Kertas Model Peperiksaan Sebenar (SPM)

1449/2

SULIT 1449/2 Matematik Kertas 2 2009 2

1 2

NAMA : TINGKATAN :

jam

JABATAN PELAJARAN NEGERI PERAK KERTAS MODEL PEPERIKSAAN SEBENAR SPM

Kod Pemeriksa MATEMATIK Kertas 2

Bahagian

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. Kertas soalan ini mengandungi dua bahagian :

A

Bahagian Bahagian A dan Bahagian Bahagian B. Jawab semua soalan daripada Bahagian A dan empat soalan dalam Bahagian B.

2.Jawa 2.Jawapan pan hendak hendaklah lah ditulis ditulis dengan dengan jelas jelas dalam dalam ruang ruang yang yang disedi disediaka akan n dalam dalam kertas kertas soalan soalan.. Tunjukkan langkah-langkah penting. Ini boleh membantu anda untuk mendapatkan markah. 3.Raj 3.Rajah ah yang yang meng mengir irin ingi gi soal soalan an tida tidakk dilu diluki kiss mengikut skala kecuali dinyatakan. 4.Satu senarai rumus disediakan di halaman 2 & 3 5.And 5.Anda a dibe dibena nark rkan an meng menggu guna naka kan n kalk kalkul ulat ator or saintifik yang tidak boleh diprogram.

B

Soalan

Markah Penuh

1

3

2

4

3

4

4

4

5

4

6

5

7

5

8

6

9

6

10

5

11

6

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12

Markah Diperoleh

Jumlah

Kertas soalan ini mengandungi

1449/2

2009 Hak Cipta JPN Perak

15 halaman bercetak.

[Lihat sebelah SULIT

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SULIT 1449/2 Untuk Kegunaan Pemeriksa

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

76

RELATIONS 1.

a

m

×

a

n

=

a

m +n

12.

.

Pythagoras Theorem c

2.

3.

a

m

÷

n

a

(a )

m n

=

=

a

a

m −n

 d − b    − c a  ad − bc   1

A−1

5.

P ( A)

6.

P ( A' )

7.

dis tan ce = ( x1 − x2 ) 2

8.

Midpoint

=

n( A)

=

n( S )

1 − P ( A)

=

+ (y − y 1

x1 + x2 y1 + y2  ,  2   2

( x, y ) =  

9.

Average speed =

10.

Mean =

distance travelled time taken

sum of data number of data

2 ) 2

= a2 + b2

se

13

m=

14.

m

mn

4.

2

y2 − y1 x2 − x1

y −intercept x −intercept

=−







SULIT 1449/2 1

×

sum of parallel sides × height

1.

Area of trapezium =

2.

Circumference of circle= π d = 2π r

3.

Area of circle =

4.

Curved surface area of cylinder = 2π rh

5.

Surface area of sphere = 4π r 2

6.

Volu Volume me of righ rightt prism rism = cro cross sect sectio ion nal area area

7.

Volume of cylinder

8.

Volume of cone =

9.

Volume of sphere =

10.

Volume of right pyramid =

11. 11.

Sum Sum of inte interi rior or angl angles es of a poly olygon gon = ( n

12.

13.

π

2

r 2

= 1

π

3

4 3

arc of length circumference of circle

area area of secto sectorr area of circle

=

π

=

r 2h

π

r 3

1 3

×

base area

3600

PA ' PA

15.

Area of image= k 2

×

×

height

−2) ×

angle subtended at centre

3600

Scale factor, k =

length

r 2h

angl anglee sub subte tend nded ed at cent centre re

14.

×

area of object

180 0

Untuk Kegunaan Pemeriksa








[ 52 marks] Answer all question in this section. 1.

On the graph provided, shade the region which satisfies the three inequalities [3 marks] y ≤ x + 5, x + y ≤ 5 and y > 1. Answer:

y

y=x+5

1-

x

O

x+y=5

2.

Calculate the values of e and f that satisfy the simultaneous linear equations. 5e − 2 f =16

e + 4 f = −10

Answer:

[4 marks]








Diagram 1

A solid cylinder of diameter 6 cm is taken out from the solid prism. Using π =

22 7

,

calculate the volume in cm3 of the remaining solid. [4 marks] Answer:

4.

Solve the quadratic equation

x( 2 x − 5) 2

=6

Answer:

[4 marks]








SULIT 1449/2 The isosceles triangle STU is the uniform cross-section of the prism. ST = SU and W is the midpoint of TU. Calculate the angle between the line PW and the base STU. marks] P

Q

13 cm

R

T

S

W 12 cm

10 cm

Answer:

6.

(a) (a)

[4

DIAGRAM 2 U

Stat Statee whet whethe herr the the foll follow owin ing g sent senten ence ce is is a sta state teme ment nt or or a non non-s -stat tatem emen ent. t. Give a reason for your answer.

2 + 7 = 1 + 6 (b) (b)

Stat Statee whet whethe herr each each of the the fol follo lowi wing ng stat statem emen ents ts is is true true or or fals false. e. (i) { 0 } is an empty set or is an empty set. (ii){ } is and empty set and is also an empty set.

(c )

Compl omplet etee the the fol follo lowi wing ng argu argume men nt.

Premise 1 : If a > 3, then 5 a > 15. Premise 2 : 5 a < 15. :__________________________________ _______________________ ______________ __ Conclusion :_______________________ [5 marks] Answer: (a) _______________________ ___________________________________ ______________ __

(b) i.___________________ i.___________________ ii.









SULIT 1449/2 Find the probability that (a) both marbles marbles are in yellow, yellow, (b) both marbles are in different color.


[5 marks]

Answer:

8.

R Q

P

150 O

T

S

Diagram 6

. Diagram 3 shows two sectors OPQ and ORS with centre O. OP = 8 cm, QR = 6 cm and OS = 2 OT. POTS is a straight line. Using

=

22

, calculate 7 a) the perimeter, in cm, of the whole diagram b) the area, in cm2, of the shaded region. Answer : π

[6 marks]








SULIT 1449/2 The following diagram shows the distance-time graph of the journeys of two particles S 9. and T .

Speed (ms-1 ) 145 E Station Y

D

80

v

Station A X 0

B

C

F 20

40

60

100

Time (minute)

Graph ABCD represents the journey of the particle S from station X to Y . The graph EF represents the journey of the particle T from station Y to X . Both particles leave station X and station Y respectively at the same time along the same road. (a) (a) Stat Statee the the len lengt gth h of of tim time, e, in minu minute te,, duri during ng which which the the part partic icle le S is stationary. (b) (b) If the the jou journ rney ey start startss at 1500 1500,, at at what what tim timee do the the part partic icles les meet? meet? (c) (c) Give Given n tha thatt the the dist distan ance ce trav travel elle led d by by par parti ticl clee S over the 60 minutes period is 186km. Calculate the value of v. (d) (d) Find Find the the dis dista tanc nce, e, in kilo kilome metr tres es,, fro from m sta stati tion on Y when the particles meet. (e) (e) Calc Calcul ulat atee the the aver averag agee spe speed ed in m/s m/s of of par parti ticl clee S for the whole journey. [ 6 marks ] Answer:








y M (−1,6)

K O

L

Diagram 4

x

N

The equation of straight line KL is 2 x + y + 4 = 0, find (a ) The equati ation of straig aight line LN , (b) The equati ation of straig aight line MN hence state its x-intercept. [ 5 marks ] Answer :

 −1 −2  2 and matrix V = v    3 2   −3

It is given matrix U = 

11.

2 

1 such that UV =   u  0

0 



1 

a) find the values of u and v. b) using the matrix method, calculate the values of x x and y that satisfy the following simultaneous equations x − 2 y = 3 3 x + 2 y = −5 −

marks] Answer :

[6








[ 48 marks] Answer any four questions from this section. 12.

(a)

Table 1 shows values of x and y for equation y x y

−3 −29

−2 v

−1 −7

=

x 3

−

2x

−

8.

0

1

2

3

4

−8

−9

−4

w

48

TABLE 1 Find the values of v and w.

( i) (ii)

[ 2 marks ] (b) For this part of the question, use graph paper. You may use a flexible curve ruler. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to10 unit on the yaxis, Draw the graph of y = x 3 − 2 x − 8 for −3 ≤ x ≤4 . [ 4 marks ] (c) From your graph, find the value of y when x = 1⋅ 5, the value of x when y = −20. [ 2 marks] (a) Draw a suitabl suitablee straight straight line on your your graph graph to find find the values values of x that satisfy satisfy the the 3 equation x −9 x + 4 =0 for −3 ≤ x ≤4 . State the values of x. [ 4 marks] Answer :








13.

Diagra Diagram m 5 shows shows Cartesian plane.

four four quadri quadrilat lateral eral ABCD, ABCD, ADEF, ADEF, GHJK and LPNM drawn drawn on

Diagram

(a )

Transformation M is a reflection about the line x = 3. Transformation N is the translation

5     3 −   

State the coordinates of the image of point (2,4) under the following transformations: MN

5










Answer:

a. i) _______________________ ___________________________________ _______________________ _________________ ______ ii) ______________________ __________________________________ _______________________ __________________ _______

b. i) ______________________ __________________________________ _______________________ ___________________ ________ ii) ______________________ __________________________________ _______________________ __________________ _______ c. i) _______________________ ___________________________________ _______________________ __________________ _______ ii) _______________________ ________________________________ _________








14.

The data in Diagram 6 shows the mass in kg of 50 students in a class. 40 43 45 38 52

52 39 53 55 48

47 54 60 62 49

64 57 38 54 46

52 53 54 45 49

35 39 62 54 48

54 55 47 36 45

43 38 45 58 54

39 46 52 44 41

54 42 53 42 46

Diagram 6 (a) (a)

Base Based d on the the data data in in Diag Diagra ram m 7 and and usin using g a clas classs inte interv rval al of of 5 kg, kg, com compl plete ete Table 2 in the answer space. [4 marks] (b) From the table in a), (i) state the modal class, (ii) (ii) calc calcul ulate ate the the mea mean n mass mass of of the the stud studen ents ts and and give give you yourr answ answer er corre correct ct to to 2 significant figures. [4 marks] (c) (c) By usi using ng a scal scalee of 2 cm to to 5 kg kg on the the x-ax x-axis is and and 2 cm to to 1 stu stude dent nt on on the the y-ax y-axis is,, draw a histogram based on the data. [4 marks] Answer: (a) Mass 35-39 40-44

Frequency

Midpoint








15.

Diagram 8 shows a right prism with a horizontal horizontal rectangular base ABCD. BCHGF is the uniform cross -section of the prism. GHIJ is a horizontal plane. ABFE and DCHI are vertical planes. I 2 cm E

J D

H

3 cm 1 cm F

G

C

A 4 cm 3 cm B

X

DIAGRAM 8(i)

(a) Draw to full scale, the elevation of the solid on a vertical plane parallel to BC as seen from X [3 marks] Answer:








(b) A solid right prism with uniform cross-section cross-section KLM is removed from the solid. The remaining solid is shown in diagram 8(ii). I 2 cm E

J D

3 cm

H M

K

1 cm G

L A

C

N

1 cm

4 cm 3 cm DIAGRAM 8(ii) Y

B

Draw at full scale ( i)

the plan of the remaining solid .

[4 marks]








16.

(650 S, 400 E), Q(650 S, 600 W), R and S are four points on the surface of the earth. PR P (65 is the diameter of the parallel of latitude 650 S. (a) (i) State the longitude of R R. (ii) Calculate the shortest distance, in nautical mile, from P to R measured along the surface of the earth. [4 marks] (b) S lies north of Q and the distance of SQ measured along the surface of the earth is 5100 nautical mile. Calculate the latitude of S . [3 marks] (c) An aero plane took off from P and flew due west to Q and then flew due north to S . The average speed for the whole flight was 560 knots. Calculate (i) (i) the the dist distan ance ce,, in naut nautic ical al mil mile, e, tak taken en by by the the aer aero o pla plane ne fro from m P to Q measured along the common parallel of latitude, (ii) the total time, in hours, taken for the whole flight. [5 marks] Answer:

Kertas Model Peperiksaan Sebenar (SPM)

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