mt 2008 zon c kuching k2

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SULIT

Jawatankuasa Kurikulum Zon C Kuching, Sarawak

PERCUBAAN SIJIL PERLAJARAN MALAYSIA 2008ADDITIONAL MATHEMATICS

3472/2

Kertas 2

Sept 200812

2 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]

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SULIT

Important Formulae

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

commonly used.

Algabra

1.2 4

2

b b ac x

a

2. m n m na a a

3. m n m na a a

4. n

m mna a

5. log log loga a a

mn m n

6. log log loga a a

mm n

n

7. log logn

a am n m

8.log

loglog

ca

c

bb

a

9. ( 1)nT a n d

10. 2 ( 1)2

n

nS a n d

11. 1nnT ar

12.)( 1) (1, 1

1 1

n n

n

a r a r s r

r r

13. 11 ,

aS r

r

Statistics

1. x

x N

2. fx

x f

3.2

2( ) x x x x

N N

4.

22( ) f x x fx

x f f

5.12

m

N F m L c

f

6.1

0

100Q

I Q

7.i i

i

W I I

W

8.!

( )!

n

r

n p

n r

9.!

( )! !

n

r

nC

n r r

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

11. ( ) , 1n r n r

r P X r C P q p q

12. mean, np µ

13 npq

14. X

Z

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

SULIT

2



Calculus

1. ,dy dv du

y uv u vdx dx dx

2.2

,

du dvv uu dy dx dx yv dx v

3.dy dy du

dx du dx

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

a x dy 5. Volume of revolution =

2b

a y dx or =

2b

a x dy

Geometry

1. Distance = 2 2

1 2 1 2 x 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 my

m 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 2 x y

6.2 2

xi y jr

x y

Trigonometry

1. Arc length, s r

2. Area of sector,21

2 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 tan

tan( )1 tan tan

A B A B

A B

11.2

2tantan21 tan

A A A

12.sin sin sin

a b c

A B C

13. a2 = b2 + c2 – 2bc cos A

14. Area of triangle = C absin21


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3



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

z 0 1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

SUBTRACT0.

0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4661 .4641 4 81

2

1

6

2

0

2

4

2

8

3

2 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 1

2

1

5

1

9

2

3

2

7

3

1

35

0.

3

.3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483 4 7 11 1

4

1

9

2

2

2

6

3

0

34

0.

4

.3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121 4 7 11 1

4

1

8

2

2

2

5

2

9

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 1

0

1

3

1

6

2

0

2

3

2

6

29

0.

7

.2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148 3 6 9 1

2

1

5

1

8

2

1

2

4

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 1

0

1

3

1

5

1

8

2

0

23

1.

0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 2 5 7 9

1

2

1

4

1

6

1

9 211.

1

.1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170 2 4 6 8 1

0

1

2

1

4

1

6

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 1

0

11 13

1.

5 .0666 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559 1 2 4 5 6 7 8

1

0 111.

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 42.

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 1

0

1

3

1

5

1

8

2

0

23

.02889 .02866 .02842 2 5 7 9 1

2

1

4

1

6

1

8

21

2.

4

.02820 .02798 .02776 .02755 .02734 2 4 6 8 11 1

3

1

5

1

7

19

.02714 .02695 .02676 .02657 .02639 2 4 6 7 9 11 1

3

1

5

17

2.

5 .02621 .02604 .02587 .02570 .02554 .02539 .02523 .02508 .02494 .02480 2 3 5 6 8 9 11

1

2 142.

6

.02466

.02

453 .02

440 .02

427 .02

415 .02

402 .02

391 .02

379 .02

168 .02

357

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

4



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 43.

1

.03968 .03935 .03904 3 6 9 1

3

1

6

1

9

2

2

2

5

28

.03874 .03845 .03816 .03739 3 6 8 11 1

4

1

7

2

0

2

2

25

.03762 .03736 .03711 2 5 7 1

0

1

2

1

5

1

7

2

0

22

3.

2

.03687 .03664 .03641 .03619 .03598 2 4 7 9 11 1

3

1

5

1

8

20

.03577 .03557 .03538 .03519 .03501 2 4 6 8 9 11 1

3

1

5

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 .03

233 .03

224 .03

216 .03

208 .03

200 .03

193 .03

185 .03

178 .03

172 .03

165 1 1 2 3 4 4 5 6 73.

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

5



SULIT

Section A

[40 marks]

Answer all questions from this section

1. Solve the simultaneous equations 2 x + y = 8 and 2 x2 + y2 – xy = 18. Give your

answers correct to three decimal places. [5 marks]

[5 marks]

2. A set of scores n y y y y y ...,,,,, 4321 has a mean of 52 and a variance of 49.

(a) Calculate the value of n if 416 y . Hence, determine the sum of the

squares of the scores 2 y .

(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.

[4 marks]

[2 marks]

3. The sum of the first n terms of an arithmetic progression is given by

23 5n

S n n . Find

(a) the first term and the common difference,

(b) the 10th term,

(c) the value of n if the sum of the first n terms is 1062.

[3 marks]

[2 marks]

[2 marks]

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

(b) Hence, using the same axes, sketch a suitable graph to find the number of

solutions to the equation 3 cos 12

x x

for 0 2 . x [3 marks]

[4 marks]

[3 marks]


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6



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

U

T S R

Q P

T is the midpoint of SR and U is the midpoint of QT . Given 4ST auuu

%and

6 .SP buu

%

(a) Express the following vectors in terms of a%

and .b%

(i) SQuuu

(ii) QRuuu

(iii) PU uuu

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

[5 marks]

[2 marks]

6.Given that 46

2

2

xdx

yd when x = 2, y = – 30 and 11

dx

dy.

(a) Find y in terms of x,

(b) Find the maximum and the minimum values of y

[5 marks]

[3 marks]


7



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

experiment. The variables x and y are related by the equationb

y axax

, where

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

(a) Plot xy against2 x , by using a scale of 2 cm to 5 units on both axes. Hence,

draw the line of best fit.

(b) Use the graph in (a) to find the values of

(i) a,

(ii) b,

(iii) y when 2 5 x .

[4 marks]

[6 marks]

8.

Diagram shows a parallelogram ABCD and vertices A ( 1, ) p , B (1, 5) ,

)

2

11,2( pC and )1,3( D .

(a) Find the value of p.

(b) The equation of straight line that passes through the midpoint of AC and

parallel to AD in general form.

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

the coordinates of P .

(d) Find the area of ABCD.

[2 marks]

[3 marks]

[3 marks]

[2 marks]


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8



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.

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]

10. (a) Given the curve3 2 1 y x x x , find the equation of normal of the curve

at the point (2, 5).

[4 marks]

(b)

Diagram above shows a curve 12 y x intersect with straight line

x y 38 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.

[6 marks]

11. (a) 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?

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

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.

[5 marks]

[5 marks]


9



SULIT

Section C

[20 marks]

Answer two questions from this section

12

.

50 °

6 cm

7.9 cm

6.4 cm

B

C

D

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) the length, in cm, of AB,

(b) ∠ ADC ,

(c) the length of DC,

(d) the area, in cm2 , of triangle ABD.

[2 marks]

[3 marks]

[3 marks]

[2 marks]

13

.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

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

based on the year 2006 is 132. Calculate

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

(ii) the total cost of the product in the year 2006 if the corresponding total

cost in the year 2007 is RM33.

[2 marks]

[2 marks]

[6 marks]


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10



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 by22 12v t t . (Assume motion to

the right is positive) Determine

(a) the maximum velocity of the particle,

(b) the distance traveled during the third second,

(c) the value of t when the particle passes point E again,

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

[3 marks]

[3 marks]

[2 marks]

[2 marks]

15. 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 the above constraints.

(b) Using a scale of 2 cm to 10 units on the two axes, construct and shade the

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.

[3 marks]

[3 marks]

[4 marks]


11



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 SoalanSoalan

dijawab

Markah

penuh

Markah Diperoleh

(Untuk Kegunaan Pemeriksa)

A

1 5

2 6

3 7

4 7

5 7

6 8

B

7 10

8 10

9 10

10 10

11 10

C

12 10

13 10

14 10

15 10

Jumlah


SULIT

12



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.


13

mt 2008 zon c kuching k2

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