add mth f4 final sbp 2008

43
3472/1 2008 Hak Cipta SBP [Lihat sebelah SULIT SEKOLAH BERASRAMA PENUH BAHAGIAN PENGURUSAN SEKOLAH BERASRAMA PENUH / KLUSTER KEMENTERIAN PELAJARAN MALAYSIA PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2008 Kertas soalan ini mengandungi 15 halaman bercetak For examiner’s use only Question Total Marks Marks Obtained 1 2 2 4 3 4 4 4 5 2 6 2 7 3 8 3 9 3 10 4 11 3 12 3 13 4 14 3 15 4 16 2 17 3 18 3 19 4 20 3 21 2 22 3 23 4 24 4 25 4 TOTAL 80 ADDITIONAL MATHEMATICS Kertas 1 Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 1. This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer for each question. 4. Write your answers clearly in the spaces provided in the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question and sub- part of a question are shown in brackets. 9. A list of formulae is provided on pages 2 and 3. 10. A booklet of four-figure mathematical tables is provided. 11. You may use a non-programmable scientific calculator. 12. This question paper must be handed in at the end of the examination. Name : ………………..…………… Form : ………………………..…… 3472/1 Additional Mathematics Kertas 1 Oktober 2008 2 Jam MOZ@C SMS MUZAFFAR SYAH , MELAKA

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Page 1: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [Lihat sebelah SULIT

SEKOLAH BERASRAMA PENUH

BAHAGIAN PENGURUSAN SEKOLAH BERASRAMA PENUH / KLUSTER

KEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2008

Kertas soalan ini mengandungi 15 halaman bercetak

For examiner’s use only

Question Total Marks Marks Obtained

1 2 2 4 3 4 4 4 5 2 6 2 7 3 8 3 9 3

10 4 11 3 12 3 13 4 14 3 15 4 16 2 17 3 18 3 19 4 20 3 21 2 22 3 23 4 24 4 25 4 TOTAL 80

ADDITIONAL MATHEMATICS

Kertas 1 Dua jam

JANGAN BUKA KERTAS SOALAN INI

SEHINGGA DIBERITAHU

1. This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer for each question. 4. Write your answers clearly in the spaces provided in the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question and sub- part of a question are shown in brackets. 9. A list of formulae is provided on pages 2 and 3. 10. A booklet of four-figure mathematical tables is provided. 11. You may use a non-programmable scientific calculator. 12. This question paper must be handed in at the end of the examination.

Name : ………………..…………… Form : ………………………..……

3472/1 Additional Mathematics Kertas 1 Oktober 2008 2 Jam

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 2: Add mth f4 final sbp 2008

SULIT 3472/1

3472/1 2007 Hak Cipta SBP [ Lihat sebelah SULIT

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

ALGEBRA

1 2 4

2b b acx

a− ± −

=

2 am × an = a m + n 3 am ÷ an = a m - n

4 (am) n = a nm

5 loga mn = log am + loga n

6 loga nm

= log am - loga n

7 log a mn = n log a m

8 logab = ab

c

c

loglog

CALCULUS

1 y = uv , dxduv

dxdvu

dxdy

+=

2 vuy = , 2v

dxdvu

dxduv

dydx −

= ,

3 dxdu

dudy

dxdy

×=

3 A point dividing a segment of a line

( x,y) = ,21

+

+

nmmxnx

+

+

nmmyny 21

4 Area of triangle

= )()(21

312312133221 1yxyxyxyxyxyx ++−++

1 Distance = 221

221 )()( yyxx −+−

2 Midpoint

(x , y) = +

221 xx ,

+

221 yy

GEOMETRY

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 3: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [ Lihat sebelah SULIT

3

STATISTIC

7 1

11

wIwI

=

1 Arc length, s = rθ

2 Area of sector , L = 212

r θ

TRIGONOMETRY

3 Cc

Bb

Aa

sinsinsin==

4 a2 = b2 + c2 - 2bc cosA

5 Area of triangle = Cabsin21

1 x = N

x

2 x =

ffx

3 σ = N

xx − 2)( =

2_2

xN

x−

4 σ =

fxxf 2)(

= 22

xf

fx−

5 m = Cf

FNL

m

−+ 2

1

6 1

0

100QIQ

= ×

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 4: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [ Lihat sebelah SULIT

Answer all questions.

1. Diagram 1 shows the linear function f . (a) State the value of k. (b) Using function notation, write a relation between set A and set B.

[ 2 marks]

Answer : (a) ……………………..

(b) ...…………………... 2. The following information above refers to the functions f and g . (a) State the value of h. (b) Find the value of )3(1−fg .

[ 4 marks ]

Answer : (a) …………………….

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

4

2

2

1

For examiner’s

use only

hxx

xxf ≠+

→ ,3

4:

xxg 21: −→

1

3

5

7

3

5

k

9

x f(x)

Set A Set B

Diagram 1

f

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 5: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP SULIT

3. Given the function 12: −→ xxf and 16: +→ xxfg . Find (a) the function g(x) (b) the value of x when 4)( =xgf .

[ 4 marks ]

Answer : (a).........…………………

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

4. Given that the roots of the quadratic equation kxx −=− 512 are 3 and p . Find the value of k and of p .

[4 marks]

Answer : k =.........………

p =....................

5. The quadratic equation 0122 2 =+−− pxx has two distinct roots. Find the range of values of p . [2 marks]

Answer : .............………

For examiner’s

use only

4

3

4

4

2

5

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 6: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [ Lihat sebelah SULIT

x

5

Diagram 2

)4,1( −+k

y

6. Diagram 2 shows the graph of the function pxy −+= 2)3( where p is a constant and )4,1( −+k is a minimum point.

Find

a) the value of k. b) the value of p.

2 marks ]

Answer : (a) ................................

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

___________________________________________________________________________

7Find the range of values of x for which )52(3 −≤ xx . [3 marks]

Answer : ..................................

8. Solve the equation 455 12 =− ++ xx . [3 marks]

Answer : …….……………….

2

6

For examiner’s

use only

3

7

3

8

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 7: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP SULIT

9. Given xp =3log and yp =2log , express 12log8 in terms of x and y.

[3 marks]

Answer : ..................................

10. Solve the equation 0)1(log)23(log 33 =−−+ xx

[4 marks]

Answer : ...................................

11. The points )2,(,),( rPttA and )2,9( −−B are on a straight line. P divides AB internally in the ratio of 3 : 4 . Find the value of t and of r . [3 marks]

Answer : t = .............................. r = ..............................

3

9

For examiner’s

use only

4

10

3

11

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 8: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [ Lihat sebelah SULIT

12. Diagram 3 shows a straight line PQ with equation 01234 =−+ yx .

Find (a) the value of h and of k (b) the equation of PQ in intercept form. [3 marks]

Answer : (a) ....……………...………..

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

13. Find the equation of the straight line that passes through a point )1,3(−P and is

perpendicular to the straight line 145=+

yx .

[4 marks]

Answer: …...…………..….......

3

12

For examiner’s

use only

4

13

P(k,0)

Q(0,h)

x

y

O

Diagram 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 9: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP SULIT

14. The coordinates of point P and Q are )1,3(− and )10,6( respectively. The point X moves such that XP : XQ = 2 : 3 . Find the equation of the locus of X . [3 marks]

Answer: …...….………..….....................

___________________________________________________________________________ 15. Table 1 shows the distribution of the weight of 40 pupils in form 4 Alpha. Table 1 (a) Find the range of the weight. (b) Without drawing an ogive, calculate the median of the distribution of weight.

[4 marks]

Answer : (a) …………………….

(b) ….……………….....

Weight (kg) Number of pupils 31 – 35 7 36 – 40 4 41 – 45 8 46 – 50 7 51 – 55 6 56 – 60 4 61 – 65 4

4

15

For examiner’s

use only

3

14

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 10: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP [ Lihat sebelah SULIT

16. The mean of ten numbers is m . The sum of squares of the number is k and the standard deviation is 4. Express k in terms of m .

[2 marks] Answer : .…………………

17. Diagram 4 shows a circle with centre O. The length of the minor arc AB is 3.9275 cm and the angle of the major sector AOB is 315o. Using 142.3=π , find (a) the value of θ , in radians, (Give your answer correct to four significant figures.)

(b) the length, in cm, of the radius of the circle.

[3 marks]

Answer: ……..…….…………... ___________________________________________________________________________

2

16

3

17

For examiner’s

use only

O

A

B

Diagram 4

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 11: Add mth f4 final sbp 2008

3472/1 2008 Hak Cipta SBP SULIT

Diagram 5 shows a sector OTV of a circle, centre O. Find the perimeter of the shaded region.

[3 marks]

Answer: ...………………………

___________________________________________________________________________ 19. Diagram 6 shows a sector of a circle OPQ with centre O and OPR is a right angle triangle. Find the area, in cm2, of the shaded region.

[4 marks]

Answer:………………………

For examiner’s

use only

3

18

O R Q 1 cm

5 cm

P

4

19

O

T

V 50°

7 cm

Diagram 5

Diagram 6

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 12: Add mth f4 final sbp 2008

SULIT 3472/1

3472/1 2008 Hak Cipta SBP [ Lihat Sebelah SULIT

20. Evaluate the following limits,

(a) 1

1lim1 +→ xx

(b) 39lim

2

3 −

−→ x

xx

[ 3 marks ] Answer: (a) ....…………..….......

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

___________________________________________________________________________ 21. The straight line 12 +−= xy is the tangent to the curve xxy 42 −= at the point P. Find the x-coordinate of the point P .

. [2 marks]

Answer: ……………………..

22. Given that xqpxy −= 2 and ,4

2xx

dxdy

+= where p and q are constants , find the

value of p and of q .

[3 marks]

Answer: ……………………..

2

21

3

22

For examiner’s

use only

3

20

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 13: Add mth f4 final sbp 2008

SULIT 3472/1

3472/1 2008 Hak Cipta SBP SULIT

23. Given the curve 221x

xy += .

(a) Find the coordinates of the turning point.

(b) Hence, determine whether it is a maximum or a minimum point.

[4 marks]

Answer: (a)…...…………..….......

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

___________________________________________________________________________ 24. A cylinder has a fixed height of 10 cm and a radius of 5 cm. If the radius decreases by 0.05 cm, find (in terms of π ) (a) the approximate change in the volume of the cylinder, (b) the final volume of the cylinder. [4 marks]

Answer: (a)……………………………

(b)……………………………

For examiner’s

use only

4

24

4

23

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 14: Add mth f4 final sbp 2008

SULIT 3472/1

3472/1 2008 Hak Cipta SBP [ Lihat Sebelah SULIT

25. Given x

xy2)2( −

= , find the value of 2

2

dxyd when 1−=x .

[4 marks]

Answer: …...…………..….......

END OF QUESTION PAPER

4

25

For examiner’s

use only

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 15: Add mth f4 final sbp 2008

2

PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2008 MARK SCHEME KERTAS 1

No. Solution and mark scheme Sub marks Full marks 1 (a) k = 7

(b) 2: +→ xxf or 2)( += xxf

1

1

2

2 (a) h = - 3 (b) - 2

B2 : 31)1(4

+−

B1 : 2

1)(1 xxg −=−

1

3

4

3 (a) 13)( += xxg B1 : 161)(2 +=− xxg (b) 1=x B1 : 41)12(3 =+−x

2

2

4

4 p = 2, k = 7 (both) B3 : p + 3 = 5 and 3p = k – 1 B2 : p + 3 = 5 or 3p = k – 1 B1 : 0152 =−+− kxx

4 4

5 21

>p

B1 : 0)1)(2(4)2( 2 >+−−− p

2 2

6 k = -4 p = 4

1

1

2

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 16: Add mth f4 final sbp 2008

3

7 21,3 −≤≥ xx (both)

B2 : 0)3)(12( ≥−+ xx B1 : 0352 2 ≥−− xx

3 3

8 x = -1 B2 : 155 −=x B1 : 45555 12 =×−× xx

3 3

9 y

yx3

2+

B2 : 2log3

2log2log3log

p

ppp ++

B1 : 32log)223(log

p

p ××

3 3

10 23

−=x

B3 : 123 −=+ xx

B2 : 03123=

+

xx

B1 : 0123log3 =

+

xx

4 4

11 r = -1 B2 : t = 5

B1 : 7

462 t+−= or

7274 −

=tr

3 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 17: Add mth f4 final sbp 2008

4

12 a) h = 4

k = 3

b) 143=+

yx

1

1

1

3

13 4

1945

+= xy or equivalent

B3 : )3(45)1( +=− xy

B2 : 45

2 =m

B1 : 54

1 −=m

4 4

14 04546210255 22 =−+++ yxyx B2 : 22 )1()3(3 −++ yx = 22 )10()6(2 −+− yx B1 : 3XP = 2XQ

3 3

15 (a) 30 (b) 46.21

B2 : m = 45.5 + 57

192

40

− (all values correct)

B1 : 45.5 ,19 , 7 , 5 (at least two are correct)

1

3

4

16 210160 mk +=

B1 : 2

1016 mk

−=

2 2

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 18: Add mth f4 final sbp 2008

5

17 a) 0.7855

b) r = 5 B1 : 3.9275 = r (0.7855)

1

2

3

18 Perimeter = 12.0263 B2 : length PQ = 2r sin 25 0 = 5.9167 B1 : arc PQ = r = 7 (0.8728) = 6.1096

3

3

19 Area = 2.045

B3 : )3)(4(21)6436.0)(5(

21 2 −

B2 : = 0.6436 rad

B1 : tan = 43

4

4

20 (a)

21

(b) 6

B1 : 3

)3)(3(lim3 −

−+→ x

xxx

1

2

3

21 1=x

B1 : 42 −= xdxdy

2 2

22 21

=p and 4=q

B2 : 21

=p or 4=q

B1 : 22xqpx

dxdy

+=

3 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 19: Add mth f4 final sbp 2008

6

23 (a)

23,1

B1 : 01 3 =− −x

(b) ,032

2

>=dx

yd minimum point

B1 : 42

2

3 −= xdx

yd

2

2

4

24 (a) π5−

B1 : )5(20π=drdV or 100π

(b) π245 B1 : ππ 5)5(10 2 −=newV

2

2

4

25 8−

B3 : 32

2

8 −= xdx

yd or 4

22 )2)(4()2(x

xxxx −−

B2 : 241 −−= xdxdy or 2

2)2()2(2x

xxx −−−

B1 : 144 −+−= xxy

4 4

END OF MARK SCHEME

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 20: Add mth f4 final sbp 2008

3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

3472/2 Additional Mathematics Kertas 2 2 ½ jam OKT 2008

SEKTOR SEKOLAH BERASRAMA PENUH BAHAGIAN PENGURUSAN

SEKOLAH BERASRAMA PENUH / KLUSTER KEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN AKHIR TAHUN

TINGKATAN 4 2008

ADDITIONAL MATHEMATICS

Kertas 2

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. This question paper consists of three sections : Section A, Section B and Section C. 2. Answer all question in Section A , four questions from Section B and two questions from

Section C.

3. Give only one answer / solution to each question..

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

5. The diagram 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 2 to 3.

8. A booklet of four-figure mathematical tables is provided.

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

Kertas soalan ini mengandungi 13 halaman bercetak

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 21: Add mth f4 final sbp 2008

2

3472/2 2008 Hak Cipta SBP SULIT

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

ALGEBRA 1

aacbbx

242 −±−

= 5 nmmn aaa logloglog +=

2 nmnm aaa +=× 6 nm

nm

aaa logloglog −=

3 nmnm aaa −=÷ 7 mnm an

a loglog = 4 ( ) mnnm aa =

8

axx

b

ba log

loglog =

KALKULUS (CALCULUS)

STATISTIK (STATISTICS)

1 uvy = , dxduv

dxdvu

dxdy

+=

2 vuy = , 2v

dxdvu

dxduv

dxdy −

=

3 dxdu

dudy

dxdy

×=

1 Nx

x =

2 =

ffx

x

3 ( ) 222

xN

xN

xx−==

−= σ

4 ( ) 2

22

xf

fxf

xxf−=

−=

σ

6 Cf

FNLM

m

−+= 2

1

5 1000

1 ×=QQI

7 =−

i

ii

WIW

I

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 22: Add mth f4 final sbp 2008

3

3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

GEOMETRI (GEOMETRY)

TRIGONOMETRI (TRIGONOMETRY) 1. Panjang lengkok, js =

Arc length, rs =

2. Luas Sektor, 21 2jL =

Area of sector, 21 2rA =

3. 4.

5.

Cc

Bb

Aa

sinsin sin ==

Abccba kos2222 −+= Abccba cos2222 −+=

12

Luas segitiga ( ) sin

Area of triangleab C=

1. Jarak (Distance) ( ) ( )221

221 yyxx −+−

2. Titik tengah (Midpoint)

++

=2

,2

),( 2121 yyxxyx

3. Titik yang membahagi suatu tembereng garis

(A point dividing a segment of a line)

+

+

+

+=

nmmyny

nmmxnxyx 2121 ,),(

4. Luas segitiga (Area of triangle) =

( ) ( )31231213322121 yxyxyxyxyxyx ++−++

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 23: Add mth f4 final sbp 2008

4

3472/2 2008 Hak Cipta SBP SULIT

SECTION A

[40 marks]

Answer all questions in this section 1. Solve the simultaneous equations

425 −=+ yx 16232 =+− yxx

[5 marks]

2. Express f (x) = 1 – 6x + 2x2 in the form f (x) = m(x + n)2 + k , where m, n and k are

constants.

(a) State the values of m, n and k.

[3 marks]

(b) Find the maximum or minimum point.

[1 marks]

(c) Sketch the graph of f (x) = 1 – 6x + 2x2.

[2 marks]

3. (a) The straight line y = 1 – tx is a tangent to the curve y2 – 3y + 3x – x2 = 0.

Find the possible value of t. [3 marks]

(b) Given p and q are the roots of the quadratic equations 2x2 + 7x = m – 5, where pq = 3 and m is a constant. Calculate the values of m, p and q.

[4 marks]

4 (a) Given that 29 . 27 1x y = , find the value of x when 4y = −

[2 marks]

(b) Solve the equation 3 23 .6 18x x x−=

[2 marks]

(c) Solve the equation 2 4log ( 2) 2 2log (4 )x x− = + −

[3 marks]

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 24: Add mth f4 final sbp 2008

5

3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

5

(a) A set of N numbers have a mean of 8 and standard deviation of 2.121. Given that the sum of the numbers, x , is 64. Find

(i) the value of N

(ii) the sum of the squares of the numbers.

[4 marks]

(b) If each of the numbers is divided by h and is added by k uniformly , the new mean

and standard deviation of the set are 5 and 1.0605 respectively.

Find the value of h and k.

[4 marks]

6. The gradient of the tangent to the curve 2qxpxy −= , where p and q are constants, at

the point (1 , 4) is 2. Find

(a) the value of p and q. [4 marks]

(b) the equation of normal to the curve at the point ( 2, 4) [3 marks]

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 25: Add mth f4 final sbp 2008

6

3472/2 2008 Hak Cipta SBP SULIT

SECTION B

Answer four questions in this section

7 Diagram 1 shows function g maps x to y and function h maps z to y.

Given g (x) = mxx

≠−

,12

1 and h (z) = 1 + 4z.

(a) State the value of m. [1 marks]

(b) Find

(i) gh -1(x) (ii) h g -1(1).

[6 marks] (c) Find the value of

(i) a (ii) b

[3 marks]

Diagram 1

x y

z

g

h

a

b

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 26: Add mth f4 final sbp 2008

7

3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

8. Diagram 2 shows a quadrilateral PQRS with vertices ( 2,5)R − and ( 1,1)S − .

Given the equation of PQ is 1474 −= xy . Find ,

(a) the equation of QR [4 marks]

(b) the coordinates of Q [2 marks]

(c) the coordinates of P [1 marks]

(d) the area of quadrilateral PQRS [3 marks]

R(−2, 5)

S(−1, 1)

P

Q

O

Diagram 2

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 27: Add mth f4 final sbp 2008

8

3472/2 2008 Hak Cipta SBP SULIT

9

Table 1 shows the total time spent on watching television by 120 students for a period of 3 weeks. Calculate ,

(a) the mean, [2 marks]

(b) the standard deviation, [3 marks]

(c) the third quartile, [5 marks]

of the distribution

Total Time (hours) Number of students

5 – 14 12

15 – 24 17

25 – 34 26

35 – 44 31

45 – 54 16

55 – 64 10

65 – 74 8

Table 1

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 28: Add mth f4 final sbp 2008

9

3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

10. Diagram 3 shows a circle ABCF with radius 6 cm and centre O.

Given that oODB 30=∠ , EBD is the tangent to the circle and OD = OE = 12 cm. Find ,

(a) the length of BD [3 marks]

(b) the area of shaded region [3 marks]

(c) the perimeter of the whole diagram [4 marks]

O

A

D

E

B

C

F

Diagram 3

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SMS MUZAFFAR SYAH , MELAKA

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3472/2 2008 Hak Cipta SBP SULIT

11

Diagram 4 shows a rectangle PQRS and a parallelogram ABCD.

(a) If L 2cm is the area of the parallelogram,

(i) Show that 22 16L x x= − +

(ii) Find the value of x when L is maximum.

(iii) Find the maximum area of the parallelogram ABCD. [7 marks]

(b)

Given that the rate of change of x is -10.1 cms , find the rate of change of L , in -1cms , when x is 2 cm.

[3 marks]

B P Q

A

S D

C

R

x cm

x cm

x cm

x cm

10 cm

6 cm

Diagram 4

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SMS MUZAFFAR SYAH , MELAKA

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3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

SECTION C Answer two questions in this section

12. Diagram 5 shows two triangles ABC and ACD . BCD is a straight line.

Find ,

(a) ADC∠ [3 marks]

(b) the length of CD [3 marks]

(c) the area of triangle ABD [4 marks]

13

Diagram 6 shows two triangles ABE and BCF , where ABC is a straight line. Given that AE = 5 cm, BE = 7 cm, BC = 8 cm, CF = 9 cm, 050=∠BAE , 0104EBF∠ = and .1000=∠BCF Calculate

(a) AEB∠ , [3 marks]

(b) the length of BF. [3 marks]

(c) if point E joint with F, find the area of the quadrilateral ACFE. [4 marks]

A

D C B 55o

8 cm 8 cm 9 cm

Diagram 5

50o 1040 100o

A B C

E F

Diagram 6

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 31: Add mth f4 final sbp 2008

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3472/2 2008 Hak Cipta SBP SULIT

14 Table 2 shows the prices and the price indices of five types of food, H, I, J, K and L

represented the cost of food. Diagram 7 shows a percentage according to the food’s pyramid.

Types of food

Price (RM) for the year Price index for the year 2008

based on the year 2006 2006 2008

H 2.20 2.75 125 I m 2.20 110 J 5.00 7.50 150 K 3.00 2.70 n

L 2.00 2.80 140

(a) Find the value of m and of n. [3 marks]

(b) Calculate the composite index for the cost of food in the year 2008 based on the year 2006.

[3 marks] (c) The price of each food increases by 30% from the year 2008 to the year 2009.

Given that the cost of food in the year 2006 is RM80, calculate the corresponding cost in the year 2009.

[4 marks]

L I

10 %

K 20 %

J

25 % H 40 %

Table 2

Diagram 7

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SMS MUZAFFAR SYAH , MELAKA

Page 32: Add mth f4 final sbp 2008

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3472/2 2008 Hak Cipta SBP [ Lihat sebelah SULIT

15

Table 3 shows the price indices and percentage of usage of four items, J, K, L and M, which are the main ingredients in the production of a brand of cake.

(a) Calculate (i) the price of item M in the year 2003 if its price in the year 2005

was RM2.50.

(ii) the price index of item J for the year 2005 based on the year 2001 if its price index for the year 2003 based on the year 2001 is 108.

[5 marks]

(b) The composite index of the cost of cake production for the year 2005 based on the year 2003 is 113. Calculate ,

(i) the value of t

(ii) the price of a cake in the year 2003 if its corresponding price in the year 2005 was RM 25,

[ 5 marks ]

END OF THE QUESTIONS

Item Price Index for the year 2005 Based on the year 2003 Percentage of usage (%)

J 118 22

K t 12

L 108 31

M 113 35

Table 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 33: Add mth f4 final sbp 2008

2

SKEMA PERMARKAHAN MATEMATIK TAMBAHAN KERTAS 2

PEPERIKSAAN DIAGNOSTIK TINGKATAN 4, 2008

Number Solution and mark scheme Sub Marks Full Marks 1 4 5 4 2@

2 5x yy x− − − −

= =

2

2

4 53 2 162

4 2 4 2@ 3 2 165 5

xx x

y y y

− − − + =

− − − − − + =

( )( ) ( )( )10 2 0@ 27 3 0x x y y− + = + − = 10, 2 and 27,3x y= − = −

P1 K1 K1 N1, N1 5

5 2

(a)

23 7

22 2

x − −

m = 2 , n = – 32

and k = – 72

K1 N 0, 1, 2 3

6

(b)

72

3( , )2

N1 1

(c)

Shape Minimum point and y-intercept

P1 P1 2

y

( 32

,– 72

) •

o x

1

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SMS MUZAFFAR SYAH , MELAKA

Page 34: Add mth f4 final sbp 2008

3

Number Solution and mark scheme Sub Marks Full Marks 3

(a)

(1 – tx)2 – 3(1 – tx) + 3x – x2 = 0 (t2 – 1)x2 + (t + 3)x – 2 = 0 (t + 3)2 – 4(t2 – 1)(–2) = 0 (3t + 1) (3t + 1) = 0

.31

−=t

K1 K1 N1 3

7

(b)

SOR or POR pq or p + q POR: pq = 3

5 32

1..

m

m

−=

= −

7:2

3 22

322

.

SOR p q

q or

p or

+ = −

= − −

= − −

K1 N1 N1 N1 4

4

(a)

3034

3)3()3( 0322

=

=+

=⋅

xyx

yx

K1 N1 2

7

(b)

3 218 183 21

x x

x xx

−=

= −

=

K1 N1 2

(c)

4log)4(log22)2(log

2

22

xx −+=−

Change base

)4(log4log)2(log 222 xx −+=−

( 2) 4(4 )3.6

x xx

− = −

=

K1 K1 N1 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 35: Add mth f4 final sbp 2008

4

Number Solution and mark scheme Sub Marks Full Marks 5

(a)

i) 648

8N

N

=

=

ii)

=

−=

99.547

88

121.2

2

22

x

x

K1 N1 K1 N1 4

8

(b)

New mean = 58=+ k

h

or

New Stnd. Deviation = 0605.1121.2=

h

h = 2

8 52

1

k

k

+ =

=

K1 N1 K1 N1 4

6

(a)

)1(22

2

qp

qxpdxdy

−=

−= ……………

Substitute ( 1, 4 ) in 2qxpxy −= 4 = p – q ………… – ; q = 2 and p = 6

K1 K1 K1 N1 4

7

(b) x = 2, )2(46 −=

dxdy

21

2

2

1

=

−=∴

m

m

3

21

)2(214

+=

−=−

xy

xy

K1 K1 N1 3

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 36: Add mth f4 final sbp 2008

5

Number Solution and mark scheme Sub Marks Full Marks

7

(a)

.

21

=m P1 1

10

(b)

(i) 4

1)(1 −=− zzh

1

412

1)(1

=−

xxgh

.3,3

2≠

−= x

x (ii) 1 = g(x)

1

1

112 11

(1) (1) 1 4(1)(1) 5

xx

hg hhg

=−

=

= = +

=

K1 K1 N1 K1 K1 N1 6

(c) (i)

1( )41

12( ) 14

2

a g

a

a

=

=−

= −

(ii)

( )1 4 2

34

h b ab

b

=

+ = −

= −

K1 N1 N1 3

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SMS MUZAFFAR SYAH , MELAKA

Page 37: Add mth f4 final sbp 2008

6

Number Solution and mark scheme Sub Marks Full Marks 8

(a)

4RSM = − 14QRM =

( )15 24

y x− = + or other suitable method

4 22y x= + or equivalent

P1

P1

K1

N1 4

10

(b)

Solve simultaneous equation 22 7 14x x+ = − (6,7)Q

K1 N1 2

(c)

(2, 0)P

N1 1

(d) Area of quadrilateral PQRS 2 6 2 1 210 7 5 1 02

1 [2(7) 6(5) ( 2)(1) ( 1)(0)] [6(0) ( 2)7 ( 1)5 2(1)]229.5

− −=

= + + − + − − + − + − +

=

K1 K1 N1 3

9

(a)

12(9.5) 17(19.5) 26(29.5) 31(39.5) 16(49.5) 10(59.5) 8(69.5)120

x + + + + + +=

= 36.5

K1 N1 2

10

(b)

2222

)5.36(120

)5.69(8...)5.19(17)5.9(12−

+++=σ

= 266 = 16.31

K1 K1 N1 3

(c)

L = 44.5 ; F = 86

3

3 (120) 86444.5 10

16Q

− = +

K1 lower boundry 44.5

K1 using formula = 44.5 + 2.5 = 47

P1, P1 K1 K1 N1 5

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SMS MUZAFFAR SYAH , MELAKA

Page 38: Add mth f4 final sbp 2008

7

Number Solution and mark scheme Sub Marks Full Marks

10

(a)

060 60

12DBDOB or Sin∠ = = °

0tan 606

10.3923

DB

DB

=

=

P1 K1 N1 3

10

(b)

2 10.392320.7846

DE = ×

=

Area of shaded region

( ) ( ) ( )( )21 120.7846 6 2.0944 62 224.6546

= −

=

N1 K1 N1 3

(c)

0240 or 4.1888 radMajor AOC∠ =

( ) 6 4.188825.13228

AFCS =

=

Perimeter 25.1328 6 6 20.784657.9174

AFCS AD CE DE= + + +

= + + +

=

P1 N1 K1 N1 4

MOZ@C

SMS MUZAFFAR SYAH , MELAKA

Page 39: Add mth f4 final sbp 2008

8

Number Solution and mark scheme Sub Marks Full Marks

11

(a)

(i)

−−−

−= )6)(10(212

212)6(10 2 xxxL

= xx 162 2 +− (ii)

164 +−= xdxdL

,0=dxdL 0164 =+− x

x = 4

(iii) 2

max 2(4) 16(4)32

L = − +

=

K1 N1 K1 K1 N1 K1 N1 7

10

(b)

0.1dxdt=

( 4(2) 16) 0.1dLdt

= − + ×

= 0.8

P1 K1 N1 3

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SMS MUZAFFAR SYAH , MELAKA

Page 40: Add mth f4 final sbp 2008

9

Number Solution and mark scheme Sub Marks Full Marks

12

(a)

5589

SinBCASin=

0 0112 51' or 67 9 'BCA ACD∠ = ∠ =

067 9 ' 67.15ADC or∠ = °

K1 N1 N1 3

10

(b)

0

0

180 2(67 9')45 42 ' 45.7

CADor

∠ = −

= °

2 2 2 08 8 2(8)(8)cos45 42 '6.213

CDCD

= + −

=

K1 K1 N1 3

(c)

0 0 0

0

180 55 112 51'12 9 '

CAB∠ = − −

=

( )( ) ( )( )0 01 1Area 9 8 sin12 9 ' 8 8 sin 45 42 '2 230.48

= +

=

N1 K1, K1 N1 4

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SMS MUZAFFAR SYAH , MELAKA

Page 41: Add mth f4 final sbp 2008

10

Number Solution and mark scheme Sub Marks Full Marks

13

(a)

5 750SinB Sin

33.17 33 10' ABE or∠ = ° °

180 50 33.17AEB∠ = − − °

= 96.83º or 96º 50’

K1 K1 N1 3

10

(b)

42.83 42 50' CBF or∠ = ° °

9sin100 sin 42.83

BF=

° °

BF = 13.04 cm OR equivalent

P1 K1 N1 3

(c)

Area AEB = 1 5 7sin 96.832× × ° or equivalent

= 17.38

Area BCF = 1 8 9sin1002× × °

or equivalent

= 35.45

Area BEF = 1 7 13.04sin1042× × °

= 44.28 Area of quadrilateral ACFE = 100.08

K1 K1 K1 N1 4

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SMS MUZAFFAR SYAH , MELAKA

Page 42: Add mth f4 final sbp 2008

11

Number Solution and mark scheme Sub Marks Full Marks

14

(a)

2.20 100 110

2.002.70 1003.00

90

mm

n

n

× =

=

× =

=

K1 N1 N1 3

10

(b)

(125 40) (110 10) (150 25) (90 20) (140 5)

10012350100

123.5

IWW

× + × + × + × + ×=

=

=

K1 K1 N1 3

(c)

080906

08

130 123.5 100 100 123.5100 100 80160.55 98.80

QI OR

Q RM

= × × × =

= =

0909

09

160.55 80 100 130100 98.80

128.44. 128.44

QQ RM

RM Q RM

= × × =

= =

K1 K1 K1 N1 4

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SMS MUZAFFAR SYAH , MELAKA

Page 43: Add mth f4 final sbp 2008

12

END OF MARK SCHEME

Number Solution and marking scheme Sub Marks Full Marks

15

(a)

(i)

11310050.203

=×P

21.203 RMP = (ii)

10810001

03 =×PP 118100

03

05 =×PP

44.127100100108

100118

K1 N1 K1, K1 N1 5

10

(b)

(i) 113100

)35(113)31(108)12()22(118=

+++=

− tI

t = 116.75

(ii) 1131002503

=×P

12.2203 RMP =

K1, K1 N1 K1 N1 5

MOZ@C

SMS MUZAFFAR SYAH , MELAKA