· pdf fileadditional mathematics paper 2 august 2008 2 ½hours jabatan pelajaran negeri...

3472/2

SULIT

3472/2ADDITIONALMATHEMATICSPAPER 2AUGUST 20082 ½HOURS

JABATAN PELAJARAN NEGERI SABAHSIJIL 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 and Section C.

2. Answer all questions in Section A, four questions from Section B and two questionsfrom 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 : _________________-____________________

MOZ@C

SMS MUZAFFAR SYAH , MELAKAhttp://tutormansor.wordpress.com/

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ALGEBRA

1.2 4

2

b b acx

a

2. m n m na 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 a

mm n

n

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. 1nnT ar

12.( 1) (1 )

, 11 1

n n

n

a r a rS r

r r

13. , 11

aS r

r

CALCULUS

1. ,dy dv du

y uv u vdx dx dx

2.2

,

du dvv u

u dy dx dxyv dx v

3.dy dy du

dx du dx

4. Area under a curve

=b

a

y dx or

=b

a

x dy

5. Volume generated

= 2b

a

y dx or

= 2b

a

x dy

STATISTICS

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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 rrP X r C p q p q

12. Mean, μ = 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 aline

1 2 1 2, ,nx mx ny my

x ym 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

ˆxi 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

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2. Area of sector, 21

2A 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 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

A B

11.2

2 tantan 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

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5

Section A

[40 marks]

Answer all questions.

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

2 Diagram 1 shows a straight line CD which meets a straight line AB at point D. The

point C lies on the y-axis.

Diagram 1

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

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

(c) Given that CD is perpendicular to AB, find the y-intercept of CD. [3 marks]

3 (a) Sketch the graph of 3sin 2 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 2x

x 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 point P is

3, find

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(a) the coordinates of P, [2 marks]

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

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

Age

(years)<30 <35 <40 <45 <50 <55 <60

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]

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Section B

[40 marks]

Answer four questions.

7 Use graph paper to answer this question.

Table 3 shows the values of two variables, x and y, obtained from an experiment.

Variables x and y 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

(a) Plot 10log y against x 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 of y which was wrongly recorded, and estimate a more

accurate value of it,

(ii) the value of a and of b,

(iii) the value of y when x = 3.5. [6 marks]

8 Diagram 2 shows a trapezium PQRS. U is the midpoint of PQ and 2PU SV

. PV and

TU are two straight lines intersecting at W where TW : WU = 1 : 3 and PW = WV.

Diagram 2

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

.

(a) Express in terms of and/ora b

,

S R

P Q

V

TW

U

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(i) SR

,

(ii) PV

,

(iii) PW

. [5 marks]

(b) Using PT : TS = h : 1, where h is a constant, express PW

in terms of h,

and/ora b

and find the value of h. [5 marks]

9 Diagram 3 shows a circle with centre C and of radius r cm inscribed in a sector OAB

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

Diagram 3

Given that rad3

AOB

, find

(a) the value of r, [2 marks]

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

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

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

Diagram 4

The curve intersects the straight line y = k at point A, where k is a constant. The

gradient of the curve at the point A is1

4.

(a) Find the value of k. [3 marks]

(b) Hence, calculate

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

(ii) the volume generated, in terms of π, when the region R which is

bounded by the curve, the x-axis and the y-axis, is revolved through

360o about the y-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 a

minimum 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

y

Ox

1y x y = kA

R S

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[20 marks]

Answer two questions.

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

v m s–1, 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 of O when t = 4.

[Assume motion to the right is positive.]

Find

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

(b) the value of p and of q, [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]

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 of w, x, y and z. [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 acre4 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 the

above constraints. [3 marks]

(b) By using a scale of 2 cm to 4 acres on both axes, construct and shade the

region R that satisfies all the above constraints. [3 marks]

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

(i) the maximum area of tapioca planted if the area of vegetables planted

is 10 acres,

(ii) the maximum profit that the farmer can get if the profit for one acre of

vegetables and one acre of tapioca planted are RM60 and RM20

respectively. [4 marks]

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15 Diagram 5 shows a quadrilateral ABCD such that ABC is acute.

Diagram 5

(a) Calculate

(i) ABC ,

(ii) ADC ,

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

(b) A triangle AB’C has the same measurement as triangle ABC, that is, AC = 15

cm, CB’ = 9 cm and ' 30B AC , but is different in shape to triangle ABC.

(i) Sketch the triangle AB C .

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

END OF QUESTION PAPER

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13

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 bukujawapan.

Kod Pemeriksa

Bahagian SoalanSoalan

DijawabMarkahPenuh

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 10

11 10

C

12 10

13 10

14 10

15 10

Jumlah

EXCEL 2PAPER 2 MARKING SCHEME

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No. Solution and Mark SchemeSub

MarksTotal

Marks

14 3y x or equivalent

34

yx

Eliminate x or y

2 ( 4 3) 3x x x or2

3 33

4 4y y

y

Solve the quadratic equation

2 5 6 0

( 2)( 3) 0

x x

x x

2 14 45 0

9 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) 1

6 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

3

(a)

K1

N1

K1

K1

N1

P1

P1

K1

K1

N1

P1

N1

xO 2π

–3

3 1x

y

y

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Shape of sin x

Maximum = 3, minimum = –3

2 periods for 0 2x

Inverted sin x

(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 xdx

2 2 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:4 3( 1)y x

3 1y x

Equation of normal:1

4 ( 1)3

y x

3 13y x

2

4 6

P1

P1

P1

P1

N1

K1

N1

N1

K1

N1

K1

K1

N1

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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

Q141 1

Q1

N-FQ L C

f

or3

Q343 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 = ar2

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

ar = a + 8dor ar2 = a + 10d

a(r2 – 1) = 10d or a(r−1) = 8d or ar(r−1)=2d

2 1 10

1 8

r

r

4

N1

K1

K1

N1 N1

P1

P1

K1

P1

K1

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1

4r

(b) (i)1

41

8a

a = 6

(ii)1

6( ) 6 84

d or 216( ) 6 10

4d

9

16d

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 against x

(Correct axes and correct scales)

6 points plotted correctly

Draw line of best fit

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

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

a = 50

b = 1.2

(iii) 10log 1.42y

y = 26.3

4

6 10

N1

N1

N1

K1

K1

N1

P1

K1

N1

N1

N1

N1

N1

N1

K1

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

7SR a

(ii) 3SV a

3 18PV a b

(iii)

13 18

2PW a b

(b)3

4PW PU UW PU UT

or equivalent

6 (18 )1

hUT UP PT a b

h

3 186 ( 6 )

4 ( 1)

hPW a a b

h

3 27

2 2( 1)

hPW a b

h

(2)

Comparing (1) & (2)27

92( 1)

h

h

h = 2

5

5 10

N1

N1

N1

N1

K1

N1

K1

K1

P1

K1

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19

9 (a) sin 3042

r

r

14r

(b) 143

or

214

3

2 228 14

Perimeter = 24.249 + 24.249 + 29.325

= 77.823 (accept 77.82)

(c) 2114

2 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 of R or Area of S

=2

2

0

( 1)y dy5

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

or 153

Area of S or Area of R

=2

2 5 43

=1

2 5 53

=1

53

= 243

Area of R : Area of S = 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 4

1 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 43 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 2 163 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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22

13(a) w = 13.33

x = 57.50

y = 187.5

z = 150

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

Use i 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

Region R 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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23

RM15 60 + RM200

= RM900

4 10

15

(a) (i)sin 30

sin 159

o

ABC

'56.44 56 27o oABC or

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

112.41 or 112 25 'ADC

(iii)1

area of 10 8 sin112.412

ACD

1area of 15 9 sin(180 56.44 30 )

2ABC

area of quadrilateral ABCD = 36.98 + 67.37

= 104.35

(b) (i)

'AB C must be obtuse

(ii) 123.56 or 123 33’

8

2 10

K1

N1

N1

K1

K1

K1

K1

N1

N1

K1

N1

N1

B

A C

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24

GRAPH FOR QUESTION 7

10log y

0

0.2

××

×

×

×

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

×

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x1 2 3 4 5 6

GRAPH FOR QUESTION 14


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0

12

16

2

24

28

3

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y

4

8

0

2

2x+ y = 30

x+ y = 20

R

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x4 8 12 16 2010x+9 y = 180


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