3472/1
3472/1
ADDITIONAL MATHEMATICS / PKertas 1
September 2011Dua jam
NAMA
TINGKATAN
ADDITIONAL MATHEMATICS
TINGKATAN 5KERTAS 1
2 JAM
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4. Calon dikehendaki membacamaklumat di halaman belakangkertas soalan ini.
Untuk Kegunaan PemeriksaKod Pemeriksa:
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Diperoleh1 3
2 3
3 2
4 3
5 3
6 3
7 3
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11 4
12 3
13 2
14 3
15 3
16 3
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18 4
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3472/2
ADDITIONAL
MATHEMATICS
Paper 2Aug/Sept 201121/2 hours
ADDITIONAL MATHEMATICS
Tingkatan 5
Kertas 2
Dua jam tiga puluh minit
3472/2
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU
1. Kertas soalan ini adalah dalam dwibahasa.
2. Soalan dalam bahasa Inggeris mendahului soalan yang sepadan dalam
Bahasa Melayu.
3. Calon dikehendafd membaca maklumat di halaman belakang kertas soalan
ini.
Kertas soalan ini mengandungi 20 halaman bercetak.
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SULIT 3472/1(PP)
[Lihat halaman sebelah] 3472/1(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
3472/1(PP) Additional Mathematics Kertas 1 Sept 2011 Peraturan Pemarkahan
MAJLIS PENGETUA SEKOLAH MENENGAH MALAYSIA CAWANGAN NEGERI PAHANG
PEPERIKSAAN PERCUBAAN
SIJIL PELAJARAN MALAYSIA 2011
ADDITIONAL MATHEMATICS
Paper 1
PERATURAN PEMARKAHAN
UNTUK KEGUNAAN PEMERIKSA SAHAJA
Peraturan pemarkahan ini mengandungi 5 halaman bercetak
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SULIT 2 3472/1(PP)
3472/1(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
ADDITIONAL MATHEMATICS PAPER 1 MARKING SCHEME Question Solution and Marking Criteria Marks 1 (a) 2 1 3
(b) 2 1 (c) {4, 8} 1
2 2=k ,
434=m (both)
2=k or 434=m
( ) mkxxfg +-= 2141)(
3
B2
B1
3
3 2
1)( -=
xxg or 21
21)( -= xxg
xfxg 1)( -= or 21
=m
2
B1
2
4 2=p ( )[ ] ( )( ) 03144 2 =+-+- pp or equivalent
( ) ( ) 0341 2 =++-+ xpxp in general form or 1+= pa , 4--= pb and 3=c (all)
3 B2
B1
3
5 3,1 ³-£ xx ( )( ) 013 ³+- xx or 21,21 -£-³- xx (both)
0322 ³-- xx or 21 ³-x or 21 -£-x (Reject 21 ±³-x )
3 B2 B1
3
6 1.677
7lg2lg42lg2
-=x or equivalent
7lg2lg)24( xx -- or equivalent (log both sides of the equation)
3 B2
B1
3
7 (a) 1=x 1 3 (b) (2, -3) 1 (c) No (real) roots 1
8 2
22 qp -+ or equivalent
4log5log3log24log
2
222 -+ or equivalent
4log2.7log
2
2 or 5log36log 44 - or 10log72log 44 - or equivalent
3
B2
B1
3
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SULIT 3 3472/1(PP)
[Lihat halaman sebelah] 3472/1(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
9 (a) 3 )102()17()9()102( +--=+-+ pppp
2 B1
4
(b) 160
[ ] [ ])4(2)12(223)4(7)12(2
28
+-+ or 24+28+32+36+40
or [ ])4(4)24(225
+
*follow through the values of a and d
2 B1
10 (a) 28 1 3 (b)
271341
311
31128*
4
-
úúû
ù
êêë
é÷øö
çèæ-
or 333
28*33
28*328*28*
´´+
´++
*follow through the value of a from part (a)
2
B1
11 (a) 1
p-+
=-+
222
4513 or
p-+-
=--
421
2523 or
242
25 +
=+p
Or 542 -=-p or 524 -=-p
2
B1
4
(b) 11
[ ]2)2(1)1()4(3)5(2)2(1)2(4)1(5)3(221
-+-++-+-+-+
Or equivalent
2 B1
12 P (0, 2) R (10,7)
41)0(4)(12
++
=x or
41)2(4)(13
++
=y
Or )02((42 -+=x , )23(43 -+=y (both)
1 2
B1
3
13 2 02 =-h
2 B1
2
14 (a) ji 99 +-
)827(23 jiji ++--
2 B1
3
(b) ji -10 1 15 12=h , 3-=k (both)
12=h or 3-=k
421 2 -= x
xy
3 B2 B1
3
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SULIT 4 3472/1(PP)
3472/1(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
16 (a) p- 1 3 (b) 212 pp --
pq2-
2 B1
17 (a) (i) (ii)
3 4
1 1
4
(b) °10.24 , °91.155 , °10.204 , °91.335 °19.48
2 B1
18 (a) 1.031 rad 1 4 (b) 21.85 cm
Perimeter = )031.1(6661010 22 +-++ Arc AB = 6(*1.031) *follow through the values from part (a)
3 B2
B1
19 (a) -2 0633 2 =-+ xx
2 B1
4
(b) 10=y
)2(*6)2(*23)2(* 23 ---+-
*follow through the values from part (a)
2 B1
20 3
76
15)2(915)4(9=
--
--
- kk or equivalent
)3(1)53( 1
-- -x
3 B2
B1
3
21 5 12103715 ´=+´ x or equivalent
2 B1
2
22 (a) 240 5! 2! or 5
52
2 PP ´ 2
B1 4
(b) 216 1
34
41
3 PPP ´´ or 3!43 ´´ 2
B1 23 (a)
94
1 3
(b) 41
121
321)1( --==XP or equivalent
2
B1
24 0.4 13.01.02.0 =+++ a
2 B1
4
0.8 )2(1 =- XP or 0.1 + 0.3 + *0.4
*follow through the values from part (a)
2 B1
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SULIT 5 3472/1(PP)
[Lihat halaman sebelah] 3472/1(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
25 30
6241 m-
=-
Z = -1
3 B2
B1
3
END OF MARKING SCHEME PERATURAN PEMARKAHAN TAMAT
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SULIT 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
3472/2(PP) Additional Mathematics Kertas 2 Sept 2011 Peraturan Pemarkahan
MAJLIS PENGETUA SEKOLAH MENENGAH MALAYSIA CAWANGAN NEGERI PAHANG
PEPERIKSAAN PERCUBAAN
SIJIL PELAJARAN MALAYSIA 2011
ADDITIONAL MATHEMATICS
Kertas 2
PERATURAN PEMARKAHAN
UNTUK KEGUNAAN PEMERIKSA SAHAJA
Peraturan pemarkahan ini mengandungi 12 halaman bercetak
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SULIT 2 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
ADDITIONAL MATHEMATICS PAPER 2 MARKING SCHEME
SECTION A [40 MARKS] Question Marking Criteria Marks 1 82 =- yx ___①
8 12 8x y
- = ___② (any two equations : one linear, one non-linear)
82 += yx or 2
8-=
xy (x in terms of y or y in terms of x)
Substitute x or y into non-linear equation …
81282
8=-
+ yy or 8
28
128=
÷øö
çèæ -
=xx
0652 =++ yy or 0862 =+- xx 0)3)(2( =++ yy or 0)4)(2( =-- xx [factorisation or formula]
2-=y , 3-=y ; 2=x , 4=x 4=x , 2=x ; 3-=y , 2-=y
[ If answers given without factorisation or using formula: K1 N1 N1 OW – 1]
P1
P1
K1
K1 N1 N1
6
2 (a) Listing : 6x, 6x + 6, 6x + 12, … or a = 6x 6=d
[ ] 750)6(9)6(22
1010 =+= xS (Equating 750=nS )
8=x
P1 N1
K1 N1
7
(b) Area = 2)(2 x , 2)1(2 +x , 2)2(2 +x , ...=> 2(8)2, 2(9)2, 2(10)2, ... AP: x , 1+x , 2+x , ... => 8, 9, 10
)110(8*10 -+=T = 17 Area of largest rectangle = 2[*8 + (10 – 1)1]2 = 578 cm2 OR Perimeter : 6(8), 6(8) + 6, 6(8) + 12, …
yT 61026)110()8(610 ==-+= 17=y [any letter except x] ___K1
Area of largest rectangle = yy 2´ )17(217 ´ ___K1
578 cm2 ___N1
K1 K1 N1
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SULIT 3 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
3 § Shape of graph of y = sin x , 0 ≤ x ≤ 2π [must begin from origin, O(0, 0)]
§ Shape of graph of y = sin 2x , 0 ≤ x ≤ 2π [exactly 2 complete cycles]
§ Amplitude of graph = 3 § Reflection of graph y = 3 sin 2x at x – axis, 0 ≤ x ≤ 2π
[all correct]
xyp
= [Equation of straight line]
Draw any straight line with positive gradient passing through (0, 0) Number of solutions = 5
N1
N1
N1 N1
P1
K1 N1
7
4 (a) [ ])2(346 --=k = 12
P1 6
(b) )3()34)(1(12* 2 ---= -x
dxdy
[ ] )3()2(34)1(12* 2 ---= -
dxdy [substituting x = 2 in his
dxdy ]
= 9
K1
K1
N1
(c) y – (–6) = 9(x – 2) or equivalent [finding equation of tangent using his dy/dx ]
249 =- xy
K1
N1
x
y
O 2π π ½ π 3/2 π
+3
-3
2
y = - 3 sin 2x
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SULIT 4 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
5 (a) Marks (Markah)
0-19 20-39 40-59 60-79 80-99
Number of students (Bilangan Murid)
3 10 14 8 5
N1
6
(b) Percentage of students scoring A+ = 100
405*
´
= 12.5%
K1
N1 (c) L = 39.5
)20(14
13)40(21
5.39-
+=m
= 49.5
P1
K1 N1
6 (a) 𝑃𝑅�����⃗ = −6𝑥 + 4𝑦
𝑃𝑆����⃗ =34 𝑃𝑅�����⃗
yx 329
+=
N1
K1
N1
8
(b) 𝑂𝑆�����⃗ = 𝑂𝑃�����⃗ + 𝑃𝑆����⃗ 𝑜𝑟 𝑂𝑅�����⃗ + 𝑅𝑆�����⃗
yx 323
+=
𝑃𝑄�����⃗ = 2𝑥 + 4𝑦
𝑃𝑄�����⃗ = ÷øö
çèæ + yx 4
23
34 And PQ is parallel to OS
OR
𝑂𝑆�����⃗ = ( )yx 4243
+ and OS is parallel to PQ
N1
N1
N1
(c) OP = 12, OR = 20 and QR = 16 [Using 2=x , 5=y to find the length] Area of OPQR = 280 units2
K1
N1
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SULIT 5 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
SECTION B [40 MARKS]
Question Marking Criteria Marks 7 (a) 2x 1.00 4.00 9.00 12.25 16.00 30.25
xy 4.50 8.90 16.20 21.53 27.00 48.40 Graph of xy against x2 : Please refer to Appendix 1 (End of Mark Scheme) Correct axes with uniform scale and a point All points correctly plotted Line of best fit
N1 N1
K1 N1 N1
10
(b) (i) hkhxxy += 22
From Graph, gradient, 50.12 »= hm 75.0»h [0.70 – 0.80]
P1
N1
(ii) xy -intercept, c = 3.0
0.375.0
»k
25.2»k [accept 2.1 – 2.4]
N1
(iii) 30=xy From graph, x2 = 18
24.4»x
N1
8 (a) 10)0)4(0()000(
21
=+-+-++ OB or 10421
=´OB
OB = 5 B (5, 9)
K1
N1
10
(b) (i) Let C be (h, k) From area ratio, AO : OC = 10 : 20
021
)3(2)(1=
+-+h or 0
21)4(2)(1
=+
-+k
OR Equation of OC
xy34
= ___K1
K1
K1
Equating 2h with its gradient OR
Equating hk with its xy-intercept
OR Construction of the line xy = 30 on the graph
K1
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SULIT 6 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
20)000()05*0(21
=++-++ k __K1
OR Area of ΔOBC
20)5(*21
=´ k ___K1
hk34
= ___K1
C (6, 8)
N1 (ii) PC = 2PA
2222 )]4([)]3([2)8*()6*( --+--=-+- yxyx [ ]2222 )4()3(4)8()6( +++=-+- yxyx 0161222 =+++ yxyx
At x-axis, y = 0
0)12(0122
=+=+
xxxx
0=x or 12-=x (0, 0), (-12, 0)
K1 N1
N1
K1
N1 9 (a) °= 37.346.0 rad
106.0tan QRrad = or
1037.34tan QR
=°
84.6»QR cm
K1 N1
10
(b) (i) °=Ð 55.63or rad 971.0RQO OR °=Ð 111.26or rad 942.1PQR
OR)90(34.37or rad )
26.0(
21
major °+°+=ÐpPQR
OR °=Ð 248.74or rad 342.4majorPQR
Arc PSR = 342.484.6 ´ = 29.70 cm
K1
N1 (ii)
Area of OPQR = 284.61021
´´´
Area of sector OPR = 2.11021 2 ´´ or 210
36074.68
´´°° p
Area of shaded region = 68.4 – 60 or 68.4 – 59.99 = 8.4 cm2 or 8.41 cm2
K1
K1
K1 N1
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SULIT 7 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
10 (a) (i) 75.0=p , 25.0=q
433
7 )25.0()75.0()3( CXP == or 43
37
41
43
÷øö
çèæ
÷øö
çèæC
= 0.05768 or 16384
945
K1
N1
10
(ii) )7()6(1)5( =+=-=£ XPXPXP 716
67 )75.0()25.0()75.0(1 +-= C
= 0.5551
K1 K1 N1
(b) (i) 16
162175 -=z
= 0.75
N1 (ii)
6.016163
-=-x
cm 4.153=x
N1 (iii) )75.01()175147( ££-=££ ZPXP
)75.0()1(1 ³-³-= ZPZP = 1 – 0.1587 – 0.2266 = 0.6147
K1 K1 N1
11 (a) (i) At P, 0123)( 21 =-=- xxxf 0)4(3 =-xx
4 ,0 == xx P(4, 0)
K1
N1
10
(ii) ( )cxx
xdxxxf
+-=
-= ò23
2
6
123)(
32=c cxxxf +-= 23 6)(
K1 N1 N1
(b) R(6, 32) Area of rectangle = 326´ OR )326*(32* 23 +-- xx = 192 ___① = 326 xx - Area under curve Area of shaded region from x = 0 to x = 6
( )ò +-=6
0
23 326* dxxx ( )ò -=6
0
326 dxxx
6
0
34
3224 ú
û
ùêë
é+-= xxx
6
0
43
436
úû
ùêë
é-=
xx
= 84 ___② 04)6(
3)6(6 43
-úû
ùêë
é-=
P1 K1
K1
K1
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SULIT 8 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
Area of shaded region = ① – ② = 192 – 84 = 108 cm2
N1
SECTION C [20 MARKS]
Question Marking Criteria Marks
12 (a) ttts 156 23 --=
15123 2 --== ttdtdsv
-1ms 15 ,0 -== vt
K1
N1
10
(b) For minimum velocity, 0126 =-= t
dtdv
2=t 15)2(12)2(3 2
min --=v = - 27 ms-1
K1
N1
N1 (c) At P, v = 0: 0)5)(1(3 =-+ tt
5=t 12)5(6 -=a
= 18 ms2
K1 N1
N1
(d) 0<v 0)5)(1(3 <-+ tt
50 << t
K1 N1
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SULIT 9 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
13 (a) (i) AB = 20 cm P1 10 (ii) 2 2 2 020 8.9 2(20)(8.9)cos 54AC = + -
AC = 16.43 cm K1 N1
(b) ACBÐ-+= cos)43.16)(*9.8(243.16*9.820 222 1739.0cos -=ÐACB
°±°=Ð 01.0 100ACB OR Using sine rule,
43.16*54sin
9.8sin °
=ÐCAB
OR
43.16*54sin
20sin °
=ÐACB
___K1
°=Ð 99.25CAB °-°-°=Ð 99.2554180ACB
= 100.01o °=Ð 100ACB ___N1
K1
N1
(c) (i) °÷
øö
çèæ=D 54sin)20)(9.8(
21 of Area ABC
= 72.00 cm2
ABCADC D´=D of Area 53 of Area
OR °99.25sin)43.16)(12(
21
= 43.20 cm2 = 43.20 cm2 OR
2.7722021
==>=´´ hh 2.712
21 of Area ´´=DADC
___K1 = 43.20 cm2 ___N1
K1
N1
K1
N1
(ii) Let h cm be the distance between AB and DC
722021
=´´ h
cm 2.7=h
N1
14 (a) 20>x 10³y
80£+ yx
xyyx41or 4 ³£
N1 N1 N1
N1
10
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SULIT 10 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
(b) § Axes correct and one *straight line correct § Draw correctly all 4 straight lines § Region R is correctly shaded and labelled
(Please refer to appendix 2)
K1 N1 N1
(c) (i) 6020 £< x N1 (ii) 200 (21) + 300 (59)
The maximum fees collected is RM 21 900 K1 N1
15 (a) Item P : 110100
52009 =´
P
5.50 RM2009 =P
K1
N1
10
(b) (i) Q2008 = RM 6.00 N1 (ii) Q2010 = RM 9.00 N1
(c) Item I2010 based on I2008 P 121 Q 150 R 120 S 104.5
[one mark each for every TWO correct answers]
N1
N1
(d) Item I2010 based on I2008 W WI P 121 80 9680 Q 150 150 22500 R 120 90 10800 S 104.5 40 4180 360=åW å = 47160WI
Weightage for item S = 40o or 4 [as seen in his formula]
40(104.5) + 90(120) + 150(150) + 80(121) =ii IWå OR 8(121) + 15(150) + 9(120) + 4(104.5)
360)5.104(40)120(90)150(150)121(80 +++
=I
OR 36
)5.104(4)120(9)150(15)121(8 +++
= 131
P1
K1
K1
N1
END OF MARKING SCHEME SKEMA PEMARKAHAN TAMAT
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SULIT 11 3472/2(PP)
[Lihat halaman sebelah] 3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
5 10 0 15 20 25 30 35 40
40
25
20
15
35
30
10
5
xy
X
X
X
X
2x
50
45
(0, 3.0)
X
18
(28, 45) X
Gradient,
45 328 0
1.50
m -=
-
=
x
Appendix 1 – Graph for Question 7
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SULIT 12 3472/2(PP)
3472/2(PP) © 2011 Hak Cipta MPSM Cawangan Pahang SULIT
10 20 0 30 40 50 60 70 80
80
50
40
30
70
60
20
10
xy
R
x
100
90
(21, 59)
x+ y = 80
X
x = 4y
x = 20
y = 10
y Appendix 2 – Graph for Question 14
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