Spm 2007 Mathematics P2

Time:zlrro.r",

Paper 2

Section A an'd Section B' Answer of 2 Instructions: ?/r.is qu,estionpaper consists -sections; B' Write your answers in' the spaces ii Section ln, Sectio n h and' fat{, ques,tions all questioru.s must be written clearly' Non-programmable stefs paper. Wo*king quurtion prouid,ed in the scientificcalculator is allou;ed' Section A Balt'agian A 152 mark'sl 152 marhahl Answer all questions in this section' Jq,wab semua so l ' y < -t r + 6, y > 2 x -4 hetak'sQ'maan "Pad,a lzetiga-tiga memLtashctn yang graf d'i ru,ang iatuapan, Iorelzhan rantct'u l3 mark'ahl > I' x a n d y < -J c * 6, Y > 2 x --4 AnswerlJawaPan:

Calculate the value of g and of h that satisfy the following simultaneous linear equations: Hitung nilai g dan nilai h yang mernuaskan persatnaan, linear serentak berikut: g+2h=L 49-Bh=-18

[4 marksl [4 markah] AnswerlJawapan:

Using factorisation, solve the following quadratic equation: Menggunakan pemfaknron, sel.esoi.hsnpersotncurn kuadratih berikut: 4f-L5=-I7x [4 rnarks] 14 markahl Answer/Jouapan:

Diagtam 1 shows a right prism. The base PQES is a horizontal rectangle. Right angled triangle QBU is the uniform cross-section of the prism. V is the midpoint of PS. Rajah L menunjukkan sebuah prisrna tegak. Tapok segicmpat tepat PQRS ad.alah mengufuk. Segitiga bersudut tegak QRU ad.alah keratan rentas serclgamprisma itu. V inlah titik tengah PS.

Diagram I Rajah 1

14r

Iderrtfy and.balculate, the,angle. between thle 'lttme'.UVa.nd the plane,&STU-' ' ' gar'is,UV'd,engqn satak'RSTU.' r,l K.enal pasti dan'la,i,tung,sud,ut,d'io.a,ntar"a [3 marksl [3 markahl Answer/Joruapani

5 In Diagram 2, O is the origin. Straight'line KL is parallel to straight'line MN.'The eQuation of straight line KL is 2x + y = 4. "f,hs:points Z and N lie on the y-axis. ' 'Dalnrn'Rajah 2, O ialah asala.n. Garis lurus KL ad.alah selari dengan garis lurus MN. Persamaan garis lurus KL ialo'h 2x + y = 4. l nfil? L dan titik N terletak pada paksi-y.

Diagram

2

Raiafu,.2 Find Cari (a) the equation of the straight line MN, perscmxaon bagi garis lurus MN, .. (b) the r-intercept of the straight line MN. pintasan-x bagi garis lurus MN.

l5 marh,sl 15markahl

AnswerlJawapqn: (a)

(b)

3 shows quadrant os? and semicircrepeR, both with centre o. ?ilq"* Rajah 3 menunjukkan suku.on bulaton OST dan sernibulatan PQR,yang kedua-d,uanya berpusatO.

Diagrarn 3 Rajah 3 OS = 2I cm and OP = L4 cm. OS = 2L cm dan OP = L4 cm.

n = 4f

rlusu tG uno 7 J Calculate Hitung (a) the area, in cm2, of the shaded region. luas, dalam cm2,kawasan yd,ng berlorek, @) the perimeter, in cm, of the whole diagram. perimeter, dalam cm, seluruh rajah itu. [6 marks] 16 markahl

AnswerlJawapan: (a)

(b)

7 (a) Complete the following statement using the quantifier "all" or "some", to make it a true statement. quadratic equations have two equal roots. Lengkapkan pernyataqn berilzut dengan menggunakan "sebilangan", untuk membentuk suatu pernyataan benar.

pengkuantiti

"semtJa" atau

persanxoan kuadratik mempunyai dua punca yang saftra. (b) Write down Premise 2 to complete the following argument: Tfu.lisPremis 2 untuk melengkapkan hujah berikut: Premise 1: Premis L:

If M is a multiple of 6, then M is a multiple of 3. Jika M ialah gandaan bagi 6, makct.M ialah gandaan bagi 3.

Premise 2lPremis 2: Conclusion: Kesimpulan:

23 is not a multiple of 6. 23 bukan gandaan bagi 6.

(c) Make a general conclusionby induction for the sequenceof numbers 7,14,27,... which follows the following pattern. Buat satu kesimpulan unxunxsecara ctruhan bagi urutan nombor 7, L4,27, ... yang mengikut pola berikut. 7 14 27

- 3(2)1+1 = 3(2)2+2 = 3(2)3+3

(d) write down two implications based on the following statement: "p - q > 0 if and only if p ) q.,, Tltlis duo irnplikasi berdasarkan pernyataan berikut: "p-q> 0 jikadanhanya jikap> q.,' [6 marhs] [6 markah] AnswetlJawapan: (a)

(b) Premise 2 I Premis 2:

(c)

(d) Implication Lllmplikasi t:

Implication 2llmplihasi 2:

Diagram 4 shows ten labelled cards in two boxes. Rajah 4 menunjukkan sepuluh kad. yang berrabel d,i dalam d.ua kotak.

A

2

B

C

D

Box P KotahP

3

E

4I FI G

Boxe

Diagram 4 Rajah 4

Kotak Q

A card is picked at random from each of the boxes. lekgoinS kad. d.ipilih secora rawq.k d.aripad,a setiap kotok itu. By listing the outcomes, find the probability that Dengan menyenoroihan kesud,ahan, cari hebarangkalian bahawa (a) both cards are labelled with a number, kedua-dua had. dilabel d.engan nombor,

(b) one card is labelled with a number and the,other card is labelled with a letter. sekepinghad,dilabel dengan nombor dan kad yang satu lasi dilabel denganhuruf. l5 marks] [5 markah] An *i,.rtJo*ooon, (a)

(b)

= orm andorn. e (a)Given *(-_t 3X; _?) (l !), n"athevarue Diberi *(-i

3X; -?)

= (l

nitain ?),"",, nitaim d.an

G) Using matrices, calculate the value of r and of y that satisfy the following matrix equation: Menggunahan kaedah matriks, hitung nilai x dan nilai y yang memuaskan percatnadn matriks berihut:

=(l) (-i 3)(;)

Answer/Joruapdn: (a)

(b)

17marksl 17markahl

10 Diagram 5 shows the distancdtime graph of the journey of a bus and a taxi. Rajah 5 menunjukkan graf jarak-rnasa bagi perjalanan sebuah bo's dan sebuah tehsi. DistancelJarob ftm) Town B Bandar B

Town A Bandar A

Time (minutes) Masa (minit\

Diagratn 5 Raiah 5 The graph PQRS represents the journey of the bus from town A to town B. The graph JI( represents the journey of the taxi from town B to town A. The bus leaves town A and the taxi leaves town B at the same time and they travel along the same road. Graf PQRS mewakili perjalaruan bas itu dari bandor A ke bandar B. Graf JK mewakili perjalanan teksi itu dari bandar B ke bandar A. ks itu bertolak dari bandar A dan teksi itu pula bertolak dari bo,ndar B pada waktu yarug sama dan melq.lui ialan yqng sa.ma. (a) State the length of time, in minutes, during which the bus is stationary. Nyatakan tempoh rnesa, dalam minit, bas itu berlwnti(b) (r) If the journey starts at 9.00 a.m., at what time do the vehicles meet? Jika perjalanan itu bermula jam 9.00 a.m., pukul berapakah kedua-d'ua kenderaan itu bertemu? (ii) Find the distance, in km, from town B when the vehicles meet. Cari jarak, dalam km, dari bandar B bila kedua-dua kenderann itu berternu. (c) Calculate the average speed, in km h-1, of the bus for the whole journey. Hitung purata laju, dalam km j-t, bas ita bagi keseluruhan perialanan. [5 marksl [5 markah] Answer/Jouapani (a)

(b) (r)

(ir)

(c)

Diagram 6 shows a soli{, formed by joining a cylinder to a right prism. Tbapezium AFGB is the uniform cross-section of the prism. AB = BC = 9 cm. The height of the cylinder is 6 cm and its diameter is 7 cm. Rajah 6 menunjukkan suatu pepejal yang terd,iri daripa.da cantuman sebuah'silinder kepad.asebwah prisma tegak. lhapeziurn AFGB ialah heratan rentas seragatn prisma itu. AB = BC = 9 cm. Ttnggt silinder ialah 6 cm dan diarnetern"yaialah 7 cm.

D

/r---_ -:l ------

L--.,, E

Diagram 6 Rajah 6

Calculate the volume, in cm3, of the solid. Hitung isipadu, dalam cm3,pepejal itu.

lusetcunan = 41 'IJ L

[4 marks] [4 marhahl Answer/Jaruapan:

Section B Bahagian B [48 rnarks] 148 rnarkahl Answer any four questions from this section. Jawab mana-nxona empat soalan daripad.a bahagian ini. L2 (a) Complete Table 1 in the answer space for the equation y = 6 - f by writing down the values of y when x = -L and x = 2. [2 marks] Lengkapkan Jadual I di rudng jawapan bagi persamoon y - 6 - f dengan menulis nilai-nilai y apabila x = -! dan x = 2. 12 markahl (b) For this part of the question, use the graph paper provided on page 1b1. You may use a flexible curve rule. Untuk ceraian soalan ini, gunahan hertas graf yang disediakan pada halaman 15L. Anda boleh menggunakan pembaris fleksibel. By using a scale of 2 cm to 1 unit the graPh of Y = 6 - f for -3 < r Denga,n menggunakan skala 2 cm pada paksi-y, Iukis graf y = $ - f

on the r-axis and 2 cm to 5 units on the y-axis, draw < 2.5. hepada L unit pada paksi-x dan 2 cm kepada 5 unit bagi -B < r < 2.5. [4 rnarksl [4 markah)

(c) From your graph, find Daripada graf anda, cari (t) the value of y when tc = L.5, nilaiyapabilax=1.5, (ii) the value of r when y = 10. nilaixapabilay=10. 12 marksl [2 markahl

(d) Draw a suitable straight line on your graph to find the values of r which satisfv the equation x3 - 8x - 6 - 0 for -3 < r < 2.5. State these values of r. 14marksl Luhis satu garis lurus yang sesuai pada graf ando, untuk mencari nilq.i-nilai x ya.ng memuaskan persaftraan f - 8x - 6 - O bagi -3 < r < 2.5. Nyatakan nilai-nilai x itu. [4 marh,ah] Answer/JawaDan: (a)

3C

v

33

otr

-2

21.63

t4

1 -l

0

I

f)

5

2

2.5 -9.63

Table 1 Jadual I (b) Refer graph on page 151. Rujulz graf di halaman 75I. (c) (i) y = (ii) r =

(d) r= 13 You are not allowed to use graph paper to answer this question. Anda tidak dibenarh.anmenggunakan kertas graf untuk menjawab soalan ini. (a) Diagram 7(i) shows a solid right prism with rectangular base ABCD on a horizontal plane. The surface ABJHGF is the uniform cross-sectionof the prism. AE HG and BJ are vertical edges. Rectangle JKLH is a horizontal plane and rectangle GMEF is an inclined plane. Rajah 7(i) menu.njukkq,nsebuah pepejal berbentakprisma tegak dengan tapak segiempat tepat ABCD terletak di atas satah mengufuk. Permukaan ABJHGF ialah heratan rentss seragoftrnyu Tbpi AE HG dan BJ adalah tegak. Segiempat tepat JKLH ialah satah mengufuk dan segiempat tepat GMEF ialah satah condong.

I I

LD

Diagram 7(i) Rajah 7(i\

Graph for Question 12 Graf untuh Soalan 12

Draw full scale, the plan of the solid. Lukis dengan skala penuh, pelan pepejal itu.

[3 marksl l3 markahl Answer/Joruapani (a)

(b) A half-cylinder solid of diameter 6 cm is joined to the prism in Diagram 7(i) at the plane SKLT. The length of 51( is 4 cm. The combined solid is shown in Diagram 7(ii). Sebuah pepejal berbentuh separuh silinder berdiarneter 6 cm dicantumkan kepada prisma pada Rajah 7(l) pada satah SKLT Panjang SK ialah 4 cm. Gabungan pepeial adalalt seperti ditunjukkan pada Raiah 7(11).

lz I

I

-___-t.---_-\_,

Diagram 7(ii) Rajah 1(ii't

Draw full scale, Lukis dengan skala penuh, (r) the elevation of the combined solid on a vertical plane parallel to AB as viewed from P. [4 marks] dongakan gabungan pepejal itu pad,a satah nxencancang yang selari dingan AE sebagaimana dilihat d.ari p. 14 iarkahj (ii) the elevation of the combined solid on a vertical plane parallel to BC as viewed from A. Ib marksl dongakan gabungan pepejal itu pada satah nxencancdng yang selari d.engan BC sebagaimana d.ilihat dari e. rnarkahl tb AnswerlJawapan: (b) (r), (il)

L4 P(65' l/, 40' W), Q(65" l/, 60' E), R and y are four points on the surface of the eafth. PR

permukaan bumipR empat titikpad,a l:,Js $:fr"$)%lH"';:"lfl ";r:"8,',::T"o{o^o ialah diameter selarian latitud 65' U. (a) (i) State the longitude of -8. Nyatakan. longitud bagi R. (ii) Calculate the shortest distance, in nautical mile, from P Io R measured along the surface of the earth. Hitung jarak terpendek, dalam batu nautika, dari P leeR diukur sepanjang permukaan bumi. 14 marhsl [4 markahl (b) y lies south of Q and the distance of VQ measured along the surface of the earth is 4500 nautical mile. Calculate the latitude of V. 13 marksl V terletah ke selq.tan Q dan jarak VQ diukur sepanjang permukaan bumi ialah 45OObatu nautika. Hitung latitud bagi V 13 markahl (c) An aeroplane took off from P and flew due east to @ and then flew due south to V. The average speed for the whole flight was 550 knots. Sebuah kapal terbang berlepas dari P arq,h ke timur ke Q dan kemudian terbang arah ke selatan he V Purata laju seluruh penerbangan kapal terbang itu idldh 550 knot. Calculate Hitung (r) the distance, in nautical mile, taken by the aeroplane from P to Q measured along the common parallel of latitude, jarak, dalarn batu nautiha, yang dilalui oleh leapal terbang itu dari P ke Q diukur sepanjang selq.rian latitud sepunyd, (ii) the total time, in hours, taken for the whole flight. jumlah n1.asqdalam jam, yang diambil bagi seluruh penerbangan itu. [5 marksl L5 markohl AnswerlJawapan: (a) (i)

(ii)

(b)

(c) (r)

(ir)

15 Diagram 8 shows quadrilaterals ABCD, EFGH and JKLM drawn on a Cartesian plane. Rajah 8 menuniukhan sisiempat ABCD, EFGH dan JKLM yang d,ilukis pa,d.a sata.h Cartesan. J

D

E

C

A

B

G

R

J I

h

L

w -6 II

^

o Diagram 8 Rajah I

(a) Tlansformation R is a rotation of g0o anticlockwise about the centre (0, 2). Tlansformation P is a reflection in the straight line r = 2. Penjelrnaan R ialah putardn 90" lq.wan arah jam pada pusat (0, 2). Penjelmaan P ialah pantulan pada garis lurus x = 2.

State the coordinates of the image of point A' under each of the following transformatisns: Nyatakan koordinat irnei titik A di bawah seti.appenielmaan berilsut: (r) R, (ii) RP' IB marhs] l3 rnarkah) b) EFGH is the image of A,ACD under the combined transformations MN. EFGH iatah imej baei ABCD di bawah gabungan penielmaan iUnN' Describe in full Huraikan selenghapnya (1) the transformation M, penjelmaan M, (ii) the transformation N, penielmaan N. 16 marksl 16 markahl (c) JKLM is the image of EFGH under an enlargement at centre (3' 0)' JKLM ialah imej bagi EFGH di bawah satu pembesaran pada pusat (3, 0). (il State the scale factor of the enlargement. Nyatakan faktor skala pembesaran itu. (ii) Given that EFGH represents a region of area lL2 m2, calculate the area, in m2, of the region represented by JKLM. Diberi, bahawa EFGH mewakili suatu kawasan yang nxenxpunyailuas LLZ m2, hitung luas, dalarn m2, kawasan yang diwakili oleh JKLM13 marks) l3 markah) Answer/Jouapan: (a) (i) (ii) (b) (r)

(il)

(c) (r)

(ir)

16 Table 2 shows the frequency distribution of the mass, in kg, of a group of g0 students. Jadual 2 menunjukhon taburan kekerapanjisim, d.alarn kg, bagi iekumpulan go murid. Mass (kg) Jisim (kg)

Frequency Kehera.pan

30-34

5

35-39

8

40-44

11

45

27

49

50-54

22

55-59

10

60-64

3 Table 2 Jad,uol 2

(a) (i) State the modal class. Nyatakan kelas mod. (ii) calculate the estimated mean of the mass of the group of students. Hitung min anggaran jisim bagi kumpulon murii itu. [4 marks] 14 marhahl (b) Based on Table 2, complete Table 3 in the anslr'er space to show the cumulative frequency distribution of the masses. lB marksl Berdqsarkq,n Jqduq,l 2, lengkapkan Jad,uol 3 pada ruong jawapan untuk menunjukkan keherapan longgokan jisim itu. Ig markahf (c) For this part of the question, use the graph paper provided on page 1b9. Untuk ceraian soalan ini, gunakan kertqs graf yang dised,iakandi halaman ISg. By using the scale of 2 -cm to 5 kg on the horizonial axis and 2 cm to 10 students on the vertical axis, draw an ogive for the data. v marksl Dengan menggurtakon sleala 2 cm kepada 5 kg pada paksi mengufuk 6an 2 cm kepada IO murid pada paksi nTencancang,Iukis satu oCtf bagl data teriebtut. [+ markah] (d) 25o/oof all the students in the group have a mass of less than p kg. These students will be supplied with nutritional food. Using the ogive you had drawn in l6(c), find the value of p. [7 mark] 25ohdaripada murid-murid dalam lzumpulan tersebut mempunyai jisim kurang daripada pkg. Murid-murid ini akan dibekalkan dengan makqnan u"rlrnoiiot. Dengan menggunakan " ogif yang telah ando lukis di t6(c), cari nilai bagi p. t{ *arhah) AnswerlJawapani (a) (i) (ii)

(b)

Upper Boundary Sempad,an Atas (kg)

Cumulative Frequency Kekerapan Longgohan

0

29.5 34.5

Table 3 Jad.ual 3 (c) Refer graph on page 159. Rujuk graf di halarna.n I59. (d)

Graph for Question 16 Graf untuk Soalan 16

SPM 2OO7EXAMINATION Paper 1 1B 28 6C 7A 11D L2 B 1 6C 17 C 2 LB 22 D 2 6C 27 C 318 32 C 36A 37 B

PAPER

3C 88 13C 188 23 A 28 B 338 388

4D 9C 14 D 19A 24 C 29 D 34 B 39A

5B 1 OD 15D 20 A 25 B 30C 35C 40 B

Paper 2 Section A 1 (a)

v I

r7

,/ \

o

//,

X

,/,/

,/r/

2

\

2 4

\

\\ \

/, .//

\

/

Ot+ @ -@

g + 2h= | .... ..... O 49- 3h = -18 ........ ....@ 49+ 8 h = 4 .. ......... @ llh = 22 ,22 n=11

- c)

(b) Length of arc ST

S u b sti tu teh= 2int oO

s + 2(2)

1

g +4

1 l -4

o

-o

"'

33cm

L z o x z x ? rt q

360 " ' "

29.33 cm

8 (a) i "Z=+,

=t.i .(?,.3) *,(+"+)

IJ

=q

t2

= 4 Y = -zrc+ 4 M(-3,

7)

= - 2( x + 3) = - 2x - 6 = | (b) At r-intercept, y = 0 = ! *2x I 2x = 1 y-7 y-7 y+2 x

I I

1 ls , ;(; -i)= -12 L-10)(; ' #(; -i)= r ( 3 -'n)

.'. r-intercept is

f

ft)(l 3X;)=(;)

(;)=(-t;)(;) =(-tl;)

.

=(i)

(a) Area of quadrant OS?

= =

9 9 ,x 4 x z t x z l

360

7

346.5 cm2

Area of sector OPQ

= =

,6 9x? , x1 4 x14

360

7

102.67 em2

Area of the shaded region = 346.5 - 102.67 = 243.83 cm2

-2 \ -;)

-2 \5 -'. tn = -2, n=3

t

6

7

(a) Some quadratic equations have two equal roots. (b) 23 is not a multiple of 3. (c) 3(2)"+ ft, tu = 1, 2,3, ... > 0,thenp> q. (d) (r) l f p-s (i i ) Ifpl q,thenp-q> O.

R

ffiMN = -2,

7

Perimeter of the whole diagram = 33+ 7 + 29.33+ 14+ 21 = 104.33cm

d = 31o 36' (a ) 2 x* y

9 0 x 2 x 4 rz t

360 -' -

Length of arc QR

=

LSUV or LWS

= *

=

--

g=-3,h=2

4*-L5 = -I7x = 0 4f+l7x-15 @x-3)(x+5) - 0 -3tx= q X=-Oo

tand

=

.'.x=

13 2,t=t

10 (a) 65 - 30 = 35 minutes (b) (r) 9.40 a.m. (ii) 90 - 30 = 60 km (c) Average speed =

Total distance Total time

" - (1rT) 36 km hr

=

11 (a) Volume of prism

=

13 (a)

E/D zcm M/L

vQ+12X 8)xe /DC) Cm " <) ,

Volume of cylinder = +x3.5x3.5x6 I

= 231 cm3 .'. Volume of the solid - 756 + 231 = 987 cm8 Section B L2 (a)

tc

-1

2

v

7

-2

F/A (b) (r)

J/B

G/H

G/M

F/E H/T/L

J/S/K

6 cm

4cm

B/C

8cm 3cm

4cm

M 2cm

S /T

E 1cm K/L

4cm

C/D

(c)

(ii) -1.6

(r) 2.5

""""'\:/6\

(d) y= 6- f

@

0 = - 6- B x + f Equation O

+ Equation @

!= - B x

0 0 .'. r = - 0. 8, - 2. 4

I

-8

:::-::::qits:

\\o

(a) (r)

l,(fE (ii) Shtest

distare

(b) IXsta'rce d f-Q = = A x 60 ^ a= = ." Latihrde d y is

of PR = A x 60 = 50x60 = 3()0() n.m. 45(X) n.m. 4500 4500 60 75" 10"s.

(c) (r) Ilistare of FQ = d x 60 cqs latitude = 10O x 60 cos 65" =

Upper boundary

0

34.5

5

39.5 44.5

13 24

49.5

45

54.5

67

59.5

77

64.5

80

Cumulative frequency

Total distance = Speed 2535.7L + 4500 550 = 12.79 hours

15 (a) (i) (-3, 5) (ii) (-3, 3) (b) (t M is an enlargement w'ith scale factor 2 at the centre (3, 0). (ii) N is a rotation of 18O" about the centre

(2, 4). (c) (r)

1

;

(ii) Area of JKLM

=

(*)'*

"'

l -xn z 2{! m2

16 (a) (i)

50 - 54

32(5)+ 37(8)+ 42(rL)+ 47(2L) + 62(3) + 52(22)+ q7-(10) (ii) Mean 80 _ 3805 80 = 47.56

Cumulative frequency

29.5

(c)

2535.71 n.m.

(ii) lbtal time

o)

ta) roo4x 8o= 20, From the ogive, p = 43.