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Matematik

Kertas 2

Mei

2007

2 jam

SEKTOR SEKOLAH BERASRAMA PENUH

BAHAGIAN SEKOLAH

KEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN PERTENGAHAN TAHUN

TINGKATAN 4 2007

MATEMATIK

Kertas 2

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1.Kertas soalan ini mengandungi dua

bahagian : Bahagian A dan Bahagian

B. Jawab semua soalan dalam

Bahagian A dan empat soalan

daripadaBahagian B.

2.Jawapan hendaklah ditulis dengan jelas

dalam ruang yang disediakan dalam

kertas soalan. Tunjukkan langkah-

langkah penting. Ini boleh membantu

anda untuk mendapatkan markah.

3.Rajah yang mengiringi soalan tidak

dilukis mengikut skala kecuali

dinyatakan.

4.Satu senarai rumus disediakan di halaman

3 dan 4.

5.Anda dibenarkan menggunakan

kalkulator saintifik yang tidak boleh

diprogram.

Pemeriksa

Bahagian SoalanMarkahPenuh

MarkahDiperoleh

A

1 5

2 4

3 4

4 7

5 3

6 5

7 6

8 6

9 3

10 4

11 5

B

12 12

13 12

14 12

15 12

16 12

Jumlah

NAMA : TINGKATAN :

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Kertas soalan ini mengandungi 22 halaman bercetak1449/2 @ 2007 Hak Cipta SBP

INFORMATION FOR CANDIDATES

1. This question paper consists of two sections: Section Aand

Section B. Answerallquestions inSection Aandfourquestions inSection B.

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

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

4. If you wish to change your answer, neatly cross out the answer that you have

done. Then write down the new answer.

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

6. The marks allocated for each question and sub-part of a question are shown in

brackets.

7. A list of formulae is provided on pages 3 and4.

8. You may use a non-programmable scientific calculator.

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

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MATHEMATICAL FORMULAE

The following formulae may be helpful in answering the questions. The symbols givenare the ones commonly used.

RELATIONS

1 am x an = a m+ n

2 am an = a m n

3 ( am )n = a mn

4 A-1

= bcad1

ac

bd

5 P ( A ) =)(

)(

Sn

An

6 P ( A ) = 1 P(A)

7 Distance = 221

2

21)()( yyxx +

8 Midpoint, (x, y ) =

++

2,

2

2121 yyxx

9 Average speed =

10 Mean =

11 Mean =

12 Pythagoras Theorem

c2 = a2 + b2

13 m =12

12

xx

yy

14 m =

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distance travelledtime taken

sum of datanumber of data

sum of (class mark frequency)

sum of frequencies

y-interceptx-intercept

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SHAPES 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 =31 base area height

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

12360

centreatsubtendedangle

circleofncecircumfere

lengtharc=

13360

centreatsubtendedangle

circleofarea

sectorofarea=

14 Scale factor , k=PA

PA'

15 Area of image = k2 area of object

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Section A

[52 marks]

Answerallquestions in this section.

1 (a) Factorise 3k2 2 ( k+ 4 ).

(b) Solve the quadratic equation 13

)4(2+=

+m

mm.

[5 marks]

Answer:

(a)

(b)

2 Calculate the value ofh and kthat satisfy the following simultaneous linear

equations:

3h 4k = 8

2

1h + 2k = 12

[4 marks]

Answer:

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5

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3 Diagram 1shows a solid formed by joining a cylinder and a cone.

The radius of the cylinder and the radius of the base of the cone

are both 6 cm. The height of the cone is 8 cm and the height of

the cylinder is 10 cm.

DIAGRAM 1

Using = 3.142, calculate the volume, in cm3, of the solid.

[4 marks]

Answer:

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4 Diagram 2 shows a quadrantADO with centre O and a sectorBEFwith centreB.

OBC is a right angled triangle andD is the midpoint of the straight line OC.Given OC= OB =BE = 14 cm.

DIAGRAM 2

Using =7

22, calculate

(a) the perimeter, in cm, of the whole diagram,

(b) the area, in cm2, of the shaded region.

. [7 marks]

Answer:

(a)

(b)

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5 The Venn diagramin the answer space shows setP, set Q and set

R with the universal set = PQ R.

On the diagram in the answer space, shade

(a) PQ R

(b) P ( Q R)

[3 marks]

Answer:

(a)

(b)

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Q

P R

R

Q

P

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6 In Diagram 3, O is the origin, pointR lies on thex-axis and point Q lies on they-axis. Straight linePQ is parallel to thex-axis and straight line RQ is parallel to

ST. The equation ofRQ is 3y x = 12.

DIAGRAM 3

(a) State the equation ofPQ.

(b) Find the equation of the straight line STand hence, state itsx-intercept.

[5 marks]

Answer:

(a)

(b)

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P Q

R

T

S(3, 10)

O

y

x

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7 (a) State whether each of the following statement is true or

false.

(i) 6 is a factor of 36 and 1 is a prime number.

(ii) 2 ( 5) = 10 or 3 ( 6) = 18.

(b) Write down two implications based on the following statement.

(c) Complete the premise in the following argument:

Premise 1 :

Premise 2 : 2m + 5 13.

Conclusion : m 4.

Answer:

(a) (i)

(ii)

(b) Implication 1 : .

Implication 2 : .

(c) Premise 1: .

[6 marks]

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x2 < 25 if and only if 5

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8 The width of a rectangle is (2y 1) cm and its length is (4y + 3) cm.

(a) Express the area of the rectangle in terms ofy.

(b) Given the area of the rectangle is 18 cm2 , find the width of the rectangle.

[6 marks]

Answer:

(a)

(b)

9 480 pieces of A4 size paper make up one ream of paper. The mass of one ream of

paper in grams is 1.26 102. Find the mass of each piece of paper in milligrams.Express your answer in standard form.

[3 marks]

Answer:

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10 Diagram 4 shows an incomplete Venn diagram that represents thenumber of students from a class of 30, according to the subjects

they have studied. Each student studied at least one of the three

subjects Physics, Chemistry and Biology.

Given the universal set, =PCB, P= {students who studied Physics}

C= {students who studied Chemistry}

B = {students who studied Biology}

DIAGRAM 4

20 students studied Chemistry and 12 students studied Biology. Based on thisinformation, find

(a) the number of students who studied Chemistry and Biology but not

Physics.

(b) the number of students who studied Biologyonly.

(c) the number of students who studied Physics.

[4 marks]

Answer:

(a)

(b)

(c)

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C P

B

1

6

25

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x

y

R(4, 6)

S

P( 6, 1)

Q(0, 2)

(a) the equation of the straight line QR,

(b) they-intercept ofthe straight linePS.

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11 In

Diagram 5,PQRSis a parallelogram. Point Q lies on they-axis.

DIAGRAM 5

Find

[5 marks]

Answer:

(a)

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(b)

Section B

[48 marks]

Answerfourquestions in this section.

12 (a) Factorise completely 2p3 50p.

[2 marks]

(b) ABCis a triangular plot of land in which angleACB is a right angle.

Given thatAB = (3x 5) m,AC= (x + 5) m andBC= 2x m.

(i) Use Pythagoras Theorem to form a quadratic equation in general

form.

(ii) Solve this equation and hence find the perimeter of the triangular plot

ABC.

[5 marks]

(c) In Diagram 6, O is the origin, the straight linesPR and STintersect at

pointM.

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2y =x

=

y = 2x + 9

=

x

=O

P

R

S

T

y

=

M

DIAGRAM 6

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(i) Find the coordinates ofM.

(ii) Given that the straight line PR passes through the point (h, h +12),

find the value of h.

[5

marks]

Answer:

12 (a)

(b) (i)

(ii)

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(c) (i)

(ii)

13 (a) Given that = {x:x is an integer and 7 x 15},A = {x: x is a multiple of 3 },

B = {x :x is an odd number} and

C= {x: 10 x 13 }.

(i) List the elements of the set C .(ii) Find n(AB).(iii) List all the possible subsets of setA.(iv) Find the elementx such thatx (BC ) andxA. [7 marks]

(b) Given that n() = 50. G andHare two sets for which n( GH) = 6,n(G) = 18 and n(H) = 14.

(i) Draw a Venn diagram to represent the relationship between sets G

andHand write the number of elements in each region.

(ii) Find n(GH) . [3 marks]

(c) On the Venn diagram in the answer space, draw a setN such that N

L andN M= .[2 marks]

Answer:

(a) (i)

(ii)

(iii)

(iv)

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A BO

210o

B

A

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(b) (i)

(ii)

(c)

14 (a) In Diagram 7, the shaded region represents the part of the flat

windscreen of a van which is being wiped by the windscreen wiperAB. The

wiper rotates through an angle of 210o about the centre O. Given that OA = 7

cm andAB = 28 cm.

DIAGRAM 7

Using =7

22, calculate

(i) the length of arcBB ,

(ii) the ratio of arc lengths , AA : BB

(iii) the area of the shaded region. [7 marks]

Answer:

(a) (i)

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L M

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(ii)

(iii)

14 (b) In Diagram 8, the pointsA,B, CandD lie on a circleABCD with centre O.

ABD = 50o and DBC = 60o.

DIAGRAM 8

Calculate

(i)

COD,

(ii)

ADC,

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6050

D

B

A

C

O

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(iii) ADO.

[5 marks]

Answer:

(b) (i)

(ii)

(iii)

15 (a) Diagram 9 shows a solid right cylinder of radius 7 cm and length 10 cm

with hemispheres of radius 7 cm attached at each end.

DIAGRAM 9

Calculate the total surface area, in cm2, of the combined solids.

(Use =7

22)

[4 marks]

(b) (i) Complete each of the following statements with the quantifier

all or some so that it will become a true statement.

(a) .. sets with 2 elements will have 4 subsets.

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(b) .. prime numbers are divisible by 3.

(ii)

(a) State the antecedent and consequent of the aboveimplication.

(b) Write the converse of the above implication and hence

determine whether its converse is true or false.

(c) Based on the information below, make a general conclusion by

induction.

12 = 2 (2) (2 + 1)

24 = 2 (3) (3 + 1)

40 = 2 (4) (4 + 1)

60 = 2(5) (5 + 1)

..

[8 marks]

Answer:

15 (a)

(b) (i) (a) ................................................................................................

(b) .....................................................................................................

(ii) (a) Antecedent: ................................................................................

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Ifx < 2, thenx < 7.

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Consequent: ...............................................................................

(b) ....................................................................................................

...................................................................................................

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

16 The data below shows the Chemistry test marks scored by a group of

40 students in Form 4 Wawasan.

TABLE 1

(a) Using the given data and a class interval of 10, complete Table 2 in the

answer space.

[4 marks]

(b) From the table in (a),

(i) state the modal class,

(ii) calculate the mean mark.

[4 marks]

Answer:

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68 40 65 66 60 46 53 56 49 58

53 57 73 53 69 51 56 64 58 62

47 65 67 66 67 36 56 53 38 37

62 61 37 74 48 41 75 63 69 55

Marks scored Midpoint Frequency

20 29

30 39

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TABLE 2

(b) (i)

(ii)

(c) For this part of the question, use the graph paper.

By using a scale of 2 cm to 10 marks on the horizontal axis and 2 cm to 2

students on the vertical axis, draw a frequency polygon for the data.

[4 marks]

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Graph for Question 16(c)[Lihat sebelah

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