jawatankuasa kurikulum zon c kuching, sarawak...jawatankuasa kurikulum zon c kuching, sarawak ......

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SULIT

Nombor Kad Pengenalan

Angka Giliran

Nama: _______________________________________________

Jawatankuasa Kurikulum Zon C Kuching, Sarawak

PERCUBAAN SIJIL PERLAJARAN MALAYSIA 2008 ADDITIONAL MATHEMATICS Kertas 1 Sept 2008 2 jam

3472/1

Dua jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU Untuk kegunaan Pemeriksa

Kod Pemeriksa: Soalan Markah Pernuh Markah Diperoleh

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1. Tuliskan angka giliran dan nombor

kad pengenalan anda pada ruang yang disediakan.

2. Kertas soalan ini adalah dalam

bahasa Inggeris sahaja. 3. Calon dikehendaki membaca arahan

di halaman belakang kertas soalan ini

Jumlah 80

Kertas soalan ini mengandungi 18 halaman bercetak 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

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2

SULIT 3472/1

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

Algabra

1. 2 4

2b b acx

a

2. m n m na a a

3. m n m na a a

4. nm mna a

5. log log loga a amn m n

6. log log loga a am m nn

7. log logna am n m

8. logloglog

ca

c

bba

9. ( 1)nT a n d

10. 2 ( 1)2nnS a n d

11. 1nnT ar

12. )( 1) (1 , 1

1 1

n n

na r a rs r

r r

13. 11 ,

aS rr

Statistics

1. x

xN

2. fx

xf

3. 2

2( )x x xx

N N

4. 2

2( )f x x fxx

f f

5. 12

m

N Fm L cf

6. 1

0

100QIQ

7. i i

i

W II

W

8. !( )!

nr

npn r

9. !( )! !

nr

nCn r r

10. ( ) ( ) ( ) ( )p A B P A P B P A B

11. ( ) , 1n r n rrP X r C P q p q

12. mean, np

13 npq

14. XZ

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3

SULIT 3472/1 Calculus

1. , dy dv duy uv u v

dx dx dx

2. 2,

du dvv uu dy dx dxyv dx v

3. dy dy dudx du dx

4. Area under a curve = ba y dx or = b

a x dy

5. Volume of revolution = 2ba y dx or = 2b

a x dy

Geometry

1. Distance = 2 21 2 1 2x x y y

2. Midpoint (x, y) = 1 2 1 2,2 2

x x y y

3. A point dividing a segment of a line, (x, y) = 1 2 1 2,nx mx ny mym n m n

4. Area of triangle = 11 2 2 3 3 1 2 1 3 2 1 32 x y x y x y x y x y x y

5. |r| = 2 2x y

6. 2 2

xi y jr

x y

Trigonometry 1. Arc length, s r

2. Area of sector, 212

A r

3. sin2 A + cos2 A = 1

4. sec2 A = 1 + tan2 A

5. cosec2 A = 1 + cot2 A

6. sin 2A = 2 sin A cos A

7. cos 2A = cos2 A sin2 A

= 2 cos2 A 1

= 1 – 2 sin2 A

8. sin (A B) = sin A cos B cos A sin B

9. cos (A B) = cos A cos B sin A sin B

10. tan tantan( )1 tan tan

A BA BA B

m

11. 2

2 tantan 21 tan

AAA

12. sin sin sin

a b cA B C

13. a2 = b2 + c2 – 2bc cos A

14. Area of triangle = Cabsin21

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Probability of upper tail Q(z) for normal distribution N(0, 1)

1 2 3 4 5 6 7 8 9 z 0 1 2 3 4 5 6 7 8 9 SUBTRACT 0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4661 .4641 4 8 12 16 20 24 28 32 36 0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247 4 8 12 16 20 24 28 32 36 0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859 4 8 12 15 19 23 27 31 35 0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483 4 7 11 14 19 22 26 30 34 0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121 4 7 11 14 18 22 25 29 32

0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776 3 7 10 14 17 20 24 27 31 0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451 3 7 10 13 16 20 23 26 29 0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148 3 6 9 12 15 18 21 24 27 0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867 3 5 8 11 14 16 19 22 25 0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611 3 5 8 10 13 15 18 20 23

1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 2 5 7 9 12 14 16 19 21 1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170 2 4 6 8 10 12 14 16 18 1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985 2 4 6 7 9 11 13 15 17 1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823 2 3 5 6 8 10 11 13 14 1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681 1 3 4 6 7 8 10 11 13

1.5 .0666 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559 1 2 4 5 6 7 8 10 11 1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455 1 2 3 4 5 6 7 8 9 1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367 1 2 3 4 4 5 6 7 8 1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294 1 1 2 3 4 4 5 6 6 1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233 1 1 2 2 3 4 4 5 5

2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183 0 1 1 2 2 3 3 4 4 2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143 0 1 1 2 2 2 3 3 4 2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110 0 1 1 1 2 2 2 3 3 2.3 .0107 .0104 .0102 0 1 1 1 1 2 2 2 2 .02990 .02964 .02939 .02914 3 5 8 10 13 15 18 20 23 .02889 .02866 .02842 2 5 7 9 12 14 16 18 21 2.4 .02820 .02798 .02776 .02755 .02734 2 4 6 8 11 13 15 17 19 .02714 .02695 .02676 .02657 .02639 2 4 6 7 9 11 13 15 17

2.5 .02621 .02604 .02587 .02570 .02554 .02539 .02523 .02508 .02494 .02480 2 3 5 6 8 9 11 12 14 2.6 .02466 .02453 .02440 .02427 .02415 .02402 .02391 .02379 .02168 .02357 1 2 3 5 6 7 8 9 10 2.7 .02347 .02336 .02326 .02317 .02307 .02298 .02189 .02280 .02272 .02164 1 2 3 4 5 6 7 8 9 2.8 .02256 .02248 .02240 .02233 .02226 .02219 .02212 .02205 .02199 .02193 1 1 2 3 4 4 5 6 6 2.9 .03968 .02181 .02175 .02169 .02164 .02159 .02154 .02149 .02144 .02139 0 1 1 2 2 3 3 4 4

3.0 .02135 .02130 .02126 .02122 .02118 .02114 .02111 .02107 .02104 .02100 0 1 1 2 2 2 3 3 4 3.1 .03968 .03935 .03904 3 6 9 13 16 19 22 25 28 .03874 .03845 .03816 .03739 3 6 8 11 14 17 20 22 25 .03762 .03736 .03711 2 5 7 10 12 15 17 20 22 3.2 .03687 .03664 .03641 .03619 .03598 2 4 7 9 11 13 15 18 20 .03577 .03557 .03538 .03519 .03501 2 4 6 8 9 11 13 15 17 3.3 .03483 .03466 .03450 .03434 .03419 2 3 5 6 8 10 11 13 14 .03404 .03390 .03376 .03362 .03349 1 3 4 5 7 8 9 10 12 3.4 .03337 .03325 .03313 .03302 .03291 .03180 .03270 .03260 .03251 .03242 1 2 3 4 5 6 7 8 9

3.5 .03233 .03224 .03216 .03208 .03200 .03193 .03185 .03178 .03172 .03165 1 1 2 3 4 4 5 6 7 3.6 .03159 .03153 .03147 .03142 .03136 .03131 .03126 .03121 .03117 .03112 0 1 1 2 2 3 3 4 5 3.7 .03108 .03104 .03100 .0496 .0492 .0488 .0485 .0482 .0478 .0475 3.8 .0472 .0469 .0467 .0464 .0462 .0459 .0457 .0454 .0452 .0450 3.9 .0448 .0446 .0444 .0442 .0441 .0439 .0437 .0436 .0434 .0433 For negative z use the relation: Q(z) = 1 – Q(–z) = P(–z) Example: if u N(0, 1), find (a) Prob (u > 2), (b) Prob (0 < u < 2), (c) Prob (|u| > 2), (d) Prob (|u| < 2). The desired probabilities are (a) Q(2) = 0.0228, (b) Q(0) – Q(2) = 0.5000 – 0.0228 = 0.4772, (c) 2Q(2) = 0.0456, (d) 1 – 2Q(2) = 0.9544. If v N(, 2), Prob (v > x) is given by Q(z) with z = (x – )/

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5

SULIT 3472/1 For

Examiner’s Use Answer all questions

(80 marks)

1. The above arrow diagram shows the relation between set A and set B. (a) Represent the above relation using ordered pairs. (b) State the type of the above relation.

[2 marks]

Answer (a)………………

(b)………………

2. Diagram below shows the function :

m xh xx

, x ≠ 0, where m is a constant.

Find the value of m. [3 marks]

Answer: m =………………..

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2

3

1

2

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SULIT 3472/1 For Examiner’s

Use

3. Given function xaxf 3: and 1 2:2 3

bf x x , where a and b are

constants. Find the value of a and of b. [3 marks]

Answer: a = …………, b = ………..

4. Given that 3 and k are the roots of quadratic equation x2 + x = p, find the values

of k and of p. [4 marks]

Answer: k = …………, p = ………..

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4

4

3

3

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7

For Examiner’s

Use

SULIT 3472/1

5. Find the range of values of x such that 4x < 3 + x2. [3 marks]

Answer: ………..…………

6. Diagram above shows the graph of a quadratic function y = f(x). The straight line y = − 4 is a tangent to the curve y = f(x).

(a) Write the equation of the axis of symmetry of the curve. (b) Express f(x) in the form of (x + b) 2 + c, where b and c are constants.

[3 marks]

Answer: (a)…………………

(b)…………………

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3

5

3

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SULIT 3472/1 For Examiner’s

Use

7. Solve the equation 6log (2 1)6 11. t [4 marks]

Answer: t = …..…………

8. Show that for all positive integer n, 4(3n+2) – 7(3n) + 3n – 1 is divisible by 8. [3 marks]

3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

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8

3

7

4

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9

For Examiner’s

Use

SULIT 3472/1

9. Find the sum of all terms of the following geometric progressions 1, 2, 4, …, 512 [2 marks]

Answer: ……………………

10. Given that 2.424242...hk is a recurring decimal, find the value of h and of k.

[3 marks]

Answer: h = …………, k = …………

3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

SULIT

10

3

9

2

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SULIT 3472/1 For Examiner’s

Use 11. The nth term of an arithmetic progression is given by 2 3nT n . Find (a) the common difference, (b) the sum of all the terms from the 3rd term to the 10th term.

[4 marks]

Answer: (a)………………

(b)………………

12. The diagram shows a part of the straight line graph of 1y

against 1x

. Express y

in terms of x. [4 marks]

Answer: ………………………

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SULIT

12

4

11

4

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11

For Examiner’s

Use

SULIT 3472/1

13. Given that the coordinates of points P and Q are (2, −1) and (3, 5) respectively, find the equation of the locus of point R which moves such that PR : RQ = 3 : 2. [3 marks]

Answer: ………………………

14. Diagram below shows two vectors, OAuuur

and BOuuur

.

Express,

(a) OAuuur

in the form

yx

(b) BOuuur

in the form xi yj% %

. [2 marks]

Answer: (a)………………

(b)………………

3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

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2

13

3

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SULIT 3472/1 For Examiner’s

Use 15.

Use the above information to find the values of h and k when 5 4r p q

% % %.

[3 marks]

Answer: h = …………, k = …………

16. In diagram below, PQRS is a parallelogram and STQ is a straight line.

If 5PQ x

uuur

%, 4PS y

uuur

%, and ST = 3TQ, express in terms of x

% and y

%,

(a) SQuuur

(b) TR

uur [4 marks]

Answer: (a)………………

(b)………………

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SULIT

16

4

15

3

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13

For Examiner’s

Use

SULIT 3472/1

17. Solve the equation 3sin 2 5cos 0x x for 0 360x . [4 marks]

Answer: …………………..

18. Diagram below shows a sector OAB with centre O and radius 6 cm. Given that the perimeter of the sector is 24 cm, find the angle of the sector in degree and minutes. [3 marks]

Answer: …………………..

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3

17

4

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SULIT 3472/1 For Examiner’s

Use 19. Given that 4( ) 5( 3)f x x , find the value of '(2)f . [2 marks]

Answer: …………………..

20. Given that 1 2x p and 21 3y p , find the rate of change of y when x is

increased at the rate of 4 units s-1 at the instant p = 6. [4 marks]

Answer: …………………..

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SULIT

20

4

19

2

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15

For Examiner’s

Use

SULIT 3472/1

21. Given that 3

,4)34(k

dxx where k > 0, find the value of k. [3 marks]

Answer: k = ……………

22. The table shows the distribution of the score a competition.

Score 2 3 4 5 6 7 Frequency 2 x 12 8 7 3

Find the range of values of x, given that the median is 4. [3 marks]

Answer: …………………

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SULIT

22

3

21

3

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SULIT 3472/1 For Examiner’s

Use 23. Seven prefects are to be selected from a group of 8 boys and 15 girls.

(a) Find the number of selection that can be carried out if 3 boys and 4 girls are to be selected.

(b) Find the number of ways these seven prefects can be arranged in a row for a group photograph if the three boys sit next to each other in the middle of the row. [4 marks]

Answer: (a) …………………

(b) …………………

24. A box contains 4 blue balls, 3 red balls and 5 white balls. Two balls are picked at random from the box. Find the probability that both balls are of the same colour. [3 marks]

Answer: …………………

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24

3

23

4

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17

For Examiner’s

Use

SULIT 3472/1

25. The marks, X, of a group of students in an examination are normally distributed with a mean of 65 and a standard deviation of 15.

(a) Find the z-score, if the mark of a student is 62 marks. (b) Find the probability of a student chosen at random obtained higher than

70 marks. [4 marks]

Answer: (a) …………………

(b) …………………

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4

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SULIT

INFORMATION FOR CANDIDATES

1. This question paper consists of 25 questions.

2. Answer all questions

3. Give only one answer for each question.

4. Write your answer 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 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 are shown in brackets.

9. A list of formulae is provided on page 2 and 3.

10. The four-figure table for probability of upper tail Q(z) for normal distribution N(0, 1) is provided on page 4.

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

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

3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

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- 1 -

Marking Scheme of Additional Mathematic Zone C Common Paper

Paper 1

(a) {(1, a), (5, a), (5, u), (9, i)}

1.

(b) Many to many 2

8 18 2

m

18 (8)2

m

2.

4m 3

33)(1 axxf

Either 32

b or a = −2 is correct.

3.

Both 32

b and a = −2 are correct

3

3 + k = 1

k = 4

3k = p

4.

p = 12 4

2 4 3 0 x x ( 1)( 3) 0 x x

5.

1, 3 x x 3 (a) 4x

(b) Either minimum value = 4 or (x – 4)2 is correct

6.

f(x) = (x − 4) 2 − 4 3

6log (2 1)6 6log 6 log 11t

6 6log 2 1 log 11t

2 1 11t

7.

5t 4

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- 2 -

2 34(3 )(3 ) 7(3 )3

nn n

1(36 7 )(3 )3

n

8.

188(3 )n . Since 88 is a multiple of 8 and (3n – 1) is an integer for all positive integer of n(n = 1, 2, 3, ….) , then 4(3n+2) – 7(3n) + 3n – 1 is divisible by 8.

3 1(2 n 1) = 512 and n = 10

9.

10

101(2 1) 1023

2 1s

2

2 0.42 0.0042 0.000042 ...hk or 0.42, 0.01a r

0.4221 0.01

hk

or 2.4

1 0.01

hk

10.

80 , 80, 3333

h h kk

3

1 22(1) 3 5 2(2) 3 7T or T

(a)

2 1 7 5 2d T T

10 2S S = 10 2(2(5) (10 1)(2)) (2(5) (2 1)(2))2 2

11.

(b)

128 *Alternative: use 3 9T as first term and find 8S

4 12

m

1 1 22y x

1 1 42

xy x

12.

21 4

xyx

4 2PR =3RQ

2222 )5()3(3)]1([)2(2 yxyx

13.

5x2 – 38x + 5y2 – 98y + 286 = 0 3

(a) 62

14.

(b) 3 5i j% %

2

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- 3 -

35 12r a b % % %

35h

15.

23k 3

(a) SQ SP PQ uuur uur uuur

5 4SQ x y

uuur

% %

(b) 14

TR TQ QR SQ PS uur uuur uuur uuur uuur

16.

1 5(5 4 ) 44 4

TR x y y x y uur

% % % 4

3(2sin cos ) 5cos 0x x x

cos (6sin 5) 0x x

5cos 0 sin6

x or x

17.

90 ,270 , 236.44 ,303.56x 4 Length of arc AB = 12 cm or 6 6 6( ) 24

2 radian

18.

2 x 180 = 114 35'

3

3'( ) 20( 3)f x x 19.

3'(2) 20(2 3) 20f 2

12

xp ,

2(1 )1 32

xy

,

(1 )32

dy xdx

When p =6, 1 2(6) 11x , (1 ( 11))3 182

dydx

20.

-118(4) 72 units sdydt

4

3 32

432

44)34(k k

xxdxx

[2(3)2 – 3(3)] – [2k2 – 3k] = 4

21.

(2k – 5)(x + 1) = 0, k = 25 ( k > 0)

3

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- 4 -

Either (2 + x) < (12 + 8 + 7 + 3) or (2 + x + 12) > (8 + 7 + 3)

Both (2 + x) < (12 + 8 + 7 + 3) and (2 + x + 12) > (8 + 7 + 3)

22.

4 < x < 28 3 (a) 8 15

3 4C C = 76440

(b) 3 4

3 4P P

23.

=144 4 4 3 3 2 5 4 or or

12 11 12 11 12 11 (any one correct)

114

125

112

123

113

124

(all correct with the summation)

24.

6619

3

(a) 62 6515

z

z = 0.2

(b) 70 65( )15

P z

25.

= 0.3696 4

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