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3472/1 Additional Mathematics Kertas 1 Dua Jam

I NAMA

KELAS

PERCUBAAN PAHANGSIJIL PELAJARAN MALAYSIA 2013

ADDTIONAKertas 1 P

Dua Jam

3472/1

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAsdaasda

1 . Kerlas soaJan ini adaJah daJam dwibahasa.

2. SoaJan daJam bahasa Jnggeris mendahuJui soaJan yang sepadan daJam bahasa MeJayu.

3. CaJon dibenarkan menjawab keseJuruhan atau sebahagian soaJan daJam bahasa Jnggeris atau bahasa MeJayu.

4. CaJon dikehendaki membaca makJumat di haJaman beJakang kerlas soalan ini.

Untuk Kegunaan Pemeriksa Ked Pemeriksa:

Soalan Markah Markah Penuh Diperoleh

1 2 2 3 3 3 4 3 5 3 6 4 7 3 8 3 9 2 10 4 11 3 12 3 13 2 14 3 15 4 16 3 17 3 18 4 19 3 20 3 21•. 4 22 4 23 4 24 4 25 3

Jumlah 80

Kertas soalan ini mengandungi 20 halaman bercetak

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SULIT 2 3472/1

THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(O, 1) KEBARANGKALIAN HUJUNG ATAS Q(z) BAG/ TABURAN NORMAL N(O, 1)

7 2 3 4 5 6 7 8 9 : (I I " 3 4 5 6 7 8 9 Minus I Tolak

I o.o 0.5000 0.4960 0.492J 0.4880 0.4840 0.4a01 0.4761 0.4721 0.46a1 0.4641 4 a 12 16 20 24 2a 32 36 01 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 4 a 12 16 20 24 28 32 36 0.2 0.4207 0.416a 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3a97 0.3a59 4 a 12 15 19 23 27 31 35 0.3 0.3821 0.37a3 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 4 7 11 15 19 l2 26 30 34 0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 4 7 11 15 18 22 25 29 32

0.5 0.30a5 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 3 7 10 14 17 20 24 27 31 0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 3 7 10 13 16 19 23 26 29 0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 3 6 9 12 15 18 21 24 27 o.a 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1a67 3 5 a 11 14 16 19 22 25 0.9 0.1a41 0.1a14 0.17a8 0.1762 0.1736 0.1711 0.16a5 0.1660 0.1635 0.1611 3 5 a 10 13 15 18 20 23

1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 2 5 7 9 12 14 16 19 21 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 2 4 6 8 10 12 14 16 18 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.103a 0.1020 0.1003 0.0985 2 4 6 7 9 11 13 15 17

1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 2 3 5 6 a 10 11 13 14 1.4 O.Oa08 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.070a 0.0694 0.0681 1 3 4 6 7 a 10 11 13

1.5 0.0668 0.0655 0.0643 0.0630 0.061a 0.0606 0.0594 0.0582 0.0571 0.0559 1 2 4 5 6 7 a 10 11

1.6 0.054a 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0 .. 047 0.0465 0.0455 1 2 3 4 5 6 7 8 9

1.7 0.0446 0.0436 0.0427 0.041a 0.0409 0.0401 0.0392 5 0.0375 0.0367 1 2 3 4 4 5 6 7 a

1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0384 0.0301 0.0294 1 1 2 3 4 4 5 6 6

1.9 0.02a7 0.0281 0.0274 0.026a 0.0262 0.0256 0.0250 0.0307 00239 0.0233 I 1 2 2 3 4 4 5 5

0.0244

2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0 , , 2 2 3 3 4 4 2.1 0.0179 o.om 0.0170 0.0166 0.0162 0.0158 O.OIS4 0.0150 0.0146 0.0143 0 , , 2 2 2 3 3 4 2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 0 I , I 2 2 2 3 3 2.3 0.0107 0.0104 0.0102 0 I I I , 2 2 2 2

0.00990 0.00964 0.00939 0.00914 3 5 8 10 13 15 18 20 23 0.00889 0.00866 0.00842 2 5 7 9 12 14 16 16 21

2.4 0.00820 0.00798 0.00776 0.00755 0.00734 2 4 6 8 II 13 15 17 19 0.00714 0.00695 0.00676 0.00657 0.00639 2 4 6 7 9 II 13 15 17

2.5 0.00621 0.00604 0.00587 000570 0.005S4 0.00539 0.00523 0.00508 0.00494 0.00480 2 3 5 6 8 9 II 12 14 2.6 0.00466 0.00453 0.00440 0.00427 0.0041 5 0.00402 0.00391 0.00379 0.00368 0.00357 I 2 3 5 6 7 9 9 10 2.7 0.00347 0.00336 0.00326 000317 0.00307 0.00298 0.00289 000280 0.00272 0.00264 , 2 3 4 5 6 7 8 9 2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 , I 2 3 4 4 5 6 8 2.9 0.00187 0.00181 0.00175 0.00169 0.001 64 0.00159 O.OOIS4 0.00149 0.00144 0.00139 0 , , 2 2 3 3 4 4

3.0 0.00135 0.00131 0.00126 0.001 22 0.00118 0.00114 0.00111 0.00107 0.001 04 0.00100 0 , I 2 2 2 3 3 4

Example I Contoh: f(z)

1 ( 1 2) f(z) = .[i;exp - 2z If X- N(O, I), then

Jika X- N(O, I), maka .,

Q(z) = f f(z) dz P(X> k) ~ Q(k) k

P(X> 2. I)= Q(2. I)= 0.0179

0 z


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SULIT 4

STATISTICS I STATIST/K

[}:_N-Fl

5. m = L+ 2

fm C

I =~xlOO 6. Qo

- "WI 7. 1= L...,; I I

Iw, 8 "P =-n_!-. r (n-r)!

9 "C = n! . r (n-r)!r!

10. P(A U B)= P(A) + P(B)- P(A u B)

11. P(X =r)="Crp' qn-r , p+q=l

12. Mean I Min , J.1. = np

13. cr=~npq

X-J.l. Z=--14. a

GEOMETRY I GEOMETRI 1.Distance I Jarak

= ~(X2 -xY +(y2-YY 2. Midpoint I Titik tengah

(x ) =(xi + Xz Y1 + Y2) ,y 2 ' 2

3. A point dividing a segment of a line

s.Jz::l=~xz+yz

1\ X~+ y j 6. r = r=======

- ~x2+y2

Titik yang membahagi suatu tembereng garis

(x,y) = (nx1 + mx2 , ny1 + my2) m+n m+n

4. Area of triangle I Luas segitiga 1 = -l(xlyz + X2Y3 + X3Y1 )- (x2Y1 + X3Y2 + X1Y3~ 2

347211


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SULIT 3 3472/1

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

Rumus-rumus berikut bo/eh membantu anda menjawab soalan. Simbo/-simbo/ yang diberi ada/ah yang biasa digunakan.

-b±.Je -4ac 1. x=------

2a

3. am +an= am-n

5. logo mn = logo m + logo n

m 6. logo-= loga m -loga n

n

ALGEBRA

1 b loge b

8. ogo =-loge a

9. T, = a+(n-l)d

n 10. S, =-[2a+(n-l)d]

2 11 . T, = ar"-1

12. sn = a(r" -1) = a(l-r") ,r :;t: 1 r-1 1-r

13. Soo =~,lrl <1 1-r

CALCULUS/KALKULUS

dy dv du 1. y=uv, -=u-+v-

dx dx dx

du dv v--u-

2. y = !!._ , dy = dx dx v dx v2

dy dy du 3. -=-X

dx du dx

4. Area under a curve Luas di bawah lengkung

b b

= Jydx or (atau) = Jxdy a a

5. Volume of revolution lsi padu kisaran b

= Jn- y 2 dx or (atau) a


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SULIT 5 3472/1

TRIGONOMETRY I TRIGONOMETRI

1. Arc length, s = rB Panjang /engkok, s = j e

1 2 2. Area of sector, A= -r B

2

L 1 ·2e Luas sector, =-J 2

3. sin 2 A + cos 2 A = 1

sin 2 A + kos 2 A = 1

4. sec 2 A = 1 + tan 2 A sek 2 A = 1 + tan 2 A

5. cosec 2 A = 1 + cot 2 A kosek 2 A = 1 + kot 2 A

6. sin2A = 2sinAcosA sin2A = 2sinA kosA

7. cos 2A =cos 2 A- sin 2 A

= 2cos 2 A -1

= 1-2 sin 2 A

kos 2 A = kos 2 A - sin 2 A = 2 kos 2 A -1 = 1- 2sin 2 A

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

9. cos( A ± B) = cos A cos B + sin A sin B kos (A± B)= kosA kosB +sin A sin B

10. tan(A ±B)= tan A± tanB l+tanAtanB

2tanA 11 . tan2A =

2 1- tan A

a b c 12. --=--=-·-

sinA sinE sinC

13. a 2 = b 2 + c 2 - 2bc cos A

a 2 = b2 + c 2- 2bc kosA

14. Area of triangle I Luas segitiga

= I.absinC 2

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SULIT

For Examiner's Use

1

[d

0

1.

6

Answer All questions. Jawab semua soa/an.

Diagram 1 shows the relation between set P and set Q. Rajah 1 menunjukkan hubungan antara set P dan set Q.

f X ------~~-------Cosx

State Nyatakan

SetP Diagram 1

Rajah 1

Set Q

(a) the type of relation between set P and set Q. jenis hubungan antara set P adan set Q.

(b) the value of kif f:x ~ cosx. nilai bagi k jika f: x ~ cos x .

Answer I Jawapan :

(a)

(b)

3472/1

[ 2 marks] [ 2 markah]

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SULIT

Given the function f: x ~ - 3-, x :t. k I find 2.

. x-1

Diberi fungsi f: x ~ - 3-, x :t. k I cari

x-1

(a) the value of kl nilai kl

(b) the function F 1•

fungsi rl .

Answer I Jawapan:

(a)

(b)

3. Given the function f: x ~ ~. x :t. 0 I find X

Diberi fungsi f: x ~ ~. x :t. 0 I cari X

(a) the value of rj /(3) 1

nilai F 1 /(3) 1

Answer I Jawapan:

(a)

(b)

7

3472/1

[3 marks] [3 markah]

[3 marks]

[3 markah]

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For Exwniner's Use

2

w

3

w 0


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SULIT

For Examiner's Use

4

w

5

w

8 3472/1

4. Given that the quadratic equation (p + 2)x2 + 2 px + p- 5 = 0 has no real roots, find the range of values of p. Diberi persamaan kuadratik (p + 2)x 2 + 2 px + p- 5 = 0 tidak mempunyai Punca nyata, cari julat nilai p.

Answer I Jawapan :

5. Find the range of the values of x for x(x- 2) < 3.

Cari ju/at nilai x bagi x(x - 2) < 3 .

Answer I Jawapan :

[3 marks) [3 markah]



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SULIT 9

6. Diagram 6 shows the graph of the function f(x) = -(x- 3) 2 + k, where k is a constant.

Rajah 6 menunjukkan gart bagi fungsi f(x) = - (x- 3) 2 + k, di mana k ialah pemalar.

Find Cari

(a) the value of h, nilai h,

(b) the value of k, nilai k,

f(x)

(h ' 15)

~~--------------· X 0

Diagram 6 Rajah 6

(c) the equation of straight line AB. persamaan garis lurus AB.

Answer I Jawapan:

(a)

(b)

(c)


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For Examiner's Use

6

[d 0

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SULIT 10

For Examiner's

Use 7. Given that 3n1 = -1- , find the value of x.

81x

7

[d

8

[d

0

Diberi, 3 x+J = -1- , cari nilai bagi x.

81x

Answer I Jawapan

8. Solve the equation : Selesaikan persamaan:

log5(x+ 9) -log 5(x + 1) = 1,

Answer I Jawapan:

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SULIT 11

9. The first term of of a geometric progression is 4 and the sum to infinity is 16. Sebutan pertama bagi suatu janjang geometri adalah 4 dan hasil tambah hingga sebutan ketakterhinggaan adalah 16.

Find the common ratio. Cari nisbah sepunya.

Answer I Jawapan:

[ 2 marks] [ 2 markah]

10. The first three terms of an arithmetic progression are k-1, k+2, 3k-5.

Tiga sebutan pertama suatu janjang arithmetik ialah k-1, k+2, 3k-5.

Find

Cari

(a) the value of k, nilai k,

(b) the sum of the 7 terms before tenth term. Hasil tambah 7 sebutan sebelum sebutan kesepuluh.

Answer I ·Jawapan:

(a)

(b)


3472/1

For Examiner'~

Use

9

Q

10

Q 0


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SULIT

For Examiner's Use

11

GJ

12

GJ 0

12

11. A triangle has vertices A(1, 3) , 8(5, 1) and C(3, k). Given that its

area is 4 cm 2 , find the two values of k.

Sebuah segi tiga mempunyai bucu-bucu A(1, 3) , 8(5, 1) dan C(3, k). Diberi /uas segi tiga ini ialah 4 cm2

, cari dua nilai k.

Answer I Jawapan

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12. The point E(1 , k) divides the line segment that connects the points P(h, 3) and Q(4, 18) internally in the ratio 2 : 3. Find the value of h and of k.

Titik E(1, k) membahagi dalam tembereng garis yang menyambungkan titik P(h, 3) dan Q( 4, 1 B) dengan nisbah 2 : 3. Cari nilai h dan nilai k.

Answer I Jawapan



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SULIT 13

13. Given PQ = (k- 3) i + j is a unit vector in the direction of AB , find the

value of k.

Diberi PQ = (k- 3) i + j ialah satu vektor unit dalam arah AB , cari nilai k.

Answer I Jawapan

[2 marks]

[2 markah]

14. Diagram 14 shows a straight line graph of .JY against x 2•

Rajah 14 menunjukkan satu graf garis Jurus .JY melawan x 2 •

.JY (4, 6)

2

0

Express y in terms of x. Ungkapkan y dalam sebutan x . .

Answer I Jawapan

Diagram 14 Rajah 14


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13

[d

6 w 0


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SULIT

For Examiner's Use

15

[d 0

14 3472/1

15 In the Diagram 15 , PQR is a triangle. E is the midpoint of PQ and F is a point on QR such that QR = 4FR .

Da/am Rajah 15, PQR ialah sebuah segitiga. E ialah titik tengah bagi PQ dan F ialah satu titik pada QR dengan QR = 4FR.

Q

Diagram 15 Rajah 15

Given that PQ=l2x and PR=8y, express the following vectors in the terms of

x andy.

Diberi PQ=I~ dan PR=8y, ungkapkan vektor yang berikut dalam sebutan ~

dan y.

(a) QF

(b) EF,

Answer I Jawapan (a)

(b)



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SULIT 15

16. Given that cos 50° = h , express in terms of h.

Diberi bahawa kos 50° = h, ungkapkan dalam seburan h.

(a) sec ( -50°),

sek(-50°)

(b) sin 100° .

Answer I Jawapan (a)

(b)

17. Diagram 17 shows the graph of a function f(x) for 0 ~ x ~ 27Z'. Rajah 17 menunjukkan graf bagi satu fungsi 0 ~ x ~ 2TC .

j(x)


Diagram 17 Rajah 17

State the function. Nyatakan fungsi itu.

Answer I Jawapan [3 marks]

[3 markah]

3472/1

For Examiner's Use

16

w

16

w 0

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SULIT

For Examiner's Use

18

~ 0

16 3472/1

18. Diagram 18 shows the sector OAB of a circle with centre 0 and radius 1 Ocm. Rajah 18 menunjukkan sektor OAB bagi sebuah bulatan berpusat 0 dengan jejari 10 em. A

8

Diagram 18

Rajah 18

Given the length of arc AB is same as radius of the circle and the area of triangle OAB is k cm 2

, find Diberi panjang lengkok AB adalah sama dengan jejari bulatan dan luas segi tiga OAB adalah k cm2

, cari

[Use/Guna 11'=3.142 ]

(a) the length of AB, panjang AB,

(b) the area of the shaded region in terms of k. Cari /uas kawasan ber/orek dalam sebutan k.

Answer I Jawapan

(a)

(b)



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SULIT 17

19. It is given y = x3 + 4x + 1 and the rate of change of xis 6 units s·1 at all time. Find the rate of change of y when x = 2.

Diberi bahawa y = x3 + 4x + 1 dan kadar perubahan bagi x ialah 6 units s·'

setiap masa. Cari kadar perubahan bagi y apabila x = 2.

Answer I Jawapan

20. Given that f(x) = 4

x + 1 , find the value of f 1 (1).

3-2x

. . h f( ) 4x + 1 . 'I . b . f I (1) Dtben ba awa x = -- , can mat agt . 3-2x

Answer I Jawapan



3472/1

For Examiner's Use

19

w

20

w 0

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SULIT

For Examiner's Use

21

GJ

22

GJ 0

18 3472/1

21. Given the mean and variance of the set data x" x2 , x3 , .. • x" is 1 0 and 6

respectively. Diberi min dan varians bagi set data x"x2,x3, ... xn masing-masing ialah 10

dan 6. Given the new set of data 2x1 - 3, 2x2 -3, 2x3 - 3, ... 2xn - 3 ,find

Diberi set data baru 2x1 -3, 2x2 - 3, 2x3 - 3, ... 2x" - 3 ,cari

(a) the mean , min,

(b) the variance. varians .

Answer I Jawapan

(a)

(b)


22. All the letters from the word 'WISDOM ' are to be arranged. Calculate the number of arrangement if they must Semua huruf daripada perkataan I WISDOM I hendak disusunkan. Hitungkan bilangan susunan jika

(a) begin with the letter' W ', bermula dengan huruf I W ',

(b) begin and end with vowels. bermula dan berakhir dengan huruf vowel.

Answer I Jawapan

(a)

(b)



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SULIT 19

23 Given that 5% of the batteries produced by a factory are defective. From a sample of 400 batteries chosen at random, calculate Dibari 5 % daripada bateri yang dihasilkan oleh sebuah kilang adalah rosak. Daripada 400 sampe/ bateri yang dipilih secara rawak, hitungkan

(a) the mean and min dan

(a) the standard deviation of the number of batteries that are defective. sisihan piawai bilangan bateri yang rosak.

Answer I Jawapan

(a)

(b)

[4 marks]

(4 markah]

3472/1

23

G 0


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SULIT

For Examiner's Usc

24

~ 0

20 3472/1

24 Lee will play against players E, F and G in a badminton competition. The

probabilities that he will beat E, F and G are .!.,..!.. and i respectively. 3 4 5

Lee akan berlawan dengan pemain E, F dan G dalam satu pertandingan badminton. Kebarangkalian bahawa dia akan mengalahkan E, F dan G

' . 'lhl ld 4 masmg- masmg 1a a - , - an - . 3 4 5

Calculate the probability that Hitungkan kebarangkalian bahawa

(a) he will beat the three players, dia akan mengalahkan ketiga-tiga orang pemain,

(b) he will beat at least two of the three players. dia akan mengalahkan sekurang-kurang dua daripada tiga orang pemain,

Answer I Jawapan

(a)

(b)

(4 marks] [4 markah]


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SULIT 21

25. Diagram 25 shows a standardized normal distribution graph. Rajah 25 menunjukkan sebuah graf taburan normal piawai.

Diagram25 Rajah 25

The area of the shaded region is 39 % of the total area under the curve. Luas rantau berlorek ialah 39 % daripada jum/ah luas di bawah /eng kung ..

(a) Find the value of k, Cari ni/ai k,

(b) P(z > -k)

Answer I Jawapan

(a)

(b)

END OF QUESTION PAPER

KERTAS SOALAN TAMAT


3472/1

25

w 0


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SULIT 22

INFORMATION FOR CANDIDATES MAKLUMAT UNTUK CALON

1. This question paper consists of 25 questions. Kertas soalan ini mengandungi 25 soalan.

2. Answer all questions. Jawab semua soa/an.

3. Give only one answer for each question. Bagi setiap soalan beri satu jawapan sahaja.

4. Write your answers in the spaces provided in this question paper. Jawapan anda hendaklah ditu/is pada ruang yang disediakan da/am kertas soalan ini.

5. Show your working. It may help you to get marks. Tunjukkan langkah-langkah penting dalam kerja mengira anda. lni boleh membantu anda untuk mendapatkan markah.

6. If you wish to change your answer, cross out the answer that you have done. Then write down the new answer. Jika anda hendak menukar jawapan, batalkan dengan kemas jawapan yang telah dibuat. Kemudian tu/is jawapan yang baru.

7. The diagrams in the questions provided are not drawn to scale unless stated. Rajah yang mengiringi soalan tidak dilukis mengikut ska/a kecuali dinyatakan.

8. The marks allocated for each question are shown in brackets. Markah yang diperuntukkan bagi setiap soalan ditunjukkan da/am kurungan.

9. A list of formulae is provided on pages 3 to 5. Satu senarai rumus disediakan di halaman 3 hingga 5.

10. A four-figure table for the Normal Distribution N(O, 1) is provided on page 2. Satujadual empat angka bagi Taburan Normal N(O, 1) disediakan di halaman 2.

11. You may use a scientific calculator. Anda dibenarkan menggunakan kalkulator saintifik.

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