tutormansor.files.wordpress.com · 3472/2 2008 hak cipta sbp [lihat sebelah sulit for examiner’s...

SEKOLAH BERASRAMA PENUHBAHAGIAN PENGURUSAN

SEKOLAH BERASRAMA PENUH / KLUSTERKEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN PERCUBAANSIJIL PELAJARAN MALAYSIA 2008

Kertas soalan ini mengandungi 15 halaman bercetak

3472/2 2008 Hak Cipta SBP [Lihat sebelah SULIT

For examiner’s use only

Question Total MarksMarks

Obtained1 32 33 34 35 36 37 38 39 310 411 412 413 414 315 216 317 418 319 320 321 322 323 324 425 3

TOTAL 80

MATEMATIK TAMBAHAN

Kertas 1 Dua jam

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1 This question paper consists of 25 questions. 2. Answer all questions. 3. Give only one answer for each question. 4. Write your answers clearly in the spaces provided in

the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work

that you have done. Then write down the new answer.

7. The diagrams in the questions provided are not

drawn to scale unless stated. 8. The marks allocated for each question and sub-part

of a question are shown in brackets. 9. A list of formulae is provided on pages 2 to 3. 10. A booklet of four-figure mathematical tables is provided. . 11 You may use a non-programmable scientific calculator. 12 This question paper must be handed in at the end of

the examination .

Name : ………………..……………

Form : ………………………..……

3472/1Matematik TambahanKertas 1Ogos 20082 Jam

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SULIT 3472/1

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

ALGEBRA

1 2 4

2

b b acx

a

2 am an = a m + n

3 am an = a m - n

4 (am) n = a nm

5 loga mn = log am + loga n

6 loga n

m = log am - loga n

7 log a mn = n log a m

8 logab = a

b

c

c

log

log

9 Tn = a + (n-1)d

10 Sn = ])1(2[2

dnan −+

11 Tn = ar n-1

12 Sn = r

ra

r

ra nn

−−=

−−

1

)1(

1

)1( , (r ≠ 1)

13 r

aS

−=∞ 1

, r <1

CALCULUS

1 y = uv , dx

duv

dx

dvu

dx

dy +=

2 v

uy = ,

2vdx

dvu

dx

duv

dy

dx−

= ,

3 dx

du

du

dy

dx

dy ×=

4 Area under a curve

= ∫b

a

y dx or

= ∫b

a

x dy

5 Volume generated

= ∫b

a

y 2π dx or

= ∫b

a

x 2π dy

3472/1 2008 Hak Cipta SBP [ Lihat sebelahSULIT

5 A point dividing a segment of a line

( x,y) = ,21

++

nm

mxnx

++

nm

myny 21

6 Area of triangle =

)()(2

1312312133221 1

yxyxyxyxyxyx ++−++

1 Distance = 221

221 )()( yyxx −+−

2 Midpoint

(x , y) = +

221 xx

, +

221 yy

3 22 yxr +=

4 2 2ˆ

xi yjr

x y

+=+

GEOMETRY


SULIT 3472/1

STATISTIC


3

1 Arc length, s = rθ

2 Area of sector , L = 21

2r θ

3 sin 2A + cos 2A = 1

4 sec2A = 1 + tan2A

5 cosec2 A = 1 + cot2 A

6 sin 2A = 2 sinA cosA

7 cos 2A = cos2A – sin2 A = 2 cos2A - 1 = 1 - 2 sin2A

8 tan 2A = A

A2tan1

tan2

−

TRIGONOMETRY

9 sin (A ± B) = sinA cosB ± cosA sinB

10 cos (A ± B) = cosA cosB sinA sinB

11 tan (A ± B) = BA

BA

tantan1

tantan

±

12 C

c

B

b

A

a

sinsinsin==

13 a2 = b2 + c2 - 2bc cosA

14 Area of triangle = Cabsin2

1

1 x = N

x∑

2 x = ∑∑

f

fx

3 σ = N

xx∑ − 2)( = 2_2

xN

x−∑

4 σ = ∑

∑ −f

xxf 2)( =

22

xf

fx−

∑∑

5 m = Cf

FNL

m

−+ 2

1

6 1

0

100Q

IQ

= ×

7 1

11

w

IwI

∑∑=

8 )!(

!

rn

nPr

n

−=

9 !)!(

!

rrn

nCr

n

−=

10 P(A ∪ B) = P(A)+P(B)- P(A ∩ B)

11 P (X = r) = rnrr

n qpC − , p + q = 1

12 Mean µ = np

13 npq=σ

14 z = σ

µ−x


Blank Page

3472/1 2008 Hak Cipta SBP [Lihat sebelah

SULIT

4


SULIT 3472/1

Answer all questions.

1. A function f maps the elements from set { 3,4,9,16,25}P m

to set Q = { 0, 3, 8, 15, 24 } as shown below in ordered pairs :

{ ( 3m , 0), (4, 3), (9, 8), (16, 15), (25, 24) }

(a) Write down the function notation for f.

(b) State the value of m.

(c) State the codomain.

[ 3 marks ]

Answer : ( a )……………………..

( b )...…………………...

( c )..................................

2. Given that function : 3, 0f x ax a and 2 : 16f x x b , find the value of a and of

b.

[ 3 marks ]


5 3

2

3

1



Answer : a =…………………….

b =..............................

3. Given that function : 3 , 0k

f x xx

, and 1 4

: ,f x x hx h

, find the value of h

and of k.

[ 3 markah ]

Answer : h =.........…………………

k =.....................................

4. Given m and -3 are the roots of a quadratic equation 22 4 1x x k , find the value of m and of k.

[ 3 marks ]


SULIT

6


3

4

3

3


SULIT 3472/1

Answer : m =.........………

k =.....................5. Given that quadratic equation x(3x – p) = 2x – 3 has no roots, find the range of values

of p. [ 3 marks ]

Answer : .................................

___________________________________________________________________________

6. Diagram 1 shows the graph of a curve qpxxf ++= 2)(2)( , where p and q are constants.

Diagram 1

Given the straight line 3−=x is the axis of symmetry of the curve and parallel to they-axis. Find

(a) the value of p,

(b) the value of q,

(c) the turning point of the curve )(xf .

[3 marks]

Answer : (a) .……........................


7

3

5

3

6


x

qpxxf ++= 2)(2)(

• 8−O3−=x

f(x)


(b) ……........................

(c)..................................

7. Find the range of the values of x for which )52(2)12(3 +≤− xxx .[3 marks]

Answer : ..................................

8. Solve the equation 23

2

18 −

+ =x

x .

[3 marks]

Answer : ...................................

9. Solve the equation xx 2log1)25(log 55 =−− .[3 marks]


SULIT

8

3

7

3

8



SULIT 3472/1

Answer : ....................................

10. Given that 2log2log 927 =− qp , express p in terms of q.

[4 marks]

Answer : ....……………...………..

11. An arithmetic progression has 9 terms. The sum of the first four terms is 24 and the

sum of all the odd number terms is 55. Find

a) the first term and the common difference,

b) the seventh term.

[4 marks]


9

3

8

3

9

4

10


4

11


Answer: ( a )…...…………..….......

( b).....................................

12. For a geometric progression, the sum of the first two terms is 30 and the third

term exceeds the first term by 15. Find the common ratio and the first term

of the geometric progression.

[4 marks]

Answer: …...….………..….......___________________________________________________________________________

13. Diagram 2 shows the graph of xy against 2x .

The variables x and y are related by the equation x

kxy +=2 , where k is a

constant. Find the value of h and of k.[4 marks]


SULIT

10

4

12

Forexaminer’s use

only

(0 , h)

2x

(7 , 2)

O

xy

Diagram 2


SULIT 3472/1

Answer : h = …………………….

k = ….……………….....

14. In Diagram 3 below, the equation of straight line LM is −6

x1

4=y

. The points

L and M lie on the x-axis and y-axis respectively.

Find the equation of a straight line which is perpendicular to LM and passes

through the point M.

[3 marks]

Answer : .…………………

15. In Diagram 4, OA a

and OB b

. On the same square grid , draw the line that

represents the vector 3OC a b

.

[ 2 marks ]


11

4

13

3

14


x

y

0

M

L

Diagram 3

O

A

B


16. Given (2, 5), (3,4)A B and ( , )C p q . Find the values of p and q such that

~ ~

2 9 5AB BC i j

.

[3 marks]

Answer: ……..…….…………... ___________________________________________________________________________

17. Solve the equation 3sin2cos4 xx for 3600 x .

[4 marks]

Answer: …...…………..….......18. Diagram 5 shows a sector BOC of a circle with centre O.


SULIT

12

2

15

4

16

4

17


3

16


B

Diagram 4


SULIT 3472/1

Given the perimeter of the sector BOC is 24.5 cm, calculate

(a) the radius of the sector

(b) the area of the sector.[3 marks]

Answer: a)...………………………

b) ……………………….___________________________________________________________________________

19. Given 3)35(

2)(

xxf

, find the value of (2)f .

[3 marks]

Answer:………………………


13

3

19

3

18


O

C

0.45 rad

Diagram 5


20. Find the equation of normal to the curve xxy 53 at the point when 1−=x .

[ 3 marks ]

Answer: …...…………..….......___________________________________________________________________________

21. Given that ∫ −=2

5

6)( dxxf , calculate dxx

xf∫

−

5

2 4)( .

[3 marks]

Answer: ……………………..

22. It is given that the sum of 5 numbers is α . The mean of the sum of the square is 200

and the variance is 3 2β . Express α in terms of .β

[3 marks]


SULIT

14

3

20

3

21

3

22


SULIT 3472/1

Answer: ……………………..

23. A pack of present containing three pens are chosen from 4 blue pens and 5 black pens. Find the probability of getting at least one black pen in the pack of present.

[3 marks]

Answer: …...…………..….......

___________________________________________________________________________

24.

Diagram 6

Diagram 6 shows six cards of different letters.

(a) Find the number of possible arrangements beginning with P.

(b) Find the number of these arrangements in which the vowels are separated.

[ 4 marks ]

Answer: ( a )……………………………

( b )……………………………


15


4

24

3

23

P O L I T E


25. X is a random variable which is normally distributed with mean µ and standard

deviation of 6. Find

(a) the value of µ, if the z-value is 1.5 when x = 42,

(b) P( X < 47 ).

[3 marks ]

Answer: ( a)…...…………..….......

( b )...................................

END OF QUESTION PAPER


SULIT

16

3

25



SULIT 3472/1


17


SULIT3472/1Additional MathematicsPaper 1Ogos 2008

SEKOLAH BERASRAMA PENUHBAHAGIAN PENGURUSAN

SEKOLAH BERASRAMA PENUH / KLUSTERKEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN PERCUBAAN SPM TINGKATAN 5

2008

ADDITIONAL MATHEMATICS

Paper 1

MARKING SCHEME

This marking scheme consists of 7 printed pages


PAPER 1 MARKING SCHEME 3472/1

Number Solution and marking schemeSub

MarksFull

Marks1 (a) 1)( −= xxf

(b) 4=m

(c) {0,3,8,15,24}

1

1

1

3

2

4−=a and 9−=b

4−=a or 9−=b

162 =a or - ba −=− 33

bxxfa −=− 163)]([

3

B2

B1

3

34−=k and 3−=h

4−=k or 3−=h

3)(1

−−=−

x

kxf

3

B2

B1

3

4 5=m and 31=k

−−=−+

2

4)3(m or

2

1)3(

+−=− km

0142 2 =+−− kxx

3

B2

B1

3

548 <<− p

0)4)(8( <−+ pp

0)3)(3(4)2( 2 <−−− p

3

B2

B1

3

6

(a) 3=p(b) 26−=q(c) )26,3( −−

1

1

13

2



MarksFull

Marks7

26

5 ≤≤− x

0)2)(56( ≤−+ xx

01076 2 ≤−− xx

3

B2

B1

3

81−=x

22

1)3(3 +−=+ xx

2

1)3(3

2+x or 22 +−x

3

B2

B1

3

9

12

5

512 =x

xx

25

25 =− or equivalent

3

B2

B1

3

103729qp =

1236

2

3 3loglog =q

p or

126

2

3=q

p

3log12log6log2 333 =− qp or

3log22

log2

3

log3

33 =−qp

27log

log

3

3 p or

9log

log2

3

3 q or 2

3 3log

4

B3

B2

B1

4

3



MarksFull

Marks 11 (a) 3=a and 2=d

2464 =+ da or 55205 =+ da

(b) 15

)2(637 +=T

2

B1

2

B1

4

12 12=a

2

3=r

30

15

)1(

)1( 2

=+−ra

ra

30)1( =+ ra or 15)1( 2 =−ra

3021 =+TT or 1513 =−TT

4

B3

B2

B1

4

13 3−=k

2

3−=h

7

2

2

1 h−=

22

2 kxxy +=

4

B3

B2

B1

4

14 y =

2

3− x – 4 or 832 −−= xy

)0(2

3))4(( −−=−− xy

Gradient LM = 3

2 or )0,6(L and )4,0( −M

3

B2

B1

3

4



MarksFull

Marks 15

2 2

16 11,1 =−= qp (both)

−=

−−

14

8

82

62

q

p

−

=

−

−

−

−

5

9

4

32

5

2

4

3

q

p or equivalent

3

B2

B1

3

17 °°= 96.298,04.241x or '0'0 57298,3241 and °= 90x

°= 90x or 1sin =x or 8

7sin −=x

0)7sin8)(1(sin =+− xx or

07sinsin8 2 =−− xx

03sin)sin21(4 2 =++− xx

4

B3

B2

B1

4

18 (a) r = 10 cm

2r + 0.45r = 24.5

(b) Area = 22.5 cm2

2

B1

13

1918

4'

))2(35(

18)2(

−=f

)3()35(6)( 4' −−−= −xxf

3

B2

B1

3

5



MarksFull

Marks

20 92 −−= xy

gradient of normal = 2

1−

53 2 +−= xdx

dy

3

B2

B1

3

21

8

33

6 - 5

2

2

8

x

dxx

dxxf∫ ∫−5

2

5

2 4)(

3

B2

B1

3

22α = 5000 - 75 2β

3 2β = 200 -

2

5

α

5

α=x or 22 )(200 x−=σ

3

B2

B1

3

23

21

20

1 -

××

7

2

8

3

9

4

1 - P (all blue pens) or any combination

3

B2

B1

3

6



MarksFull

Marks24

(a) 120

5P5

(b) 144

3P3 or 4P3

2

B1

2

B1

4

25 (a) 51

5.142 =−

σµ

(b) 0.3284

2

B1

1

3

7


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