Kertas 2 Maths Peperiksaan Percubaan PMR 2011 · PDF fileSULIT 2 50/2 50/2 © 2011 Hak...

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50/2 © 2011 Hak Cipta JABATAN PELAJARAN SARAWAK SULIT

MATHEMATICAL FORMULAE

RUMUS MATEMATIK

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

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

digunakan.

RELATIONS

PERKAITAN

1. nmnmaaa

+=×

2. nmnmaaa

−=÷

3. ( ) mnnm aa =

4. Distance / Jarak = ( ) ( )2

12

2

12 yyxx −+−

5. Midpoint / Titik tengah

( )

++=

2,

2,

2121yyxx

yx

6. Average speed = takentime

travelleddistance

Purata laju =

jarak yang dilalui

masa yang diambil

7. Mean = data ofnumber

data of sum

Min =

hasil tambah nilai data

bilangan data

8. Pythagoras’ Theorem / Teorem Pithagoras

222bac +=

1. Area of rectangle = length × width

Luas segiempat tepat = panjang × lebar

2. Area of triangle = ×2

1base × height

Luas segitiga = ×2

1tapak × tinggi

3. Area of parallelogram = base × height

Luas segiempat selari = tapak × tinggi

SHAPE AND SPACE

BENTUK DAN RUANG

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50/2 © 2011 Hak Cipta JABATAN PELAJARAN SARAWAK [Lihat halaman sebelah

SULIT

4. Area of trapezium = ×2

1 sum of parallel sides × height

Luas trapezium = ×2

1 hasil tambah dua sisi selari × tinggi

5. Circumference of circle = πd = 2πr

Lilitan bulatan = πd = 2πj

6. Area of circle = πr2

Luas bulatan = πj2

7. Curved surface area of cylinder = 2πrh

Luas permukaan melengkung silinder = 2πjt

8. Surface area of sphere = 4πr2

Luas permukaan sfera = 4πj2

9. Volume of right prism = cross sectional area × length

Isipadu prisma tegak = luas keratan rentas × panjang

10. Volume of cuboid = length × width × height

Isipadu kuboid = panjang × lebar × tinggi

11. Volume of cylinder / Isipadu silinder = πr2h

12. Volume of cone / Isipadu kon = 3

1 πj2h

13. Volume of sphere = 3

4 πr

3

Isipadu sfera =

3

4 πj

3

14. Volume of right pyramid = 3

1 × base area × height

Isipadu piramid = 3

1 × luas tapak × tinggi

15. Sum of interior angles of a polygon = ( n – 2 ) × 180o

Hasil tambah sudut pedalaman poligon = ( n – 2 ) × 180o

16. Arc length Angle subtended at centre

Circumference of circle 360=

°

360

Panjang lengkok Sudut pusat

Lilitan bulatan =

°

17. Area of sector Angle subtended at centre

Area of circle 360=

°

360

Luas sektor Sudut pusat

Luas bulatan=

°

18. Scale factor / Faktor skala, PA

PAk

'=

19. Area of image = k2 × area of object

Luas imej = k2 × Luas objek


SULIT 4 50/2


For

Examiner’s

Use

Answer all questions.

Jawab semua soalan

1 Calculate the value of :

Hitung nilai bagi:

3 × (– 18 + 20 ÷ 4 ) [2 marks]

[2 markah]

Answer / Jawapan:

2 Calculate the value of ( 4 − 0.22 ) ×2

11 and express the answer as a decimal. [2 marks]

Hitung nilai bagi ( 4 − 0.22 ) × 2

11 dan ungkapkan jawapan sebagai nombor

perpuluhan. [2 markah]

Answer / Jawapan:

3 (a) Find the value of :

Cari nilai bagi :

4

16

(b) Calculate the value of:

Hitung nilai bagi :

−1

64

81 3

[3 marks]

[3 markah]

Answer / Jawapan:

(a)

(b)

2

2

3

3

1

2


SULIT 5 50/2


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4 Solve each of the following linear equations :

Selesaikan tiap-tiap persamaan linear berikut :

(a) 4 − x = 1

(b) 4

x − 3 = x + 3

[3 marks]

[3 markah]

Answer / Jawapan

(a)

(b)

For

Examiner’s

Use

5 Diagram 5 in the answer space shows quadrilateral PQRS and straight line MN drawn on a

grid of equal squares. R′S′ is the image of RS under a reflection in the straight line MN. On

Diagram 5, complete the image of quadrilateral PQRS.

Rajah 5 di ruang jawapan menunjukkan sisi empat PQRS dan garis lurus MN yang dilukis

pada grid segi empat sama yang sama besar. R′S′ ialah imej bagi RS di bawah pantulan

pada garis lurus MN. Pada Rajah 5, lengkapkan imej bagi sisi empat PQRS.

[2 marks]

[2 markah]

Answer / Jawapan:

Diagram 5

Rajah 5

4

3

5

2

P

Q R

S

R′

S′ M

N


SULIT 6 50/2


For

Examiner’s

Use

6 Diagram 6 shows two quadrilaterals, PQRS and P′Q′R′S′, drawn on a Cartesian plane.

P′Q′R′S′ is the image of PQRS under an enlargement.

Rajah 6 menunjukkan dua sisi empat, PQRS dan P′Q′R′S′, yang dilukis pada satah

Cartesan. P′Q′R′S′ ialah imej bagi PQRS di bawah suatu pembesaran.

Diagram 6

Rajah 6

(a) State the coordinates of the centre of enlargement. [1 mark]

Nyatakan koordinat bagi pusat pembesaran itu. [1 markah]

(b) State the scale factor of the enlargement. [1 mark]

Nyatakan faktor skala bagi pembesaran itu. [1 markah]

Answer / Jawapan:

(a)

(b)

6

2

y

x O 2 – 2

2

P′

S′

Q′

R′

Q

S

R

P

4

4

–2


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7 Diagram 7 in the answer space shows a parallelogram PQRS drawn on a grid of equal

squares with sides of 1 unit. M and N are the midpoints of straight lines PQ and RS

respectively. X, Y and Z are three moving points inside the parallelogram.

Rajah 7 di ruang jawapan menunjukkan sebuah segi empat selari PQRS yang dilukis pada

grid segi empat sama bersisi 1 unit. M dan N adalah titik tengah bagi garis lurus PQ dan RS

masing-masing. X, Y dan Z adalah tiga titik yang bergerak di dalam segi empat selari itu.

(a) X is the point which moves such that it is always equidistant from point R and S.

By using the letters in Diagram 7, state the locus of X.

X ialah titik yang bergerak dengan keadaan jaraknya adalah sentiasa sama dari titik R

dan S.

Dengan menggunakan huruf di dalam Rajah 7, nyatakan lokus bagi X.

(b) On Diagram 7, draw

Pada Rajah 7, lukis

(i) the locus of the point Y which moves such that its distance is constantly 4 units

from point S.

lokus bagi titik Y yang bergerak dengan keadaan titik itu sentiasa berjarak 4 unit

dari titik S.

(ii) the locus of the point Z which moves such that it is always equidistant from the

straight lines PS and SR.

lokus bagi titik Z yang bergerak dengan keadaan jaraknya adalah sentiasa sama

dari garis lurus PS dan SR.

(c) Hence, mark with the symbol ⊗ the intersections of the locus of Y and the locus of Z.

Seterusnya, tandakan dengan simbol ⊗ kedudukan bagi semua persilangan lokus Y dan

lokus Z itu.

[5 marks]

[5 markah]

Answer / Jawapan:

(a)

(b) (i), (ii) (c)

Diagram 7

Rajah 7

For

Examiner’s

Use

7

5

P Q

S R N

M


SULIT 8 50/2


For

Examiner’s

Use

8 Table 8 shows the number of students obtaining 8As in the PMR examinations of a school.

Jadual 8 menunjukkan bilangan murid yang memperolehi 8A dalam peperiksaan PMR di

sebuah sekolah.

Year / Tahun 2006 2007 2008 2009 2010

Number of students

Bilangan murid 6 11 18 14 20

Table 8

Jadual 8

On Diagram 8 in the answer space, draw a line graph to represent all the information in

Table 8. Use the scale 2 cm to 5 students on the vertical axis.

[3 marks]

Pada Rajah 8 di ruang jawapan, lukiskan suatu graf garis untuk mewakili semua

maklumat dalam Jadual 8. Gunakan skala 2 cm kepada 5 orang murid pada paksi

mencancang.

[3 markah]

Answer / Jawapan :

Diagram 8

Rajah 8

8

3

Number of students

Bilangan pelajar

2006 2007 2008 2009 Year

Tahun

2010


SULIT 9 50/2


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9 (a) Expand :

Kembangkan :

4(t – 2) + 3

(b) Simplify: Permudahkan:

( ) ( )mm 23552

−−−

[3 marks]

[3 markah]

Answer / Jawapan:

(a)

(b)

For

Examiner’s

Use

10 Factorize completely each of the following expressions:

Faktorkan selengkapnya tiap-tiap ungkapan berikut:

(a) 10g – 2gk

(b) 12 – 27p2

[3 marks]

[3 markah]

Answer / Jawapan:

(a)

(b)

.

10

3

9

3


SULIT 10 50/2


For

Examiner’s

Use

11 Diagram 11 shows a right prism with a rectangular base.

Rajah 11 menunjukkan sebuah prisma bertapak segi empat tepat.

Complete the net of the prism in Diagram 11. [3 marks]

Lengkapkan bentangan prisma itu dalam Rajah 11 [3 markah]

Answer / Jawapan:

11

3

6 units

6 unit

3 units

3 unit

4 units

4 unit

Diagram 11

Rajah 11


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12 (a) Diagram 12.1 shows a quadrilateral PQRS drawn not to scale.

Rajah 12.1 menunjukkan sebuah sisi empat PQRS yang dilukis tidak mengikut skala.

Diagram 12.1

Rajah 12.1

Diagram 12.2 in the answer space shows two straight lines PQ and QR.

Rajah 12.2 dalam ruang jawapan menunjukkan dua garis lurus PQ dan QR.

Use only a ruler and a pair of compasses, construct

Menggunakan pembaris dan jangka lukis sahaja, bina

(i) quadrilateral PQRS to the measurements shown in Diagram 12.1,

sisi empat PQRS menggunakan ukuran seperti yang ditunjukkan dalam Rajah

12.1,

(ii) the perpendicular line to the straight line QR which passes through the point S.

garis serenjang kepada garis lurus QR yang melalui titik S.

[4 marks]

[4 markah]

(b) Measure the perpendicular distance, in cm, between straight line QR and point S.

Ukur jarak tegak, dalam cm, di antara garis lurus QR dan titik S.

[1 mark]

[1 markah]

Answer / Jawapan:

(a) (i), (ii)

Diagram 12.2

Rajah 12.2

(b)

For

Examiner’s

Use

12

5

Q R

P

60°

4 cm

S

Q R

P


SULIT 12 50/2


For

Examiner’s

Use

13 Given ddc 32 =+ , express c in terms of d .

Diberi ddc 32 =+ , ungkapkan c dalam sebutan d.

[3 marks]

[3 markah]

Answer / Jawapan

14 (a) Evaluate:

Nilaikan:

2

35

p

pp −×

.

(b) Given 9 43 3 +−=

xx , calculate the value of x .

Diberi 9 43 3 +−=

xx , cari nilai x.

[3 marks]

[3 markah]

Answer / Jawapan

(a)

(b)

13

3

14

3


SULIT 13 50/2


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15 The data in Table 15 shows the grades obtained by 20 candidates in an examination.

Data dalam Jadual 15 menunjukkan gred yang diperolehi oleh 20 orang calon dalam

suatu peperiksaan.

B D B B C B A C D C

E A C C D B D B B B

Table 15

Jadual 15

(a) Using the data above, complete the frequency table in the answer space.

Dengan menggunakan data di atas, lengkapkan jadual kekerapan pada ruang

jawapan.

(b) State the mode.

Nyatakan mod. [3 marks]

[3 markah]

Answer / Jawapan :

(a)

Grades

Gred A B C D E

Frequency

Kekerapan

(b)

For

Examiner’s

Use

16 Express 2

6

65

2

3

p

p

p

−− as a single fraction in its simplest form.

Ungkapkan 2

6

65

2

3

p

p

p

−− sebagai satu pecahan tunggal dalam bentuk termudah.

[3 marks]

[3 markah]

Answer / Jawapan :

15

3

16

3


SULIT 14 50/2


For

Examiner’s

Use

17 Solve each of the following inequalities :

Selesaikan tiap-tiap ketaksamaan berikut :

(a) 3 + q ≤ 7

(b) 6x − 2 ≥ 2 ( 3 + x )

[3 marks]

[3 markah]

Answer / Jawapan :

(a)

(b)

18 In Diagram 18, triangles PQR and TSR are similar.

Dalam Rajah 18, segi tiga PQR dan TSR adalah serupa.

Diagram 18

Rajah 18

Given that PQ = 2 cm, calculate the length, in cm, of ST.

Diberi PQ = 2 cm, hitung panjang, dalam cm, bagi ST.

[2 marks]

[2 markah]

Answer / Jawapan:

17

3

S

T

Q

2 cm

P

R

3 cm

8 cm

18

2


SULIT 15 50/2


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19 In Diagram 19, PQR and PST are right-angled triangles. PRS is a straight line.

Dalam Rajah 19, PQR dan PST adalah segi tiga bersudut tegak. PRS ialah garis lurus.

Diagram 19

Rajah 19

(a) Find the value of cos xo.

Cari nilai bagi kos xo

(b) Calculate the length, in cm, of RS.

Hitung panjang, dalam cm , bagi RS.

[3 marks]

[3 markah]

Answer / Jawapan:

(a)

(b)

For

Examiner’s

Use

20 Use the graph paper on page 16 to answer this question.

Gunakan kertas graf di halaman 16 untuk menjawab soalan ini.

Table 20 shows the values of two variables, x and y, of a function.

Jadual 20 menunjukkan nilai-nilai dua pembolehubah, x dan y, bagi suatu fungsi.

x – 3 – 2 – 1 0 1 2 3

y – 10 5 8 5 2 5 20

Table 20

Jadual 20

The x-axis and the y-axis are provided on the graph paper on page 16.

Paksi-x dan paksi-y telah disediakan pada kertas graf di halaman 16.

(a) By using a scale of 2 cm to 5 units, complete and label the y-axis.

Dengan menggunakan skala 2 cm kepada 5 unit, lengkap dan labelkan paksi-y.

(b) Based on the Table 20, plot all the points on the graph paper.

Berdasarkan Jadual 20, plot semua titik pada kertas graf itu.

(c) Hence, draw the graph of the function.

Seterusnya, lukiskan graf fungsi itu.

[4 marks]

[4 markah]

19

3

17 cm

10 cm 8 cm

8 cm

R P

Q

S

T

x°


SULIT 16 50/2


20

4

For

Examiner’s

Use

x

y

– 3 – 2 – 1 1 2 3 O


SULIT 17 50/2


SULIT

INFORMATION FOR CANDIDATES

MAKLUMAT UNTUK CALON

1. This question paper consists of 20 questions.

Kertas soalan ini mengandungi 20 soalan.

2. Answer all questions.

Jawab semua soalan.

3. Write your answers clearly in the spaces provided in the question paper.

Tulis jawapan anda pada ruang yang disediakan dalam kertas soalan ini.

4. Show your working. It may help you to get marks.

Tunjukkan langkah-langkah penting dalam kerja mengira anda. Ini boleh membantu anda untuk

mendapatkan markah.

5. If you wish to change your answer, neatly cross out the answer that you have done. Then write down

the new answer.

Jika anda hendak menukar jawapan, batalkan jawapan yang telah dibuat. Kemudian tulis jawapan

yang baru.

6. The diagrams in the questions provided are not drawn to scale unless stated.

Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan.

7. The marks allocated for each question are shown in brackets.

Markah yang diperuntukkan bagi setiap soalan ditunjukkan dalam kurungan.

8. A list of formulae is provided on pages 2 to 3.

Satu senarai rumus disediakan di halaman 2 hingga 3.

9. The usage of calculator is not allowed.

Penggunaan kalkulator tidak dibenarkan.

10. This question paper must be handed in at the end of the examination.

Serahkan kertas soalan ini kepada pengawas peperiksaan pada akhir peperiksaan.


- 1 -

PEPERIKSAAN PERCUBAAN PMR 2011 MATHEMATICS PAPER 2

ANSWER

No Solution and Mark Scheme Sub Mark

Full Mark

1 3 × (– 18 + 20 4 ) = 3 × ( –18 + 5 ) = 3 × ( – 13 ) = – 39

1 1

2

2

(4− 0.22) × 121

= 3.78 × 23

= 5.67

1 1

2

3

. a. 416

= 425

= 25 OR 2

21

b. . (6481 − 1 )3

= ( 89 − 1 )3

= (81 )3

= 5121

1

1 1

3

Note: Award 1 mark for – 13 seen

Note: Award 1 mark for 3.78 seen

Note:

Award 1 mark for 89 or

81 seen


- 2 -

4 a. 4 − x = 1 − x = 1 − 4 − x = − 3 x = 3

b. 4x − 3 = x + 3

. 4x − x = 3 + 3

. 43x = 6

−3x = 24 x = − 8

1 1 1

3

5 Complete diagram drawn

2

6 (a) (4,3) (b) 3

1 1

2

P

Q R

S

R′

S′ M

N

Q′ P′


- 3 -

7 (a) PN Note: Award 1 mark for PN drawn on the diagram.

(b)

Locus Y drawn correctly Locus Z drawn correctly marked correctly

2 1 1 1

5

8 Uniform scale marked correctly on vertical axis All the five points marked correctly All the points joined with straight line

Note: If scale is not written or different scale used, but all the points plotted correctly, award 1 mark. If scale is not written or different scale used, but line graph drawn correctly, award 2 marks.

1 1 1

3

Number of students Bilangan pelajar

2006 2007 2008 2009 Year Tahun

2010

25

20

15

10

5

P Q

S R N

M

Locus Z

Locus Y


- 4 -

9 a) 4t + 5

b) mmm 101525102 = 102 m

1 1 1

3

10 (a) 2g( 5 – k)

(b) 12 – 27p2 = 3 ( 4 – 9p2) = 3 ( 2 + 3p ) ( 2 – 3p)

1 1 1

3

11

Complete diagram drawn Note: Any 2 sides drawn correctly, award 1 mark Any 3 sides drawn correctly, award 2 marks

3


- 5 -

12 (a) (i), (ii) Correct construction of 60°. Line SR = 4 cm correctly drawn. Complete quadrilateral drawn. Correct construction of perpendicular line from point S to QR. (b) 3.5 cm ± 0.1

1 1 1 1 1

5

13

22 )3()2( ddc

2

92 ddc OR 2232 ddc

ddc 2

92

29 2 ddc

OR 219 )(

ddc

1 1 1

3

14 a) 235 p

= 0p

= 1

b) 4)3(2 33 xx

4)3(2 xx

462 xx

642 xx

10x

1 1 1

3

Q R

P

S


- 6 -

15 a) Grades Gred A B C D E

Frequency Kekerapan 2 8 5 4 1

Note: 3 or 4 answers in the frequency table correctly filled, award 1 mark.

b) Grade B

2 1

3

16 26

6523

pp

p

26

659p

pp

2664

pp

26

322pp )(

2332

pp

1 1 1

3

17 a. 3 + q ≤ 7

q ≤ 7 − 3 q ≤ 4

b. 6x − 2 ≥ 2(3 + x )

6x − 2 ≥ 6 + 2 x 6x − 2x ≥ 6 + 2 4 x ≥ 8 x ≥ 2

1 1 1

3

18

STPQ =

SRQR

ST2 =

83

ST = 3

82

= 3

16 OR 5 31 cm

1 1

2


- 7 -

19 (a) cos xo =

106 =

53

(b) PS = 22 817 = 15 OR PR = 22 810 = 6 RS = PS – PR =15 – 6 = 9 cm Note : If 15 or 6 is marked correctly in the diagram, award 1 mark

1 1 1

3

20

x

y

5

х

х

х

х

х

х

х

10

15

20

1 2 3 O – 1 – 2 – 3

– 5

– 10


- 8 -

1. Uniform scale on y-axis marked correctly in the range 2010 y

2. All 7 points correctly plotted. Note: 5 or 6 points marked correctly or curve passes through 5 or 6 points, award 1 mark. 3. Smooth curve passes through all the 7 points. Note: If scale is not written but all the points plotted correctly, award 2 marks. For other scale on the y-axis, deduct 1 mark from the total marks obtained.

1 2 1

4


Kertas 2 Maths Peperiksaan Percubaan PMR 2011 · PDF fileSULIT 2 50/2 50/2 © 2011 Hak...

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