan bagi p dalam sebutan q.

It is given that p ∝ q. (ii) Find the value of p when q = 2.7.
Diberi bahawa p ∝ q . Cari nilai p apabila q = 2.7.

(i) Write the equation of p in terms of q. (i) p ∝ q
Tuliskan persamaan bagi p dalam sebutan q. p = kq

(ii) Find the value of x. p = 3 when / apabila q = 18
Cari nilai x.
3 = k(18)

(i) p ∝ q k = 138
p = kq
= 1
6

p = 9.1 when / apabila q = 1.69 Therefore / Maka, p = q
6
9.1 = k1.69
9.1
k =  1.69 (ii) When / apabila q = 2.7,

= 7 p= 2.7
6

Therefore / Maka, p = 7q = 0.45

(ii) When / Apabila p = 84,

84 = 7x
84
x = 7

x = 122

= 144

7. Write the relation of each of the following joint variations using the symbol ∝. PL 2
Tuliskan hubungan setiap ubahan tercantum berikut dengan menggunakan simbol ∝.

Example (a) m varies directly as n and p.
G varies directly as H and L . m berubah secara langsung dengan n dan p.

G berubah secara langsung dengan H dan L  . m ∝ np

G ∝ HL

(b) P varies directly as R and the square of Q. (c) y varies directly as the cube of x and the square
P berubah secara langsung dengan R dan kuasa dua Q. root of z.
y berubah secara langsung dengan kuasa tiga x dan punca
P ∝ RQ2 kuasa dua z.

y ∝ x3z

© Penerbitan Pelangi Sdn. Bhd. 6

Mathematics  Form 5  Chapter 1  Variation    

8. For each of the following, express r in terms of s and t. PL 4
Bagi setiap yang berikut, ungkapkan r dalam sebutan s dan t.

Example (a) r varies directly as s and t 2. It is given that

r varies directly as s and the cube of t. It is given that r = 490 when s = 1.96 and t = 5.
r = 0.6 when s = 25 and t = 0.2. r berubah secara langsung dengan s  dan t 2. Diberi r = 490
r berubah secara langsung dengan s dan kuasa tiga t. Diberi apabila s = 1.96 dan t = 5.

r = 0.6 apabila s = 25 dan t = 0.2. r ∝ s t2

r ∝ st 3 r = ks t2
r = kst 3
r = 490 when / apabila s = 1.96, t = 5
r = 0.6 when / apabila s = 25, t = 0.2 490 = k1.96 (5)2
k = 490
0.6 = k(25)(0.2)3
0.6  1.96 (5)2
k = (25)(0.2)3 = 14

= 3

Therefore / Maka, r = 3st 3 Therefore / Maka, r = 14s t2

(b) It is given that r ∝ st 3 and r = 384 when s = 9 (c) The mass, r kg, of a cylindrical sheet of metal

and t = 4. varies directly as the square of its radius, s cm,

Diberi r ∝ st 3 dan r = 384 apabila s = 9 dan t = 4. and its height, t cm. It is given that r = 12.5 when

r ∝ st 3 s = 5 and t = 50.
r = kst 3 Jisim, r kg, sekeping lembaran logam berbentuk silinder
berubah secara langsung dengan kuasa dua jejarinya,
r = 384 when / apabila s = 9, t = 4 s cm, dan tingginya, t cm. Diberi r = 12.5 apabila s = 5 dan
t = 50.
384 = k(9)(4)3

k = 384 r ∝ s2t
(9)(4)3 r = ks2t

= 2 r = 12.5 when / apabila s = 5, t = 50
3
12.5 = k(5)2(50)
2
Therefore / Maka, r = 3 st 3 k = 12.5
(5)2(50)

= 0.01

Therefore / Maka, r = 0.01s2t

9. Solve each of the following problems. PL 4
Selesaikan setiap masalah yang berikut.

Example

It is given that y ∝ xz2 and y = 432 when x = 15 and z = 3. Find the positive value of z when y = 921.6
and x = 2.
Diberi y ∝ xz2 dan y = 432 apabila x = 15 dan z = 3. Cari nilai positif z apabila y = 921.6 dan x = 2.

y ∝ xz2 Alternative Method
y = kxz2

y = 432 when / apabila x = 15, z = 3 Concept of proportion:

432 = k(15)(3)2 Konsep kadaran:

k = 432 y1 = y2 when / apabila y1 = 432, x1 = 15,
(15)(3)2 x1z12 x2z22

= 3.2 z1 = 3 and / dan y2 = 921.6, x2 = 2

Therefore / Maka, y = 3.2xz2 (1453)(23)2 = 921.6
(2)z22
When / Apabila y = 921.6, x = 2,
(921.6)(15)(3)2
921.6 = 3.2(2)z2 z22 = (432)(2)
921.6
z2 = (3.2)(2) z2 = 144
= 12
z = 144

= 12

7 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(a) It is given that y varies directly as x and the cube (b) It is given that y ∝ xz and y = 84 when x = 21

root of z and y = 243 when x = 30 and z = 0.729. and z = 25. Find the value of x when y = 51.2

Find the value of z when y = 21.6 and x = 0.6. and z = 256.
Diberi y ∝ xz dan y = 84 apabila x = 21 dan z = 25. Cari
Diberi y berubah secara langsung dengan x dan punca kuasa

tiga z dan y = 243 apabila x = 30 dan z = 0.729. Cari nilai nilai x apabila y = 51.2 dan z = 256.

z apabila y = 21.6 dan x = 0.6.

y ∝ x3z y ∝ xz
y = kxz
y = kx3z
y = 84 when / apabila x = 21, z = 25

y = 243 when / apabila x = 30, z = 0.729 8 4 = k(21)25

243 = k(30)30.729  k = 84
243 (21)25
k = (30)30.729  4
5
= 9 =

Therefore / Maka, y = 9x3z Therefore / Maka, y = 4 xz
5

When / Apabila y = 21.6, x = 0.6, When y = 51.2, z = 256

21.6 = 9(0.6)3z 5 1.2 = 54(x5)(25516.2)
21.6 x = (4)256
3z = (9)(0.6)

z = 43 = 4
= 64

10. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example (a) P varies directly as Q3 and R and P = 680.4 when

P varies directly as Q and the square of R and P = 6.6 Q = 3 and R = 1.8.
when Q = 55 and R = 0.4. P berubah secara langsung dengan Q3 dan R dan P = 680.4
P berubah secara langsung dengan Q dan kuasa dua R dan apabila Q = 3 dan R = 1.8.

P = 6.6 apabila Q = 55 dan R = 0.4. (i) Write the equation of P in terms of Q and R.

(i) Write the equation of P in terms of Q and R. Tuliskan persamaan bagi P dalam sebutan Q dan R.

Tuliskan persamaan bagi P dalam sebutan Q dan R. (ii) Find the value of Q when P = 240.1 and

(ii) Find the value of P when Q = 5.2 and R = 9. R = 0.05.
Cari nilai P apabila Q = 5.2 dan R = 9. Cari nilai Q apabila P = 240.1 dan R = 0.05.

(i) P ∝ QR2 (i) P ∝ Q3R
P = kQR2 P = kQ3R

P = 6.6 when / apabila Q = 55, R = 0.4 P = 680.4 when / apabila Q = 3, R = 1.8

6.6 = k(55)(0.4)2 680.4 = k(3)3(1.8)
680.4
k = 6.6 k = (3)3(1.8)
(55)(0.4)2
= 14
= 0.75

Therefore / Maka, P = 0.75QR2 Therefore / Maka, P = 14Q3R

(ii) When / Apabila Q = 5.2, R = 9, (ii) When / apabila P = 240.1, R = 0.05,
P = 0.75(5.2)(9)2
= 315.9 240.1 = 14Q3(0.05)

Q3 = 240.1
(14)(0.05)

Q = 3343
= 7

© Penerbitan Pelangi Sdn. Bhd. 8

Mathematics  Form 5  Chapter 1  Variation    

(b) The table below shows the values of P, Q and R. (c) It is given that P ∝ Q3R and P = 112 when
Jadual di bawah menunjukkan nilai-nilai P, Q dan R.
Q = 20 and R = 0.343.

P 576 51.2 Diberi P ∝ Q3R dan P = 112 apabila Q = 20 dan R = 0.343.
(i) Write the equation of P in terms of Q and R.

Q 225 x Tuliskan persamaan bagi P dalam sebutan Q dan R.

R 12 0.8 (ii) Find the value of P when Q = 0.8 and R = 27.
Cari nilai P apabila Q = 0.8 dan R = 27.
It is given that P ∝ QR .
Diberi bahawa P ∝ QR . (i) P ∝ Q3R
P = kQ3R
(i) Write the equation of P in terms of Q and R.
Tuliskan persamaan bagi P dalam sebutan Q dan R. P = 112 when / apabila Q = 20, R = 0.343
112 = k(20)30.343 
(ii) Find the value of x.
Cari nilai x. k = 112
(20)30.343
(i) P ∝ QR
P = kQR = 8

P = 576 when / apabila Q = 225, R = 12 Therefore / Maka, P = 8Q3R

576 = k225 (12) (ii) When / Apabila Q = 0.8, R = 27,
P = 8(0.8)327 
k = 576 = 19.2
225 (12)

= 3.2

Therefore / Maka, P = 3.2QR

(ii) When / Apabila P = 51.2, R = 0.8,

51.2 = 3.2x (0.8)

x = 51.2
(3.2)(0.8)

x = 202

= 400

11. Solve each of the following. PL 5 Daily Application
Selesaikan setiap yang berikut,

(a) The distance, D m, required to stop a car varies directly as the square of its speed, v km h−1. It is given
that 64 m is required to stop a car with 80 km h−1.
Jarak, D m, diperlukan untuk memberhentikan sebuah kereta berubah secara langsung dengan kuasa dua laju kereta itu,
v km j−1. Diberi bahawa 64 m diperlukan untuk memberhentikan sebuah kereta dengan laju 80 km j−1.

(i) Write the equation of D in terms of v.
Tuliskan persamaan bagi D dalam sebutan v.

(ii) Find the value of D when v = 110.
Cari nilai D apabila v = 110.

(i) D ∝ v 2 (ii) When / Apabila v = 110,
D = kv2 D = 0.01(110)2
= 121
D = 64 when / apabila v = 80

64 = k(80)2

k = 64
802

= 0.01

Therefore / Maka, D = 0.01v2

9 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(b) The kinetic energy, E joules, of an object varies directly as the mass, m kg, and the square of the velocity,
v m s–1. A van of a mass of 1 200 kg moving with a velocity of 10 m s–1 has a kinematic energy of
60 000 joules.
Tenaga kinetik, E joule, bagi suatu objek berubah secara langsung dengan jisim, m kg, dan kuasa dua halaju, v m s–1. Sebuah
van dengan jisim 1 200 kg bergerak dengan laju 10 m s–1 mempunyai tenaga kinetik 60 000 joule.

(i) Write the equation of E in terms of m and v.
Tuliskan persamaan bagi E dalam sebutan m dan v.

(ii) Find the value of v when E = 108 000 and m = 1 500.
Cari nilai v apabila E = 108 000 dan m = 1 500.

(i) E ∝ mv2 (ii) When / Apabila E = 108 000, m = 1 500,
E = kmv2
108 000 = 0.5(1 500)v2

E = 60 000 when / apabila m = 1 200, v = 10 v 2 = 108 000
(0.5)(1 500)
60 000 = k(1 200)(10)2
v = 144
k = 60 000 = 12
(1 200)(102)

= 0.5

Therefore / Maka, E = 0.5mv2

(c) The price, RMP, of a cable varies directly as its length, L m, and its diameter, d mm. It is given that a
10-metre cable with a diameter of 12 mm costs RM25.
Harga, RMP, bagi suatu kabel berubah secara langsung dengan panjangnya, L m, dan diameternya, d mm. Diberi bahawa
segelung kabel yang panjangnya 10 meter dengan diameter 12 mm berharga RM25.

(i) Mr Hamid paid RM48.75 to buy a cable with a radius of 7.5 mm. Find the length, in m, of the
cable he bought.
Encik Hamid membayar RM48.75 untuk membeli segelung kabel dengan jejari 7.5 mm. Cari panjang, dalam m, kabel
yang dibeli olehnya.

(ii) If Mr Hamid uses the same amount of money to buy a cable longer than in (i), would he get a
cable with bigger diameter? Explain your answer.
Jika Encik Hamid menggunakan jumlah wang yang sama untuk membeli kabel yang lebih panjang daripada (i), adakah
dia akan mendapat kabel yang diameternya lebih besar? Jelaskan jawapan anda.

HOTS Analysing

(i) P ∝ Ld (ii) No. He would not get a cable with bigger
P = kLd diameter because the price varies directly as
the product of the length and the diameter
P = 25 when / apabila L = 10, d = 12 of the cable.
Tidak. Dia tidak akan mendapat kabel yang diameternya
25 = k(10)(12) lebih besar kerana harga berubah secara langsung
dengan hasil darab panjang dan diameter kabel.
k = 25
(10)(12)

= 5
24

Therefore / Maka, P = 5 Ld
24

When / Apabila P = 48.75, d = 2 × 7.5 = 15,
5
48.75 = 24 L(15)

L = (48.75)(24)
(5)(15)

= 15.6 m

© Penerbitan Pelangi Sdn. Bhd. 10

1.2 Inverse Variation Mathematics  Form 5  Chapter 1  Variation    
Ubahan Songsang Textbook
pg. 17 – 25

SMART Notes

1. In an inverse variation, the value of a variable y increases 4. When y ∝ 1 , the graph of y against xn is a hyperbola.
when the value of variable x decreases with the same xn
rate, and vice versa. 1
Apabila y ∝ xn , graf y melawan xn ialah hiperbola.
Bagi ubahan songsang, nilai suatu pemboleh ubah y bertambah
apabila nilai pemboleh ubah x berkurang dengan kadar yang sama y
dan sebaliknya.

2. The relation between the variables can be written as y O xn
varies inversely as xn.

Hubungan antara pemboleh ubah ini boleh ditulis sebagai y berubah
secara songsang dengan xn.

y ∝ 1 (variation relation / hubungan ubahan) 5. When y ∝ 1 , the graph of y against 1 is a straight line
xn xn xn
k
y = xn (equation form / bentuk persamaan) that starts from the origin.

where / dengan keadaan n = 1, 2, 3, 1 , 1 , Apabila y ∝ 1 , graf y melawan 1 ialah satu garis lurus yang
and k is a constant / dan 2 3 xn xn
bermula daripada asalan.
k ialah pemalar
y
3. When y varies inversely as xn, the value of xny = k, where
k is called the constant of proportionality.

Apabila y berubah secara songsang dengan xn, nilai xny = k, dengan
keadaan k dikenali sebagai pemalar perkadaran.

1
O xn

12. For each of the following situations, state the change on the given quantity. PL 1
Bagi setiap situasi berikut, nyatakan perubahan pada kuantiti yang diberikan.

Example (a) The renovation of a room will take 6 days to

Teacher divides the members of Mathematics Club complete if 4 people are working. State the
into groups. 6 groups are formed with 20 members
in each group. State the change on change on
Cikgu membahagikan ahli Kelab Matematik kepada beberapa Kerja pengubahsuaian sebuah bilik mengambil masa 6 hari
kumpulan. 6 kumpulan dibentuk dengan 20 orang ahli dalam untuk disiapkan jika terdapat 4 orang bekerja. Nyatakan
perubahan pada

setiap kumpulan. Nyatakan perubahan pada (i) the time needed to complete the renovation

(i) the number of members in a group if the number if the number of people working is halved.
masa yang diperlukan untuk menyiapkan kerja
of groups is increased by 2 times. pengubahsuaian jika bilangan orang yang bekerja
bilangan ahli dalam setiap kumpulan jika bilangan dikurangkan separuh.
kumpulan bertambah 2 kali ganda.

(ii) the number of groups if the number of members (ii) the number of people working if the time
needed to complete the renovation is
in a group is decreased by 5 times. decreased by 50%.
bilangan kumpulan jika bilangan ahli dalam setiap bilangan orang yang bekerja jika masa yang diperlukan
kumpulan berkurang 5 kali ganda. untk menyiapkan kerja pengubahsuaian berkurang

(i) the number of members in a group is decreased 50%.

by 2 times (i) the time needed is doubled
bilangan ahli dalam setiap kumpulan berkurang 2 kali ganda masa yang diperlukan berganda dua

(ii) the number of groups is increased by 5 times (ii) the number of people working is increased
bilangan kumpulan bertambah 5 kali ganda by 50%

bilangan orang yang bekerja bertambah 50%

11 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(b) The table below shows the relation between the price and the number of pens that can be purchased

with a fixed amount of money.
Jadual di bawah menunjukkan hubungan antara harga dengan bilangan pen yang boleh dibeli dengan sejumlah wang yang
tertentu.

Price of a pen / Harga sebatang pen (RM) 1 2 2.50 4

Number of pens / Bilangan pen 40 20 16 10

State the change on
Nyatakan perubahan pada

(i) the number of pens if the price of a pen is decreased by 20%.
bilangan pen jika harga sebatang pen berkurang 20%.

(ii) the price of a pen if the number of pens is increased by 2 times.
harga sebatang pen jika bilangan pen bertambah 2 kali ganda.

(i) the number of pens is increased by 20%
bilangan pen bertambah 20%

(ii) the price of a pen is decreased by 2 times
harga sebatang pen berkurang 2 kali ganda

13. Determine whether the relation is an inverse variation. If yes, write the relation in the form of variation. PL 2

Tentukan sama ada hubungan yang diberikan ialah ubahan songsang atau bukan. Jika ya, tuliskan hubungan tersebut dalam bentuk
ubahan.

Example (a) The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y.
The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y. x4 9 16 25

x8 27 64 125 y 18 12 9 7.2

y 21 13 9 7 Determine whether y varies inversely as x .
Tentukan sama ada y berubah secara songsang dengan x .
Determine whether y varies inversely as 3x .

Tentukan sama ada y berubah secara songsang dengan 3x .

3xy 42 39 36 35 xy 36 36 36 36
xy is a constant, therefore y varies inversely as
x and y ∝ 1 .
3xy is not a constant, therefore y does not vary
inversely as 3x.  x

3xy bukan satu pemalar, maka y tidak berubah secara songsang xy ialah satu pemalar, maka y berubah secara songsang
dengan 3x. 1
dengan x dan y ∝  x  .

(b) The table below shows the values of x and y. (c) The table below shows the relationship between
Jadual di bawah menunjukkan nilai-nilai x dan y. the height, H cm, and the radius, R cm, of a
cylinder.
x3 4 9 15 Jadual di bawah menunjukkan hubungan antara tinggi, H
y9 12 27 35 cm, dan jejari, R cm, bagi suatu silinder.

Determine whether y varies inversely as x. R (cm) 2 45 6
Tentukan sama ada y berubah secara songsang dengan x. H (cm) 18 4.5 2.88 2

xy 27 48 243 525 Determine whether H varies inversely as R2.
Tentukan sama ada H berubah secara songsang dengan R2.

xy is not a constant, therefore y does not vary

inversely as x. R2H 72 72 72 72

xy bukan satu pemalar, maka y tidak berubah secara R2H is a constant, therefore H varies inversely as

songsang dengan x. 1
R2
R2 and H ∝

R2H ialah satu pemalar, maka H berubah secara songsang
1
dengan R2 dan H ∝ R2

© Penerbitan Pelangi Sdn. Bhd. 12

Mathematics  Form 5  Chapter 1  Variation    

14. Determine whether each of the following variations is an inverse variation using graph method. PL 3
Tentukan sama ada setiap ubahan berikut ialah ubahan songsang atau bukan dengan menggunakan kaedah graf.

Example (a) The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y.
The table below shows the values of x and y.
Jadual di bawah menunjukkan nilai-nilai x dan y. x5 10 20 25

x2 3 4 5 y 24 12 6 4.8
7 3
y 22 14 Determine whether y varies inversely as x.
Tentukan sama ada y berubah secara songsang dengan x.
Determine whether y varies inversely as x2.
Tentukan sama ada y berubah secara songsang dengan x2. 1
x
1 0.2 0.1 0.05 0.04
x2 12 6 4.8
0.25 0.11 0.06 0.04 y 24

y 22 14 7 3 y

y 30

30

20

20

10

10

1 1
O 0.1 0.2 0.3 x2 O 0.1 0.2 x

The graph of y against 1 shows a straight line
The graph of y against 1 does not show a straight x
that starts from the origin. Therefore, y varies
x2
line. Therefore, y does not vary inversely as x2. inversely as x. 1
x
Graf y melawan 1 bukan satu garis lurus. Maka, y tidak berubah Graf y melawan menunjukkan satu garis lurus yang
x2
bermula daripada asalan. Maka, y berubah secara langsung

secara songsang dengan x2. dengan x.

15. For each of the following, express p in terms of q. PL 4
Bagi setiap yang berikut, ungkapkan p dalam sebutan q.

Example (a) p varies inversely as q and p = 5 when q = 2.56.

p varies inversely as the square of q and p = 0.05 p berubah secara songsang dengan q dan p = 5 apabila

when q = 4. q = 2.56.

p berubah secara songsang dengan kuasa dua q dan p = 0.05

apabila q = 4. p ∝ 1
 q
p ∝ 1
q2 p= k
 q
p= k
q2 p = 5 when / apabila q = 2.56

p = 0.05 when / apabila q = 4 5 = k
k 2.56
0.05 = 42
k = (5)2.56 
k = (0.05)(42)
4 = 8
= 5 Therefore / Maka, p = 8

4 q
5q2
Therefore / Maka, p =

13 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(b) It is given that p ∝ 1 and p = 36 when q = 0.5. (c) The electrical resistance, p ohms, of a wire varies
Diberi p ∝ apabila q = 0.5. inversely as the square of its radius, q cm. It is
1 dan p = q3 given that the resistance of a wire is 1 ohm when
q3 36 its radius is 0.2 cm.

p ∝ 1 Rintangan elektrik, p ohm, bagi satu dawai berubah
q3 secara songsang dengan kuasa dua jejarinya, q cm. Diberi

p= k rintangan elektrik bagi suatu wayar ialah 1 ohm apabila
q3 jejarinya ialah 0.2 cm.

p = 36 when / apabila q = 0.5 p ∝ 1
k q2
36 = 0.53

k = (36)(0.53) p = k
q2
= 4.5

Therefore / Maka, p= 4.5 p = 1 when / apabila q = 0.2
q3 k
1 = 0.22

k = (1)(0.22)

= 0.04

Therefore / Maka, p = 0.04
q2

16. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example

It is given that y ∝ 1 and y = 7.5 when x = 0.512. Find the value of x when y = 20.
 3x
1
Diberi y ∝  3x dan y = 7.5 apabila x = 0.512. Cari nilai x apabila y = 20.

y ∝ 1
 3x
Alternative Method
y= k Concept of proportion:
 3x Konsep kadaran:

y = 7.5 when / apabila x = 0.512 3x1y1 = 3x2y2 when y1 = 7.5, x1 = 0.512 and y2 = 20

7.5 = k 3x1y1 = 3x2y2 apabila y1 = 7.5, x1 = 0.512 dan y2 = 20
30.512

k = (7.5)30.512   30.512 (7.5) = 3x2 (20)

= 6

Therefore / Maka, y = 6 3x2 = 30.512 (7.5)
 3x 20

When / Apabila y = 20, x2 = 0.33
20 = 6
= 0.027
 3x
6
3x = 20

x = 0.33

= 0.027

© Penerbitan Pelangi Sdn. Bhd. 14

Mathematics  Form 5  Chapter 1  Variation    

(a) It is given that y varies inversely as the square root of x and y = 15 when x = 0.64. Find the value of x
when y = 0.8.
Diberi y berubah secara songsang dengan punca kuasa dua x dan y = 15 apabila x = 0.64. Cari nilai x apabila y = 0.8.

y∝ 1 When / apabila y = 0.8,
 x
0.8 = 12
y= k x
 x
x = 12
y = 15 when / apabila x = 0.64 0.8
15 = k
x = 152
0.64
k = (15)0.64  = 225
= 12

Therefore / Maka, y = 12
 x

(b) It is given that y ∝ 1 and y = 0.25 when x = 6. Find the value of y when x = 1.2.
= x3 Cari nilai y apabila x = 1.2.
Diberi y ∝ 1 dan y 0.25 apabila x = 6.
x3

y ∝ 1 When / Apabila x = 1.2,
x3 54
y = 1.23
y= k
x3 = 31.25

y = 0.25 when / apabila x = 6
k
0 .25 = 63

k = (0.25)(63)

= 54

Therefore / Maka, y = 54
x3

17. Solve each of the following. PL 4 (i) D ∝ 1 (ii) When D = 750,
Selesaikan setiap yang berikut. E3
Apabila D = 750,
Example k
D varies inversely as the cube of E and D = 6 D = E3 750 = 48
when E = 2. E3

D berubah secara songsang dengan kuasa tiga E dan D = 6 D = 6 when E = 2 E3 = 48
apabila E = 2. 750
D = 6 apabila E = 2
(i) Write the equation of D in terms of E. k 8
Tuliskan persamaan bagi D dalam sebutan E. 6 = 23 = 125

(ii) Find the value of E when D = 750. k = (6)(23)  E = 3 8
Cari nilai E apabila D = 750. 125

= 48

Therefore / Maka, = 2
48 5
D = E3

15 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(a) D varies inversely as E2 and D = 25 when E = 0.8. (i) D ∝ 1 (ii) When D = 0.64,
D berubah secara songsang dengan E2 dan D = 25 apabila E2
Apabila D = 0.64,
0.64 = 16
E = 0.8. D= k E2
(i) Write the equation of D in terms of E. E2
16
Tuliskan persamaan bagi D dalam sebutan E. D = 25 when E = 0.8 E2 = 0.64

(ii) Find the positive value of E when D = 0.64. D = 25 apabila E = 0.8 = 25
25 = k
Cari nilai positif E apabila D = 0.64. 0.82 E = 25

k = (25)(0.82) = 5

= 16

Therefore / Maka,
D = 16

E2

(b) The table below shows the values of D and E. (i) D ∝ 1 (ii) When D = 125,
Jadual di bawah menunjukkan nilai-nilai D dan E. E
Apabila D =125,
1
D 0.0625 125 D = k 125 = 4x
E
1
E4 x D = 0.0625 when E = 4 x = (125)(4)

It is given that D ∝ 1 . D = 0.0625 apabila E = 4 = 0.002
E k
1 0.0625 = 4
E
Diberi bahawa D ∝ . k = (0.0625)(4)

(i) Write the equation of D in terms of E. = 1
Tuliskan persamaan bagi D dalam sebutan E. 4
Therefore / Maka,
(ii) Find the value of x. 1
Cari nilai x. D = 4E

(c) It is given that D ∝ 1 and D = 45 (i) D ∝ 1 (ii) When D = 11.25,
 3E
when E = 0.008.  3E Apabila D =11.25,
D= k 11.25 = 9
Diberi D ∝ 1 dan D = 45 apabila E = 0.008.  3E  3E
 3E
D = 45 when E = 0.008 3E = 9
(i) Write the equation of D in terms of E. D = 45 apabila E = 0.008 11.25
Tuliskan persamaan bagi D dalam sebutan E.
45 = k E = 0.83
(ii) Find the value of E when D = 11.25.  30.008
Cari nilai E apabila D = 11.25. = 0.512
k = (45)30.008
= 9

Therefore / Maka,
D= 9

 3E

© Penerbitan Pelangi Sdn. Bhd. 16

Mathematics  Form 5  Chapter 1  Variation    

18. Solve each of the following. PL 5 Daily Application
Selesaikan setiap yang berikut.

(a) The radius, r cm, of a cylinder of fixed volume varies inversely as the square root of its height, h cm.
It is given that the radius of a cylinder is 2 cm when its height is 49 cm.
Jejari, r cm, bagi satu silinder yang mempunyai isi padu yang tetap, berubah secara songsang dengan punca kuasa dua
tingginya, h cm. Diberi jejari suatu silinder ialah 2 cm apabila tingginya ialah 49 cm.

(i) Write the equation of r in terms of h.
Tuliskan persamaan bagi r dalam sebutan h.

(ii) Find the value of h when r = 3.5.
Cari nilai h apabila r = 3.5.

(i) r ∝ 1 (ii) When / Apabila r = 3.5,
 h
  3.5 = 14
r= k  h
 h 14
h = 3.5
r = 2 when / apabila h = 49
2 = k h = 42
 49
k = (2)49  = 16

= 14
Therefore / Maka, r = 14
 h

(b) A fixed quantity of a metal is melted to make a number of spheres. The number of spheres, p, varies
inversely as the cube of its diameter, x mm. It is given that 40 spheres each with a diameter of 12 mm
can be made from a certain quantity of metal.
Kuantiti logam yang tetap dileburkan untuk membuat sejumlah sfera. Bilangan sfera, p, berubah secara songsang dengan
kuasa tiga diameternya, x mm. Diberi bahawa sebanyak 40 sfera masing-masing dengan diameter 12 mm boleh dibuat dengan
sejumlah kuantiti logam tertentu.
(i) Write the equation of p in terms of x.
Tuliskan persamaan bagi p dalam sebutan x.
(ii) If 135 spheres can be made from the same quantity of the metal, find the diameter, in mm, of each

sphere.
Jika 135 sfera boleh dibuat dengan kuantiti logam itu, cari diameter, dalam mm, setiap sfera.

(i) p ∝ 1 (ii) When / apabila p = 135,
x3
135 = 69 120
p= k x3
x3
x3 = 69 120
p = 40 when / apabila x = 12 135

40 = k x = 3512
123
k = (40)(123) = 8 mm

= 69 120

Therefore / Maka, p = 69 120
x3

17 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

(c) The volume, V, m3, of a gas varies inversely as the pressure, P N m–2. It is given that the volume of a
gas is 400 m3 under a pressure of 13 N m–2.
Isi padu, V m3, bagi gas berubah secara songsang dengan tekanannya, P N m–2. Diberi isi padu bagi suatu gas ialah
400 m3 di bawah tekanan 13 N m–2.

(i) Write the equation of V in terms of P.
Tuliskan persamaan bagi V dalam sebutan P.

(ii) The volume of the gas is decreased to 260 m3. Find the pressure, in N m–2, of the gas.
Isi padu gas itu dikurangkan kepada 260 m3. Cari tekanan, dalam N m–2, gas itu.

(i) V∝ 1 (ii) When / apabila V = 260,
P
260 = 5 200
V= k P
P
P = 5 200
V = 400 when / apabila P = 13 260

400 = k = 20 N m–2
13
k = (400)(13)

= 5 200

Therefore / Maka, V = 5 200
P

1.3 Combined Variation Textbook
Ubahan Bergabung pg. 26 – 29

SMART Notes

1. A combined variation involves a combination of direct 3. The relation between the variables can be written as y
variation or joint variation and inverse variation. varies directly as xm and inversely as zn.

Ubahan bergabung melibatkan gabungan ubahan langsung atau Hubungan antara pemeboleh ubah ini boleh ditulis sebagai y berubah
ubahan tercantum dan ubahan songsang. secara langsung dengan xm dan secara songsang dengan zn.

2. In a combined variation, a variable varies directly/ jointly y ∝ xm (variation relation/ hubungan ubahan)
and inversely as two or more other variables. zn
kxm
Bagi suatu ubahan bergabung, suatu pemboleh ubah berubah secara y = zn (equation form / bentuk persamaan)
langsung/ tercantum dan songsang dengan dua atau lebih pemboleh
ubah yang lain. where / dengan keadaan m = 1, 2, 3, 1 , 1
2 3
1 1
and/ dan n = 1, 2, 3, 2 , 3

and k is a constant / dan k ialah pemalar

19. Write the relation of each of the following combined variations using the symbol ∝. PL 2
Tuliskan hubungan setiap ubahan bergabung berikut dengan menggunakan simbol ∝.

Example (a) P varies directly as Q and inversely as the cube

D varies directly as E and inversely as F. of R.
P berubah secara langsung dengan Q dan secara songsang
D berubah secara langsung dengan E dan secara songsang dengan kuasa tiga R.
dengan F.

D∝ E P ∝ Q
F R3

(b) y varies directly as x and z, and inversely as the (c) s varies directly as t2 and inversely as 3u.
s berubah secara langsung dengan t2 dan secara songsang
square of w. dengan 3u.
y berubah secara langsung dengan x dan z, dan secara
songsang dengan kuasa dua w. s ∝ t2
3u
y ∝ xz
w2

© Penerbitan Pelangi Sdn. Bhd. 18

Mathematics  Form 5  Chapter 1  Variation    

20. For each of the following, express m in terms of n and p. PL 4
Bagi setiap yang berikut, ungkapkan m dalam sebutan n dan p.

Example (a) m varies inversely as 3n and p. It is given that

m varies directly as n and inversely as the square root m = 2.5 when n = 0.125 and p = 16.
of p. It is given that m = 21 when n = 7 and p = 144. m berubah secara songsang dengan 3n dan p. Diberi
m berubah secara langsung dengan n dan secara songsang m = 2.5 apabila n = 0.125 dan p = 16.

dengan punca kuasa dua p. Diberi m = 21 apabila n = 7 dan 1
3n p
p = 144. m ∝

m∝ n m = k
 p 3n p

m = kn m = 2.5 when / apabila n = 0.125, p = 16
 p k
2.5 =
m = 21 when / apabila n = 7, p = 144 30.125 (16)

21 = k(7) k = (2.5)30.125 (16)
 144
= 20
(21)144 
k =  7 Therefore / Maka, m = 20
3n p
= 36

Therefore / Maka, m = 36n
 p

(b) It is given that m ∝ n2 and m = 10.8 when (c) m varies directly as n and inversely as p. It is given
n = 18 and p = 6. p
that m = 51 when n = 1.7 and p = 5.
Diberi m ∝ n2 dan m = 10.8 apabila n = 18 dan p = 6. m berubah secara langsung dengan n dan secara songsang
p dengan p. Diberi m = 51 apabila n = 1.7 dan p = 5.

n2
p
m ∝ m ∝ n
p

m = kn2 m= kn
p p

m = 10.8 when / apabila n = 18, p = 6 m = 51 when / apabila n = 1.7, p = 5

1 0.8 = k(18)2 5 1 = k(1.7)
6 5

k = (10.8)(6) k = (51)(5)
182 1.7
= 150
= 1
5 150n
Therefore / Maka, m = p
Therefore / Maka, m = n2
5p

19 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

21. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example

It is given that y ∝ x2 and y = 12.8 when x = 0.8 and z = 0.25. Find the positive value of x when y = 67.5
 z
and z = 144.
Diberi y ∝ x2 dan y = 12.8 apabila x = 0.8 dan z = 0.25. Cari nilai positif x apabila y = 67.5 dan z = 144.
 z

y ∝ x2 When y = 67.5, z = 144, Alternative Method
 z
Apabila y = 67.5, z = 144, Concept of proportion:
y = kx2
 z 6 7.5 = 10x2 Konsep kadaran:
 144
y1z1 = y2z2 when /apabila y1 = 12.8,
(67.5)144 x12 x22
y = 12.8 when x = 0.8, z = 0.25 x2 = 10 x1 = 0.8, z1 = 0.25, y2 = 67.5, z2 = 144

y = 12.8 apabila x = 0.8, z = 0.25 x = 81

12.8 = k(0.8)2 = 9 (12.80).820.25 = (67.5)144 
 0.25 x22

k = (12.8)0.25  x22 = (67.5)144 (0.82)
0.82 (12.8)0.25

= 10 x2 = 81
= 9
Therefore / Maka, y = 10x2
z

(a) It is given that y varies inversely as x and the cube (b) It is given that y ∝  x and y = 32 when x = 2.56
root of z, and y = 0.25 when x = 6 and z = 0.512. z
and z = 0.4. Find the value of y when x = 81 and
Find the value of z when x = 0.4 and y = 5.
z = 6.
Diberi y berubah secara songsang dengan x dan punca kuasa  x
tiga z, dan y = 0.25 apabila x = 6 dan z = 0.512. Cari nilai Diberi y ∝ z dan y = 32 apabila x = 2.56 dan z = 0.4.
z apabila x = 0.4 dan y = 5.
Cari nilai y apabila x = 81 dan z = 6.

y ∝ 1
x3z y ∝  x
z

y= k y =  kx
x3z z

y = 0.25 when / apabila x = 6, z = 0.512 y = 32 when / apabila x = 2.56, z = 0.4

0.25 = k 32 =  k 2.56 
0.4
(6)30.512 

k = (0.25)(6)30.512  k = (32)(0.4)
2.56
= 1.2
= 8
Therefore / Maka, y = 1.2
x3z Therefore / Maka, y =  8x
z
When / Apabila x = 0.4, y = 5,
When / Apabila x = 81, z = 6,
5 = 1.2
(0.4)3z  y =  881
6
3z = 1.2 = 12
(5)(0.4)

z = 0.63

= 0.216

© Penerbitan Pelangi Sdn. Bhd. 20

Mathematics  Form 5  Chapter 1  Variation    

22. Solve each of the following. PL 4
Selesaikan setiap yang berikut.

Example (a) p varies inversely as q2 and r, and p = 24 when

p varies directly as the cube root of q and inversely q = 0.5 and r = 12.
as r, and p = 80 when q = 27 and r = 0.6. p berubah secara songsang dengan q2 dan r, dan p = 24
p berubah secara langsung dengan punca kuasa tiga q dan secara apabila q = 0.5 dan r = 12.

songsang dengan r, dan p = 80 apabila q = 27 dan r = 0.6. (i) Write the equation of p in terms of q and r.

(i) Write the equation of p in terms of q and r. Tuliskan persamaan bagi p dalam sebutan q dan r.

Tuliskan persamaan bagi p dalam sebutan q dan r. (ii) Find the positive value of q when p = 0.625

(ii) Find the value of p when q = 0.216 and r = 4. and r = 0.8.
Cari nilai p apabila q = 0.216 dan r = 4. Cari nilai positif q apabila p = 0.625 dan r = 0.8.

(i) p ∝  3q (ii) q = 0.216, r = 4, (i) p∝ 1 (ii) p = 0.625, r = 0.8,
r q2r
72
p= k3q p =  1630.216 p= k 0.625 = q2(0.8)
r 4 q2r
72
p = 80, q = 27, r = 0.6 = 2.4 p = 24, q = 0.5, q2 = (0.625)(0.8)

k327  r = 12 q = 144
80 = 0.6 k
24 = (0.5)2(12) = 12

k = (80)(0.6) k = (24)(0.5)2(12)
327
= 72
= 16
Therefore / Maka,
72
Therefore / Maka, p= q2r

p =  163q
r

(b) The table below shows the values of p, q and r. (c) It is given that p ∝  q and p = 75 when q = 324
Jadual di bawah menunjukkan nilai-nilai p, q dan r. and r = 0.3. r2

p 10 6.25 Diberi p ∝  q dan p = 75 apabila q = 324 dan r = 0.3.
r2
q 35 x
(i) Write the equation of p in terms of q and r.
r 0.2 4
Tuliskan persamaan bagi p dalam sebutan q dan r.

It is given that p ∝ 1 . (ii) Find the value of q when p = 2.4 and r = 0.5.
qr3 Cari nilai q apabila p = 2.4 dan r = 0.5.

Diberi bahawa p ∝ 1 . (i) p ∝  q
qr3 r2
(ii) p = 2.4, r = 0.5,
(i) Write the equation of p in terms of q and r.
Tuliskan persamaan bagi p dalam sebutan q dan r. p =  kq   2.4 = 3q
r2 8(0.5)2
(ii) Find the value of x.
Cari nilai x. p = 75, q = 324, q = (2.4)(8)(0.5)2
3
r = 0.3
q = 1.62
1  k324
(i) p∝ qr3 (ii) p = 6.25, q = x, r = 4, 7 5 = 0.32 = 2.56

p= k 6.25 = 2.8 k = (75)(0.32)
qr3 x(4)3 324
2.8
p = 10, q = 35, x = (6.25)(4)3 3
r = 0.2 8
= 0.007 =

10 = k Therefore / Maka,
(35)(0.2)3
p =  3q
k = (10)(35)(0.2)3 8r2

= 2.8

Therefore / Maka,
2.8
p = qr3

21 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

23. Solve each of the following. PL 5 Daily Application
Selesaikan setiap yang berikut.

(a) The time needed, T minutes, to boil water varies directly as the weight, w kg, of the water and inversely
as the power, P watts, of a kettle. Shamsiah uses 4 minutes to boil 2 kg water using a kettle with the
power of 1 500 watts.
Masa, T minit, untuk mendidih air berubah secara langsung dengan jisim air, w kg, dan secara songsang dengan kuasa cerek,
P watt. Shamsiah menggunakan 4 minit untuk mendidih 2 kg air dengan sebuah cerek 1 500 watt.

(i) Write the equation of T in terms of w and P.
Tuliskan persamaan bagi T dalam sebutan w dan P.

(ii) If the power of the kettle used is 1 200 watts, what is the time needed, in minutes, to boil 1.8 kg
water?
Jika kuasa cerek yang digunakan ialah 1 200 watt, berapakah masa, dalam minit, yang diperlukan untuk mendidih
1.8 kg air?

(i) T ∝ w (ii) When / Apabila w = 1.8, P = 1 200,
P
T = 3 000(1.8)
1 200
T= kw = 4.5 minutes / minit
P

T = 4 when / apabila w = 2, P = 1 500

4 = k(2)
1 500

k = (4)(12500)
= 3 000

Therefore / Maka, T = 3 000w
P

(b) The time taken, T hours, to harvest pineapple in a farm varies directly as the area, A m2, of the farm
and inversely as the number of workers, P, working on it. It is given that 5 workers take 2 hours to
harvest pineapple in a farm with an area of 1 000 m2.
Masa yang diambil, T jam, untuk menuai nanas di satu ladang berubah secara langsung dengan luas, A m2, ladang dan secara
songsang dengan bilangan pekerja, P, yang bekerja. Diberi 5 pekerja mengambil 2 jam untuk menuai nanas di sebuah ladang
yang seluas 1 000 m2.

(i) Pak Samad has a farm with an area of 840 m2 which is harvested by 3 workers. Find the time
taken, in hours, to complete the work.
Pak Samad mempunyai satu ladang yang seluas 840 m2 yang dituai oleh 3 orang pekerja. Cari masa yang diperlukan,
dalam jam, untuk menyiapkan kerja.

(ii) What will happen if 2 workers are added to complete the harvesting work? Explain your answer.
Apakah yang akan berlaku jika 2 orang pekerja ditambahkan untuk menyiapkan kerja penuaian? Jelaskan jawapan anda.

(i) T ∝ A (ii) The time taken will be decreased when the
P number of workers increased because the
time taken varies inversely as the number
T = kA of workers.
P M asa yang diambil akan berkurang apabila bilangan
pekerja bertambah kerana masa yang diambil berubah
T = 2 when / apabila A = 1 000, P = 5 secara songsang dengan bilangan pekerja.

2 = k(1 000)
5

k = (120)(050)

= 1
100

Therefore / Maka, T = A
100P

When / Apabila A = 840, P = 3,

T = 840
100(3)

= 2.8 hours / jam

© Penerbitan Pelangi Sdn. Bhd. 22

SPM Practice Mathematics  Form 5  Chapter 1  Variation    

1

Paper 1 6. It is given that p ∝ 1 and q = 2r – 1. If p = 0.25 when
q
r = 2, express p in terms of q. HOTS Applying
1
1. It is given that y varies inversely as 3x – 2 and Diberi p ∝ q dan q = 2r – 1. Jika p = 0.25 apabila r = 2,

SPM y = 4 when x = 2. Find the value of x when y = – 4. ungkapkan p dalam sebutan q. 3
2017 Diberi y berubah secara songsang dengan 3x – 2 dan y = 4 2 4q
apabila x = 2. Cari nilai x apabila y = –4. A p= q C p=

A –   2 C 2 B p= 1 D p= 1
3 3 3q 4q

B –2 D 2

2. The table below shows the values of E, F and G. 7. The table below shows the values of variables M and N.
Jadual di bawah menunjukkan nilai-nilai pemboleh ubah M
SPM Jadual di bawah menunjukkan nilai-nilai E, F dan G. dan N.
2017

E6 3.5 M 345.6 5.4
F 0.4 7
G 0.008 p N 12 x

Given M varies directly as the square of N, find the value

It is given that E varies directly as F and inversely as the of x.

cube root of G. Find the value of p. Diberi M berubah secara langsung dengan kuasa dua N, cari
nilai bagi x.
Diberi E berubah secara langsung dengan F dan secara songsang A 1.5 C 2.25
dengan punca kuasa tiga G. Cari nilai p.
A 27 C 0.125 B 1.6 D 2.56

B 216 D 0.027 8. It is given that y varies directly as x and inversely as the

3. It is given that m varies directly as the square root of n and cube of z. If y = 2 when x = 108 and z = 3, write an

SPM inversely as the cube of p. If m ∝ nx , state the value of x equation to relate y, x and z.
2018 and of y. py
Diberi y berubah secara langsung dengan x dan secara songsang
dengan kuasa tiga z. Jika y = 2 apabila x = 108 dan z = 3, tulis
Diberi m berubah secara langsung dengan punca kuasa dua n satu persamaan yang menghubungkan y, x dan z.
nx
dan secara songsang dengan kuasa tiga p. Jika m ∝ py , nyatakan A y = x C y= x
nilai x dan nilai y. 2z3 z3
1
A x = 2, y = 3 C x = 2, y = 3 B y= 2x D y = x
z3 4z3
B x= 1 , y = 1 D x= 1 , y = 3
2 3 2 9. It is given that p varies inversely as q and r. If p = 5.25

4. The table below shows the time needed to clean hotel when q = 256 and r = 0.5, express p in terms of q and r.
SPM rooms, x and the number of workers, y.
2018 Jadual di bawah menunjukkan masa yang diperlukan untuk Diberi p berubah secara songsang dengan q dan r. Jika
mengemaskan bilik hotel, x dan bilangan pekerja, y. p = 5.25 apabila q = 256 dan r = 0.5, ungkapkan p dalam
sebutan q dan r.
2 24
x5 m A p =  q r C p =  q r
y3 5
B p = 3 D p = 42
 q r  q r
It is given that x varies inversely as the square of y. Find
10. It is given that m varies inversely as n2. If m = 0.25 when
the value of m.
n = 12, find the value of n when m = 100.
Diberi x berubah secara songsang dengan kuasa dua y. Cari
nilai m. Diberi m berubah secara songsang dengan n2. Jika m = 0.25
A 1.8 C 2.5 apabila n = 12, cari nilai n apabila m = 100.
A 36 C 0.6
B 2 D 3
B 25 D 0.5
n
5. It is given that m ∝  3p and m = 20 when n = 15 and 11. The weight, w g, of a small metal cube varies directly as

p = 216. Calculate the value of p when m = 36 and n = 1.8. the cube of its side, x cm. If w = 360 when x = 2, find the

Diberi m ∝ n dan m = 20 apabila n = 15 dan p = 216. value of x when w = 2.88.

 3p Berat, w g, sebuah kubus logam kecil berubah secara langsung
Hitung nilai p apabila m = 36 dan n = 1.8. dengan kuasa tiga sisinya, x cm. Jika w = 360 apabila x = 2,
A 0.027 C 0.064
cari nilai x apabila w = 2.88.
B 8 D 125 A 0.2 C 0.25

B 0.45 D 0.4

23 © Penerbitan Pelangi Sdn. Bhd.

  Mathematics  Form 5  Chapter 1  Variation

12. Hamid divides a certain amount of flour into several plastic It is given that Q varies inversely as P and the value of
bags. What is the change of the weight of each plastic bag
if the number of plastic bags is doubled? m × n is 6. Calculate the value of x.

Hamid membahagikan sejumlah tepung ke dalam beberapa beg Diberi bahawa Q berubah secara songsang dengan P dan nilai
plastik. Apakah perubahan pada berat setiap plastik beg jika m × n ialah 6. Hitung nilai x
bilangan plastik beg berganda dua? A 0.3 C 1.2
A the weight of each plastic bag increased by two
berat setiap plastik beg ditambah dengan dua B 0.6 D 1.6
B the weight of each plastic bag decreased by two
berat setiap plastik beg ditolak dengan dua 14. The height, h, of a cone varies directly as its volume, V,
C the weight of each plastic bag is halved SPM and inversely as the square of the radius, r, of its base.
berat setiap plastik beg berkurang separuh 2019 It is given that h = 5.7 cm, V = 150 cm3 and r = 5 cm.
D the weight of each plastic bag is doubled Calculate the value of r, in cm, when h = 76 cm and
berat setiap plastik beg berganda dua V = 180 cm3.

13. The table below shows the values of variables P and Q. Tinggi, h, bagi satu kon berubah secara langsung dengan isi
padunya, V, dan secara songsang dengan kuasa dua jejari,
SPM Jadual di bawah menunjukkan nilai-nilai pemboleh ubah P dan r, tapaknya. Diberi bahawa h = 5.7 cm, V = 150 cm3 dan
2019 pemboleh ubah Q. r = 5 cm. Hitung nilai r, dalam cm, apabila h = 76 cm dan
V = 180 cm3.
Pm x A 4
Qn 20 B 1.5
C 2.5
D 2

HOTS Challenge

I t is given that P varies directly as Q and inversely as R. If P = 10 when Q = 2 and R = 0.8,
Diberi P berubah secara langsung dengan Q dan secara songsang dengan R. Jika P = 10 apabila Q = 2 dan R = 0.8,

(a) express P in terms of Q and R,
ungkapkan P dalam sebutan Q dan R,

(b) find the percentage of the change of P when Q increases by 8% and R decreases by 10%. HOTS Analysing
cari peratus ubahan bagi P apabila Q bertambah sebanyak 8% dan R berkurang sebanyak 10%.

Answer / Jawapan:
Q
(a) P∝ R (b) Percentage of change of P

kQ Peratus ubahan bagi P
R
P= = 1.08 × 100%
0.9

P = 10 when / apabila Q = 2, R = 0.8 = 120%

10 = k(2)
0.8

k = (10)(0.8)
2

= 4

Therefore / Maka, P = 4Q HOTS
R

Quiz 1

21PAK- 24

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