Mt 2008 Zon C Kuching K1

5 SULIT Nombor Kad Pengenalan Angka Giliran

Nama:

_______________________________________________

Jawatankuasa Kurikulum Zon C Kuching, Sarawak 3472/1

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

Dua jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU 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

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

Jumlah

80

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

[lihat sebelah SULIT

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 b  b 2  4ac 2a

1. x 

8. log a b 

9. Tn  a  (n  1)d

2. a m  a n  a m  n 3. a m  a n  a m  n 4.

a 

m n

10. S n 

 a mn

n  2a  (n  1)d  2

n 1 11. Tn  ar

5. log a mn  log a m  log a n 6. log a

log c b log c a

m  log a m  log a n n

n 7. log a m  n log a m

12. sn 

a (r n  1) a (1  r n )  ,r 1 r 1 1 r

13. S 

a r 1 1  r,

Statistics 1. x  2. x 

x

7. I 

N

 fx f

3.  

 ( x  x)   x

4.  

  

6. I 

x

N

N

f ( x  x)

 f

 1 2

5. m  L  

2

f



N F  c fm 

Q1  100 Q0

fx 2

2

W I

i i

Wi

8.

n

pr 

n! ( n  r )!

9.

n

Cr 

n! (n  r )!r !

10. p( A  B )  P ( A)  P ( B )  P ( A  B ) x

2

11. P( X  r )  nCr P r q n  r , p  q  1 12. mean, µ  np 13   npq 14. Z 

X  

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

SULIT 3472/1

3

Calculus 1. y  uv,

dy dv du u v dx dx dx

3.

du dv u u dy 2. y  ,  dx 2 dx v dx v

dy dy du   dx du dx

4. Area under a curve =

v



b a

5. Volume of revolution =

y dx or =



b a



b a

x dy

 y 2 dx or =



b a

 x 2 dy

Geometry 1. Distance =

 x1  x2 

2

  y1  y2 

2

 x1  x2 y1  y2  ,  2 2  

2. Midpoint (x, y) = 

 nx1  mx2 ny1  my2  ,  mn   mn

3. A point dividing a segment of a line, (x, y) =  4. Area of triangle = 5. |r| = 

6. r 

1 2

 x1 y2  x2 y3  x3 y1    x2 y1  x3 y2  x1 y3 

x2  y 2 xi  y j x2  y 2

Trigonometry 1. Arc length, s  r

8. sin (A  B) = sin A cos B  cos A sin B 1 2

2. Area of sector, A  r 2

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

3. sin2 A + cos2 A = 1

10. tan( A  B) 

4. sec2 A = 1 + tan2 A 11. tan 2 A 

5. cosec2 A = 1 + cot2 A 6. sin 2A = 2 sin A cos A 7. cos 2A = cos A  sin A 2

2

12.

tan A  tan B 1 tan A tan B

2 tan A 1  tan 2 A

a b c   sin A sin B sin C

= 2 cos2 A  1

13. a2 = b2 + c2 – 2bc cos A

= 1 – 2 sin2 A

14. Area of triangle =

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

1 ab sin C 2

[lihat sebelah SULIT

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

z 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 2. 0 2. 1 2. 2 2. 3

2

3

4

5

6

7

8

9

.5000 .4602

.4960 .4562

.4920 .4522

.4880 .4483

.4840 .4443

.4801 .4404

.4761 .4364

.4721 .4325

.4661 .4286

.4641 .4247

4 4

8 8

.4207

.4168

.4129

.4090

.4052

.4013

.3974

.3936

.3897

.3859

4

8

.3821

.3783

.3745

.3707

.3669

.3632

.3594

.3557

.3520

.3483

4

7

1 2 1 2 1 2 11

.3446

.3409

.3372

.3336

.3300

.3264

.3228

.3192

.3156

.3121

4

7

11

.3085 .2743

.3050 .2709

.3015 .2676

.2981 .2643

.2946 .2611

.2912 .2578

.2877 .2546

.2843 .2514

.2810 .2483

.2776 .2451

3 3

7 7

.2420

.2389

.2358

.2327

.2296

.2266

.2236

.2206

.2177

.2148

3

6

1 0 1 0 9

.2119

.2090

.2061

.2033

.2005

.1977

.1949

.1922

.1894

.1867

3

5

8

.1841

.1814

.1788

.1762

.1736

.1711

.1685

.1660

.1635

.1611

3

5

8

1 0

.1587 .1357

.1562 .1335

.1539 .1314

.1515 .1292

.1492 .1271

.1469 .1251

.1446 .1230

.1423 .1210

.1401 .1190

.1379 .1170

2 2

5 4

7 6

9 8

.1151

.1131

.1112

.1093

.1075

.1056

.1038

.1020

.1003

.0985

2

4

6

.0968

.0951

.0934

.0918

.0901

.0885

.0869

.0853

.0838

.0823

2

3

.0808

.0793

.0778

.0764

.0749

.0735

.0721

.0708

.0694

.0681

1

.0666 .0548

.0655 .0537

.0643 .0526

.0630 .0516

.0618 .0505

.0606 .0495

.0594 .0485

.0582 .0475

.0571 .0465

.0559 .0455

.0446

.0436

.0427

.0418

.0409

.0401

.0392

.0384

.0375

.0359

.0351

.0344

.0336

.0329

.0322

.0314

.0307

.0287

.0281

.0274

.0268

.0262

.0256

.0250

.0228 .0179

.0222 .0174

.0217 .0170

.0212 .0166

.0207 .0162

.0202 .0158

.0139

.0136

.0132

.0129

.0125

.0122

.0107

.0104

.0102 .02964

.02939

.02820

.02621 .02466

.02798

.02776

.02755

2 8 2 8 2 7 2 6 2 5 2 4 2 3 2 1 1 9 1 8 1 6 1 4 1 3 11

7

5

6

8

3

4

6

7

1 0 8

1 1

2 2

4 3

5 4

6 5

7 6

8 7

1 0 8

11 9

.0367

1

2

3

4

4

5

6

7

8

.0301

.0294

1

1

2

3

4

4

5

6

6

.0244

.0239

.0233

1

1

2

2

3

4

4

5

5

.0197 .0154

.0192 .0150

.0188 .0146

.0183 .0143

0 0

1 1

1 1

2 2

2 2

3 2

3 3

4 3

4 4

.0119

.0116

.0113

.0110

0

1

1

1

2

2

2

3

3

0

1

1

1

1

2

2

2

2

3

5

8

5

7

2

4

6

8

1 3 1 2 11

1 8 1 6 1 5 1 3

23

2

1 0 9

1 0

3 2 3 2 3 1 3 0 2 9 2 7 2 6 2 4 2 2 2 0 1 9 1 6 1 5 1 3 11

9

2 4 2 4 2 3 2 2 2 2 2 0 2 0 1 8 1 6 1 5 1 4 1 2 11

.02914

1 6 1 6 1 5 1 4 1 4 1 4 1 3 1 2 11

8

2 0 2 0 1 9 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 0 9

36 36 35 34 32 31 29 27 25 23 21 18 17 14 13

.02714

.02695

.02676

.02657

.02639

2

4

6

7

9

1 5 1 4 1 3 11

2 1

3 2

5 3

6 5

8 6

9 7

11 8

2 0 1 8 1 7 1 5 1 2 9

1

2

3

4

5

6

7

8

9

1

1

2

3

4

4

5

6

6

.02889

2. 5 2. 6 2. 7 2. 8

4 5 6 7 SUBTRACT

1

.02990

2. 4

1 2 3

0

.02866

.02842

.02734

.02604

.02587

.02570

.02554

.02539

.02523

.02508

.02494

.02480

.02453

.02440

.02427

.02415

.02402

.02391

.02379

.02168

.02357

.02336

.02326

.02317

.02307

.02298

.02189

.02280

.02272

.02164

.02248

.02240

.02233

.02226

.02219

.02212

.02205

.02199

.02193

.02347 .02256

21 19 17 14 10

5 2. 9 3. 0 3. 1

.03968

.02181

.02175

.02169

.02164

.02159

.02154

.02149

.02144

.02139

0

1

1

2

2

3

3

4

4

.02135 .03968

.02130 .03935

.02126 .03904

.02122

.02118

.02114

.02111

.02107

.02104

.02100

0 3

1 6

1 9

.03845

.03816

.03739

3

6

8

2

5

7

2

4

7

1 0 9

2 1 6 1 4 1 2 11

2

4

6

8

9

2 1 9 1 7 1 5 1 3 11

2

3

5

6

8

3 2 2 2 0 1 7 1 5 1 3 11

4 28

.03874

2 1 3 11

7 5

.03337

.03325

.03313

.03302

.03291

.03180

.03270

.03260

.03251

.03242

1

2

3

4

5

6

7

3 2 5 2 2 2 0 1 8 1 5 1 3 1 0 8

.03233 .03159

.03224 .03153

.03216 .03147

.03208 .03142

.03200 .03136

.03193 .03131

.03185 .03126

.03178 .03121

.03172 .03117

.03165 .03112

1 0

1 1

2 1

3 2

4 2

4 3

5 3

6 4

.03108

.03104

.03100

.0496

.0492

.0488

.0485

.0482

.0478

.0475

.0472

.0469

.0467

.0464

.0462

.0459

.0457

.0454

.0452

.0450

.0448

.0446

.0444

.0442

.0441

.0439

.0437

.0436

.0434

.0433

.03762 3. 2

.03687

.03664

.03641

.03619

3. 4 3. 5 3. 6 3. 7 3. 8 3. 9

.03483

.03466

.03450

.03434

.03711

.03598 .03577

3. 3

.03736

.03557

.03538

.03519

.03501

.03419 .03404

.03390

.03376

.03362

.03349

1

3

4

5

7

1 0 8

9

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(µ, σ2), Prob (v > x) is given by Q(z) with z = (x – µ)/σ

25 22 20 17 14 12 9

5 For Examiner’s Use

SULIT

3472/1 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]

1 Answer (a)………………

2 2.

(b)……………… mx Diagram below shows the function h : x  , x ≠ 0, where m is a constant. x Find the value of m. [3 marks]

2 3

Answer: m =……………….. 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

SULIT

SULIT

3472/1

For

7 Examiner’s Use

3.

b 2 1 Given function f : x  a  3 x and f : x  x  , where a and b are 2 3 constants. Find the value of a and of b. [3 marks]

3

4.

3

Answer: a = …………, b = ……….. Given that 3 and k are the roots of quadratic equation x2 + x = p, find the values of k and of p.

[4 marks]

4 Answer: k = …………, p = ……….. 3472/1 [lihat sebelah © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR] SULIT

For Examiner’s

SULIT

4

3472/1

8 Use

5.

Find the range of values of x such that 4x < 3 + x2.

[3 marks] [3 marks]

5 3 6.

Answer: ………..………… 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]

6 Answer: (a)…………………

3

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

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3472/1

For

9 Examiner’s Use

7.

Solve the equation

6log6 (2t 1)  11.

[4 marks]

7

8.

4

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

8 3 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

For Examiner’s

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3472/1

10 Use

9.

Find the sum of all terms of the following geometric progressions 1, 2, 4, …, 512

[2 marks]

9 2

Answer: …………………… 10. Given that

h  2.424242... is a recurring decimal, find the value of h and of k. k [3 marks]

10 3

Answer: h = …………, k = ………… 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

SULIT

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3472/1

For Examiner’s Use

11 11.

The nth term of an arithmetic progression is given by Tn  2n  3 . Find (a) the common difference, (b) the sum of all the terms from the 3rd term to the 10th term. [4 marks]

11 Answer: (a)………………

4

(b)……………… 12.

The diagram shows a part of the straight line graph of

1 y

against

in terms of x.

1 . x

Express y [4 marks]

12 Answer: ………………………

4

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For Examiner’s Use

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3472/1

12 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]

13 3 14.

Answer: ……………………… uuur uuur Diagram below shows two vectors, OA and BO .

Express, uuur  x (a) OA in the form    y uuur (b) BO in the form xi%  yj . %

[2 marks]

14 Answer: (a)………………

2

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

SULIT 15.

SULIT

3472/1

For Examiner’s Use

13

Use the above information to find the values of h and k when r  5 p  4q . % % % [3 marks]

15 3

Answer: h = …………, k = ………… In diagram below, PQRS is a parallelogram and STQ is a straight line.

16.

uuur uuur If PQ  5 x , PS  4 y , and ST = 3TQ, express in terms of x and y , % % uuur % % (a) SQ uur (b) TR [4 marks]

16 Answer: (a)……………… (b)………………

4

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3472/1

14 17.

Solve the equation 3sin 2 x  5cos x  0 for 0  x  360 .

[4 marks]

17 4 18.

Answer: ………………….. 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]

18 3

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

SULIT 19.

Given that f ( x)  5( x  3) 4 , find the value of f '(2) .

SULIT

3472/1 [2 marks]

For Examiner’s Use

15

19

20.

2

Answer: ………………….. 2 x  1  2 p Given that and y  1  3 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]

20 Answer: …………………..

4

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3472/1

16 3

21.

Given that ∫ (4 x − 3) dx = 4, where k > 0, find the value of k.

[3 marks]

k

21 3 22.

Answer: k = …………… The table shows the distribution of the score a competition. Score 2 3 4 5 6 Frequency 2 x 12 8 7 range of values of x, given that the median is 4.

7 3

Find the [3 marks]

22 3

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

SULIT 23.

SULIT

3472/1 Seven prefects are to be selected from a group of 8 boys and 15 girls.

For Examiner’s Use

17 (a) (b)

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

23 Answer: (a) ………………… 24.

4

(b) ………………… 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]

24 Answer: …………………

3

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

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3472/1

18 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) …………………

25 4 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

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19 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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