SPM 2007 Mathematics P2

Time: zlrro.r",Paper 2

Instructions: ?/r.is qu,estion paper consists of 2 -sections;

Section A an'd Section B' Answer

all questioru.s ln, Sectio n h and' fat{, ques,tions ii Section B' Write your answers in' the spaces

prouid,ed in the quurtion paper. Wo*king stefs must be written clearly' Non-programmable

scientific calculator is allou;ed'

Section ABalt'agian A

152 mark'sl152 marhahl

Answer all questions in this section'

Jq,wab semua so<llctn daktm bahagian ini'

On the graph in the answer space, shade the

y < - t r + 6, y > 2x-4 and x)> l '"Pad,a

graf d'i ru,ang iatuapan, Iorelzhan rantct'u

y < -Jc * 6, Y > 2x--4 and x > I '

AnswerlJawaPan:

region which satisfy the three inequalitresl3 marksl

yang memLtashctn lzetiga-tiga hetak'sQ'maanl3 mark'ahl

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=L49-Bh=-18

[4 marksl[4 markah]

AnswerlJawapan:

Using factorisation, solve the following quadratic equation:Menggunakan pemfaknron, sel.esoi.hsn persotncurn 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 angledtriangle 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 serclgam prisma itu. V inlah titik tengahPS.

Diagram IRajah 1

14r

Iderrtfy and.balculate, the,angle. between thle 'lttme'.UV a.nd the plane,&STU-' ' '

K.enal pasti dan'la,i,tung,sud,ut, d'io.a,ntar"a gar'is,UV'd,engqn satak'RSTU.' r,l[3 marksl

[3 markahl

Answer/Joruapani

5 In Diagram 2, O is the origin. Straight'line KL is parallel to straight'line MN.'The eQuationof 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.nfil? L dan titik N terletak pada paksi-y. l

Diagram 2Raiafu,.2

FindCari(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,sl15 markahl

(b)

AnswerlJawapqn:

(a)

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

Diagrarn 3Rajah 3

OS = 2I cm and OP = L4 cm.OS = 2L cm dan OP = L4 cm.

lusutGuno n =4fr 7JCalculateHitung(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)

7 (a) Complete the following statement using the quantifier "all" or "some", to make it a truestatement.

quadratic equations have two equal roots.

Lengkapkan pernyataqn berilzut dengan menggunakan pengkuantiti "semtJa" atau"sebilangan", untuk membentuk suatu pernyataan benar.

persanxoan kuadratik mempunyai dua punca yang saftra.

(b) Write down Premise 2 to complete the following argument:Tfu.lis Premis 2 untuk melengkapkan hujah berikut:

Premise 1: If M is a multiple of 6, then M is a multiple of 3.Premis L: Jika M ialah gandaan bagi 6, makct. M ialah gandaan bagi 3.

Premise 2lPremis 2:

Conclusion: 23 is not a multiple of 6.Kesimpulan: 23 bukan gandaan bagi 6.

(c) Make a general conclusion by induction for the sequence of numbers 7,14,27,... whichfollows the following pattern.Buat satu kesimpulan unxunx secara ctruhan bagi urutan nombor 7, L4,27, ... yangmengikut pola berikut.

7 - 3(2)1 +114 = 3(2)2+227 = 3(2)3+3

(b)

(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 j ikadanhanya j ikap> q. , '

[6 marhs][6 markah]

(a)

AnswetlJawapan:

(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.

Box PKotahP

D 3 E 4IFIG

Box eKotak Q

Diagram 4Rajah 4

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 thatDengan menyenoroihan kesud,ahan, cari hebarangkalian bahawa(a) both cards are labelled with a number,

kedua-dua had. dilabel d.engan nombor,

A 2 B C

(b) one card is labelled with a number and the, other card is labelled with a letter.sekeping had, dilabel dengan nombor dan kad yang satu lasi dilabel dengan huruf.

l5 marks][5 markah]

An *i,.rtJo*ooon,

(a)

(b)

e (a) Given *(-_t 3X; _?) = (l !), n"a the varue or m and or n.

Diberi *(-i 3X; -?) = (l ?),"",, nitai m d.an nitai n

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

(-i 3)(;) = (l)17 marksl

17 markahl

Answer/Joruapdn:

(a)

(b)

DistancelJarob ftm)

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.

Town BBandar B

Town ABandar A

Time (minutes)Masa (minit\

Diagratn 5Raiah 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 thetaxi 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 mewakiliperjalanan teksi itu dari bandar B ke bandar A. ks itu bertolak dari bandar A dan teksiitu 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 kenderaanitu 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)

Diagram 6 shows a soli{, formed by joining a cylinder to a right prism. Tbapezium AFGBis 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'silinderkepad.a sebwah prisma tegak. lhapeziurn AFGB ialah heratan rentas seragatn prisma itu.AB = BC = 9 cm. Ttnggt silinder ialah 6 cm dan diarnetern"ya ialah 7 cm.

Diagram 6Rajah 6

(ir)

(c)

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

L---. , , E

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

lusetcuna n = 41L ' IJ

[4 marks][4 marhahl

Answer/Jaruapan:

Section BBahagian 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 thevalues of y when x = -L and x = 2. [2 marks]Lengkapkan Jadual I di rudng jawapan bagi persamoon y - 6 - f dengan menulisnilai-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 on the r-axis and 2 cm to 5 units on the y-axis, drawthe graPh of Y = 6 - f for -3 < r < 2.5.Denga,n menggunakan skala 2 cm hepada L unit pada paksi-x dan 2 cm kepada 5 unitpada paksi-y, Iukis graf y = $ - f bagi -B < r < 2.5.

[4 rnarksl[4 markah)

(c) From your graph, findDaripada graf anda, cari(t) the value of y when tc = L.5,

ni la iyapabi lax=1.5,

(ii) the value of r when y = 10.ni la ixapabi lay=10.

12 marksl[2 markahl

(d) Draw a suitable straight line on your graph to find theequation x3 - 8x - 6 - 0 for -3 < r < 2.5.State these values of r.

values of r which satisfv the

Luhis satu garis lurus yang sesuai pada graf ando, untukmemuaskan persaftraan f - 8x - 6 - O bagi -3 < r < 2.5.Nyatakan nilai-nilai x itu.

Answer/JawaDan:

14 markslmencari nilq.i-nilai x ya.ng

[4 marh,ah]

Table 1Jadual 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.an menggunakan kertas graf untuk menjawab soalan ini.

(a) Diagram 7(i) shows a solid right prism with rectangular base ABCD on a horizontalplane. The surface ABJHGF is the uniform cross-section of the prism. AE HG and BJ arevertical edges. Rectangle JKLH is a horizontal plane and rectangle GMEF is an inclinedplane.Rajah 7(i) menu.njukkq,n sebuah pepejal berbentak prisma tegak dengan tapak segiempattepat ABCD terletak di atas satah mengufuk. Permukaan ABJHGF ialah heratan rentssseragoftrnyu Tbpi AE HG dan BJ adalah tegak. Segiempat tepat JKLH ialah satahmengufuk dan segiempat tepat GMEF ialah satah condong.

Diagram 7(i)Rajah 7(i\

(a) 3C otr -2 1- l 0 I 2 2.5

v 33 21.63 t4 f) 5 -9.63

IILD

Graph for Question 12Graf untuh Soalan 12

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

[3 marksll3 markahl

Answer/Joruapani

(a)

(b) A half-cylinder solid of diameter 6 cm is joined to the prism in Diagram 7(i) at the planeSKLT. 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 prismapada Rajah 7(l) pada satah SKLT Panjang SK ialah 4 cm. Gabungan pepeial adalaltseperti ditunjukkan pada Raiah 7(11).

Diagram 7(ii)Rajah 1(ii't

lz

I

I

-___-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 AEsebagaimana dilihat d.ari p.

14 iarkahj

(ii) the elevation of the combined solid on a vertical plane parallel to BC as viewed fromA. Ib marksldongakan gabungan pepejal itu pada satah nxencancdng yang selari d.engan BCsebagaimana d.ilihat dari e. tb rnarkahl

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

l:,Js $:fr"$)%lH"';:"lfl ";r:"8,',::T"o{o^o empat titik pad,a permukaan bumi pRialah 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 thesurface of the earth.Hitung jarak terpendek, dalam batu nautika, dari P lee R diukur sepanjang permukaanbumi.

14 marhsl[4 markahl

(b) y lies south of Q and the distance of VQ measured along the surface of the earth is4500 nautical mile.Calculate the latitude of V. 13 markslV terletah ke selq.tan Q dan jarak VQ diukur sepanjang permukaan bumi ialah45OO batu 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. Theaverage speed for the whole flight was 550 knots.Sebuah kapal terbang berlepas dari P arq,h ke timur ke Q dan kemudian terbang arahke selatan he V Purata laju seluruh penerbangan kapal terbang itu idldh 550 knot.

CalculateHitung(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 diukursepanjang selq.rian latitud sepunyd,

(ii) the total time, in hours, taken for the whole flight.jumlah n1.asq dalam jam, yang diambil bagi seluruh penerbangan itu.

[5 markslL5 markohl

AnswerlJawapan:

(a) (i)

(ii)

(b)

(c) (r)

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.hCartesan.

Diagram 8Rajah 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.

(ir)

G

J

D

E C

A B

R J

Ih L

w

-6II

^ o

State the coordinates of the image of point A' under each of the fol lowing

transformatisns:Nyatakan koordinat irnei titik A di bawah seti.ap penielmaan 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 fullHuraikan selenghapnya(1) the transformation M,

penjelmaan M,

(ii) the transformation N,penielmaan N.

16 marksl16 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 nxenxpunyai luas LLZ m2, hitung

luas, dalarn m2, kawasan yang diwakili oleh JKLM-13 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 kekerapan jisim, d.alarn kg, bagi iekumpulan go murid.

Mass (kg)Jisim (kg)

FrequencyKehera.pan

30-34 535-39 840-44 1145 49 2750-54 2255-59 1060-64 3

Table 2Jad,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 frequencydistribution of the masses. lB markslBerdqsarkq,n Jqduq,l 2, lengkapkan Jad,uol 3 pada ruong jawapan untuk menunjukkankeherapan 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,iakan di halaman ISg.By using the scale of 2 -cm to 5 kg on the horizonial axis and 2 cm to 10 students onthe vertical axis, draw an ogive for the data. v markslDengan menggurtakon sleala 2 cm kepada 5 kg pada paksi mengufuk 6an 2 cm kepadaIO murid pada paksi nTencancang, Iukis satu oCtf bagl data teriebtut. [+ markah]

(d) 25o/o of all the students in the group have a mass of less than p kg. These students willbe supplied with nutritional food.Using the ogive you had drawn in l6(c), find the value of p. [7 mark]25oh daripada murid-murid dalam lzumpulan tersebut mempunyai jisim kurang daripadapkg. Murid-murid ini akan dibekalkan dengan makqnan u"rlrnoiiot. Dengan menggunakanogif yang telah ando lukis di t6(c), cari nilai bagi p. "

t{ *arhah)

AnswerlJawapani

(a) (i)

(ii)

(b)

Table 3Jad.ual 3

(c) Refer graph on page 159.Rujuk graf di halarna.n I59.

(d)

Upper BoundarySempad,an Atas

(kg)

Cumulative FrequencyKekerapan Longgohan

29.5 0

34.5

Graph for Question 16Graf untuk Soalan 16

Paper 11B 28 3C 4D 5B6C 7A 88 9C 1OD

11D L2 B 13C 14 D 15D16C 17 C 188 19A 20 A2LB 22 D 23 A 24 C 25 B26C 27 C 28 B 29 D 30C318 32 C 338 34 B 35C36A 37 B 388 39A 40 B

SPM 2OO7 EXAMINATION PAPER

Paper 2Section A

1 (a)

v

I

r7\

,/

//, X,/,/ ,/r/ \

2 \ \\

2 4\

o / ,.// \

/

g + 2h = | . . . . . . . . . O49 - 3h = -18 . . . . . . . . . . . . @49 + 8h = 4 . . . . . . . . . . . @Ot+

@-@ l lh = 22,22n=-

11

-c)

Subst i tuteh=2intoO

s + 2(2)g+4

o

" ' g=-3,h=2

4*-L5 = - I7x4f+l7x-15 = 0@x-3)(x+5) - 0

-3X=-Ootx=q

LSUV or LWS

1

1

l-4

-o

(b) Length of arc ST

= 90 x2x4rzt360 - ' - 7

= 33cm

Length of arc QR

- Lzoxzx?rtq- 360 " ' " 7

= 29.33 cm

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

= 104.33 cm

(a) Some quadratic equations have two equalroots.

(b) 23 is not a multiple of 3.(c) 3(2)" + f t , tu = 1, 2,3, . . .(d) ( r ) l f p-s >0,thenp>q.

( i i ) I fp lq, thenp-q>O.

Rtand = *

IJ

d = 31o 36'

(a) 2x*y = 4

Y = -zrc+ 4

ffiMN = -2, M(-3, 7)

y-7 = -2(x+3)y-7 = -2x-6

y+2x = |

(b) At r-intercept, y =

!*2x =2x=

.'. r-intercept is f

.

6 (a) Area of quadrant OS?

= 99, x4xztxzl360 7

= 346.5 cm2

Area of sector OPQ

= ,69, x?x14x14360 7

= 102.67 em2

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

=qt2

1 ls -2\= -12 L-10)(; -;)

-'n)

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

= r(3-2 \5

n=3

, ' ;(; -i)#(; -i)-'. tn = -2,

0II

I

1t

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

= (-t l;)= (i)

13. ' .x= 2, t=t

10 (a) 65 - 30 = 35 minutes(b) (r) 9.40 a.m.

(ii) 90 - 30 = 60 km

Total distanceTotal time

" - (1rT)36 km hr

(c) Average speed =

11 (a) Volume of prism =

=

Volume of cylinder =

=

.'. Volume of the solid

Section B

L2 (a)

vQ+12X8)xe/DC) Cm"

<),

+x3.5x3.5x6I

231 cm3- 756 + 231= 987 cm8

13 (a) E/D zcm M/L

F/A G/H

(b) (r) G/M

F/E

8cm

3cm 4cm

S/T

J/B

J/S/K

4cm

B/C

M

2cm

E1cmK/L

4cm

C/D

(c)

(d)

(r) 2.5

y=6-f

0=-6-Bx+f

Equation O +

!=-Bx

(ii) -1.66\"""" ' \ : /

@Equation @

tc -1 2

v 7 -2

H/T/L 6 cm

:::-::::qits:\ \o

0 I

0 -8

. ' . r=-0.8,-2.4

(r) l,(fE(ii) Shtest distare of PR = A x 60

= 50x60= 3()0() n.m.

IXsta'rce d f-Q = 45(X) n.m.Ax60 = 4500

^ 4500a=

60= 75"

." Latihrde d y is 10"s.

(r) Ilistare of FQ= d x 60 cqs latitude= 10O x 60 cos 65"= 2535.71 n.m.

Total distance(ii) lbtal time =

Speed

2535.7L + 4500550

= 12.79 hours

15 (a) (i) (-3, 5)(ii) (-3, 3)

(b) (t M is an enlargement w'ith scale factor2 at the centre (3, 0).

(ii) N is a rotation of 18O" about the centre

o)(a)

(b)

(c)

(c)

Cumulative frequency

(2, 4).1

(c) (r) ;

(ii) Area of JKLM = (*) ' * " '

l -xnz

2{! m2

16 (a) (i) 50 - 54

32(5) + 37(8) + 42(rL) + 47(2L)

(ii) Mean - + 52(22) + q7-(10) + 62(3)

80_ 3805

80= 47.56 ta) roo4 x 8o = 20,

From the ogive, p = 43.

Upperboundary

Cumulativefrequency

29.5 0

34.5 5

39.5 13

44.5 24

49.5 45

54.5 67

59.5 77

64.5 80