1449/2MatematikKertas 2Okt20062 ½ jam

SEKTOR SEKOLAH BERASRAMA PENUHBAHAGIAN SEKOLAH

KEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN AKHIR TAHUNTINGKATAN 4 2006

Kertas soalan ini mengandungi 22 halaman bercetak1449/2 @ 2006 Hak Cipta SBP

MATEMATIK

Kertas 2

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

Pemeriksa

Bahagian SoalanMarkahPenuh

MarkahDiperoleh

1 4

2 4

3 4

4 5

5 5

6 4

7 7

8 5

9 3

10 5

A

11 6

12 12

13 12

14 12

15 12

B

16 12

1. Kertas soalan ini mengandungi duabahagian : Bahagian A dan BahagianB.

2. Jawab semua soalan Bahagian A danempat soalan daripada Bahagian B.

3. Jawapan hendaklah ditulis dengan jelasdalam ruang yang disediakan dalamkertas soalan.

4. Tunjukkan langkah-langkah pentingdalam kerja mengira anda. Ini bolehmembantu anda untuk mendapatkanmarkah.

5. Rajah yang mengiringi soalan tidakdilukis mengikut skala kecualidinyatakan.

6. Satu senarai rumus disediakan dihalaman 2 dan 3.

7. Anda dibenarkan menggunakankalkulator saintifik yang tidak bolehdiprogram.

Jumlah

NAMA : …………………………………………TINGKATAN : ………………

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2

The following formulae may be helpful in answering the questions. The symbolsgiven are the ones commonly used.

RELATIONS

1 am an = am + n

2 am an = am - n

3 (am)n = am n

4 A-1 =

ac

bd

bcad

1

5 P(A) = Sn

An

6 P(A’) = 1 – P(A)

7 Distance = 2

122

12 )( yyxx

8 Midpoint

(x, y) =

2,

22121 yyxx

9 Average speed =

10 Mean =

11 Mean =

12 Pythagoras Theoremc2 = a2 + b2

13

12

12xx

yym

14erceptintx

erceptintym

distance traveledtime taken

sum of datanumber of data

Sum of (midpoint of interval frequency)Sum of frequencies

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3

SHAPE AND SPACE

1 Area of trapezium =2

1 sum of parallel sides height

2 Circumference of circle = d = 2r

3 Area of circle = r2

4 Curved surface area of cylinder = 2rh

5 Surface area of sphere = 4r2

6 Volume of right prism = cross sectional area length

7 Volume of cylinder = r2h

8 Volume of cone =3

1r2h

9 Volume of sphere =3

4r3

10 Volume of right pyramid =3

1 base area height

11 Sum of interior angles of a polygon = (n – 2) 180°

12

13

14 Scale factor, k =PA

PA'

15 Area of image = k2 area of object.

arc length angle subtended at centre

circumference of circle 360°

area of sector angle subtended at centre

area of circle 360°

=

=

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4

Section A

[52 marks]

Answer all questions in this section.

1 Solve the quadratic equation2

1

3

52 2

p

p.

[4 marks]Answer:

2 Calculate the value of p and of q that satisfy the following simultaneous linearequations:

3p – 4q = –2

1322

1 qp

[4 marks]Answer:

ForExaminer’s

Use

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5

3 Diagram 1 shows a right prism. Right angled triangle PQR is the uniform cross-section of the prism.

DIAGRAM 1

(a) Name the angle between the plane STP and the plane STQR,

(b) Calculate the angle between the plane STP and the plane STQR.

[4 marks]

Answer:(a)

(b)

ForExaminer’s

Use

[Lihat sebelah

P

Q R

ST

U

9 cm18 cm

12 cm

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6

4 A straight line ST is parallel to line y = –2x + 5 and passes through point (4, –2).

Find(a) the gradient of the straight line ST,

(b) the equation of the straight line ST and hence, state its y-intercept.. [5 marks]

Answer:

(a)

(b)

5 A bag contains 50 pens. 15 of them are blue and the rest are red and green.A pen is chosen at random from the bag.

(a) Find the probability that a blue pen is chosen.

(b) The probability of choosing a green pen is5

1. How many green pens

are there in the bag?

(c) Another 10 green pens are put into the bag. Find the probability that agreen pen is chosen from the bag.

[5 marks]

Answer:

(a)

(b)

(c)

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7

6 Diagram 2(a) shows a container in the form of a right pyramid fully filled withwater. Diagram 2(b) shows an empty cylindrical container. The height of thepyramid is 15 cm.

DIAGRAM 2(a) DIAGRAM 2(b)

All the water from the right pyramid container is poured into the cylindricalcontainer.

By using =7

22, calculate the height, in cm, of the water level in the cylinder.

[4 marks]

Answer:

ForExaminer’s

Use

[Lihat sebelah

14 cmA B

CD

E

12 cm

8 cm

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8

7 In Diagram 3, OPQS is a quadrant with the centre O and OSQR is a semicirclewith the centre S.

DIAGRAM 3

Given that OP = 14 cm. Using π =7

22, calculate

(a) the area, in cm2, of the shaded region,

(b) the perimeter, in cm, of the whole diagram.

[6 marks]

Answer:

(a)

(b)

O P

Q

R S

60°

T

ForExaminer’s

Use

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9

8 (a) Determine whether the following is a statement or not. Give a reason toyour answer.

(b) Rewrite the following statemoriginal statement. State the

(c) Construct a true statement uand the property.

Answer:

(a) ………………………………

(b) ………………………………

(c) ………………………………

ForExaminer’s

Use

3 is

Object :Property :

ent by inserting the word ‘not’ into thetruth value of your new statement.

64 = 42

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sing a suitable quantifier for the given object

[5 marks]

……………………………………………..

……………………………………………..

…………………………………………….

[Lihat sebelah

the factor of 15

TrianglesHave a right angle

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10

9 The Venn diagram in the answer space shows set P, set Q and set R with theuniversal set = P Q R.

On the diagrams provided in the answer space, shade(a) the set Q (P R),(b) the set Q R’ P.

. [3 marks]Answer:

(a) (b)

10 In Diagram 4 , AE is a tangent to the circle centre O, at A. CDE is a straight line.

DIAGRAM 4Find the value of

(a) x

(b) y

(c) z

[5 marks]

Answer:

(a)

(b)

(c)

ForExaminer’s

Use

P

R

Q

A

20°

B

CD

70°

z°x°

y° O E

P

R

Q

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11

11 (a) Diagram 5 shows a unit circle.

DIAGRAM 5Find the value of

(i) sin v°

(ii) cos v° + sin 270°

[3 marks]

(b) Diagram 6 shows a rhombus PQRS.

DIAGRAM 6

It is given that TSQ is a straight line and tan x° =12

5 .

Find

(i) the length of RP

(ii) sin x°

[3 marks]Answer:

(a) (i)

(ii)

(b) (i)

(ii)

ForExaminer’s

Use

[Lihat sebelah

–1 1

–1

1

y

x

v°

(– 0.42, – 0.91)

O

P

Q

R

ST

x°26 cm

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12

Section B

[48 marks]

Answer four questions in this section.

12 (a) In Diagram 7, a straight line RS is parallel to the straight line PQ. Theequation of PQ is 2y = x – 4.

DIAGRAM 7

Find(i) the y-intercept of PQ,(ii) the value of m,

(iii) the equation of RS.

[6 marks]

Answer:

(a) (i)

(ii)

(iii)

ForExaminer’s

Use

y

xO

R Q(8, m)

P

S(4, 5)

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13

(b) In Diagram 8, EFGH is a parallelogram and O is the origin. The gradientof EF is 2.

DIAGRAM 8

Find

(i) the equation of line EF,

(ii) the coordinates of H,

(iii) the gradient of HF.

[6 marks]

Answer:

(b) (i)

(ii)

(iii)

[Lihat sebelah

ForExaminer’s

Use

H

xO G

E

F(4, 10)y

12

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14

13 (a) Diagram 9 shows a right prism with a horizontal rectangular base PQRS.

DIAGRAM 9

Given that E and F are midpoints of WV and SR respectively.

(i) Find the length of PF,

(ii) Calculate the angle between the line PE and the plane PQRS,

(iii) Name the angle between the plane PQVE and the plane PQUT.

[6 marks]

Answer:

(a) (i)

(ii)

(iii)

ForExaminer’s

Use

8 cm

5 cm

7 cm

12 cm

P

QR

U

W

T

E

F

S

V

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15

(b) In Diagram 10, MP and LQ are two vertical flagpoles standing onhorizontal ground KLM .

DIAGRAM 10

Given that the angle of depression of P from Q is 38, MP = 5 m,LQ = h m and KM = 10 m.

Calculate

(i) the angle of elevation of P from K,

(ii) the value of h.

[6 marks]Answer:

(b) (i)

(ii)

ForExaminer’s

Use

400

5 m

P

70 M

Q

h m

K

L

13

[Lihat sebelah

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14 (a) (i) Fill the blanks with ‘and’ or ‘or’ in order to form a true statement.

(a) sin 30˚ =2

1cos 30˚ =

2

1

(b) All the multiples of 3 are multiples of 6 all multiplesof 6 are multiples of 3.

(ii) Complete the argument below.Premise 1 : If a A, then a A B.Premise 2 : __________________________________Conclusion : 5 A

(iii) Make a general conclusion by induction for the sequence13, 28, 49, 76,… which follows the following pattern:

13 = 3 (2)2 + 128 = 3 (3)2 + 149 = 3 (4)2 + 176 = 3 (5)2 + 1………………………………

[6 marks]Answer:

(a) (i) (a) ……………………………………………………………….

(b) ……………………………………………………………….

(ii) ..........................................................................................................

(iii) ..........................................................................................................

ForExaminer’s

Use

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(b) Diagram 11 is an incomplete Venn diagram. A group of 45 students

ForExaminer’s

Use

14

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involved in their favourite games.

DIAGRAM 11

Given that = B S T and the number of students who involve inbadminton only is the same as the students who involve in tennis only.

Find the number of students who involve in(i) all the three games,(ii) neither badminton nor tennis,(iii) two games only.(iv) tennis only.

[6 marks]

Answer:

(b) (i)

(ii)

(iii)

(iv)

[Lihat sebelah

B = Badminton

2

5

3

6

11

T = Tennis

S = Squasy

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15 The daily distances, in km, travelled in 40 days by a salesman are shown inDiagram 12.

(a) Usinin th

Answer:

(a)

(b) Dete(c) Base

trave

Answer:(b)

(c) (i)

(ii)

(d) For t

By u1 day

Answer:(d) Refer

ForExaminer’s

Use

42 62 51 37 70 14 12 93

53 45 54 27 70 76 91 72

68 58 57 65 21 18 22 87

64 63 57 59 70 19 41 55

DIAGRAM 12

g data in Diagram 12 and a class interval of 10 km, complete Table 1e answer space.

[4 marks]

rd

s

36 56 50 57 73 75 13 39

Distance (km) Midpoint Frequency

11 – 20

21 – 30

31 – 40

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TABLE 1

mine the range of the given data. [1 mark]on your table in (a), calculate the estimated mean distance

lled.[3 marks]

his part of the question, use the graph paper on page 19.

ing a scale of 2 cm to 10 km on the x-axis and a scale of 2 cm toon the y-axis, draw a frequency polygon for the data above.

[4 marks]

graph on page 19.

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Graph for Question 15(d)

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16 The histogram in Diagram 13 shows the marks achieved in a test by a group ofstudents in a particular school.

DIAGRAM 13

(a) Find the total number of students who sat for the test. [1 mark](b) Find the modal class. [1 mark](c) Based on the histogram, complete the Table 3 below. [3 marks]

Answer:

(a)

(b)

(c)

Marks Upper boundary Cumulativefrequency

20 – 29 29.5 030 – 39 39.5 4

TABLE 3(d) For this part of the question, use the graph paper on page 22.

By using the scale of 2 cm to 10 marks on the x-axis, and 2 cm to 10students on the y-axis, draw an ogive for the data.

[4 marks](e) From the ogive, find

(i) the median,(ii) the number of students who passed the test if the passing mark is

45%. [3 marks]

ForExaminer’s

Use

29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.5

4

8

12

16

20

Marks

Frequency

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Answer:(d) Refer graph on page 22.

(e) (i)

(ii)

ForExaminer’s

Use

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Graph for Question 16(d)

END OF QUESTION PAPER