addmath2 q&a(sbh)

8/14/2019 AddMath2 Q&A(Sbh)

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SULIT

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ADDITIONAL

MATHEMATICS

PAPER 2

AUGUST 20082 HOURS

JABATAN PELAJARAN NEGERI SABAH

SIJIL PELAJARAN MALAYSIA TAHUN 2008

EXCEL 2

___________________________________________________________________________

ADDITIONAL MATHEMATICSPAPER 2 (KERTAS 2)

TWO HOURS THIRTY MINUTES (DUA JAM TIGA PULUH MINIT)

___________________________________________________________________________

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. This question paper consists of three sections: Section A, Section B andSection C.2. Answerall questions in Section A, four questions from Section B andtwo questions

from Section C.

3. Give only one answer / solution for each question.

4. Show your working. It may help you to get marks.

5. The diagrams in the questions provided are not drawn to scale unless stated.

6. The marks allocated for each question and sub-part of a question are shown inbrackets.

7. A list of formulae is provided on pages 2 to 4.

8. A booklet of four-figure mathematical tables is provided.

9. You may use a non-programmable scientific calculator.

___________________________________________________________________________This question paper consists of 13 printed pages.

(Kertas soalan ini terdiri daripada 13 halaman bercetak.)[Turn over (Lihat sebelah)

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

NAMA : ____________________

KELAS : _____________________

NO K.P : _____________________

A. GILIRAN : _________________-


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ALGEBRA

1.2 4

2

b b acx

a

=

2.m n m n

a a a + =

3. m n m na a a =

4. ( )m n mna a=

5. log log loga a amn m n= +

6. log log loga a am

m nn

=

7. log logna am n m=

8.log

loglog

ca

c

bb

a=

9. ( 1)nT a n d = +

10. [2 ( 1) ]2

n

nS a n d = +

11. 1n

nT ar=

12.( 1) (1 )

, 11 1

n n

n

a r a r S r

r r

= =

13. , 11

aS r

r =


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3

1.x

xN

=

2.fx

xf

=

3.

2 2

2( )x x x

xN N

= =

4.

2 2

2( )f x x fx

xf f

= =

5.

1

2

m

N Fm L c

f

= +

6. 1 100o

QI

Q=

7.i i

i

W I

I

W

=

8.( )

!!r

nnn rP=

9.( )

!

! !r

nnn r rC

=

10. ( ) ( ) ( ) ( )P A B P A P B P A B = +

11. ( ) , 1n r n r rP X r C p q p q= = + =

12. Mean, = np

13. npq =

14.x

Z

=

GEOMETRY

1. Distance

= ( ) ( )2 2

1 2 1 2x x y y +

2. Midpoint

( ) 1 2 1 2, ,2 2

x x y yx y

+ + =

3. A point dividing a segment of a

line

( )1 2 1 2

, ,

nx mx ny my

x y m n m n

+ +

= + +

4. Area of triangle =

1 2 2 3 3 1 2 1 3 2 1 3

1( ) ( )

2x y x y x y x y x y x y+ + + +

5. 2 2r x y= +%

6.2 2

xi yj

rx y

+=

+% %

%

TRIGONOMETRY

1. Arc length, s r= 8. sin ( ) sin cos cos sinA B A B A B = 9. cos ( ) os os sin sinA B c Ac B A B = m


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2. Area of sector, 212

A r=

3. 2 2sin cos 1A A+ = 4. 2 2sec 1 tanA A= + 5. 2 2cosec 1 cotA A= + 6. sin 2 2sin cosA A= 7. 2 2cos 2 cos sinA A A=

2

2

2 os 1

1 2sin

c A

A

=

=

10.tan tan

tan ( )1 tan tan

A BA B

B

=

m

11.2

2tantan 2

1 tan

AA

A=

12.sin sin sin

a b c

A B C = =

13. 2 2 2 2 cosa b c bc A= +

14. Area of triangle1

sin2

ab C=

Section A

[40 marks]

Answerall questions.

1 Solve the simultaneous equations 24 3x y x x y+ = + = . [5 marks]


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2 Diagram 1 shows a straight line CD which meets a straight lineAB at pointD. The

point Clies on they-axis.

Diagram 1

(a) State the equation ofAB in the intercept form. [1 mark]

(b) Given that 2AD =DB, find the coordinates ofD. [3 marks]

(c) Given that CD is perpendicular toAB, find they-intercept ofCD. [3 marks]

3 (a) Sketch the graph of 3sin2 for 0 2y x x = . [4 marks]

(b) Hence, using the same axes, sketch a suitable straight line to find the number

of solutions for the equation 3sin 2 =1 for 0 2xx x

+ . State the number

of solutions. [3 marks]

4 Given that the gradient of the tangent to the curve 3 22 6 9 1y x x x= + at pointPis

3, find

(a) the coordinates ofP, [2 marks]

(b) the equation of the tangent and normal to the curve atP. [4 marks]

5 Table 1 shows the distribution of the ages of 100 teachers in a secondary school.

Age

(years)


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Number of

teachers8 22 42 68 88 98 100

Table 1

(a) Based on Table 1, copy and complete Table 2.

Age

(years)25 - 29

Frequency

Table 2

[2 marks]

(b) Without drawing an ogive, calculate the interquartile range of the distribution.

[5 marks]

6 The first three terms of a geometric progression are also the first, ninth and eleventh

terms, respectively of an arithmetic progression.

(a) Given that all the term of the geometric progressions are different, find the

common ratio. [4 marks]

(b) If the sum to infinity of the geometric progression is 8, find

(i) the first term,

(ii) the common difference of the arithmetic progression. [4 marks]

Section B

[40 marks]

Answerfour questions.

7 Use graph paper to answer this question.Table 3 shows the values of two variables,x andy, obtained from an experiment.

Variablesx andy are related by the equation xy ab= , where a and b are constants.

x 1 2 3 4 5 6

y 41.7 34.7 28.9 27.5 20.1 16.7

Table 3


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(a) Plot 10log y againstx by using a scale of 2 cm to 1 unit on the x-axis and 2 cm

to 0.2 unit on the 10log y -axis.

Hence, draw the line of best fit. [4 marks]

(b) Use your graph from (a) to find

(i) the value ofy which was wrongly recorded, and estimate a more

accurate value of it,

(ii) the value ofa and ofb,

(iii) the value ofy whenx = 3.5. [6 marks]

8 Diagram 2 shows a trapeziumPQRS. Uis the midpoint ofPQ and 2PU SV =uuur uuur

.PVand

TUare two straight lines intersecting at Wwhere TW: WU= 1 : 3 andPW= WV.

Diagram 2

It is given that 12 , 18 and QR 18 5PQ a PS b b a= = = uuur uuur uuur

% % % %.

(a) Express in terms of and/ora b% %

,

(i) SRuur

,

(ii) PVuuur

,

(iii) PWuuuur

. [5 marks]

(b) UsingPT: TS= h : 1, where h is a constant, express PWuuuur

in terms ofh,

and/ora b% %

and find the value ofh. [5 marks]

9 Diagram 3 shows a circle with centre Cand of radius rcm inscribed in a sectorOAB

of a circle with centre O and of radius 42 cm. [Use = 3.142]

S R

P Q

V

TW

U


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Diagram 3

Given that rad3

AOB

= , find

(a) the value ofr, [2 marks]

(b) the perimeter, in cm, of the shaded region, [4 marks]

(c) the area, in cm2, of the shaded region. [4 marks]

10 Diagram 4 shows part of the curve 1y x= .

Diagram 4

The curve intersects the straight liney = kat pointA, where kis a constant. The

gradient of the curve at the pointA is1

4.

(a) Find the value ofk. [3 marks]

(b) Hence, calculate

(i) area of the shaded regionR : area of the shaded region S.

y

Ox

1y x= y = k

A

R S


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(ii) the volume generated, in terms of, when the regionR which is

bounded by the curve, thex-axis and they-axis, is revolved through

360o

about they-axis. [7 marks]

11 (a) A committee of three people is to be chosen from four married couples. Find

how many ways this committee can be chosen

(i) if the committee must consist of one woman and two men,

(ii) if all are equally eligible except that a husband and wife cannot both

serve on the committee. [5 marks]

(b) The mass of mango fruits from a farm is normally distributed with a mean of

820 g and standard deviation of 100 g.

(i) Find the probability that a mango fruit chosen randomly has aminimum mass of 700 g.

(ii) Find the expected number of mango fruits from a basket containing

200 fruits that have a mass of less than 700 g. [5 marks]

Section C

[20 marks]

Answertwo questions.

12 A particle moves along a straight line and passes through a fixed point O. Its velocity,

v m s1

, is given by 2 16v pt qt = + , where t is the time, in seconds, after passing

through O, p and q are constants. The particle stops momentarily at a point 64 m to

the left ofO when t= 4.

[Assume motion to the right is positive.]

Find

(a) the initial velocity of the particle, [1 mark]

(b) the value ofp and ofq, [4 marks]

(c) the acceleration of the particle when it stops momentarily, [2 marks]

(d) the total distance traveled in the third second. [3 marks]


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13 Table 4 shows the prices of four types of book in a bookstore for three successive

years.

Book

Price in year (RM) Price index in2001

based on 2000

Price index in2002

based on 2000

Weightage2000 2001 2002

P w 20 30 150 225 6

Q 50 x 65 115 130 5

R 40 50 56 125 140 3

S 80 z 150 y y 2

Table 4

(a) Find the values ofw,x,y andz. [4 marks]

(b) Calculate the composite index for the year 2002 based on the year 2001.

[4 marks]

(c) A school spent RM4, 865 to buy books for the library in the year 2002. Find

the expected total expenditure of the books in the year 2003 if the composite

index for the year 2003 based on the year 2002 is the same as for the year

2002 based on the year 2001.

[2 marks]


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14 Use graph paper to answer this question.

A farmer wants to plant x-acres of vegetables and y-acres of tapioca on his farm.

Table 5 shows the cost of planting one acre and the number of days needed to plant

one acre of vegetable and one acre of tapioca.

Vegetables Tapioca

Cost of planting

per acreRM100 RM 90

Number of days

needed per acre 4 2

Table 5

The planting of the vegetables and tapioca is based on the following constraints:

I The farmer has a capital of RM1800.

II The total number of days available for planting is 60.

III The area of his farm is 20 acres.

(a) Write down three inequalities, other than 0 and 0x y , which satisfy all theabove constraints. [3 marks]

(b) By using a scale of 2 cm to 4 acres on both axes, construct and shade theregionR that satisfies all the above constraints. [3 marks]

(c) By using your graph from (b), find


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(i) the maximum area of tapioca planted if the area of vegetables plantedis 10 acres,

(ii) the maximum profit that the farmer can get if the profit for one acre ofvegetables and one acre of tapioca planted are RM60 and RM20

respectively. [4 marks]

15 Diagram 5 shows a quadrilateralABCD such that BC is acute.

Diagram 5

(a) Calculate

(i) BC ,(ii) DC ,

(iii) the area, in cm2, of quadrilateralABCD. [8 marks]

(b) A triangleABChas the same measurement as triangleABC, that is,AC= 15

cm, CB= 9 cm and ' 30B AC = o , but is different in shape to triangleABC.

(i) Sketch the triangle AB C .

(ii) State the size of 'B C . [2 marks]

END OF QUESTION PAPER


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NO. KAD PENGENALAN

ANGKA GILIRAN

Arahan Kepada Calon

1 Tulis nombor kad pengenalan dan angka giliran anda pada ruang yang disediakan.2 Tandakan ( ) untuk soalan yang dijawab.3 Ceraikan helaian ini dan ikat sebagai muka hadapan bersama-sama dengan buku

jawapan.

Kod Pemeriksa

Bahagian SoalanSoalan

Dijawab

Markah

Penuh

Markah Diperoleh

(Untuk Kegunaan Pemeriksa)

A

1 5

2 6

3 5

4 9

5 7

6 8

B

7 10

8 10

9 10

10 1011 10

C

12 10

13 10

14 10

15 10


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Jumlah

EXCEL 2

PAPER 2 MARKING SCHEME

No. Solution and Mark SchemeSub

Marks

Total

Marks

1

4 3y x= or equivalent3

4

yx

=

Eliminatex ory

2 ( 4 3) 3x x x+ = or2

3 33

4 4

y yy

+ =

Solve the quadratic equation2

5 6 0( 2)( 3) 0x xx x

+ + =+ + =

2

14 45 09 5 0

y y( y )( y )

+ = =

3, 2x = for both values of x. y = 5, 9

9,5y = x = 3, 2

5 5

2

(a) 16 3

x y

=

(b) : 1: 2AD DB =

1(6) 2(0) 1(0) 2( 3),

3 3

+ +

( )2, 2

(c) 2CDm =

( 2) 2( 2)y x =

2 2y x= +

intercept 2y =

1

3

3 7

K1

N1

K1

K1

N1

P1

P1

K1

K1

N1

P1

N1


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3

(a)

Shape of sinx

Maximum = 3, minimum = 3

2 periods for 0 2x

Inverted sinx

(b) 1x

y

= or equivalent

Draw the straight line 1x

y

=

No. of solutions = 5

4

3 7

4 (a)26 12 9 3dy x x

dx= + =

22 1 0x x + =

( 1)( 1) 0x x =

1x = 3 22(1) 6(1) 9(1) 1

4

y = +

=

(1,4)P

(b) Equation of tangent:

2

P1

P1

P1

P1

xO 23

3 1x

y

=

N1

K1

N1

y

N1

K1

K1


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4 3( 1)y x =

3 1y x= +

Equation of normal:

14 ( 1)

3

y x =

3 13y x= + 4 6

5(a)

Age (years) Frequency,f

25 29 8

30 34 14

35 39 20

40 44 26

45 49 20

50 54 10

55 59 2

(b)1 Q1

1 Q1

L 34.5, F 22

or L =34.5 , f 20

= =

=

3 Q3

3 Q3

L 44.5, F 68

or L 44.5, f 20

= =

= =

Use1

Q14

1 1

Q1

N-FQ L C

f

= +

or3

Q34

3 3

Q3

N-FQ L C

f

= +

Interquartile Range = 46.25 35.25

= 11

2

5 7

6(a) GP : T1 = a, T2 = ar, T3 = ar

2

AP : T1 = a, T9= a + 8d, T11 = a + 10d

N1

K1

N1

N1

K1

K1

N1 N1

P1

P1


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ar= a + 8dorar2 = a + 10d

a(r2

1) = 10d ora(r1) = 8dorar(r1)=2d

2 1 10

1 8

r

r

= 1

4r=

(b) (i)1

4

1

8a

=

a = 6

(ii)1

6( ) 6 84

d= + or 21

6( ) 6 104

d= +

9

16d=

4

4 8

7(a)

x 1 2 3 4 5 6

10log y 1.620 1.540 1.461 1.439 1.303 1.223

Plot 10log y againstx

(Correct axes and correct scales)

6 points plotted correctly

Draw line of best fit

(b) (i) y = 27.5 should bey = 24.0

(ii) 10 10 10log (log ) logy b x a= +

a = 50

b = 1.2

4

K1

P1

N1

K1

N1

N1

K1

K1

N1

P1

K1

N1

N1

N1

N1

N1


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(iii) 10log 1.42y =

y = 26.3

6 10

8(a) (i) Use Triangle Law to find orSR PV

uur uuur

7SR a=uur

(ii) 3SV a=uuur

%

3 18PV a b= +uuur

% %

(iii) ( )1

3 18

2

PW a b= +uuuur

% %

(b)3

4PW PU UW PU UT = + = +uuuur uuur uuuur uuur uuur

or equivalent

6 (18 )1

hUT UP PT a b

h= + = +

+

uuur uuur uuur

% %

3 186 ( 6 )

4 ( 1)

hPW a a b

h= + +

+

uuuur

% % %

3 272 2( 1)

hPW a bh

= ++

uuuur

% %(2)

Comparing (1) & (2)27

92( 1)

h

h=

+

h = 2

5

5 10

N1

N1

N1

N1

N1

K1

N1

K1

K1

P1

K1

K1


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9 (a) sin3042

r

r=

o

14r=

(b) 143

or

214

3

2 228 14

Perimeter = 24.249 + 24.249 + 29.325

= 77.823 (accept 77.82)

(c) 21

142 3

114 588

2

Area = 2 ( 169.741 102.639)

= 134.204

Accept 134.2

2

4

4 10

10

(a)

1

2 1

dy

dx x=

1 1

42 1x=

x = 5,

k = 2

(b) (i) Area ofR or Area of S

=

2

2

0

( 1)y dy+ 5

11x dx=

3

P1

K1

N1

K1

K1

N1

K1

K1

K1

N1

K1

N1

K1

K1


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20

=

23

03

yy

+

=

( )5

32

1

1

32

x

=2

43

or1

53

Area ofS or Area of R

=2

2 5 43

=1

2 5 53

=1

53

=2

43

Area ofR : Area ofS= 7 : 8

(ii)

2

2 2

0

( 1)V y dy= +

25

3

0

2

5 3

1113

15

yV y y

V

= + +

=

7 10

11(a) (i) 4 41 2C C

= 24

(ii) If 4 4 4 2 4 3 4 43 0 2 1 1 2 0 3C or or or CC C C C C C is shown

4 4 4 2 4 3 4 4

3 0 2 1 1 2 0 3C + + + CC C C C C C

= 32

or

8 6 4 3!

8 6 4

3!

32

or

=

(b) (i)700 820

100

P( 1.2)X >

5

N1

K1

N1

K1

K1

K1

K1

N1

K1

N1

K1

N1

K1

K1

K1

N1


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21

=1 0.1151

= 0.8849

(ii) 200 x 0.1151

= 23

5 10

12(a) 16 m s

1

(b) Integrate 2 16pt qt + with respect to t

3 216

3 2

p qs t t t = +

t= 4, v = 0

16p + 4q = 16 or64

8 03

pq+ =

p = 3

q = 8

(c) a = 6t 8

t= 4, a = 16

(d) 3 24 16s t t t =

Find3

2dtv or 3 2t tS S= =

Substitute 2 or 3 into st t= =

d= |[ 3 23 4(3 ) 16(3)s = ] [ 3 22 4(2 ) 16(2)s = ]|

d= 17 m

1

4

2

3 10

K1

N1

N1

N1

K1

N1

N1

N1

K1

K1

N1

K1

K1


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13(a) w = 13.33

x = 57.50

y = 187.5

z= 150

(b) I2002 / 2001 : 150 , 113.04, 112, 100

Usei i

i

W II

W=

150 6 113.04 5 112 3 100 2

6 5 3 2I

+ + + =

+ + +

2001.2

16=

= 125.08

(c)125.08

4865100

=6085.14

4

4

2 10

14(a) 100 90 1800x y+ or equivalent

4 2 60x y+ or equivalent

20x y+ or equivalent

(b) Draw correctly at least one straight line

Draw correctly all the three straight lines

RegionR shaded correctly

(c) (i) y = 8.0 9.0

(ii) maximum point (15, 0)

3

3

N1

N1

K1

N1

N1

N1

N1

N1

N1

N1

N1

N1

K1

K1

P1

N1

K1

N1


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RM15 60 + RM200

= RM900

4 10

15

(a) (i)sin 30

sin 159

o

ABC =

'56.44 56 27

o oABC or =

(ii) 2 2 215 10 8 2(10)(8)cos ADC= +

112.41 or 112 25'ADC = o o

(iii)1

area of 10 8 sin112.412ACD = o

1area of 15 9 sin(180 56.44 30 )

2ABC = o o o

area of quadrilateral ABCD = 36.98 + 67.37

= 104.35

(b) (i)

'AB C must be obtuse

(ii) 123.56o or 123 33

8

2 10

K1

N1

N1

K1

K1

K1

K1

N1

N1

K1

N1

N1

B

A C

GRAPH FOR QUESTION 7

10log y

1.6

1.8


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24

y

32

GRAPH FOR QUESTION 14


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0

4

8

12

16

20

24

282x+ y = 30

x4 8 12 16 20

10x+9 y = 180

x+ y = 20

R

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