2. Matematik Soalan modul PLUS

SET SET

1 1

The Venn diagram in the answer space shows sets P, Q and R such that the universal set,  = = P  Q  R Gambarajah Venn di ruang jawapan menunjukkan set P, Q dan R dengan keadaan set semesta,  = = P  Q  R. On the diagrams in the answer space, shade: Dalam diagram di ruang jawapan, lorek : (a) (b)

set P  R set Q  ( P  R ) [3 marks] [3 markah] markah]

Answer/Jawapan Answer/Jawapan : : (a) Q

R

P

(b) Q

R

P

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PLUS JABATAN PENDIDIKAN SELANGOR

2.

The Venn diagram in the answer space shows set P, Q and R with the universal set  = P  Q  R. Gambar rajah Venn di ruang jawapan menunjukkan set P, Q dan R dengan keadaan set semesta,  = P  Q  R. On the diagram in the answer space, shade Pada rajah di ruang jawapan, lorekkan (a) (b)

PQ R P   ( Q  R) [3 marks] [3 markah] markah]

Answer/Jawapan Answer/Jawapan : : (a) Q

P

R

(b) P

Q

R

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3.

Each Venn diagram in the answer space shows sets P, Q and R such that the universal set, ξ = P ∪ Q ∪ R. Setiap gambar rajah Venn di ruang jawapan menunjukkan set P, Q dan R dengan keadaan set semesta, R. ξ = P ∪ Q ∪ R. On the diagrams in the answer space, shade Pada rajah di ruang jawapan, lorek (a) Q' ∩ R (b) P ∩ (Q ∪ R') [3 marks] [3 markah] markah] Answer/Jawapan Answer/Jawapan : :

(a) P

Q

R

(b) P

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Q

R


4.

On the diagrams provided in the answer space, shade the region which represent the following operation of set. Pada rajah yang disediakan di ruang jawapan, lorek rantau yang mewakili operasi set berikut (a) (b)

A  B' A  (B  C)' [3 marks] [3 markah]

Answer/Jawapan : (a) B

C

A

(b) C

B

A

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5.

(a)

Set R is the set of even numbers and set S is the set of multiples of 7. Complete the Venn Diagram in the answer space to show the relationship between set R and set S. Set R ialah set nombor genap dan set S ialah set gandaan 7. Lengkapkan gambar rajah Venn di ruang jawapan untuk menunjukkan hubungan antara set R dan set S.

(b)

Given three set, K, L, and M such that the universal set,  = K ∪ L ∪ M K ∩ L =  and K ∪ L ⊂ M Diberi tiga set K, L, dan M dengan keadaan set semesta,  = K ∪ L ∪ M K ∩ L =  dan K ∪ L ⊂ M Draw the Venn Diagram in the answer space to show the relationship between sets K, L and M. Lukis gambar rajah Venn di ruang jawapan untuk menunjukkan hubungan antara set K, L dan M. [3 marks] [3 markah]

Answer/Jawapan : (a) R

(b)

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2 1

QUADRATIC EQUATIONS PERSAMAAN KUADRATIK Solve the equation x 2 Selesaikan persamaan



2( x 1)  3 x  2 . 2

x



2( x 1)  3 x  2

. [ 4 marks ] [ 4 markah ]

Answer / Jawapan :

2

Using factorisation, solve the following quadratic equations Menggunakan pemfaktoran, selesaikan persamaan kuadratik berikut: ( x  3)( x  3)



2 x ( x  4)  6

[4 markah] [4 marks] Answer / Jawapan :

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3

Solve the following quadratic equation: Selesaikan persamaan kuadratik berikut : x  3 

8 x  4

[4 marks] [4 markah] Answer / Jawapan :

4

Solve the following quadratic equation: Selesaikan persamaan kuadratik yang berikut :

5 x  2 x

 x  6 [ 4 marks ] [4 markah]

Answer / Jawapan :

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5

Solve the following quadratic equation: Selesaikan persamaan kuadratik berikut: 9x2 – 8x = 8(2 – x) [ 4 marks ] [ 4 markah ] Answer / Jawapan:

6

Faizal rides his bicycle at a speed of 3x kmj-1 for ( x - 3 ) hours. After that, he increased his speed to (3 x - 5) kmj-1 for ( x - 4 ) hours. Find the distance travelled by him and express your anwer in the form of quadratic expression Faizal mengayuh basikal dengan kelajuan 3x km/j selama ( x - 3 ) jam.Kemudian,dia menambah kelajuannya kepada (3 x - 5 ) kmj -1 selama ( x - 4 ) jam. Cari jarak yang dilaluinya dan ungkapkan jawapan anda dalam bentuk ungkapan kuadratik [ 3 marks ] [ 3 markah ] Answer / Jawapan :

7

A rectangular piece of paper is folded to make the curve surface of a cylinder. The length of the paper, x cm, is 6 cm longer than its breadth. If the area of the curved surface is 135 cm 2, find the value of x. Sekeping kertas berbentuk segi empat tepat dilipat untuk membentuk permukaan lengkung sebuah silinder. Panjang kertas itu, x cm , adalah 6 cm lebih panjang daripada lebarnya. Jika luas permukaan lngkung itu ialah 135 cm2, cari nilai x. [ 4 marks ] [ 4 markah ] Answer / Jawapan :

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3

MATHEMATICAL REASONING PENAAKULAN MATEMATIK

1. Determine whether each of the following statements is true of false. Tentukan sama ada setiap pernyataan berikut adalah benar atau palsu. a) 75% = b)

3 4

9 59 < 50 100

c) 8.2333 × 103 = 8 233.3 d) 50 000 m = 5 km Answer: Jawapan: a)……………………………………………………………………………………..……………………….. b)………………………………………………………………………………… .….……………………….. c)………………………………………..……………………………………… .……………………………. d)……………………………………………………………………………………………………………… 2. a) Construct a true statement by using the quantifier "all" or "some" in each of the following statements. Bina suatu pernyataan benar dengan menggunakan pengkuantiti "semua" atau "sebilangan" dalam setiap pernyataan berikut. i) Odd numbers are prime numbers. Nombor ganjil adalah nombor perdana. ii) Pentagon have five sides. Pentagon mempunyai lima sisi. b) State the converse of the following implication. Hence, state whether the converse is true or false. Nyatakan akas untuk implikasi berikut. Seterusnya, nyatakan sama ada akas tersebut adalah benar atau palsu. If q > 9, then q > 7. Jika q > 9, maka q > 7.

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Answer/ Jawapan: a)

i)……………………………………………………………………...………………………………. ii) ……………………………………………………………………….…………………………….

b)…………………………………………………………………………………………………………. 3.

(a)

(b)

For each of the following statements, determine whether the statement is true or false. Untuk setiap pernyataan berikut, tentukan sama ada pernyataan ini benar atau palsu . (i)

1 is a prime number and 11 is an odd number. 1 ialah nombor perdana dan 11 ialah nombor ganjil.

(ii)

5 is a factor of 40 or 14 is a multiple of 4. 5 ialah faktor bagi 40 atau 14 ialah gandaan bagi 4.

State the converse of the following statement and hence determine whether its converse is true or false. Nyatakan akas bagi pernyataan berikut dan seterusnya nyatakan sama ada akas itu benar atau palsu.

If x  10, then x  12 Jika x  10, maka x  12

(c)

Make a general conclusion by induction for the sequence of numbers 3, 10, 21… which follows the following pattern. Buat satu kesimpulan umum secara aruhan bagi urutan nombor mengikut pola berikut. 3 =

2(1)2 + 1

10 =

2(2) 2 + 2

21 =

2(3) 2 + 3

….…..

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=

…………..


3, 10, 21…

yang

Answer / Jawapan:

(a)

(i)

………………………………………………………………………….……………………

(ii)

……………………………………………………………………….………………………

………………………………………………………………………..………………………………

(b)

…………………………………………………………………..…………………………………… …………………………………………………………………………..……………………………

(c)

………………………………………………………………..……………………………………… 4. (a)

Complete each of the following statement with the quantifier “ all ” or “some” to make it a true statement. Lengkapkan pernyataan berikut dengan menggunakan pengkuantiti “ semua” atau “ s ebilangan ”, untuk membentuk suatu pernyatan benar. …..……..…… ………………

prime numbers are odd numbers. nombor perdana adalah nombor ganjil .

(b) Write down two implication based on the following compound statement: Tulis du a implikasi berdasarkan pernyataan berikut. if and only if 3 x > 18 x > 6 x > 6 jika dan hanya jika 3 x > 18

(c) Polygon

Number of sides

Sum of interior angles

Triangle Quadrilateral Pentagon Hexagon

3 4 5 6

( 3  2 ) x 180  ( 4  2 ) x 180  ( 5  2 ) x 180  ( 6  2 ) x 180 

Based on the information in the table, make on e conclusion by induction regarding the sum of interior angles of a polygon with an n sides. Berdasarkan maklumat dalam jadual, buat satu kesimpulan secara induksi tentang jumlah sudut pedalaman sebuah poligon bersisi n.

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Answer / Jawapan :

(a) …………………………….. (b)

Implication 1 / Implikasi 1

(i)

……………….…………………………………………………………..…………………… Implication 2 / Implikasi 2

(ii)

………………………………….………………………………………..…………………… (c)

Conclusion / Kesimpulan :

……………………..…………………………………………………………….. 5(a)

Complete the following statements using quantifier ‘all’ or ‘some’, to form a

true statement. Lengkapkan setiap pernyataan berikut dengan menggunakan pengkuantiti ‘semua’ atau ‘sebilangan’ untuk membentuk suatu pernyataan benar .

(i)

………… even numbers are divisible by 2. ………… nombor genap boleh dibahagi dengan 2.

(ii)

………….. triangles have right angles

.…………. segi tiga mempunyai sudut tegak.

(b)

State the converse of the following statement and hence determine whether its converse is true or false. Nyatakan akas bagi pernyataan berikut dan seterusnya nyatakan sama ada akas itu benar atau palsu.

Jika x = 5, maka x + 2 = 7 If x = 5, then x + 2 = 7

(c) Write down Premise 2 to complete the following argument: Tulis Premis 2 untuk melengkapkan hujah berikut: Premise 1 : Premis 1 :

If Q  R = Q, then R  Q. Jika Q  R = Q, maka R  Q.

Premise 2 / Premis 2: Kesimpulan: Conclusion:

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…………………………………….

R  Q. R  Q.


Jawapan/ Answer:

(a)

(b)

(i)

……………………………..……………………………………………………………

(ii)

……………………………………………………………..……………………………

……………………………………………………………………………………………… ………………………………………………………………………………………………

(c)

………………………………………………………………………………………………

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THE STRAIGHT LINES GARIS LURUS

4 1

Diagram 4.1 shows two parallel lines, OP and RQ. Straight line PQ is parallel to y-axis and O is the origin. Rajah 4.1 menunjukkan dua garis selari OP dan RQ. Garis lurus PQ adalah selari dengan paksi-y dan O ialah asalan. y 2

R

Q x

O

P (2, – 3) Diagram 4.1 Rajah 4.1 Find Cari (a) the equation of the straight line PQ, persamaan garis lurus PQ, (b) the equation of the straight line QR, persamaan garis lurus QR, (c) x-intercept of the straight line QR. pintasan-x bagi garis lurus QR. [6 marks] [6 markah] Answer / Jawapan: (a)

(b)

(c)

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2

In Diagram 4.2, O is the origin. ABCD is a parallelogram. The equation of straight line AB is 3y = x + 17. Dalam Rajah 4.2, O ialah asalan. ABCD ialah sebuah segi empat selari. Persamaan bagi AB ialah 3y = x + 17. y

B (h, 8)

A x

O C D (6, 10) Diagram 4.2 Rajah 4.2 Find cari (a) the value of h, Nilai h, (b) the equation of the straight line CD, persamaan garis lurus CD, (c) the x-intercept of straight line CD. pintasan-x bagi garis lurus CD.

[6 marks] [6 markah] Answer / Jawapan: (a)

(b)

(c)

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3

Diagram 4.3 shows a trapezium PQRS drawn on a Cartesian plane. PQ is parallel to SR. The equation of straight line SR is y = x + 5. Rajah 4.3 menunjukkan sebuah trapezium PQRS dilukis pada satah Cartes. PQ adalah selari dengan SR. Persamaan garis lurus SR ialah y = x + 5. R

y S

Q (5, 4)

P

x

O Diagram 4.3 Rajah 4.3 Find Cari (a) the equation of the straight line PQ, persamaan garis lurus PQ, (b) the x-intercept of straight line PQ. pintasan-x bagi garis lurus PQ.

[5 marks] [5 markah] Answer / Jawapan: (a)

(b)

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4

Diagram 4.4 shows a trapezium PQRS on a Cartesian plane such that RS is parallel to xaxis. Rajah 4.4 menunjukkan sebuah trapezium PQRS pada suatu satah Cartes di mana RS adalah selari dengan paksi-x. y R S(8, 5)

Q(1, 3)

P O

2

x Diagram 4.4 Rajah 4.4

Find Cari (a) the equation of the straight line RS persamaan garis lurus RS (b) the equation of the straight line QR. persamaan garis lurus QR. (c)

the x –intercept of the the straight line QR. pintasan-x bagi garis lurus QR. [6 marks] [6 markah]

Answer / Jawapan: (a)

(b)

(c)

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5

Diagram 4.5 shows a trapezium PQRS on a Cartesian plane such that PQ parallel to RS, QR parallel to y-axis and PS is parallel to x-axis. Rajah 4.5 menunjukkan sebuah trapezium PQRS pada suatu satah Cartes di mana PQ selari dengan RS, QR adalah selari dengan paksi-y dan PS selari dengan paksi-x. Q y

R 4

O P

x

S Diagram 4.5 Rajah 4.5

Given that QR = 5 unit and OS = 3 unit. Diberi bahawa QR = 5 unit and OS = 3 unit. Find Cari (a) the equation of the straight line QR persamaan garis lurus QR (b) the equation of the straight line PQ. persamaan garis lurus PQ. (c)

the x –intercept of the the straight line PQ. pintasan-x bagi garis lurus PQ. [6 marks] [6 markah]


(b)

(c)

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5

LINES AND PLANES IN 3 DIMENSIONS GARIS DAN SATAH DAL AM 3 DIMENSI

1. Diagram 5.1, shows a right prism. ABFE is the uniform cross section of the prism. ABCD is the horizontal base of the right prism. AE = 10 cm, AB = 5 cm and BC = 12 cm Rajah 5.1 menunjukkan sebuah prisma tegak. ABFE ialah keratan rentas seragam prisma itu. ABCD adalah tapak mengufuk bagi prisma itu. AE = 10 cm, AB = 5 cm dan BC = 12 cm H G E

D

F

A

C

B

Diagram 5.1 Rajah 5.1

(a)

Name the angle between the line EC and plane ABCD Namakan sudut di antara garis EC dengan satah ABCD.

(b)

Calculate the angle between the line EC and the base ABCD. Hitung sudut di antara garis EC dengan tapak ABCD. [3 marks] [3 markah]

Answer/Jawapan : (a)

(b)

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2

Diagram 5.2 shows a right prism with rectangle CDEF as its horizontal base . Triangle AED is the uniform cross section of the prism. Rajah 5.2 menunjukkan sebuah prisma tegak dengan segiempat tepat CDEF sebagai tapak mengufuknya Segitiga AED ialah keratan rentas seragam prisma itu. B

A F

C

7 cm 8 cm E

12 cm

D Diagram 5.2 Rajah 5.2

(a) Name the angle between the plane BED and the base CDEF. Namakan sudut di antara satah BED dengan tapak CDEF. (b) Hence, calculate the angle between plane BED and the base CDEF Seterusnya,hitung sudut diantara satah BED dengan tapak CDEF [3 marks] [3 markah] Answer/Jawapan : (a)

(b)

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3. Diagram below shows a cuboid with a horizontal base PQRS. Rajah di bawah menunjukkan sebuah kuboid dengan tapak mengufuk PQRS.

Diagram 5.3 Rajah 5.3 M is the midpoint of GR. M ialah titik tengah bagi GR. a) Name the angle between the plane MPQ and the base PQRS. Namakan sudut di antara satah MPQ dengan tapak PQRS. b) Calculate the angle between the plane MPQ and the base PQRS. Hitung sudut di antara satah MPQ dengan tapak PQRS [3 marks] [3 markah] Answer/Jawapan :

(a)

(b)

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4. Diagram below shows a cuboid with horizontal base EFGH. P is the midpoint of LM. Rajah di bawah menunjukkan sebuah kuboid dengan tapak mengufuk EFGH . P ialah titik tengah bagi LM.


a) Name the angle between the plane FGP and the plane FGLK. Namakan sudut di antara satah FGP dengan satah FGLK. b) Calculate the angle between the plane FGP and the plane FGLK. Hitung sudut di antara satah FGP dengan satah FGLK. [3 marks]

[3 markah] Answer/Jawapan : (a)

(b)

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5. Salsabila has a container shape as a cuboid with the square base PQRS. She puts a straw in inclided position as shown in diagram 5.5. Salsabila mempunyai sebuah bekas berbentuk kuboid dengan tapak segi empat sama PQRS. Dia memasukkan sebatang penyedut minuman dalam kedudukan condong seperti ditunjukkan dalam Rajah 5.5. (a) Name the angle between the straw KQ and the horizontal base. Namakan sudut di antara penyedut minuman KQ dan tapak mengufuk. (b) Hence, calculate the angle between the straw KQ and the plane PQRS. Seterusnya, hitung sudut di antara straw KQ dengan satah PQRS .

K

12 cm

M

N


16 cm

R

S

P

L

Q

[3 marks] [3 markah] Answer/Jawapan : (a)

(b)

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6 1.

STATISTICS STATISTIK The data in the Diagram 1 shows the mass, in kg, for 40 students of Form Six from a certain school. Data dalam Rajah 1 menunjukkan jisim, dalam kg, untuk 40 orang murid Tingkatan Enam di sebuah sekolah tertentu. 74

60

65

69

73

64

56

48

55

75

68

66

46

70

54

47

68

57

65

49

62

48

47

55

59

66

45

65

50

63

53

73

64

65

68

48

63

54

42

49

Diagram 6.1/ Rajah 6.1 (a)

Based on the data in Diagram 6.1, complete Table 1 in the answer. Berdasarkan data di Rajah 6.1, lengkapkan Jadual 1 di ruang jawapan. [4 marks / 4 markah]

(b)

Based on the Table 1, calculate the estimated mean of the mass of the students. Berdasarkan Jadual 1, hitung min anggaran bagi jisim murid-murid tersebut. [3 marks / 3 markah]

(c)

For this part of the question, use the graph paper provided. Untuk ceraian soalan ini, gunakan kertas graf yang disediakan. By using a scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 1 student on the vertical axis, draw a histogram for the data. Dengan menggunakan skala 2 cm kepada kg pada paksi mengufuk dan 2 cm kepada 1 orang murid pada paksi mencancang, lukis satu histogram untuk data itu. [4 marks / 4 markah]

(d)

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State one information based on the histogram in 1(c). Nyatakan satu maklumat berdasarkan histogram di 1( c).


[1 mark / 1 markah]

Answer/Jawapan: (a) Mass (kg) Jisim(kg)

Frequency Kekerapan

Mid point Titik tengah

42 – 46 47 – 51

(b)

(c) Refer to the graph Rujuk graf. (d)

2.

The data below shows the scores obtained by 36 participants in a competition. Data di bawah menunjukkan markah yang diperolehi oleh 36 peserta dalam satu pertandingan. 76 68 28 52 39 31

22 45 44 63 56 34

13 33 78 53 15 15

26 65 43 43 32 30

46 48 54 24 58 44

38 41 49 45 36 28

Diagram 6.2/ Rajah 6.2 (a)

Based on the data, complete the table 2 in the answer space. Berdasarkan data itu, lengkapkan jadual 2 di ruang jawapan. [4 markah/marks]

(b)

State the modal class of the set of data. Nyatakan kelas mod bagi set data itu. [1 markah/marks]

(c)

Based on the table , calculate the estimated mean of the score of the participants. Berdasarkan jadual itu, hitung min anggaran markah peserta . [3 markah/marks]

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(d)

By using a scale of 2 cm to 10 marks on the horizontal axis and 2 cm to 1 participant on the vertical axis, draw a frequency polygon for the data. Dengan menggunakan skala 2 cm kepada 10 markah pada paksi mengufuk dan 2 cm kepada 1 peserta pada paksi mencancang, lukis satu poligon kekerapan bagi data tersebut. [4 markah/marks]

Answer/Jawapan: (a)

Class interval Selang Kelas

Mid point Titik tengah

0 – 9 10 – 19 20 – 29

4.5 14.5

Frequency Kekerapan

(b)

(c)

(d)

3.

Refer to the graph. Rujuk graf.

The data in Diagram 6.3 shows the number of books read by a group of 45 students during NILAM Project. Data di Rajah 6.3 menunjukkan bilangan buku yang dibaca oleh sekumpulan 45 murid semasa Projek NILAM.

7

10

12

33

12

23

13

8

12

15

21

19

11

18

16

25

31

7

10

19

28

20

9

25

9

10

11

18

27

34

26

18

24

32

21

15

12

14

13

34

16

6

11

28

29

Diagram 6.3/Rajah 6.3 (a)

Based on the data in Diagram 6.3 and by using a class interval of 5, complete Table 3(a) in the answer space. Berdasarkan data di Rajah 6.3 dan dengan menggunakan selang kelas 5 buah buku, lengkapkan Jadual 3 di ruang jawapan. [4 marks/markah]

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(b)

(c)

(d)

Based on Table 3, calculate the estimated mean of the number of books read by the students. Berdasarkan Jadual 3, hitung min anggaran bilangan buku yang dibaca oleh kumpulan murid itu. [3 marks/markah] For this part of the question, use the graph paper provided. Untuk ceraian soalan ini, gunakan kertas graf yang disediakan. By using a scale of 2 cm to 5 books on the horizontal axis and 2 cm to 1 student on vertical axis, draw a frequency polygon for the data. Dengan menggunakan skala 2 cm kepada 5 buah buku pada paksi mengufuk dan 2 cm kepada 1 murid pada paksi mencancang, lukis satu poligon kekerapan bagi data tersebut. [4 marks/markah] Based on the frequency polygon in 3(c), state one information about the number of books read by students. Berdasarkan polygon kekerapan di 3(c), nyatakan satu maklumat berkaitan dengan bilangan buku yang dibaca oleh kumpulan murid tersebut . [1 mark/markah]

Answer/Jawapan: (a)

Number of books Bilangan buku

Midpoint Titik tengah

Frequency Kekerapan

1 – 5 6 – 10

(b)

(c)

Refer to the graph Rujuk graf

(d)

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4.

The data in Diagram 4 show the scores obtained by 40 students in a mathematics quiz in a school. Data di dalam Rajah 4 menunjukkan markah diperolehi oleh 40 murid di dalam kuiz Matematik di sebuah sekolah. 40

24

18

30

23

27

35

20

24

29

34

34

40

35

40

29

28

35

45

22

39

23

28

35

30

35

32

38

40

33

15

29

20

33

30

27

33

23

39

45

Diagram 6.4/Rajah 6.4 (a)

(b)

(c)

Based on the data in Diagram 6.4 and by using a class interval of 5, complete Table 4 in the answer space. Berdasarkan data di dalam Rajah 6.4 dan dengan menggunakan saiz kelas 5, lengkapkan Jadual 4 pada ruang jawapan [4 marks/markah] For this part of the question, use the graph paper provided. Untuk ceraian soalan ini, gunakan kertas graf yang disediakan By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 5 pupil on the vertical axis, draw an ogive for the data. Dengan menggunakan skala 2 cm kepada 5 markah pada paksi mengufuk dan 2 cm kepada 5 murid pada paksi mencancang, lukiskan ogif bagi data tersebut. [4 marks/markah] Based on the ogive in (b), find Berdasarkan ogif pada soalan (b), cari (i) the interquartile range . julat antara kuartil. (ii) the number of students who score 40 marks and above. Bilangan pelajar yang memperolehi 40 markah ke atas. [4 marks/markah]

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Answer/Jawapan: (a) Score Skor

Frequency Kekerapan

Cumulative Frequency Kekerapan Longgokan

6 - 10 11 - 15 16 - 20 21 – 25 26 – 30 31 – 35 36 – 40 41 - 45

(c)

(i) (ii)

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Upper Boundary Sempadan Atas

5.

Table 5 shows the waiting time, in minutes of a group of 100 patients seeking treatment from the outpatient department of a hospital. Jadual 5 menunjukkan masa menunggu dalam minit bagi sekumpulan 100 orang pesakit yang menerima rawatan daripada jabatan pesakit luar hospital. WaitingTime (minute) Masa Menunggu (minit)

Number of patients Bilangan pesakit

10 – 14

2

15 – 19

8

20 – 24

14

25 – 29

25

30 – 34

27

35 – 39

19

40 – 44

5 Table 5 / Jadual 5

(a)

(b)

(c)

Calculate the estimated mean of the waiting time for treatment from the outpatient department of a hospital. Kira min anggaran bagi masa menunggu untuk menerima rawatan dari jabatan pesakit luar hospital. [3 marks/markah] Based on the Table 5, complete Table 5(a) in the answer space to show the cumulative frequency distribution of the patients. Berdasarkan Jadual 5, lengkapkan jadual 5 (a) dalam ruang jawapan untuk menunjukkan taburan kekerapan pesakit. [2 marks/markah] For this part of the question, use the graph paper provided. By using the scales of 2 cm to 5 minutes on the horizontal axis and 2 cm to 10 patients on vertical axis, draw an ogive for the data. Untuk ceraian soalan ini, guna kertas graf yang disediakan. Dengan menggunakan skala 2 cm kepada 5 minit pada satah mengufuk dan 2 cm kepada 10 pesakit pada satah menegak, lukis graf ogif untuk data tersebut. [4 marks/markah]

(d)

From your ogive, find Dari ogif yang anda lukis, cari (i) (ii)

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the interquartile range julat antara kuartil the number of patients who have to wait for more than 38 minutes. bilangan pesakit yang menunggu lebih daripada 38 minit. [3 marks]


Answer/Jawapan: (a)

(b) Waiting Time (minute) Masa (minit)

Number of patients Bilangan pesakit

Upper Boundry Sempadan Atas

0 10 – 14

2

15 – 19

8

20 – 24

14

25 – 29

25

30 – 34

27

35 – 39

19

40 – 44

5

(c)

Refer the graph Rujuk graf

(d)

(i)

(ii)

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Cumulative Frequency Kekerapan longgokan

7 1

LINEAR INEQUALITIES KETAKSAMAAN LINEAR On the graph in the answer space, shade the region which satisfies the three inequalities y   x, y  x + 3, and x   5. Pada graf di ruang jawapan, lorek rantau yang memuaskan ketiga-tiga ketaksamaan y   x, y  x + 3, dan x   5. Answer / Jawapan :

3

3

x

O

y = x + 3 y = x


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2.

On the graph in the answer space, shade the region which satisfy the three inequalities y  x  2, y  2 x  4 and y  4 . Pada graf di ruang jawapan, lorekkan rantau yang ditakrifkan oleh ketaksamaan y  x  2, y  2 x  4 and y  4

Answer / Jawapan :

y

y = x - 2

4

2

[3 X

0

2

-2 y + 2 x = 4


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3.

On the graph in the answer space, shade the region which satisfies the three inequalities y  2 x  3, y  2 x and y  4.

Pada graf di ruang jawapan, lorek rantau yang memuaskan ketiga-tiga ketaksamaan y  2 x  3, y  2 x dan y  4. Answer / Jawapan :

y=4

4 y = – 2 x – 3

– 2

x O

– 3


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4.

Diagram shows a region which satisfies the three inequalities. Wr ite the inequalities. Rajah menunjukkan rantau yang memuaskan ketiga-tiga ketaksamaan. Tulis ketiga-tiga ketaksamaan.

y

6

4

2

−2

−4

0

2

4

x

−2

−4

Jawapan/ Answer

a) _____________________________________________________

b) _____________________________________________________

c) _____________________________________________________

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5.

Diagram shows a region which satisfies the three inequalities. Write the inequalities. Rajah menunjukkan rantau yang memuaskan ketiga-tiga ketaksamaan. Tulis ketiga-tiga ketaksamaan. Answer / Jawapan: y

=+

4

2

x

-4

-2

0

2

4

-2

Jawapan/ Answer

a) _____________________________________________________

b) _____________________________________________________

c) _____________________________________________________

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8 1.

GRAPH OF FUNCTIONS GRAF FUNGSI (a)

Complete Table 1 in the answer space, for the equation y 

36 x

by writing down the values of y when x = 4.5 and x = 12. Lengkapkan Jadual 1 di ruang jawapan, bagi persamaan y 

36 x

dengan menulis

nilai-nilai y apabila x = 4.5 dan x = 12. [2 marks/2 markah] (b)

For this part of the question, use the graph paper. You may use a flexible curve rule. By using a scale of 2 cm to 2 unit on the x-axis and 2 cm to 2 unit on the y-axis, 36 draw the graph of y  for 2  x  14. x Untuk ceraian soalan ini, guna kertas graf. Anda boleh guna pembaris fleksibel. Dengan menggunakan skala 2 cm kepada 2 unit pada paksi-x dan 2 cm kepada 2 36 unit pada paksi-y, lukis graf y  untuk 2  x  14. x [4 marks/4 markah]

(c)

From the graph drawn in 1 (b), find Dari graf yang dilukis di 1 (b), cari

(i)

the value of x when y = 15.2, nilai x apabila y = 15.2

(i)

the value of y when x = 7. nilai y apabila x = 7. [2 marks/2 markah]

(d)

Draw a suitable straight line on graph in 1(b) to find the values of x which satisfy 36 4 the equation  x  10  0 for 2  x  14. 7 x State the values of x.

Lukis satu garis lurus yang sesuai pada graf 1(b) untuk mencari nilai-nilai x yang 36 4 memuaskan persamaan  x  10  0 untuk 2  x  14. 7 x Nyatakan nilai-nilai x itu. [4 marks/4 markah]

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Answer/Jawapan : (a) x

2

3

4

y

18

12

9

4.5

6

8

10

6

4.5

3.6

12

14 2.6

Jadual 1

(b)

Refer graph. Rujuk graf.

(c)

(d)

(i) x =

…………….................………………………

(ii) y =

………………………………………………

The equation of the straight line: Persamaan garis lurus:

……………………………………………………. x = …………………

2.

(a)

,

……….………………

Complete Table 2 in the answer space for the equation y

 2 x 2  x  3 .

Lengkapkan Jadual 2 di ruang jawapan bagi persamaan y

 2 x 2  x  3 . [2 marks/2 markah]

(b)

For this part of the question, use the graph paper provided. You can use a flexible curve rule. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y

 2 x 2  x  3 for

2  x  5.

Untuk ceraian soalan ini, gunakan kertas graf yang disediakan . Anda boleh menggunakan pembaris fleksibel. Dengan menggunakan skala 2 cm kepada 1 unit pada paksi-x dan 2 cm kepada 5 unit pada paksi-y, lukiskan graf y

 2 x 2  x  3 bagi

2 

x 

5.

[4 marks/4 markah] (c)

From your graph, find Daripada graf anda, carikan

(i) (ii)

the value of y when x  3  6 , nilai y apabila x  3  6 , the value of x when y = 37. nilai x apabila y = 37. [2 marks/2 markah]

38 | M O D U L


(d)

Draw a suitable straight line on your graph to find all the values of x which satisfy the equation 2 x 2  3x  10 for  2  x  5 . State these values of x. Lukiskan satu garis lurus yang sesuai pada graf anda untuk mencari semua nilai x yang memuaskan persamaan 2 x 2  3x  10 bagi  2  x  5 . Nyatakan nilai-nilai x itu. [4 marks/4 markah]


x

2

y

7

1

−0.5

1

2

3

2

2

3

12

Jadual 2 (b)

Refer graph Rujuk graf

(c)

(i)

y = ……………………......................………

(iii)

x = ……………………………….……………

(d)

39 | M O D U L

x=

…………………… , ……………………………


4

45

5

33

42

3.

(a)

Complete Table 3 in the answer space for the equation y = −x3 +8x − 5 by writing down the values of y when x = − 4 and x = 1 Lengkapkan Jadual 3 di ruang jawapan bagi persamaan y = −x3 +8x − 5 dengan menulis nilai y apabila x = − 4 dan x = 1 [2 marks/2 markah]

(b)

For this part of the question , use the graph paper provided. You may use a flexible curve rule. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph y = −x3 +8x − 5 for  4  x  4. Untuk ceraian soalan ini, gunakan kertas graf yang disediakan . Anda boleh menggunakan pembaris fleksibel. Dengan menggunakan skala 2 cm kepada 1 unit pada paksi-x dan 2 cm kepada 10 unit pada paksi-y, lukiskan graf y = −x3 +8x − 5 bagi  4  x  4. . [4 marks/4 markah]

(c)

From your graph , find

(i) (ii)

the value of y when x = −1.5 nilai y apabila x = −1.5 the value of x when y = − 20 nilai x apabila y = −20 [2 marks/2 markah]

(d)

Draw a suitable straight on your graph to find a value of x which satisfies the 3

equation  x  13x  7  0 . State the values of x. Lukiskan satu garis lurus yang sesuai pada graf anda untuk mencari semua nilai x yang memuaskan persamaan Nyatakan nilai-nilai x itu.

 x3  13x  7  0 [4 marks/4 markah]

40 | M O D U L



−4

x y

−3

−2

−1

0

−2

−13

−12

−5

1

2

3

4

3

−8

−37

Jadual 3 (b)

(c)

(d)

Rujuk graf

(i)

x

= ……………………………………………………..………

(ii)

y

= ………………………………………………...……………

x

= ………………………………………………………………………

41 | M O D U L


4.

(a)

Complete Table 4 in the answer space for the equation y  Lengkapkan Jadual 4 di ruang jawapan bagi persamaan y 

5 x 5 x

. . [2 marks/2 markah]

(b)

For this part of the question, use the graph paper provided. You may use a flexible curve rule. By using a scale of 2 cm to 1 unit on both axis, draw the graph of

y 

5

for 0 < x  8.

x

Untuk ceraian soalan ini, guna kertas graf yang disediakan. Anda boleh guna pembaris fleksibel. Dengan menggunakan skala 2 cm kepada 1 unit pada kedua-dua paksi, lukis graf bagi

y 

5

x

untuk 0 < x  8. [4 marks/4 markah]

(c)

From the graph in 4(b), find Dari graf di 4(b), cari

(i) the value of x when y = 4.5 nilai x apabila y = 4.5

(ii) the value of y when x = 3.6 nilai y apabila x = 3.6 [2 marks/2 markah] (d)

Draw a suitable straight line on your graph to find the values of x which satisfy the equation x27x +5 = 0 for 0 < x  8. State the values of x. Lukis satu garislurus yang sesuai pada graf anda untuk mencari nilai-nilai x yang memuaskan persamaan x 27x +5 = 0 untuk 0 < x  8. Nyatakan nilai-nilai x itu. [4 marks/4 markah]

42 | M O D U L


Answer/Jawapan : (a) x

0.5

y

1

2

2.5

4

5

2.5

2

1.25

5

6

7

8

0.83

0.7

0.6

Jadual 4 (b)

(c)

Rujuk graf

(i)

x

= ………………………………….…………………

(ii)

y

= ………………………………….…………………

(d)

The equation of the straight line: Persamaan garis lurus:

…………………………………………………………………….

x

= ……………………………… , ………………………………

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PROBABILITY II KEBARANGKALIAN II

9

1. Two fair dice are tossed Dua biji dadu adil dilambung serentak

(a) List the sample space Senaraikan ruang sampel.

(b) List all the possible outcome of the events and find the probability that Senaraikan semua kesudahan yang mungkin bagi peristiwa itu dan cari kebarangkalian bahawa (i) The both numbers are same Kedua-dua nombor adalah sama (ii) First number is even and second number is multiple of three Nombor pertama ialah nombor genap dan nombor kedua ialah nombor gandaan 3 (iii)Numbers outcome is perfect square. Nombor yang terhasil ialah suatu nombor kuasa dua [5Marks/ Markah]


…………………….……………………………………………………………………………

(b) (i)

……………………………………………………………………………………………

(ii)

……………………………………………………………………………………………

(iii)

……………………………………………………………………………………………

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2. Table 1 shows the way a group of students come to school. Jadual 1 menunjukkan cara sekumpulan pelajar hadir ke sekolah.

Walking/ Berjalan Kaki Motorcyle Motosikal Car Kereta Bicycle Basikal School Bus Bas Sekolah

65 87 25 125 58

Table 1/Jadual 1 (a) Find the probability for student that come to school by car. Cari keberangkalian bahawa pelajar itu kesekolah dengan menaiki kereta

(b) Which transportation that student prefer the most? Apakah cara utama menjadi pelajar pilihan untuk ke sekolah?

Give your reason Berikan alasan anda.

[5 Marks/Markah]

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3. Diagram 9.1 shows 7 labelled cards in two boxes P and Q Rajah 9.1 menunjukkan 7 keping kad yang berlabel di dalam dua buah kotak P dan Q

B 5 E

8

U

W

3

Box P/ Kotak P

Box Q/Kotak Q Diagram 9.1/Rajah 9.1

A card is picked at random from each boxes. Sekeping kad dipilih secara rawak dari Kotak P dan diikuti sekeping kad lagi dari Kotak Q. (a) List the sample space Senaraikan ruang sampel

(b) Find the probability that Hitung kebarangkalian bahawa (i) The first card labelled vowels and the second card labelled number Kad pertama berlabel huruf vokal dan kad kedua berlabel nombor

(ii)

The first card labelled with a number or the second card is labeled with an alphabet Kad pertama berlabel nombor atau kad kedua berlabel huruf

[5 Marks/Markah]

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4.

Diagram 9.2 shows a spinner labelled by numbers. The spinner is spins twice. Rajah 9.2 menunjukkan roda berputar yang dilabelkan dengan nombor. Roda itu kemudian diputar dua kali.

Diagram 9.2/Diagram 9.2 (a) List the sample space Senaraikan ruang sampel

(b) By listing down all the possible outcomes of the event Dengan menyenaraikan semua kesudahan yang mungkin bagi peristiwa (i)

Number where the sum of the digits is more than 12.Calculate the probability. Nombor yang mana hasil tambah digit-digitnya adalah nombor lebih besar daripada 12. Hitung keberangkalian tersebut.

(ii)

The first number is multiple of three and the second number is prime number . Nombor pertama adalah nombor gandaan 3 dan nombor adalah kedua nombor perdana.

[5 Marks/Markah]

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5.

Box A contain six cards labelled “RECORD”.Two card are drawn from box A without replacement Kotak A mengandungi enam kad berlabel “RECORD”.Dua keping kad dipilih secara rawak satu persatu tanpa penggantian. (a) List the sample space Senaraikan ruang sampel

(b) By listing the outcomes of the event, find the probability that (i) first card is vowel kad pertama huruf vokal

(ii)

f irst card is labelled with “R” and second card is labeled with “C”. kad pertama berlabel huruf “R” dan kad kedua berlabel huruf “C”

(iii) both card are same labelled or second card is D. kedua-dua kad dengan label yang sama atau kad kedua huruf D.

(7 Marks/Markah)

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10

GRADIENT AND UNDER A GRAPH KECERUNAN DAN LUAS DI BA WAH GRAF

1. Farid went back to his hometown in Ayer Keroh, Malacca by bus. The distance from Petaling Jaya Bus Terminal to its stop in Malacca is 120 km. The note of the bus journey are shown below. Farid telah pulang ke kampungnya di Ayer Keroh, Melaka dengan menaiki bas. Jarak perjalanan dari Terminal Bas Petaling Jaya ke perhentiannya di Melaka ialah 120 km. Catatan perjalanan bas tersebut ditunjukkan di bawah .



The bus left the Terminal in Petaling Jaya at 7.30 a.m. Bas bertolak dari Terminal di Petaling Jaya pada jam 7.30 pagi.



Made a stop at the R & R Seremban, Negeri Sembilan at 8.15 a.m, a distance of 50 km from the terminal. Singgah sarapan pagi di R & R Seremban, Negeri Sembilan tepat pada jam 8.15 pagi, dengan jarak 50 km dari terminal.



Continued with the journey to Ayer Keroh at 8.50 a.m. Meneruskan perjalan ke Ayer Keroh pada pukul 8.50 pagi.



Arrived at Ayer Keroh, Malacca at 9.20 a.m. Tiba di Ayer Keroh Melaka pada pukul 9.20 pagi.

The graph in the answer space below shows a distance time graph. Graf di ruangan jawapan di bawah adalah graf jarak – masa. a) Refer to the graph, Rujuk kepada graf, i)

Complete the graph to represent Farid’s whole journey. Lengkapkan graf itu bagi seluruh perjalanan Farid.

ii)

Calculate the speed, in km h -1, from R & R Seremban to Ayer Keroh. Hitung laju, dalam km h -1, dari R & R Seremban ke Ayer Keroh.

b) State the value of t. Nyatakan nilai bagi t. [6 marks] [6 markah]

49 | M O D U L


Answer : Jawapan : a) i)

Distance / Jarak 120 km

50

45

t = ___


a) ii)

b)

50 | M O D U L


110

Time / Masa ( minute/ minit )

2. Diagram 10.2 shows a speed-time graph for the movement of a particle for a period of 70 seconds. Rajah 10.2 menunjukkan graf laju-masa bagi pergerakan suatu zarah dalam tempoh 70 saat. Speed ( ms-1) La u ( −)

24

0

12

t

70

Time (s) Masa (s)

(a) Calculate the rate of change of speed, in ms −of the particle for the first 12 second. Hitung kadar perubahan laju, dalam  −, zarah itu untuk tempoh 12 saat yang pertama. (b) Calculate the value of t, if the distance travelled by the particle for the last 58 minutes is 1032m. Hitung nilai t, jika jarak yang dilalui oleh zarah itu dalam tempoh 58 saat terakhir ialah 1032 m. [5 marks] [5 markah] Answer / Jawapan : (a)

(b)

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3. Sarah drives the car for 58 km from her house to her office. Table 3 shows the note of her journey. Sarah memandu kereta sejauh 58 km dari rumah ke pejabatnya. Jadual 3 menunjukkan catatan perjalanannya. Thursday / Khamis

Time / Masa 8.30 pagi /a.m

Start journey. Memulakan perjalanan.

9.05 pagi /a.m

Breakfast in the Pak Ali’s restaurant after driving 28 km.

9.55 pagi /a.m 10.30 pagi / a.m

Sarapan pagi di restoran Pak Ali selepas memandu sejauh 28 km. Continue journey for another 30 km. Meneruskan perjalanan sejauh 30 km. Arrives at her office. Tiba di pejabatnya. Table 3 / Jadual 3

(a)

Complete the graph in the answer space to represent Sarah whole journey. Lengkapkan graf di ruang jawapan untuk mewakili keseluruhan perjalanan Sarah .

(b) State the value of p and q. Nyatakan nilai p dan q. Calculate the average speed of the car, in kmh -1, the journey from house to the office. Hitung purata laju, dalam kmj−1, perjalanan dari rumah ke pejabatnya. [6 marks / 6 markah] Answer/ Jawapan : (c)

(a)

Distance/ Jarak (km)

58

p

O

35

q

120

(b)

(c)

52 | M O D U L


Time /Masa (min)

4. Diagram 10.3 shows the speed-time graph for the movement of two particles, J and K, for a period of t s. The graph ABCD represents the movement of J and the graph AE represents the movement of K. Both particles start at the same point and move along the same route. Rajah 10.3 menunjukkan graf laju-masa bagi pergerakan dua zarah, J dan K, dalam tempoh t s. Graf ABCD menunjukkan pergerakan zarah J dan graf AE menunjukkan pergerakan zarah K. Kedua-dua zarah bermula dari titik yang sama melalui laluan yang sama. Masa (ms-1 )

E B

26

C D

A 0

13

t

Masa (Saat)

Diagram 10.3 / Rajah 10.3

(a) State the uniform speed, in ms -1, of particle J. Nyatakan laju seragam, dalam ms -1, zarah J. (b) Calculate the rate of change of speed, in ms -2, of particle J for the first 13 s. Hitung kadar perubahan laju, dalam ms -2, zarah J dalam 13 s pertama. (c) At t s, the difference between the distance travelled by J and K is 169 m. Calculate the value of t. Pada t s, beza antara jarak yang dilalui oleh J dan K ialah 169 m. Hitung nilai t. [ 6 marks ]/[ 6 markah ] Answer / Jawapan : (a)

(b)

(c)

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5. Diagram 10.4 shows a speed-time graph for the movement of a particle for a period of t seconds. Rajah 10.4 menunjukkan graf laju-masa bagi pergerakan satu zarah dalam tempoh t saat. Speed ( m s –1)

22

Laju ( m s –1)

18 10

O

5

12

t

Time (s) Masa (s)

Diagram 10.4 Rajah 10.4 (a) State the length of time, in s, that the particle moves uniform speed. Nyatakan tempoh masa, dalam s, zarah itu bergerak dengan laju seragam. (b) Calculate the rate of change of speed, in m s –2 , in the first 5 seconds. Hitungkan kadar perubahan laju, dalam m s –2, dalam tempoh 5 saat yang pertama. (c) Calculate the value of t, if the total distance traveled for the period of t seconds is 188 metres. Hitungkan nilai t, jika jumlah jarak yang dilalui dalam tempoh t saat itu ialah 188 meter. [6 marks] [6 markah] Answer / Jawapan : (a)

(b)

(c)

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11 1

MATRIX MATRIKS  6

Find the inverse of the matrix 

2 



  4  1 

 6

Cari matriks songsang 

2 



  4  1  [1 marks/markah]

2

 5  4    3 3  

Find the inverse of the matrix 

 5 Cari matriks songsang   3

 4    3 

[1 marks/markah]

3

 6  2   4 1   

Find the inverse of the matrix 

 6  2    1 

Cari matriks songsang   4

[1 marks/markah]

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4

Given that

1  5

 k  4

 2   2 2   1 0     . 2    4 m   0 1 

Find the value of k and of m Diberi bahawa

1  5

 k  4

 2   2 2   1 0    . 2    4 m   0 1 

Cari nilai k dan m. [2 marks/markah]

5

 2 Given that P    1

 4  3   1 0   such that PQ    and Q  m   . 0 1 4  n 2     3 

Find the value of m and of n

 2 Diberi bahawa P    1

 1 0   4  3   di mana PQ    dan Q  m   . 4  n 2 0 1     3 

Cari nilai m dan n [2 marks/markah]

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6

 4  2   is  1 

It is given that the inverse matrix of P =   3

1   1 2 



m  n

 .

4 

 4  2  1   1 2   ialah  .  1  m  n 4 

Diberi bahawa matriks songsang bagi P =   3

(a) Find the value of m and of n. Carikan nilai m dan nilai n.

(b) Using matrices, calculate the value of x and of y that satisfy the following simultaneous linear equations: Menggunakan kaedah matriks, hitung nilai x dan nilai y yang memuaskan persamaan matriks berikut :

4 x  2 y  6 3 x  y  3 [ 6 marks /markah]

Answer / Jawapan (a)

(b)

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 6

Given that P =   4

7

  1 m   1 0   and PQ =    , Q = 1  0 1  1  k   4 6    2 

 6

Diberi bahawa P =   4

  1 m   1 0   dan PQ =    , Q = 1  0 1  1  k   4 6    2 

(a) Find the value of k and m. Cari nilai k dan m.

(b) Write the following simultaneous linear equations as a matrix equation: Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks : 6x + 2y = 1 4x  y = 3 Hence, using matrix method, calculate the values of x and y. Seterusnya, menggunakan kaedah matriks, hitung nilai x dan nilai y. [6 marks/markah] Answer / Jawapan : (a)

(b)

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8

The inverse matrix of

 4 3  is m  2 n .     2 2  2 4    

 2 n   4 3      Matriks songsang bagi  2 2  ialah m   2 4  .

(a)

Find the value of m and of n. Cari nilai m dan nilai n.

(b)

Hence, using matrix method, calculate the value of x and of y. Seterusnya, dengan menggunakan kaedah matriks, hitung nilai x dan nilai y. 4x + 3y = 6 2x + 2y = 8 [ 6 marks/markah ]


(b)

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 5 6   . 2 3  

(a) Find the inverse matrix of 

9

 5 6 

 . Cari matriks songsang bagi   2 3  (b) Write the following simultaneous linear equations as matrix equations: Tulis persamaan linear serentak berikut dalam persamaan matriks : 5m + 6n = 4 2m + 3n = 2 Hence, by using the matrix method, calculate the value of m and of n. Seterusnya, menggunakan kaedah matriks, hitung nilai m dan nilai n. [6 marks/markah] Answer / Jawapan: (a)

(b)

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10

(a)

 2  2    4 q   is 1   . 3 4 3 2    p    

The inverse matrix of 

Matriks songsang bagi

 2  2    4 q    ialah 1   3 4 3 2    p    

Find the values of p and q. Cari nilai bagi p dan q. (b) Write the following following simultaneous linear as matrix equation. Tulis persamaan linear serentak berikut dalam bentuk persamaan matrik. 2x − 2y =7 =7 3x − 4y =13 =13 Hence, using matrix method, calculate the value of x and of y. Seterusnya, Seterusnya, dengan menggunakan kaedah matriks, matriks , hitung nilai x dan nilai y. [6marks/markah [6marks/markah]] Answer / Jawapan: (a)

(b)

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11

Diberi bahawa matriks R ialah suatu matriks 2 x 2 dengan keadaan

It is given that matrix R isa 2 x 2 matrix such that

(a)

 3  2    R = 5 4   

 3  2    R =  5 4  

 1 0    .  0 1 

 1 0    . 0 1  

Cari matriks R. Find matrix R.

(b)

Tulis persamaan linear serentak berikut dalam bentuk matriks: Write the following simultaneous linear equations as matrix equations : 3 p  2q 5 p



4q





2 12

Seterusnya, dengan menggunakan kaedah matriks, hitung nilai p dan nilai q. nilai q. Hence, Hence, using matrix method, method, calculate the value of p of p and of q of q. [6 marks/ markah ] Jawapan / Jawapan / Answer: (a)

(b)

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12 1

PLANS PLA NS AND ELEVATIONS ELEVATIONS PELA PELAN N DAN DONGAKA DONGAKAN N

You are not no t allowed to use graph paper to answer this question. Anda tidak dibenarkan tidak dibenarkan menggunakan kertas graf untuk menjawab soalan ini. (a)

Diagram 1 (i) shows a solid consisting of a cuboid and a half-cylinder which joined at the plane plane JKGH. JKGH. The base ABCD base ABCD is is on a horizontal plane and JH = JH = 4 cm. Rajah 1(i) menunjukkan sebuah pepejal terdiri dari sebuah kuboid dan separuh silinder yang bercantum pada satah JKGH.Tapak ABCD terletak pada satah mengufuk dan JH = 4 cm.

K J

G

3 cm H

D F

A E

6 cm

C B

4 cm

Diagram 1 (i) Rajah 1 (i)

Draw full scale, the plan of the solid. Lukis dengan skala penuh, pelan pepejal itu. [3 marks] [3markah] Answer/ Jawapan: Jawapan: (a)

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(b)

A solid right prism is joined to the solid in Diagram 1(ii) at the vertical plane GFCV. The combined solid is shown in Diagram 1(ii). Trapezium PVTS is the uniform cross-section of the prism and PQRS is an inclined plane and V is the midpoint of DC. The base BCTUWVDA lies on a horizontal plane whereby CT = 2 cm and PG = 3 cm. Sebuah pepejal prisma tegak dicantumkan kepada pepejal dalam Rajah 1(i) pada satah mencancang GFCV. Gabungan pepejal adalah seperti yang ditunjukkan dalam Rajah 6(ii). Trapezium PVTS ialah keratan rentas seragam prisma itu dan PQRS ialah satah condong dan V adalah titik tengah bagi DC. Tapak BCTUWVDA terletak pada satah mengufuk di mana CT = 2 cm dan PG = 3 cm. P

K J

G

3 cm

H

D

w A

E

F R

V S

6 cm

C X

4 cm

3 cm

2 cm

U

B T

Diagram 1 (ii) Rajah 1 (ii)

Y

Draw to a full scale, Lukis dengan skala penuh, (i)

the elevation of the combined solid on a vertical plane parallel to AB as viewed from X. dongakan gabungan pepejal itu pada satah mencancang selari dengan AB sebagaimana dilihat dari X. [5 marks] [5 markah ]

(ii) the elevation of the combined solid on a vertical plane parallel to BC as viewed from Y. dongakan gabungan pepejal itu pada satah mencancang selari dengan BC sebagaimana dilihat dari Y. [4 marks] [4 markah]

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Answer / Jawapan : (b) (i), (ii)

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2

You are not allowed to use graph paper to answer this question. Anda tidak dibenarkan menggunakan kertas graf untuk menjawab soalan ini. (a) Diagram 2.1 shows a solid right prism with rectangular base EFGH on a horizontal table. The surface EFRJKL is the uniform cross-section of the prism.Rectangle KLMN and JPQR are horizontal planes. Rectangle JKNP is an inclined plane. RF and LE are vertical edges. Rajah 2.1 menunjukkan sebuah pepejal berbentuk prisma tegak dengan tapak segiempat tepat EFGH terletak di atas meja mengufuk. Permukaan EFRJKL ialah keratan rentas seragamnya. Segiempat tepat KLMN dan JPQR ialah satah mengufuk dan segiempat tepat JKNP ialah satah condong. Tepi RF dan LE adalah tegak. P

J

2 cm

Q

R

M

L

3 cm

7 cm

N

6 cm

K

H

G

4 cm

F

8 cm

E

X


Draw full scale, the elevation of the solid on a vertical plane parallel to EF as viewed from X . Lukiskan dengan skala penuh , dongakan pepejal itu pada satah mencancang yang selari dengan EF sebagaimana dilihat dari X. [ 3 marks ] [3 markah ] Answer / Jawapan : 2 (a)

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2

(b)

A solid cuboid is removed from the solid in Diagram 2.1. The remaining solid is as shown in Diagram 2.2. Rectangle TUVW is a horizontal plane. JT and BW are vertical edges. UF = 4 cm dan UV = 3 cm. Sebuah pepejal berbentuk kuboid dikeluarkan daripada pepejal dalam Rajah 2.1 Pepejal yang tinggal adalah seperti dalam Rajah 2.2. Segiempat tepat TUVW ialah satah mengufuk. Tepi JT dan BW adalah sisi tegak. UF = 4 cm dan UV = 3 cm. P B

2 cm

Q

A

J M N

6 cm

L

3 cm

7 cm W

T

K

V

U

H

G

4 cm Y E

F

8 cm

Diagram 2.2 Rajah 2.2 Draw to full scale, Lukiskan dengan skala penuh, (i) the plan of the remaining solid, pelan pepejal yang tinggal itu, [4 marks] [4 markah] (ii) the elevation of the remaining solid on a vertical plane parallel to FG as viewed from Y. dongakan pepejal yang tinggal itu pada satah mencancang yang selari dengan FG sebagaimana dilihat dari Y . [5 marks ] [5 markah]

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Answer / Jawapan : 2 (b) (i)

2(b) (ii)

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3

You are not allowed to use graph paper to answer this question. Anda tidak dibenarkan menggunakan kertas graf untuk menjawab soalan ini. (a)

Diagram 3.1 shows a solid combined consist of a right prism and a cuboid which are joined at the plane EBHG. ABH is the uniform cross-section of the prism. Rectangle AHGF is an inclined plane. The rectangle base ACDF is placed on a horizontal table. JKLI is a horizontal plane. LC, IB, KD and JE are vertical edges. Rajah 3.1 menunjukkan sebuah gabungan pepejal yang terdiri daripada sebuah prisma tegak dan sebuah kuboid yang tercantum pada satah EBHG. Permukaan ABH ialah keratan rentas seragam bagi prisma. Segi empat tepat AHGF ialah satah condong. Segi empat tepat ACDF terletak di atas meja mengufuk. Segi empat tepat JKLI ialah satah mengufuk. Tepi LC, IB, KD dan JE adalah tegak. K

G

6 cm

I

L

H

E D

F

5 cm 4 cm

C

2m

B

6 cm

Diagram 3.1 Rajah 3.1 Draw to full scale, the plan of the solid. Lukiskan dengan skala penuh, pelan pepejal itu. [3 marks] [3 markah ] Answer / Jawapan : 3 (a)

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(b)

Another solid in the shape of quarter of a cylinder is joined to the solid in Diagram 3.1 at the vertical plane DCPS. The combined solid is as shown in Diagram 3.2. CPQ is a quadrant of a circle with centre C. Sebuah pepejal lain berbentuk suku silinder digabungkan dengan pepejal dalam Rajah 3.1 pada satah mencancang DCPS. Gabungan pepejal adalah seperti dalam Rajah 3.2. CPQ adalah sukuan bulatan berpusat di C.

K

2 cm

J

G

I

L S

E R

Y

D

F

H

5 cm

P

4 cm Q

2 cm

C

6 cm

B

A

X


Draw to full scale, Lukis dengan skala penuh, (i) the elevation of the combined solid on a vertical plane parallel to AQ as viewed from X. dongakan gabungan pepejal itu pada satah mencancang yang selari dengan AQ sebagaimana dilihat dari X . [4 marks] [4 markah] (ii) the elevation of the solid on a vertical plane parallel to RQ as viewed from Y. dongakan gabungan pepejal itu pada satah mencancang yang selari dengan RQ sebagaimana dilihat dari Y. [5 marks] [5 markah]

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3(b) (ii)

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4

You are not allowed to use graph paper to answer this question. Anda tidak dibenarkan menggunakan kertas graf untuk menjawab soalan ini. (a) Diagram 4.1 shows a composite solid consist of a right prism and a quarter of cylinder on a horizontal plane. The surface ABLKJIH is the uniform crosssection of the prism. HG, IF, JE, LD, BC, KN and AM are vertical edges. KL= 4 cm and HI = LB = DC = GF = 2 cm. Rajah 4.1 menunjukkan sebuah gabungan pepejal yang terdiri daripada sebuah prisma tegak dan sebuah sukuan silinder terletak di atas satah mengufuk. Permukaan ABLKJIH ialah keratan rentas seragam prisma itu. Tepi HG, IF, JE, LD, BC, KN dan AM adalah tegak. KL= 4 cm dan HI = LB = DC = GF = 2 cm. A

8 cm

6 cm K B

J

H

L

I

M

6 cm N C G

D

E F


Draw to full scale, the plan of the solid. Lukiskan dengan skala penuh, pelan pepejal itu. [3 marks] [3 markah ] Answer / Jawapan : 4 (a)

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(b)

Part of the composite solid in Diagram 4.1 is removed. The remaining solid is shown in Diagram 4.2. Sebahagian daripada gabungan pepejal dalam rajah 4 .1 dikeluarkan. Pepejal yang tinggal ditunjukkan di Rajah 4.2.

A T

8 cm

K

J

H

L

I P M

Q

2cm

N

C G

D

E F

Y


X

Draw to full scale, dengan skala penuh, (i) the elevation of the remaining solid on a vertical plane parallel to DC as viewed from X, dongakan pepejal yang tinggal itu pada satah mencancang yang selari dengan DC sebagaimana dilihat dari X , [4 marks] [4markah]

(ii) the elevation of the remaining solid on a vertical plane parallel to GF as viewed from Y. dongakan pepejal yang tinggal itu pada satah mencancang yang selari dengan GF sebagaimana dilihat dari Y. [5 marks] [5 markah]

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4(b) (ii)

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5

You are not allowed to use graph paper to answer this question. Anda tidak dibenarkan menggunakan kertas graf untuk menjawab soalan ini. (a)

Diagram 5.3 shows a combine solid consist of a right prisms and a half cylinder which are joined at the plane FGHI. AJEG is a horizontal plane and the surface BCDH is the uniform cross-section of the prism. Given DH = 9 cm and the diameter of the cylinder is 4 cm. Rajah 5.3 menunjukkan sebuah gabungan pepejal yang terdiri daripada satu prisma tegak dan separuh silinder yang bercantum pada satah FGHI. AJEG ialah satah mengufuk dan permukaan BCDH ialah keratan rentas seragam bagi prisma itu. Diberi DH = 9 cm dan diameter silinder ialah 4 cm. C

2 cm D I

B

H

J

6 cm

E

A

F

5 cm G

Diagram 5.3 Rajah 5.3 Draw to full scale, the plan of the solid. Lukiskan dengan skala penuh, pelan pepejal itu. [3 marks] [3 markah ] Answer / Jawapan : 5 (a)

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(b)

Part of the composite solid in Diagram 5.3 is removed. The remaining solid is as shown in Diagram 5.4. ILK is a quadrant of a circle with centre L. Given OG = 2 cm. Sebahagian daripada pepejal dalam rajah 5.3 dikeluarkan. Pepejal yang tinggal adalah seperti yang ditunjukkan dalam Rajah 5.4. ILK ialah satu sukuan bulatan berpusat L. Diberi OG = 2 cm. C D

I

B L

H

K

J E

M O A

Y

N F

G

P

X


Draw to full scale, Lukis dengan skala penuh, (i) the elevation of the remaining solid on a vertical plane parallel to GE as viewed from X, dongakan pepejal yang tinggal itu pada satah mencancang yang selari dengan GE sebagaimana dilihat dari X , [4 marks] [4 markah ] (ii) the elevation of the remaining solid on a vertical plane parallel to AG as viewed from Y. dongakan pepejal yang tinggal itu pada satah mencancang yang selari dengan AG sebagaimana dilihat dari Y. [5 marks] [5 markah ]

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(b) (ii)

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13

TRANSFORMATIONS PENJELMAAN

1) TRANSLATION (TRANSLASI) Instruction / Arahan : a) Draw the image for each of the following object. Lukis imej bagi setiap gambar rajah yang berikut. b) State the coordinates of the image for each of the points under the given translation. Nyatakan koordinat bagi setiap imej titik di bawah translasi yang diberi. EXAMPLE / CONTOH

 4 

 a) Translation  2   

1) a) Translation

 4 

 / Translasi  2   

 3   3    / Translasi    0   0 

(7,5) b) b)

  2   / Translation 1  

1) a) Translasi 

b)

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  2     1 

3 a) Translation

3 3 / Translasi 0 0    

b)


  4   2   

 2   1   

5) a) Translation 

4 a) Translation 

  4   2   

 2   1   

Ttranslasi 

Translasi 

b)

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b)


2) REFLECTION (PANTULAN) Instruction / Ar ahan Draw and label the image under a reflection for each of the following Lukis dan labelkan imej di bawah pantulan yang berikut. EXAMPLE / CONTOH

2)

2)

3)

Axis of reflection Paksi pantulan

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Instruction / Ar ahan Draw and state the axis of reflection. Lukis dan nyatakan paksi pantulan. EXAMPLE / CONTOH

1)

Paksi pantulan : x = 2 Axis of replection : x = 2

Axis of replection : Paksi pantulan :

2)

3)



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3) ROTATION (PUTARAN) Instruction / Ar ahan Draw the image for each of the object under the rotation given : Lukis imej bagi setiap putaran yang diberi: EXAMPLE / CONTOH M under a Rotation of 90 0 clockwise about A. M di bawah putaran 90 0arah jam pada titik A.

2) Rotation of 900 clockwise about B Putaran 900 arah jam pada titik B

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1) Rotation of 900 anticlockwise about A. Putaran 900 lawan arah jam pada titik A.

3) Rotation of 1800 anticlockwise about H Putaran 1800 lawan arah jam pada titik H


Instruction / Ar ahan Determine centre,angle and direction of rotation Kenal pasti pusat, sudut dan arah putaran berikut. EXAMPLE / CONTOH

Centr e / Pusat : (0,0) Angl e / Sudut/ : 900 , Direction / Arah : Clockw ise / Arah jam

1)

Centr e / Pusat : Angl e / Sudut/ : Direction / Arah :

2)

3)



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4) ENLARGEMENT (PEMBESARAN) To determine the centre of enlargementand the scale factor Mengenalpasti pus at pembesaran d an faktor skala Instruction / Ar ahan i) Mark the centre of enlargement Tandakan pusat pembesaran ii) Nyatakan faktor skala bagi pembesaran berikut State the factor scale of enlargment EXAMPLE / CONTOH 1)

Scale factor / Faktor skala : 2 Scale factor / Faktor skala : _____ 3)

2)

Scale factor / Faktor skala : _______

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Scale factor / Faktor skala : _____


To draw an image under enlargement Melukis imej di bawah pembesaran

Instruction / Ar ahan: Draw and label an image under the following enlargement Lukis dan label imej di bawah pembesaran berikut.

EXAMPLE / CONTOH

1) Centre (1,2) and scale factor 2 Pusat (1,2) dan faktor skala 2

Centre (7,2) and scale factor 2 Pusat ( 7,2) dan factor skala 2

2) Centre (2,2) and scale factor Pusat (2,2) dan faktor skala

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1 2

1 2

.

3) Centre (2,4) and scale factor Pusat ( 2,4) dan faktor skala


1 2 1 2

TRANSFORMATIONS (PENJELMAAN) 1

Diagram 13.1 shows the point (4 ,1) and the straight line y = x drawn on a Cartesian plane. Rajah 13.1 menunjukkan titik (4 ,1) dan garis lurus y = x dilukis pada suatu satah Cartesan.

y

6

y = x

4

2 ●

(4,1)

4

6

x

O

6

4

2

2

4,

Diagram 13.1 Rajah 13.1 (a) Transformation P is a translation

3 2

. Transformation Q is a reflection in the straight

line y = x. Penjelmaan P ialah satu translasi

 3 . Penjelmaan Q ialah satu pantulan pada 2

garis lurus y = x.

State the coordinates of the image of point (4, 1) under each of the following transformations: Nyatakan koordinat imej bagi titik (4, 1) di bawah setiap penjelmaan berikut : (i) P (ii) Q (iii) PQ [4 marks] [4 markah]

86 | M O D U L


(b)

Diagram 13.2 shows three triangles PQR, ACG and EFG on a Cartesian plane. Rajah 13.2 menunjukkan tiga segitiga PQR, ACG dan EFG pada satu satah Cartesian. y F 8

6

A 4

P G

E

C

2

Q O

2

R 4

6

8

10

12

14

x 16


Triangle ACG is the image of triangle PQR under transformation V. Triangle EFG is the image of triangle ACG under transformation W. Segitiga ACG adalah imej bagi segitiga PQR di bawah penjelmaan V. Segitiga EFG adalah imej bagi segitiga ACG di bawah penjelmaan W. (i) Describe in full transformation : Huraikan selengkapnya penjelmaan: (a) V (b) W [6 marks] [6 markah]

(ii) Given that triangle EFG represents a region of area 72 unit 2. Calculate the area, in unit 2, of the region represented by triangle PQR. Diberi bahawa segitga EFG mewakili suatu kawasan berlorek yang mempunyai luas 72 unit 2. Hitung luas, dlm unit 2, kawasan yang diwakili oleh segitiga PQR. [2 marks] [2 markah]

87 | M O D U L


2

(a)

Diagram 13.3 shows a point (2,5) and a straight line y = 4 drawn on a Cartesian plane. Rajah 13.3 menunjukkan titik (2,5) dan satu garis lurus y = 4 dilukis pada suatu satah Cartesan. y

6

(2,5) 

y=4

4

2

x

4

2

0

2

4

2


  2 

 . Transformation Q is a reflection in the Transformation P is a translation  3    straight line y = 4.   2   . Penjelmaan Q ialah satu pantulan pada   3 

Penjelmaan P ialah satu translasi  garis lurus y = 4 .

State the coordinates of the image of point (2,5) under each of the following transformations: Nyatakan koordinat imej bagi titik (2,5) di bawah setiap penjelmaan berikut : (i) Q (ii) QP [3 marks] [3 markah]

88 | M O D U L


(b)

Diagram 13.4 shows three pentagons, ABCDE, HJKLM and PQRST, drawn on a Cartesian plane. Rajah 13.4 menunjukkan tiga pentagon , ABCDE, HJKLM dan PQRST, dilukis pada suatu satah Cartesan. y

A

E

10 B

D

8 C

6 J

H Q

4

P

R

K

2

T

S L

0

2

4

6

8

M x 10

Diagram 13.4 Rajah 13.4 (i)

HJKLM is the image of ABCDE under a combined transformation VU. HJKLM ialah imej bagi ABCDE di bawah gabungan penjelmaan VU. Describe in full the transformation Huraikan selengkapnya penjelmaan

(ii)

(a)

U,

(b)

V.

It is given that the shaded region represents a region of area 105 m 2. Calculate the area, in m 2, of the pentagon ABCDE. Diberi bahawa kawasan yang berlorek mewakili luas 105 m2. Hitung luas, dalam m2, pentagon ABCDE. [9 marks] [9 markah]

89 | M O D U L


3 Diagram 13.5 shows point J drawn on a Cartesian plane. Rajah 13.5 menunjukkan titik J dilukis pada suatu satah Cartesan. y

4

2

J

x

2

0

2

4

6

2


  2   . 3  

(a) Transformation T is a translation 

Transformation R is a rotation of 90o anticlockwise about the centre (2, 0).

  2    . Penjelmaan T ialah translasi   3  Penjelmaan R ialah satu putaran 90o lawan arah jam pada pusat (2, 0). State the coordinates of the image of point J under each of the following transformations: Nyatakan koordinat imej titik J di bawah setiap penjelmaan berikut: (i) RT, (ii)

TR. [4 marks] [4 markah]

90 | M O D U L


b)

Diagram 13.5 shows three pentagons, ABCDE, PQRST and PKLMN drawn on a Cartesian plane. Rajah 13.5 menunjukkan tiga pentagon , ABCDE, PQRST dan PKLMN yang dilukis pada satah Cartesan. y 8

S

6 R M

44 Q

2

2

L

0

2

P

K

4

2

x

6

B

A C D

4

E


PQRST is the image of ABCDE under the transformation VU. Describe in full the transformation: PQRST ialah imej bagi ABCDE di bawah penjelmaan VU. Huraikan selengkapnya penjelmaan:

(i)

(a)

U,

(b)

V. [5 marks] [5 markah]

(ii) It is given that ABCDE represents a region of area 50 cm 2. Calculate the area, in cm 2, of the region represented by the shaded region. Diberi bahawa ABCDE mewakili suatu kawasan yang mempunyai keluasan 50 cm2. Hitung luas, dalam cm2, kawasan yang diwakili oleh kawasan berlorek. [3 marks] [3 markah]

91 | M O D U L


4

Diagram 13.6 shows point H drawn on a Cartesian plane. Rajah 13.6 menunjukkan titik H dilukis pada suatu satah Cartesan.

y

6

4 ●

H 2

2

0

2

4

6

x

2


 3   . (a) Transformation T is a translation  1    Transformation R is a rotation of 90 o anticlockwise about the centre (1, 1) .  3  Penjelmaan T ialah translasi   .   1  Penjelmaan R ialah satu putaran 90o lawan arah jam pada pusat (1, 1). State the coordinates of the image of point H under each of the following transformations: Nyatakan koordinat imej bagi titik H di bawah setiap penjelmaan berikut : (i)

R2,

(ii)

RT. [4marks] [4markah]

92 | M O D U L


(b) Diagram 13.7 shows three pentagons, ABCDE, EFGHI and PQRST drawn on a Cartesian plane. Rajah 13.7 menunjukkan tiga pentagon, ABCDE, EFGHI dan PQRST yang dilukis pada satah Cartesan. y 8

D

6

4 S

Q

R

C

I

2

B

C G

2 P

0 E

2

F

4

2

H

6

x

A

4

Diagram 13.7 Rajah 13.7 (i) ABCDE is the image of PQRST under the transformation VU. Describe in full the transformation: ABCDE ialah imej bagi PQRST di bawah penjelmaan VU. Huraikan selengkapnya penjelmaan : (a) U, (b) V. [5 marks] [5markah] (ii) It is given that PQRST represents a region which has of area 28 cm 2. Calculate the area, in cm 2, of the region represented by the shaded region. Diberi bahawa PQRST mewakili suatu kawasan yang mempunyai keluasan 28 cm2. Hitung luas, dalam cm2, kawasan yang diwakili oleh kawasan berlorek. [3 marks] [3markah]

93 | M O D U L


5

(a) Diagram 13.8 shows a Cartesian plane. Rajah 13.8 menunjukkan suatu satah Cartesan.

y

4

2

x

2

0

2

4

6

2

4

Rajah 13.8 Diagram 13.8

 5   . Transformation R is a clockwise rotation of 90° 3   

Transformation T is a translation  about centre ( 1, 1).

 5   . Penjelmaan R ialah satu putaran 90°arah jam  3  

Penjelmaan T ialah satu translasi 

pada pusat (1, 1) . State the coordinates of the image of point (3, 1) under each of the following transformations: Nyatakan koordinat imej bagi titik (3, 1) di bawah setiap penjelmaan berikut: : (i)

T2

(ii)

TR [4 markah] [4 marks]

94 | M O D U L


(b) 5..

Diagram 13.9 shows three pentagons, EKLMW, EJIHG and GBFDE and drawn on a Cartesian plane. Rajah 13.9 menunjukkan tiga pentagon, EKLMW, EJIHG dan GBFDE dilukis pada suatu satah Cartesan.

y

B

6

G

4

F

2 D

W

H I

4

2

0

4

2

M

6

8

x

L

2

E K

4

J


(i) GBFDE is the image of EKLMW, under a combined transformation VU. GBFDE ialah imej bagi EKLMW di bawah gabungan penjelmaan VU Describe in full the transformation Huraikan selengkapnya penjelmaan (a)

U,

(b)

V.

(ii) It is given that the shaded region represents a region of area 168 m2. Calculate the area, in m2, of the pentagon EKLMW. Diberi bahawa suatu kawasan yang berlorek mewakili luas 168 m 2. Hitung luas, dalam m 2, pentagon EKLMW.

[8 markah]

95 | M O D U L


[8 marks]

CIRCLE III BULATAN III

14 1

Diagram 1 shows a quadrant OQR with radius 14 cm. RUV and PTQ are semicircles. P and V are midpoints of OQ and OR. Rajah 1 menunjukkan sukuan bulatan OQR dengan jejari 14 cm. RUV dan PTQ adalah semibulatan. P dan V adalah titik tengah bagi OQ dan OR.

R U

V T

Q

P

O

Diagram 14.1 Rajah 14.1 Using  =

22 7

, calculate

Dengan menggunakan  = (a)

22

, hitungkan 7 the perimeter, in cm, of the shaded region, perimeter , dalam cm, kawasan yang berlorek,

(b) the area, in cm 2, of the shaded region. luas, dalam cm2, kawasan yang berlorek. [6 marks] [6 markah] Answer / Jawapan : (a)

(b)

96 | M O D U L


2

Diagram 14.2 shows a sector OEC with the centre O. ACD is a semicircle with centre B. OABC is a straight line. Rajah 14. 2 menunjukkan sektor bulatan OEC berpusat O. ACD ialah semibulatan berpusat di B. OABC ialah garis lurus.

O

60 A B C

E D


Given that OE = 7 cm and OA = 3 cm. Diberi bahawa OE = 7 cm dan OA = 3 cm. Use  

22

, calculate 7 22 Guna   , hitung 7

(a)

the area, in cm 2, of the shaded region and give the answers correct to two decimal places, luas, dalam cm2, kawasan berlorek dan beri jawapan betul kepada dua tempat perpuluhan,

(b)

the perimeter, in cm, of the shaded region. perimeter , dalam cm, kawasan berlorek. [6 marks] [6 markah]

Answer / Jawapan : (a)

(b)

97 | M O D U L


3

Diagram 14.3 shows a sector OMN and a quadrant OQR with the same centre O. OMQ is a straight line. Rajah 14.3 menunjukkan satu sektor bulatan OMN dan satu sukuan bulatan OQR, keduaduanya berpusat di O. OMQ ialah garis lurus.

Q N

M

N 10 cm 60

R

14 cm

O

Diagram 14.3 Rajah 14.3 Using  

22 7

, calculate

Dengan menggunakan   (a)

(b)

22

, hitung

7 the perimeter, in cm, of the shaded region, perimeter , dalam cm, kawasan yang berlorek,

the area, in cm 2, of the shaded region. luas, dalam cm2, kawasan yang berlorek. [6 marks] [6 markah]

Answer / Jawapan : (a)

(b)

98 | M O D U L


4

Diagram 14.4 shows two quadrants WOX WOX and VOU, VOU, with common centre O. W is the midpoint of OV. OV. Rajah 14.4 14. 4 menunjukkan dua sukuan bulatan WOX dan VOU, VOU , dengan pusat sepunya di O. W adalah titik tengah OV.

V

W

X

O

U

Diagram 14.4 Rajah 14.4 Rajah 14.4 It is given that OV = OV = 14 cm. 22 Using   , calculate 7 Diberi bahawa OV = OV = 14 cm. 22 Menggunakan   , hitung 7 (a) the perimeter, in cm, of the whole diagram, perimeter , dalam dalam cm, seluruh rajah, rajah, (b)

the area, in cm 2, of the shaded shaded region. 2 luas, luas, dalam cm dalam cm , kawasan berlorek. [6 marks] marks] [6 markah] markah]

Answer / / Jawapan : (a)

(b)

99 | M O D U L


5

Diagram 14.5 shows four quadrants and two semicircles, with radius 7 cm drawn on a canvas. Amy wanted to paint the quadrants and the semicircles in blue. Rajah 14.5 Rajah 14.5 menunjukkan empat sukuan bulatan dan dua semibulatan, semibulatan , dengan jejari 7 cm dilukis di atas sebuah kanvas. Amy mahu mengecat sukuan dan semibulatan itu dengan warna biru.

D

C

E

F

H

B

A

Using  

22 7

(b)

L

K

J

I

Diagram 14.5 Rajah 14.5 Rajah 14.5 , calculate

Menggunakan   (a)

G

22

, hitung

7 the perimeter, in cm, of the painted region, perimeter , dalam dalam cm, kawasan bercat, bercat ,

the area, in cm 2, of the unpainted region. region. 2 luas, luas, dalam cm dalam cm , kawasan tidak bercat bercat.. [6 marks] marks] [6 markah] markah]

Answer / / Jawapan : (a)

(b)

100 | M O D U L


15

REVISION ULANG KAJI KA JI ALGEBRAIC EXPRESSIONS │ UNGKAPAN ALGEBRA

1.

Express

4u  v

uv

Ungkapkan

A.

B.

2. 2.



8  v

as a single fraction in its simplest form.

2v

4u  v

uv

8  v



2v

sebagai satu pecahan tunggal dalam bentuk termudah.

1

u  2

C.

2u

u  2

D.

2u

2  n Express m

n



2

n



 u  2


4

2

2m

4  n  n


2

A.

B.

3. 3.

n  2 )

D.

2m

Express

1  w

wk



1  w

Ungkapkan

A.

C.

m 4  n )(

wk

k  w  kw

1  k

k 2



1  k

k 2

wk 2

101 | M O D U L

2m

( 2  n ) n 2m


C.

wk

k   2kw

 n ( n  2 )


2

B.

2u

4

2m

2  n Ungkapkan m

2u

w  k 2

wk

D.

k   w 2

wk


4.

2

Express

3

7m

4m

2

Ungkapkan

A.

m  5





3


m  5


4m

5

C.

5 (

m  3 ) 12m

D.

( 5

m  3 ) 12m

12m

B.

7m



5

12m

3

1

5.

Express



2

2p

3

3

A.

2

2p


6p

1

Ungkapkan

p



p

6p

 3 p  p 2


C.

1  3p

 p 2 2 6p

D.

p 2  3p  1

2

6 p

p 2  3p  3

B.

6p

6.

Express

( n 2

2

2

2

2p

 9 )  n  3 as a single fraction in its simplest form. 7

Ungkapkan 2( n  9 )  2

A.

B.

2

14(n + 3)

( n  3 ) 2 7

102 | M O D U L

n  3 7


C.

D.

2 (

n  3 )2 7

14(n – 3)


ALGEBRAIC FORMULAE │ RUMUS ALGEBRA

1.

5s  4t

Given that

3 5s  4t

Diberi

A.

B.

2.

 2t , ungkapkan s dalam sebutan t.

3

s

 2t , express s in terms of t.

2t



C.

2t

s



Given that

x  y 

B.

y 

4t ( 3t



y

4t ( 9t



x  y  y

5

, ungkapkan y dalam sebutan x.





4 x  1

2m

2m

Diberi,

A.

t

3

C.

y 

D.

y 

4x

x  4



3 3

2m  3 )

2

 2m  3     3 

103 | M O D U L



4 x  1

x

 3 express

t in terms of m

ungkapkan t dalam sebutan m.

C.

2m  9

2

9

3

B.

1)

4

x  1

t



, express y in terms of x.

4x

Given that,

1)

5

x

3.



4

x 

Diberi bahawa

y 

s

5

x 

A.

D.

s

2

D.

 2m  3     9 

2


4.

5m 

Given that

Diberi

n

6

5m 

n

 4,

6

 4,

then n =

maka n =

5m  24 )

A.

2

C.

2

25m

 24 2

B.

5.

Given that Diberi

A.

B.

 24

2

25m

p 

p  1 2



1 2





4qr  12 , express q in terms of p and r .



4qr  12 , ungkapkan q dalam sebutan p dan r.

p 2  12

C.

r p

2

6

D.

B.

p

2

6

3

n  1

n  1 ,

, then n =

maka n =

2

 1   1   2m 

104 | M O D U L

3

3

Diberi 6m 

A.

2

2r

Given that 6m 

 2     3m 

p

r

2r

6.

 5   m   4  6 

D.

2

C.

 1     2m 

D.

 1   1   2m 

2

2


INDICES │ INDEKS

1

1.

2.

( 18 3 9

3

1

)2 (

2 4

2

)2

A.

54

C.

108

B.

162

D.

216

Given that 36

Diberi 36





216 36

216 36

2

, find the value of x.

x

2

x

cari nilai bagi x.

3

A.

3

C.

4

B.

3.

x

x

2

D.

1

2

1

a

3

b

6

)3 

5

1

A.

a b

B.

a b

5

a

6

b

5

=

7

C. a

3

5

7

b 7

D. a b 2

4.

3  3   5  42  Simplify   2  16   

.

3  3   5  42  Permudahkan   2  16   

105 | M O D U L

2


5.

A.

5

B.

6

 45

5

6

 4

C.

5

6



D.

5

6

 4

1

3

4

5

1

Given that

2k )

 103 , find the value of k and x.

x

1

Diberi bahawa

2k )

6.

 103 , cari nilai k dan x.

x

A.

k = 5, x = 3

C.

k = 3, x = 5

B.

k = 5, x = 3

D.

k = 3, x = 5



Find the value of 9  4



Cari nilai bagi 9  4

A.

B.

1 54

2 9

106 | M O D U L

4

4



1 2



1 2

 25  34 .

 25  3 4 .

C.

D.

2 27 9 64


LINEAR EQUATIONS │ PERSAMAAN LINEAR x  2

1. Solve the equation

3

Selesaikan persamaan



A.





2 3

2x  1 4

3

3p  2 3





2  p

2  p 2

2

3

D.

4

 2p  3(

Diberi bahawa

3

1

1 2

107 | M O D U L

1 2

3 2

4

3 2

maka p =

3

2

3

then p =

C.

3

3



D.

3p  2

2



C.

4

Given that

B.

3

2

B. 

A.

x  2

4

Diberi

3.

4

5

Given that

A.

2x  1

1

B.

2.



2

 2p  3(

1 4 3 2

p ) , calculate the value of p . 2  p ) , hitung

C.

D.

nilai p. 9 10

3

1 2


4.

3

Given that

Diberi bahawa

5.

3 3

value of x .  3x , calculate the value

6x  2 3

 3x , hitung nilai x.

A.

7 15

C.

11 15

B.

1 3

D.

1

Given

Diberi

6.

6x  2

w 4

 5  w  1, then w =

w 4

 5  w  1, maka w =

A.

2

C.

8

B.

6

D.

10

Given

Diberi

3n  1 2 3n  1 2 9

A. A.



B.

1

5

108 | M O D U L

 4n  5, then n=  4n  5, then n= C.

D.

9 5

1


LINEAR INEQUALITIES │KETAKSAMAAN LINEAR 1.

2.

List all the integer x which satisfy the inequalities 2 x + 2 ≤ 3x and −2 < 5− x Senaraikan semua integer x yang memuaskan ketaksamaan 2x + 2 ≤ 3x dan −2 < 5− x A.

3, 4, 5, 6

C.

3, 4, 5, 6, 7

B.

2, 3, 4, 5, 6

D.

2, 3, 4, 5, 6, 7

Which number line represents the solution of the simultaneous linear inequalities x 2

> 2x 2x −3 and

9 ≥ 3  3x .

Garis nombor manakah yang mewakili penyelesaian bagi kedua-dua ketaksamaan linear serentak

x 2

> 2x 2x −3 and

9 ≥ 3  3x . 

A.

•

2

2



•

2

6

C. 2

2



B.

6

2

3.



D.

List all the integers x which satisfy both the inequalities

x 3

≤ 2 and

1 2

Senaraikan semua integer x yang memuaskan kedua-dua ketaksamaan 1 2

A. B.

( x + 4 ) > 1 x 3

( x + 4 ) > 1

─2, ─1, 0, 1, 2, 3, 4, 5 ─2, ─1, 0, 1, 2, 3, 4, 5, 6

109 | M O D U L

C. D.

─1, 0, 1, 2, 3, 4, 5 ─1, 0, 1, 2, 3, 4, 5, 6


≤ 2

dan

4.

9

1

List all the integers x which satisfy the inequality

3

x <

2x 2x +

2

< 11 .

Senaraikan semua integer x yang memuaskan ketaksamaan linear A. B.

5.

C. D.

 5  m  2

The solution for 5



Penyelesaian bagi

A.

x <

B.

x >

110 | M O D U L

5 6 5 6

3 x

5



 x  3x



1 3

D.

2  m  5

C.

x >

D.

x

is

x



1 3

3

x <

−2, −1, 0, 1, 2, 3 −3, −2, −1, 0, 1, 2, 3

The solution for 3  m  8 and 6  2m  2 is Penyelesaian untuk 3  m  8 dan 6  2m  2 ialah A. C.  5  m  2  2  m  5 B.

6.

−3, −2, −1, 0, 1, 2 −2, −1, 0, 1, 2

9

1

ialah

4 3



4 3


2x 2x +

2

< 11.

16

LINEAR EQUATIONS PERSAMAAN SERENTAK

1. Calculate the value of x and of y that satisfy the following simultaneous linear equations: Hitung nilai x dan nilai y yang memuaskan persamaan linear serentak berikut: 1 x  2 y  1 2 3 x  4 y  26 [4 marks/markah]

2.

Calculate the value of x and of y that satisfy the following simultaneous linear equations: Hitung nilai x dan nilai y yang memuaskan persamaan linear serentak berikut: 1 x  2 y  3 3 x  6 y  15 [4 marks/markah]

111 | M O D U L


3. Calculate the value of x and of y that satisfy the following simultaneous linear equations: Hitung nilai x dan nilai y yang memuaskan persamaan linear serentak berikut: 3 x  y  3 2 4 x  y  16 [4 marks/markah]

4. Calculate the value of x and of y that satisfy the following simultaneous linear equations: Hitung nilai x dan nilai y yang memuaskan persamaan linear serentak berikut:

4 x  y  11 2 x  3 y  5 [4 marks/markah]

5. The sum of age of two siblings is 30 years and the difference of their age is 6 years . Find their age. Hasil tambah umur bagi dua orang adik beradik ialah 30 tahun dan beza umur mereka ialah 6 tahun. Cari umur mereka. [4 marks/markah]

112 | M O D U L


6. Azmi and Chong have 50 sweets. If Azmi has 4 sweets more than Chong, find the number of sweets that each of them has. Azmi dan Chong berkongsi 50 biji gula-gula. Jika Azmi mendapat 4 biji gula-gula lebih daripada Chong, cari bilangan gula- gula yang diperoleh oleh setiap orang. [4 marks/markah]

7. Bernas shop sell 1 kg of bananas and 1 kg of mangoes with the price of RM17. Puan Najwa pays RM44 for 2 kg of bananas and 3 kg of mangoes. Calculate the price, in RM, of 1 kg of bananas and 1 kg of mangoes sold in Bernas Shop.Kedai Bernas menjual 1 kg pisang dan 1 kg mangga dengan harga RM17. Puan Najwa membayar sebanyak RM44 untuk 2 kg pisang dan 3 kg mangga. Hitung harga, dalam RM, 1 kg pisang dan 1 kg mangga yang dijual di Kedai Bernas. [4 marks/markah]

8. Rozi bought 3 t-shirts and a pair of socks with price of RM45. The difference of price between 2 t-shirts and 3 pair of socks are RM8. What is the price, in RM, of pair of socks? Rozi membeli 3 helai baju-t dan sepasang stokin dengan harga RM45. Beza harga antara 2 helai baju-t dan 3 pasang stokin ialah RM8. Berapakah harga, dalam RM, bagi sepasang stoking? [4 marks/markah]

113 | M O D U L


17 1

SOLID GEOMETRY III PEPEJAL GEOMETRI III

Diagram 17.1 shows the remaining solid after a cuboid is removed from a cylinder. The radius of the cylinder is 4 cm and the height of the cylinder is 14 cm. Rajah 17.1 menunjukkan pepejal yang tinggal setelah sebuah kuboid dikeluarkan daripada sebuah silinder. Jejari silinder itu ialah 4 cm dan tinggi silinder itu ialah 14 cm.

8 cm 14 cm 4 cm

3 cm

Diagram 17.1 Rajah 17.1 Using  

22 7

3 , calculate the volume, in cm , of the remaining solid.

Menggunakan  

22 7

3 , hitungkan isipadu, dalam cm , bagi pepejal yang tinggal itu.

[ 4 marks ] [4 markah] Answer / Jawapan:

114 | M O D U L


2

Diagram 17.2 shows a combined solid formed by joining a cone and a hemisphere. Rajah 17.22 menunjukkan sebuah gabungan pepejal yang dibentuk daripada cantuman sebuah kon dan sebuah hemisfera.

Diagram 17.2 Rajah 17.2 The radius of the the cone = the radius of the hemisphere = 7 cm. The volume of the solid is 1155 cm 3. 22 Using  = , calculate the height, in cm, of t he cone. 7 Jejari kon = jejari hemisfera = 7cm. Isipadu pepejal itu ialah 1155 cm3. 22 Menggunakan  = , hitung tinggi, dalam cm, kon itu. 7 [4 markah] [4 marks] Answer / Jawapan :

115 | M O D U L


3

Diagram 17.3 shows a combined solid consits of a cube and a right pyramid which are joined at the plane EFGH. V is vertically above the base EFGH. Rajah 17.3 menunjukkan sebuah gabungan pepejal yang terdiri daripada sebuah kiub dan sebuah piramid tegak yang tercantum pada satah EFGH. V terletak tegak di atas tapak EFGH.

V

H

E F

6 cm

G

D

A

C B

Diagram 17.3 Rajah 17.3 The volume of the solid is 324 cm 3. Calculate the height, in cm, of the right pyramid. Isipadu pepejal itu ialah 324 cm3. Hitung tinggi, dalam cm, piramid tegak itu. [ 4 marks ] [4 markah] Answer / Jawapan:

116 | M O D U L


4

Diagram 17.4 shows a solid right prism. Trapezium ABCD is the uniform cross section of the prism. A cylinder with diameter 6 cm and height 14 cm is taken out from the solid. Rajah 17.4 menunjukkan sebuah pepejal yang berbentuk prisma tegak. Trapezium ABCD ialah keratan rentas seragam prisma itu. Sebuah silinder berdiameter 6 cm dan tinggi 14 cm dikeluarkan dari pepejal itu. P

A

S

D Q

R

14 cm B

C

Diagram 17.4 Rajah 17.4 The area of the cross section ABCD is 108 cm2. Calculate the volume, in cm 3 of the remaining solid. Luas keratan rentas seragam ABCD ialah 108 cm2. Hitungkan isipadu dalam cm 3, pepejal yang tinggal. 22 [Using/ Guna  = ] 7 [ 4 marks ] [4 markah] Answer / Jawapan:

117 | M O D U L


5.

Diagram 17.5 shows a pencil. It is made up of a cylinder and a cone. The radius and the length of the cylinder are 0.4 cm and 16 cm respectively. The radius of the base of the cone is 0.4 cm and the length of the cone is 2 cm. Rajah 17.5 menunjukkan sebatang pensel yang terdiri daripada silinder dan kon. Jejari dan panjang silinder ialah 0.4 cm dan 16 cm masing-masing. Jejari tapak kon ialah 0.4 cm dan panjang kon ialah 2 cm. 2 cm

16 cm

Diagram 17.5 Rajah 17.5 Using

π  22 , 7

Menggunakan

calculate

π  22 , hitung 7

(a) the slant height, in cm, of the cone, panjang, dalam cm, tepi condong kon, (b) the total surface area, in cm 2, of the pencil. jumlah luas, dalam cm2, permukaan pensel. [ 4 marks ] [4 markah] Answer / Jawapan: (a)

(b)

118 | M O D U L


18

ANSWER SCHEME JAWAPAN SET │ SET

SOALAN

SKEMA JAWAPAN

MARKAH SUB JUMLAH MARKAH MARKAH

1. (a)

1

3 (b) 2

2.(a) Q

P

1

P

(b)

3

Q

2

3.(a)

1

3 (b) 2

119 | M O D U L


4. (a)

1

3 (b) 2

5.(a) R

S

1

3 (b) K

L

2

QUADRATIC EQUATIONS │ PERSAMAAN KUADRATIK

SOALAN

1

SKEMA JAWAPAN

x2 – 5x + 4 = 0 or equivalent (x - 4) ( x - 1 ) = 0 x= 1 x= 4

120 | M O D U L

MARKAH SUB JUMLAH MARKAH MARKAH 1 1 1 4 1


2

2 x  8 x  15  0

1

( x  5)( x  3)  0

1

x  5

1

x  3

1

4x  2y = 12 or 2x  y = 6 or y = 6  4x or equivalent 6x = 12 or 3y = – 6 or 3x = 6 or equivalent x= 2 y = –2

1 1 1 1

3

4

5

x

2

 x  2  0

or equivalent ( x  1)( x  2)  0 or equivalent

x  1 , x  2 9 x 2  16  0

or equivalent

1

or equivalent

1

(3 x  4)(3 x  4)  0 x 

1

4 3

x   6

4

3 x  9 x

1

3 x  17 x  20

1

3 x - 26x  20

1

x(x + 6) = 135 x2 + 6x 135 = 0 (x  9)(x + 15) = 0 x = 9, x = 15 x =9

1 1 1

2

1

MATHEMATICS REASONING │ PENAAKULAN MATEMATIK

No 2 a)

b)

a) True/Benar b) True/Benar c) True/Benar d) False/Palsu

i) ii)

4

4

3

2

No1

4

1

2

7

1 1 1 1

4

All / Semua Some / sebilangan

If x+2 = 7, then x = 5 True / benar

121 | M O D U L


4

4

3(a) (i)

False // Palsu

(ii)

True // Benar

(b)

If x < 12 then x < 10 // Jika x < 12 maka x < 10 False //Palsu 2 (n)2 + n , n = 1, 2, 3,...

(c)

4(a)

some / sebilangan (i) If

(b)

x > 6 then 3x > 18 Jika x > 6 maka 3x > 18

(ii) If 3x > 18 then x > 6 Jika 3x > 18 maka x > 6 (c)

The sum of interior angles of a polygon with n sides is (n  2 ) x 180 , n = 3, 4, 5, …… Jumlah sudut pedalaman sebuah polygon n sisi ialah (n  2 ) x 180, n = 3, 4, 5, ……

5(a)(i) All / Semua (ii)

Some / sebilangan

(b)

If x+2 = 7, then x = 5 True / benar

(c)

QR=Q

122 | M O D U L


THE STRAIGHT LINES │ GARIS LURUS MARKAH SOALAN 1 (a)

SKEMA JAWAPAN x=2

 

 

2 =  (0) + c

1

c=2  y =  x + 2

1



(c)

 

0 =  x + 2 x =

2 (a)

h = 2

(b)

mCD =

 

 

(b)

1

1  

 x – 8 

x-intercept = 24 3 (a)

6

1

 y = x – 8 

(c)

1

1

10 = (6) + c

0=

JUMLAH MARKAH

1

mPO = mQR=  (b)

SUB MARKAH

1 6 1

1 1

mPO = mSR = 1 4 = 1(5) + c c = 1 y = x – 1

1 1

y = 0 x-intercept = 1

1 1

123 | M O D U L

1


5

MARKAH SOALAN 4 (a) (b)

SKEMA JAWAPAN y=5 − mQR= mPS = −

=

 

 (1) + c   c =    y = x +    0 = x  5 

3=

(c)

5 (a)

(b)

1 1

6 1 1 1

x = 4

1

mPQ= mRS =  

JUMLAH MARKAH

1

x = 10

5=

SUB MARKAH

−

−

=

 

1 1

(4) + c

6

c = 2  

1

 

1

y = x + 2 (c)

0 = x + 2  

x = 

1

LINES AND PLANES IN 3 DIMENSIONS │ GARIS DAN SATAH DALAM 3 DIMENSI

BIL 1. (a) (b)

JAWAPAN      

1 1

   

1

tan   =  

2. (a) (b)

MARKAH Submarkah Jumlah Markah 1

= 37.57

tan   =  

124 | M O D U L

 

= 41.19

3

3 1 1


3. (a) (b)

   

tan   =  

4. (a) (b)

(b)

   

1

 Tan



(b)

= 32

tan   =

 .

1 1

= 43.31

WPT =

WPT = 31 

8 5  12 2

2

36′ atau

atau setara

3

1 3

31.61 

1

 MRD

1

 MRD = − 

1 1

50.19°

3

1

WPT dikenalpasti



3

1 1 1

(b)

7 (a)

1 1

 

tan   =

 

6 (a)

 

= 14.04    

 

5. (a)

1

3

8 (a)

1 3

8.5

125 | M O D U L




tan

EDC =

3 1

45 .

(b)

33.7o or 33o 41’

or 33o 4

1

 MFL

9 (a) (b)

Tan

10 (a)

 MFL =  

1

33.7o atau 33o 41’

1

 BEF

1

Tan

(b)

1

3

1

 BEF =  

3

1

41.19o atau 41o 11’

STATISTICS │ STATISTIK No 1.

Cara Pemarkahan

Markah

a)

(b)

Jisim (kg) Mass (kg)

Frekuensi Frequency

42 – 46

3

Titik tengah Midpoint 44

47 – 51

8

49

52 – 56

6

54

57 – 61

3

59

62 – 66

11

64

67 – 71

5

69

72 – 76

4

74

3  44  8  49  6  54  3  59  11 64  5  69  4  74 3  8  6  3  11  5  4



Column I All correct: 1 mark Column II : All correct: 1 mark Column III All correct: 2 marks note: (7 correct: 1 mark) 2

2370 40

1

= 59.25 d) graph: uniform scale and correct axis direction all points are plotted and joined correctly 2 additional midpoints for one class before and one class after

126 | M O D U L

1 mark 2 marks 1 mark


127 | M O D U L


No . 2

Cara Pemarkahan

Markah

a) Class interval Selang Kelas 0-9

Mid point Titik tengah 4.5

Frequenc y Kekerapa 0

10 - 19

14.5

3

20 - 29

24.5

5

30 - 39

34.5

8

40 - 49

44.5

10

50 - 59

54.5

5

60 - 69

64.5

3

70 - 79

74.5

2

Column I All correct: 1 mark Column II : All correct: 1 mark Column III All correct: 2 marks note: (7 correct: 1 mark)

b) 40-49 1 mark c) 4.5(0)  14.5(3)  24.5(5)  34.5(8)  44.5(10)  54.5(5)  64.5(3)  74.5(2)

36



1502 36

2 marks

= 41.72 d) graph: uniform scale and correct axis direction all points are plotted and joined correctly 2 additional midpoints for one class before and one class after

128 | M O D U L


1 mark

1 mark 2 marks 1 mark

129 | M O D U L


No

Markah Submarkah Total

Cara pemarkahan

3. (a)

Column I Number of books

Column II

Column III

Midpoint

Frequency

Bilangan buku

Titik tengah

Kekerapan

1 – 5

3

0

6 – 10

8

9

11 – 15

13

12

16 – 20

18

8

21 – 25

23

6

26 – 30

28

5

31 – 35

33

5

Column I (all class interval correct)

P1

Column II (all midpoint correct)

P1

Column III (all frequency correct)

P2

Note : Allow two mistakes in column III for P1 (b)

3(0)  8(9)  13(12)  18(8)  23(6)  28(5)  33(5) 45 or

(c)

(d)

815

K2

45

18.11 Refer graph: Axes drawn in the correctt directions with uniform scales, for 3 ≤ x ≤ 38 and 0 ≤ x ≤ 10.

N1

7 points correctly plotted. Note : 6 or 5 points correctly plotted , award K1 Frequency polygon passes all 8 correct points. The modal class is 11 – 15 or any correct information about numbers of books read by the students

K2

130 | M O D U L

P1

N1 K1


12

131 | M O D U L


No

Markah Submarkah Total

Cara pemarkahan

4. Score

Frequency

Cumulative Frequency

Upper Boundary

6 - 10

0

0

10.5

11 - 15

1

1

15.5

16 - 20

3

4

20.5

21 – 25

6

10

25.5

26 – 30

10

20

30.5

31 – 35

11

31

35.5

36 – 40

7

38

40.5

41 - 45

2

40

45.5

(a)

Column I

Column II

Column III

Column I Column I Column III

K2 K1 K1

Ogive : (b) All the axes are drawn in the correct direction with uniform scale 10.5  x  45.5 and 0  y  40 , the x-axis labelled with upper boundaries.

K1

All 9 points plotted correctly.

N1

Plot the point (10.5 ,0)

N1

Smooth and continous curve within the range

N1

10.5.5  x  45.5

through all the 9 points. Note : (1) If 7 or 8 points were plotted give only K1. (2) If other scales are used , deduct 1 mark from K1 given. i. (c)

= 10.5 ii.

K1

35 - 25.5

 0.5

‘40’ marked on the y -axis

Number of students chosen = 2 persons

N1 K1 N1 12

132 | M O D U L


Marks No

5. (a)

Marking Scheme

Submar Total ks Marks

(12x2)+(17x8)+ (22x14) + (27x25) + (32x27) + (37x19) +(42x5) 100

N1

29.2

(b)

K2

Upper boundary / sempadan atas

Cumulative frequency / kekerapan longgokan

9.5

0

14.5

2

19.5

10

24.5

24

29.5

49

34.5

76

39.5

95

44.5

100 P1

Column 1 ( all upper boundary correct) P1 Column II ( all cumulative frequency correct ) (c)

Refer the graph Ogive Axis drawn in the correct direction, uniform scale for 9.5  x  44 .5 and 0  y  100 Plot 8 points correctly

(d)

Smooth and continuous curve without straight line(s) and passes through all the 8 points (i) 34  24.5 = 9.5 ± 1.0

K2 N1

K1 N1 P1

(ii) 10 ± 1

133 | M O D U L

P1


12

LINEAR

INAQUALITIES │ KETAKSAMAAN LINEAR

No1

y

3

x

3

O y =  x

y = x + 3

Dotted line x = – 5 Shaded area No2. The line y = 4 correctly drawn

134 | M O D U L


No 3 y

y=4

4

y = – 2 x – 3 y =2 x

– 4

– 2

O

2

4

x

– 3

Solid or Dotted line y = 2 x Shaded area (with dotted line y = 2 x)

No 4 i)

y=5

ii)

y=x+2

iii)

y = 2x + 5

No 5 i) ii) iii)

y ≤ x + 3, y < 2 x < 4.

135 | M O D U L


GRAPH OF FUNCTIONS / GRAF FUNGSI

MARKAH NOMBOR

SKEMA JAWAPAN

1.(a)

(b)

(c)

x

4.5

12

SUB MARKAH 1

y

8

3

1

Graph Axes are drawn in the correct direction, uniform scale for 2  x  14. 7 points and 2 points* plotted accurately Smooth and continuous curve without straight line(s) and passes through all the 9 correct points for 2  x  14. Notes : (1) 7 or 8 points plotted correctly, award 1 mark (2) Other scale being used, subtract 1 mark (i) 2.30 < x < 2.45 (ii) 5.0 < y < 5.3

JUMLAH MARKAH

1 2 1

1 1

Note : Do not accept the values of x and y obtained by calculation. (d)

y 

0

36 x 36 x 4

or

subtitute y 

4 7

4

 x 10 7

y   x  10 7

4 y   x  10 7

4 Identify the equation y   x 10 or equivalent 7 4 The straight line y   x  10 drawn correctly 7 The values of x : 5 < x < 5.2 12.4 < x < 12.6

Notes : (1) Award mark(s) if the value(s) of x shown on the graph. (2) Do not accept the value(s) of x obtained by calculation.

136 | M O D U L

12

x  10  0


1 1

1 1

137 | M O D U L


NO 2. (a)

(b)

MARKING SCHEME 0 25 Axes drawn in correct direction and uniform scales in - 2 ≤ x ≤ 5 and - 5 ≤ y ≤ 42 .

MARKS K1 K1 P1

7 points and *2 points correctly plotted or curve passes through the points for - 2 ≤ x ≤ 5.

K2

A smooth and continuous curve without any straight line and passes through all 9 correct points for - 2 ≤ x ≤ 5.

N1

(c)

(i) 18 ≤ y ≤ 20 (ii) 4.65 ≤ x ≤ 4.8

P1 P1

(d)

Identify equation y = 2x + 7 or 2x2 – x - 3 = 2x + 7

K1 K1

Straight line y = 2x + 7 correctly drawn Values of x : 1.65 ≤ x ≤ - 1.50

138 | M O D U L

, 3.0 ≤

x

≤ 3.2


N1N1

139 | M O D U L


NO 3. (a)

(b)

MARKING SCHEME 1

y

27

2

K1

All the axes drawn in the correct direction with uniform scales for -4  x  4 dan –37  y  80.

K1

7 points and 2 extra points being ploted correctly within the range - 4  x  4. Smooth and continous curve without any straight line between any two points within −4 ≤ x ≤ 4

K2

(1)

(d)

− 4

Graph

Note: (1)

(c)

x

MARKS K1

N1

If 7-8 points plotted correctly only K1 will be given. Deduct 1 mark if other scales are used.

(i)

y = -14.5

(ii)

x = 3.5

 0.5

N1

 0.05

N1

The line y = − 5x − 12 drawn or its equation is written − 3.3 ≤ x1 ≤ −3.2 - 0.5 3.7

 x  - 0.4

N1 N1

2

 x  3.8 2

Nota : (1) N marks will be given if the values of x are shown in the graph. (2) If the values of x are obtained by calculation , N0 .

140 | M O D U L

K1


N1

NO 4.

MARKING SCHEME (a)

(b)

x

0.5

5

1 mark

y

10

1

1 mark

Graph Axes are drawn in the correct direction, uniform scale for 0  x  8. 7 points and 2 points* plotted accurately Smooth and continuous curve without straight line(s) and passes through all the 9 correct points for 0  x  8.

(c)

(i) (ii)

MARKS

1.0 < x < 1.2 1.3 < y < 1.5

1 mark

2 marks 1 mark

1 mark 1 mark

Note : Do not accept the values of x and y obtained by calculation. (d)

y = –x + 7 line y = –x + 7 correctly drawn The values of x : 0.7 ≤ x ≤ 0.9 , 6.1 ≤ x ≤

6.3

Notes :(1) Award mark(s) if the value(s) of x shown on the graph. (2) Do not accept the value(s) of x obtained by calculation.

141 | M O D U L


1 mark 1 mark

10

9

8

7

6

5

4

3

2

1

0

1

142 | M O D U L

2

3

4

5

6


7

8

PROBABILITY │ KEBARANGKALIAN

BIL

SKEMA PEMARKAHAN SUB MARKAH

MARKAH JUMLAH

1 (a) {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66}

2

(b)

3

(i) {11, 22, 33, 44, 55, 66} (ii) {23, 26, 43, 46, 63, 66} (iii) {16, 25, 36, 64}

5

2.

5

(a)

2

72

(b) Basikal kerana seramai 125 daripada 360 pelajar atau

25 72

2

menaiki basikal untuk ke sekolah. 5

3.

4

(a) {(B5, BU, BW, E5, EU, EW, 85, 8U, 8W, 35, 3U, 3W} 1 (b) (i) {E5}  12 (ii) {85, 8U, 8W, 35, 3U, 3W, BU, BW, 10 5 EU, EW}  @ 12 6

1

1. (a) {33, 35, 37, 39, 55, 57, 59, 53, 77, 73, 75, 79, 99, 93, 95, 97}

1

(b) (i) {57, 59, 77, 79, 99, 95, 97}  (ii) {33, 35, 37, 93, 95, 97} 

143 | M O D U L

6 16

@

1 4

2 2

5

7 16

2 2


5

5

(a) {RE, RC, RO, RR, RD, ER, EC, EO, ER, ED, CR, CE, CO, CR, CD, OR, OE, OC, OR, OD, RR, RE, RC, RO, RD, DR, DE, DC, DO, DR}

1

2 (b) (i) {ER, EC, EO. ER, ED, OR, OE, OC, 10 1 OR, OR} = @ 30 3 (b) (ii) {RC, RC} =

2 30

@

1 15

2 2

(b) (iii) {RR, RR, RD, ED, CD, OD, RD}



7 30

144 | M O D U L


7

GRADIENT

AND UNDER A GRAPH │ KECERUNAN DAN LUAS DI BAWAH GRAF

SOALAN


SKEMA JAWAPAN

1a)i) Distance 120

50

2 45

t=

110

Time / Masa

ii) 0.5 hour

1

70 atau setara 0.5

1 6

140 km h-1

1

Note : Award P1 for (9.20 – 8.50) or (110 – 80) b)

2(a)

t = 80

1

− atau setara −

1

2 Nota : Tanpa kerja, beri K1N1 (b)

 × (5 8 +   1 2) × 24 

= 1032 atau setara

40

145 | M O D U L

1

5 2 1


3(a)

Distance/Jarak (km)

58

p

1

O

(b)

(c)

35

120

q

6

Time/Masa (min)

p = 28 q = 85

1

58

1

2

1

29

1

4(a)

16 − 10 (b)

=6s

1

Rate of decrese of speed) Kadar pengurangan laju = 26-0/13-0

1 1

= 2 m s −2 (c)

334 = × (v + 48) × 8 + × (48 + 30) × 2 v = − 48 = 16

6 2 1

5 (a) 7 saat 10  18 (b) 5 8

–

(c)

5

1 1 or

1.6 m s –2

1

1 1 (18  10)  5 + (7 x 10) + (10  22)(t  12) =188 2 2 70 + 70 + 16t – 192 = 188 t = 15 saat MATRIX / MATRIKS

146 | M O D U L


2

1

BIL

MARKAH SUB JUMLAH MARKAH MARKAH 1

SKEMA JAWAPAN

1

1    1   2   3 2  

2

4   1   3 M   5 1   3  

1

3

1   1

2    2   4 6 

1

4 5

k = 18, m = – 4

2 2

m

6 (a)

1 5

, n  1

m = 2 n = -3

2 1

Note : 4 ( - 1) – 3 ( - 2 ) seen award P1

(b)

4 3

-2 -1

x y

=

x y

=

6 3

1 4 ( -1 ) – 3 ( -2 )

1

-1 -3

2 4

6 3

or equivalent 1 7

x = 0

1

y = - 3

1

Note : 1. Accept only matrix method. 2.

x y

=

0 -3

award N1

7

1 1

(a) k = −14 m = −2 (b)

1

147 | M O D U L


 6 2   x   1         4 1     y   3   x    1  2   1  1        or equivalent y 4 6        6 1 4 2      3 

1 6

x = 1 , 2

y

= −1

1 1

Note : 3. Accept only matrix method.  x   1 / 2   award N1. 4.      y    1 

8 (a)

(b)

9

1

m

2

1

n  3

1

 4 3  x   6        2 2   y   8 

1

 x  1  2  3  6         y  2   2 4  8 

1

x =  6

1

y = 10

1

(a)

 3  6    53  6 2    2 5 

6

1

 6   or equivalent =  3   2 5  1  3

 5 6  m    4       2 3   n   2   m   3  6    4  1        or equivalent 2 5 2  n 5  3   6  2        

(b) 

1

1

1

1

m = 8 , n = 6 Note : 5. Accept only matrix method.

148 | M O D U L

1


1

6

 m    8        n   6 

2.

award 1 mark

10

1 1

(a) p = 2 q = 2

 2  2  x   7       3 4    y   13 

(b) 

 x    4 2   7  1        or equivalent y 3 2       2 4 2 3         13  x = 1 y = 

1

1 1

5 2

6

1

Note : Accept only matrix method. 1   x    5    award 1 mark    y    2  11

  4 2       (3)( 4) ( 2)(5)   5 3   2  1    5  3   2   2

a) R 

1

1

1

b)

 3  2  p   2        5 4    q   12    4 2  2   p  1       5 312 q ( 3 )( 4 ) ( 2 )( 5 )         p  8 q  13

PLANS AND ELEVATIONS │ PELAN DAN DONGAKAN

149 | M O D U L


1

1 1 1

6

NO

MARKING SCHEME

1

(a)

MARKS

A

D

H

G

C B Correct shape with rectangles AHGD. All solid lines.

1

AB > BC, AH > HB 1 Measurements correct to ± 0.2 cm and all angles at vertices of rectangles = 90º ± 1º 1

1

b (i) P, Q

J

E

H

S, R

B, C A V T U Correct shapes with hexagon PSTBEH, rectangle ABEJ and a semicircle with radius 2 cm. All solid lines. H and V joined with dotted line.

1

AT> JE >AV > JA = PH > ST and semicircle drawn correctly Measurements correct to ± 0.2 cm and ± 1º 1

b(ii)

150 | M O DMU L

P

 A,  B,  T and  H = 90º

1 1

Q

P L U S J A B A T ALN P E N D I D I K A N S E L A N G O R

2

Correct shape with rectangles BTSE, TURS, PQRS and MLSE. All solid lines.

1

BU = UQ > BM > BT > CM Measurements correct to ± 0.2 cm and all angles at vertices of rectangles = 90º ± 1º

1

2 2 (a)

Untuk soalan ini, bentuk betul tetapi ukuran yang dilukis tidak mengikut skala sepatutnya. Pastikan calon melukis mengikut skala sebenar. 2 cm R J

L

3 cm

K

7 cm

4 cm

E

8 cm

Shape visually correct and all lines are solid.

F

1 EF  FR  EL  LK  JR 1 Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1o

151 | M O D U L


1

2

(b) (i)

M

P

N

Q

3 cm

6 cm

A

B

3 cm

L

2

3 cm

3 cm

2 cm U

J

K

Shape visually correct with rectangles EKNH, KJPN, BAGP and ABJF , all lines are solid.

1

FE  EH  HN = NP = PB  BA

1

Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1

2

(b) (ii)

J

3 cm

A

3 cm

Q

3 cm

K

V

N

4 cm

6 cm

F

G

Shape visually correct with solid lines rectangles FGPJ and TWBJ 1 W and N joined with a dashed line. 1 PB = BJ = JT  TF  FG 1 Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1o

152 | M O D U L


2

3(a) K

F

J

4 cm

L

2 cm

6 cm

I

A

Correct shape with rectangles ILKJ and AIJF. All solid lines.

1

AL > AI > AF > KJ

1

Measurement correct to  0.2 cm (one way) and all angles at the vertices of rectangle = 90º  1º. 3

1

(b)(i) L

2 cm

I

1 cm H 4 cm

P

Q

2 cm

C

2 cm

B

6 cm

A

Correct shape with rectangle BCLI, triangle ABH and quadrant of circle CPQ. All solid lines.

1

AB = BI > BH > PL > PC and A, B, C, Q collinear

1

Measurement correct to  0.2 cm (one way) and all angle at the vertices of rectangle = 90º  1º.

2

153 | M O D U L


3

b(ii)

K

L

1 cm G

H

3 cm

S

P

2 cm R

Q

4 cm

Correct shape with rectangles SPLK and SRQP. All solid lines.

1 1

G is joined to H by a dashed line to form rectangle GHQK. KR > RQ > PH > PQ > HL

1 Measurement correct to of rectangle = 90º  1º. 4

 0.2 cm

(one way) and all angles at vertices 2

Untuk soalan ini, bentuk betul tetapi ukuran yang dilukis tidak mengikut skala sepatutnya. Pastikan calon melukis mengikut skala sebenar. 8 cm

H

(a)

A

2 cm

I

2 cm J

K

4 cm

6 cm

4 cm

2SLK. cm AllBlines are L Shape visually correct with L shape and quadrant solid.

1

HA  AB=IK  JK=KL  LB = HI=IJ

1

Measurements accurate up to  0.2 cm (one way) and all right angles = 90 o  1o

1

154 | M O D U L


4

(b)(i)

L

J

I

T

4 cm 6 cm Q

P

2 cm F

2 cm

E

4 cm

D

2 cm

C

Shape visually correct with rectangle ITCF, ILEF, LKDE, KTPQ and QPCD. All lines are solid.

1

IT  TC  TP=ED >PC, IF = FD and QD = DC 1 Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1 4

(b) (ii)

H

2 cm

I

2

L

6 cm Q

2 cm

G

2 cm

F

4 cm

D

Shape visually correct with rectangle HLDG, HIFG and ILKF. All lines are solid.

1

I joined to Q by a dashed line to form triangle ILQ.

1

GH = HL  LQ = IL  QD = FG

1

Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1o

2

155 | M O D U L


5

Untuk soalan ini, bentuk betul tetapi ukuran yang dilukis tidak mengikut skala sepatutnya. Pastikan calon melukis mengikut skala sebenar. 5 cm

C

(a)

D

5 cm

I

4 cm

B

2 cm H

Shape visually correct and all lines are solid.

1 1

BC  DH  CD  BH and CD = DI Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1o 5

(b)(ii)

H

2 cm

K

2 cm

I

5 cm

D

F

5 cm

P

1

4 cm

O

2 cm

N

2 cm G

4 cm

Shape visually correct with rectangle DEGH, GFIH, DEFI and HKNO. All lines are solid.

1

DH  DE  EF  FG > GO and HK = KI 1 Measurements accurate up to  0.2 cm (one way) and all right angles = 90 o  1

156 | M O D U L


2

5

(b)(ii) B

3 cm

C

2 cm

H

2 cm

K

4 cm 6 cm

O

N

2 cm A

J

P

G

1 mark

Shape visually correct with solid lines rectangles BKPA, GONP and HKNO. 1 mark C and J joined with a dashed line. 1 mark BK > BA  KN > NP = NO Measurements accurate up to  0.2 cm (one way) and all right angles = 90o  1o

TRANSFORMATION │ PENJELMAAN 1)

TRANSLASI a)

PANTULAN (BAHAGIA N 1) 1)

b) (6,2) 2)

a)

2)

b) (1,6)

157 | M O D U L


2 marks

3)

a)

3)

4)

b) (8,6) a)

4)

5)

b) (7,1) a)

b) ( 2,5) PANTULAN (BAHAGIAN 2) 1)

2

Paksi pantulan : y=1 Paksi pantulan : x=3 3)

Paksi pantulan : x=y

158 | M O D U L


PUTARAN ( BAHAGIAN 1) 1)

2

3)

1)

3)

PUTARAN (BAHAGIAN 2) Pusat/Centre : (-1,5) Sudut/Angle : 90 0 Arah/Direction : arah jam

2

Pusat/Centre : (-1,5) Sudut/Angle : 900 Arah/Direction : lawan jam

2

Faktor skala/scale factor :

Pusat/Centre : (0,0) Sudut/Angle :180 0 Arah/Direction :arah jam

1)

PEMBESARAN (BAHAGIAN 1) Faktor skala/scale factor 3

3)

Faktor skala/scale factor :

4

PEMBESARAN (BAHAGIAN 2) 1)

159 | M O D U L

2


 

3)

BENTUK SOALAN SPM 1 (a)

(b)

(i)

(1, 3 )

1

(ii) (1, 4)

1

(iii) (2 , 6)

2

(i) (a)

V = rotation of 90o clockwise about point (7, 0)

Note : (1) rotation only, get 1 mark. (2) rotation of 90o clockwise only, get 2 marks (3) rotation about point (7, 0) only, get 2 marks (b)

Area PQR = 8 unit2

(ii) (b)(i)

2

3

Area EFG = k2 x (Area PQR) 72 = 32 x (Area PQR)

2(a)(i )

1

W = Enlargement, centre G or (7,3), scale factor 3

Note : (1) Enlargement, centre G or (7, 3), get 2 marks (2) Enlargement, scale factor 3 , get 2 marks (3) Enlargement only, get 1 mark (ii)

1

1 1

(2, 3)

1

(0, 6) Note : (0, 2) award P1

2

U = A rotation of 90° clockwise about the centre (2, 4)

3

Note: Rotation of 90° clockwise or Rotation about the centre (2, 4), award P2 Rotation, award P1

160 | M O D U L


12

V = An enlargement with scale factor 2 at the centre ( 8, 3)

3

Note: Enlargement Enlargement with scale factor 2 or Enlargement Enlargement at the centre ( 8,3), award P2 Enlargement, Enlargement, award P1 (ii) ii)

3 (a (a)

105 = 22 × Area ABCDE Area ABCDE = 35 m2

– Area ABCDE

2 1

12

2

(i) (3, 3) Note: Note: (1, 5) get 1 mark (ii) (2, 2)

2

Note: Note: (0, 1) get 1 mark (b)

(i) (a (a) U = Reflection at the line y = 1. Note : (1) Reflection Reflection only, only, get 1 marks. (b)

V = Enlargement, centre P, scale factor 2 (note: accept centre (6, 3)

2

3

Note : (1) Enlargement, Enlargement, centre (6, 3), get 2 marks (2) Enlargement, scale factor 2, get 2, get 2 marks (3) Enlargement only, only , get 1 get 1 mark (ii)

4

Area of JKLMN = 22 x 50

1

Area of the shaded shaded region = (2 2 x 50) − 50 = 150

1 1

(a) (i) (0, −1) Note : Note : (−1, 2) 2) award 1 mark

12

2

(ii) (0, 5) Note : Note : (5, 2) award 1 mark

2

(b) (i) (a) U = a reflection in the line x = 0 or y-axis Note : (1) reflection only reflection only,, get 1 marks. (b) V = Enlargement, centre E, scale factor of 2 Note : accept centre (1,1)

2

3

Note : (1) Enlargement, Enlargement, centreE or (1, 1), get 2 marks (2) Enlargement,scale factor 2, get 2, get 2 marks (3) Enlargement only, only , get1 get1 mark 1 (ii)

Area of ABCDE of ABCDE= = 22 x 28 Area of shaded shaded region = (2 2 = 84

161 | M O D U L

x 28) −28


1 1

12

5

(i) (7, − 5) Note : (2, 2) award 1 mark mark

2

(ii) (8, 0) Note : (3, (3, 3) award 1 mark

2 12

(b) (i) U = An enlargement, centre E, scale factor of 2 Note : (1) enlargement only, only, get 1 mark (2) enlargement , centre E only, only, get 2 marks (3) enlargement , scale factor of 2 only, only, get 2 marks

2

3

V = a 180 clockwise rotation at centre W Note: Note: (1) accept an 180  anticlockwise rotation (2) accept centre (5, 1) (3) Rotation, centre W only, get 2 marks (4) Rotation 180 clockwise,get clockwise,get 2 marks (5) Rotation only, get 1 mark NOTE: Accept for others others U and V Equivalent (ii)

Area of image = 2 ²  area of of object = 2 ²  A 1

Area of shaded shaded region = 2(2 ²  A) – A = 168 168 Area of object object = 168

1

7 = 24

1

ALGEBRAIC │ ALGEBRA Linaer inequalities / Ketaksamaan Linear 1B

2A

3. D

4.C

5. A

6 .C .C

INEAR LINEAR EQUATION │ KESAMAAN L INEAR 1D

2A

3. B

4C

5C

6C

5B

6C

INDICES │ INDEKS 1D

2C

3. B

4D

ALGEBRAIC EXPRESSION │ UNGKAPAN ALGEBRA 1C

2B

162 | M O D U L

3.D

4. D

5. B

6.D


ALGEBRAIC FORMULAE │ RUMUS ALGEBRA 1. C

2. C

3. C

4. A

5.C

6. D

SIMULTANEOUS EQUATIONS │ PERSAMAAN SERENTAK SOALAN

SKEMA JAWAPAN


1

 + 4 = 2 atau setara 4 = 24 atau setara  = 6, =2

1 1 1 1

4

2

 + 6  = 9 atau setara 12 = 24 atau setara  = 2, =3

1 1 1 1

4

3

4    = 1 6 atau setara 7 = 28 atau setara  = 4, =3

1 1 1 1

4

4

12  3 = 33 atau setara 14 = 28 atau setara  = 2, =3

1 1 1 1

4

5

+=30 =6 Dapat menulis kedua-dua persamaan

2 = 24 =12, =14 6

4 1 1 1 1

Azmi ada x biji gula-gula Chong ada y biji gula-gula

+=50 =4+ Dapat menulis kedua-dua persamaan

2 = 46 =23 =27 7

4 1 1 1 1

+=17 2 + 3 = 44

4

2 + 2 = 34

1 1

Harga sekilogram pisang = RM7 Harga sekilogram mangga = RM10

1 1

Dapat menulis kedua-dua persamaan

163 | M O D U L


8

3+=45 23=8

4 1 1 1

Dapat menulis kedua-dua persamaan

11 = 143 =13,=6 Harga sepasang stoking = RM6

1

SOLID GEOMETRY III │ PEPEJAL GEOMETRI III SOALAN

1

SKEMA JAWAPAN

MARKAH SUB JUMLAH MARKAH MARKAH 1

4 × 3 × 8 or 96 22 7

22

1

 4 2  14

4

 4 2  14  4  3  8

1

7 608 2

2 3 1

3 2 3

3







1 22 7 22

7 22 7

777

1

 7  7  t

1

 7  7  7 +

3



22 7

4

 7  7  t = 1155

1

8.5

1

6×6×6 1  6  6  t 3

1

666 

4

1

1

1 3

 6  6  t  324

4 1

9

1

108 × 14

1

22

1

7

 33 1 4

108 x 14 1116 cm

164 | M O D U L



22 7

 3 3 1 4


1 1

4

2. Matematik Soalan modul PLUS

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