Mt 2008 Zon C Kuching K2

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SULIT

Jawatankuasa Kurikulum Zon C Kuching, Sarawak 3472/2

PERCUBAAN SIJIL PERLAJARAN MALAYSIA 2008 ADDITIONAL MATHEMATICS Kertas 2 Sept 2008 2 12 jam

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. Kertas soalan ini adalah dalam bahasa Inggeris sahaja. 2. Calon dikehendaki membaca arahan di halaman belakang kertas soalan ini. 3. Calon dikehendaki menceraikan halaman 11 dan ikat sebagai muka hadapan bersama-sama kertas jawapan.

Kertas soalan ini mengandungi 12 halaman bercetak 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

[lihat sebelah SULIT

2 SULIT

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

7. log a m  n log a m n

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.  

 f ( x  x)   fx f f

N



5. m  L  

1 2



N F  c fm 

Q1  100 6. I  Q0

N

2

x

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 )

2

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

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 absin C 2

[lihat sebelah SULIT

4 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

4 5 6 7 SUBTRACT

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

3

1

.02990

2. 4

1 2

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

2

.0 347 .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

6 SULIT Section A [40 marks] Answer all questions from this section

1.

Solve the simultaneous equations 2x + y = 8 and 2x2 + y2 – xy = 18. Give your answers correct to three decimal places.

2.

3.

[5 marks]

[5 marks]

A set of scores y1 , y 2 , y 3 , y 4 , ..., y n has a mean of 52 and a variance of 49. (a)

Calculate the value of n if y  416 . Hence, determine the sum of the squares of the scores  y 2 .

[4 marks]

(b)

Each score is multiplied by 3, then 5 is added to it. Find, for the new set of scores, (i) the mean, (ii) the standard deviation.

[2 marks]

The sum of the first n terms of an arithmetic progression is given by S n  3n 2  5n . Find

4.

[3 marks]

(a)

the first term and the common difference,

(b)

the 10th term,

[2 marks]

(c)

the value of n if the sum of the first n terms is 1062.

[2 marks]

(a) Sketch the graph of y = 3 |cos x| for 0  x  2 . (b)

[4 marks]

Hence, using the same axes, sketch a suitable graph to find the number of [3 marks] x  1 for 0  x  2 . solutions to the equation 3 cos x  [3 marks] 2

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

[lihat sebelah SULIT

7 SULIT 5. In the diagram below, PQRS is a trapezium and PQTS is a parallelogram. P

Q

U

S

T

R

uuur T is the midpoint of SR and U is the midpoint of QT. Given ST  4a and uur % SP  6b. %

6.

(a) Express the following vectors in terms of a and b. % uuur % (i) SQ uuur (ii) QR uuur (iii) PU

[5 marks]

(b) Hence, determine whether the points P, U and R are collinear or not.

[2 marks]

Given that

dy d2y   11 .  6 x  4 when x = 2, y = – 30 and 2 dx dx

(a)

Find y in terms of x,

[5 marks]

(b)

Find the maximum and the minimum values of y

[3 marks]

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

SULIT

8 SULIT Section B [40 marks] Answer four questions from this section 7.

Use the graph paper to answer this question. Table below shows the values of two variables, x and y, obtained from an b experiment. The variables x and y are related by the equation y  ax  , where ax a and b are constants. x 1.0 2.0 3.0 4.0 5.0 6.0 y 4.3 4.4 5.5 6.8 8.2 9.6 2 (a) Plot xy against x , by using a scale of 2 cm to 5 units on both axes. Hence, draw the line of best fit. [4 marks] (b)

Use the graph in (a) to find the values of (i) a, (ii) b, (iii) y when x  2 5 .

[6 marks]

8.

Diagram shows a parallelogram ABCD and vertices A (1, p ) , B (1, 5) , 1 C ( 2 p ,1 ) and D(3,  1) . 2 (a) Find the value of p.

[2 marks]

(b) The equation of straight line that passes through the midpoint of AC and parallel to AD in general form.

[3 marks]

(c) AB is extended to point P such that point B divides AP in the ratio 1 : 2. Find the coordinates of P.

[3 marks]

(d) Find the area of ABCD.

[2 marks]

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

[lihat sebelah SULIT

9 SULIT 9.

The above diagram shows a sector PQTS with centre P inscribed in a circle with centre O. Given PQ = 10 cm and the length of arc QTS is 12 cm.

10.

a) Find QPS in radians. b) Find the radius of the circle with centre O. c) Find the area of the shaded region.

[2 marks] [3 marks] [5 marks]

(a)

[4 marks]

Given the curve y  x 3  x 2  x  1 , find the equation of normal of the curve at the point (2, 5).

(b)

[6 marks]

Diagram above shows a curve x  y 2  1 intersect with straight line 8 y  3x at A. Find (i) the coordinates of point A, (ii) the volume generated, in terms of π , when the shaded region is revolved through 360 about the y-axis. 11.

(a)

(b)

Senior citizens make up 25 % of the population of a settlement. (i)

If 7 people are randomly selected from the settlement, find the probability that at least two of them are senior citizens.

(ii)

If the variance of the senior citizens is 375, what is the population of the settlement?

The mass of the workers in a factory is normally distributed with a mean of

[5 marks]

[5 marks]

61 kg and a variance of 121 kg2. 200 of the workers in the factory weigh between 53 kg and 72 kg. Find the total number of workers in the factory. 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

SULIT

10

SULIT Section C [20 marks] Answer two questions from this section 12 .

D 50°

6.4 cm

C

7.9 cm

6 cm

B

A

In the above diagram, BCD is a straight line, BC = 6 cm, AC = 7.9 cm, AD = 6.4 cm, ∠ACD = 50° and ∠ADC is obtuse. Calculate (a) (b) (c) (d) 13 .

the length, in cm, of AB, ∠ADC, the length of DC, the area, in cm2 , of triangle ABD.

[2 marks] [3 marks] [3 marks] [2 marks]

A product is made of 4 components A, B, C and D. The table shows, for each component, the price index for the year 2007, using 2006 as the base year and thier respective weightages.

(a)

Calculate the cost for component A in the year 2007 if its cost in the year 2006 is RM4.50.

(b)

The price index for component C in the year 2006 based on the year 2004 is [2 marks] 110. Calculate the corresponding price index in the year 2007 based on the year 2004.

(c)

The composite index for the cost of making the product in the year 2007 [6 marks] based on the year 2006 is 132. Calculate (i)

[2 marks]

the values of x and y, given that x = 2y.

(ii) the total cost of the product in the year 2006 if the corresponding total cost in the year 2007 is RM33. 3472/1 © 2008 Hak Cipta Jawatankuasa Kurikulum Zon C, Pelajaran Gabungan Kuching [SMKTAR]

[lihat sebelah SULIT

11

SULIT 14. A particle moves in a straight line from a fixed point E. Its velocity, v ms-1, at t seconds after leaving the point E is given by v  2t 2  12t . (Assume motion to the right is positive) Determine

15.

(a)

the maximum velocity of the particle,

[3 marks]

(b)

the distance traveled during the third second,

[3 marks]

(c)

the value of t when the particle passes point E again,

[2 marks]

(d)

the time after leaving E when the particle reverses its directions of motion.

[2 marks]

Use the graph paper to answer this question. A factory produces x units of toy P and y units of toy S per day. The production of the toys is based on the following constraints. I

:

The total number of both types of toys produced in a day is not more than 80 units.

II

:

The number of toy S produced in a day is not more than three times the number of toy P produced in a day.

III :

The number of toy S produced in a day is at least 30 units.

(a) Write down three inequalities, other than x  0 and y  0, which satisfy all of [3 marks] the above constraints. (b) Using a scale of 2 cm to 10 units on the two axes, construct and shade the [3 marks] region R which satisfies all of the above constraints. (c) By using your graph from (b), (i)

the maximum units of toy S produced if 35 units of toy P are produced in a day,

(ii)

the maximum profit obtained given the profit from the sale of one unit of toy P and one unit of toy S are RM45 and RM 60 respectively.

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

[4 marks]

SULIT

12

SULIT Nombor Kad Pengenalan Angka Giliran Nama : _______________________________________________ Arahan kepada calon 1. Tuliskan angka giliran dan nombor kad pengenalan anda pada ruang yang disediakan. 2. Tanda () untuk soalan yang dijawab. 3. Ceraikan halaman ini dan ikat sebagai muka hadapan bersama-sama kertas jawapan. Kod Pemeriksa Bahagian

A

B

C

Soalan

Soalan dijawab

Markah penuh

1

5

2

6

3

7

4

7

5

7

6

8

7

10

8

10

9

10

10

10

11

10

12

10

13

10

14

10

15

Markah Diperoleh (Untuk Kegunaan Pemeriksa)

10 Jumlah

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

SULIT

13

INFORMATION FOR CANDIDATES 1.

This question paper consists of three sections: Section A, Section B and Section C.

2.

Answer all questions in Section A, four questions from Section B and two questions from Section C.

3.

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

4.

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

5.

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

6.

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

7.

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

8.

You may use a non-programmable scientific calculator.

9.

Tie page 11, graph paper and answer sheet together hand in at the end of the examination.

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

SULIT

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