Additional Mathematics Module

Exercise 7 d) [ 70.2 o ii ) 34.66 cm 2 ]
c) [Answer i) 25.8o ii) 15.82 cm2 ]
Base on the diagram
ABE and DCF are given on the left
equilateral triangle Calculate
BCFE and ADFE
are rectangle. i) PUQ
Calculate ii) area of PUQ

i) ACE
ii) Area of AEB

Homework Text Book Exercise 10.3.2 page 255

145

SPM – QUESTIONS

PAPER 2
(SOLUTION OF TRIANGLE)

YEAR 2003

1. Diagram 5 shows a tent VABC in the shape of a pyramid with triangle ABC as the horizontal base. V
is the vertex of the tent and the angle between the inclined plane VBC and the base is 50.
V

AC

B

Diagram 5

Given that VB = VC = 2.2 m and AB = AC = 2.6 m, calculate [3 marks]
(a) the length of BC if the area of the base is 3 m2, [3 marks]
[4 marks]
(b) the length of AV if the angle between AV and the base is 25,

(c) the area of triangle VAB.

146

YEAR 2004

2. Diagram 6 shows a quadrilateral ABCD such that ABC is acute.

D
5.2 cm

9.8 cm 12.3 cm C

A 40.5 9.5 cm

Diagram 6 B

(a) Calculate [8 marks]

(i) ABC,

(ii) ADC,
(iii) the area, in cm2, of quadrilateral ABCD.

(b) A triangle A’B’C’ has the same measurements as those given for triangle ABC, that is, A’C’ =

12.3 cm, C’B’ = 9.5 cm and B’A’C’ = 40.5, but which is different in shape to triangle ABC.

(i) Sketch the triangle A’B’C’,

(ii) State the size of A’B’C’. [2 marks]

147

YEAR 2005
3. Diagram 7 shows triangle ABC.

A

20 cm

B C
15 cm

Diagram 7

(a) Calculate the length , in cm, of AC. [2 marks]

(b) A quadrilateral ABCD is now formed so that AC is a diagonal, ACD = 40 and

AD = 16 cm. Calculate the two possible values of ADC. [2 marks]

(c) By using the acute ADC from (b), calculate

(i) the length, in cm, of CD, [6 marks]
(ii) the area, in cm2, of the quadrilateral ABCD.

148

YEAR 2006 [2 marks]
[2 marks]
4. Diagram 5 shows a quadrilateral ABCD.
[3 marks]
D 5 cm [3 marks]
C

40
6 cm

B
9 cm
A Diagram 5

The area of triangle BCD is 13 cm2 and BCD is acute.
Calculate
(a) BCD,
(b) the length, in cm, of BD,
(c) ABD,
(d) the area, in cm2, quadrilateral ABCD.

149

YEAR 2007

5. Diagram 7 shows quadrilateral ABCD.

A

5.6 cm
B 105o

16 .4 cm

50o D
C 6 cm
Diagram 7 [4 marks]
(a) Calculate [6 marks]
(i) the length, in cm, of AC,
(ii) ACB.

(b) Point A’ lies on AC such that A’B = AB.
(i) Sketch A’BC.

(ii) Calculate the area, in cm2, of A’BC.

150

YEAR 2008

6. In Diagram 14, ABC is a triangle. ADFB, AEC and BGC are straight lines.
The straight line FG is perpendicular to BC.
A

80o
DE

F

B 45o C
G
Diagram 14

It is given that BD = 19 cm, DA = 16 cm, AE = 14 cm, DAE = 80o and FBG = 45o.

a) Calculate the length , in cm, of

(i) DE,

(ii) EC. [5 marks]

b) The area of triangle DAE is twice the area of triangle FBG.

Calculate the length , in cm, of BG. [4 marks]

c) Sketch triangle A’B’C’ which has a different shape from triangle ABC such that

A’B’ = AB, A’C’ = AC and  A’B’C’ = ABC. [1 marks]

151

1. (a) 2.70 cm Answer PAPER 2
(b) 3.149 cm C’
(c) 2.829 cm2

2. (a) (i) 57.23
(ii) 106.07
(iii) 80.96 cm2

(b) (i)

A’ B’ B

(ii) 122.77

3. (a) 19.27 cm
(b) AD1C  129.27, AD2C  50.73
(c) (i) 24.89 cm
(ii) 290.1 cm2

4. (a) 60.07 or 60 4’
(b) 5.573 cm
(c) 116.55 or 116 33’
(d) 35.43 cm2

5. (a) (i) AC = 13.36 (i) 23 53’
(b) (i) B (ii) 13.80 cm

A’ A’

C

6. (i) 19.43 (ii) 16.21 (b) BG = 10.5 cm B’ C’

152

CHAPTER 11 – INDEX NUMBER

Students will be able to:

Understand and use the concept of index number to solve problems.
1.1 Calculate index number.
1.2 Calculate price index.
1.3 Find Q or Q given relevant information.

01

1.1 Calculating index number.

Index number Is a measure used to show the change of a certain quantity for a stated period of time by
choosing a specific time as the base year. In general an index number is the comparison of a quantity at two
different times and is expressed as a percentage.

I  Q1 100
Q2

I = index number
Q1 = quantity at specific time
Qo = quantity at base time

Example 1 b) The number of cars sold by a company in the year
a) The number of visitors of national museum in the 2000 and 2002 was 6000 unit and 8400 unit
year 2000 is 1 . 4 million compared to 1.7 million in respectively. Calculate the index number of the cars
the year 2003. The index number that shows the sold in the year 2002 based on the year 2000
difference in the number of visitors in the year 2003 [ Answer 140 ]
based on the year 2000 is [ Answer 121.43 ]

Exercise 1 b) . The table below shows the number of computers

a) The number of cars sold by a company in the year sold by a company Z for year 2002 and 2004.
2000 and 2004 was 5000 unit and 8500 unit
respectively. Calculate the index number of the cars Year 2002 2004
sold in the year 2004 based on the year 2000.[ Answer
170.00] Number of computers 800 1 000

Calculate the index of the computers sold in year

2004 using 2002 as a base year [ Answer 125 ]

Homework Text Book Exercise 11.1.1 page 261

153

1.2 Calculating price index Exercise 2 [ answer 300 ]

Example 2 [ answer 125} a) The price of a particular goods in year 1997 is RM15.

a) The price of a particular goods was RM16.00 and Its price in the year 2005 is 3 times its price in year 1997.
Calculate price index of this good for year 2005 based on
RM20.00 in year 1998 and 2002 respectively. Calculate year 1997.
the price index for this goods, taking 1998 as base year

b) The table below shows the prices of an item from 2000 b) The following table shows the prices of 3 types of fruits

to 2004. for year 2001 and 2003.

Type of food Price /kilogram

Year 2000 2001 2002 2003 2004 2000 = 100 2003
Price 72 88 100 108 126
(RM) Papaya RM 1.00 RM 1.00

Durian RM 2.00 RM 2.40

Calculate the price index for each year of the given item, Banana RM 1.50 RM 1.60
using a. 2000 as a base year b. 2002 as a base year.
Calculate price index for each fruit above
Ans a). 122.2, 138.9, 150, 175 b). 113.6, 122.7, 143.2
[ Answer 100, 120, 106.7 ]

Homework Text Book Exercise 11.1.2 page 262 b Price for one tine of biscuit in year 1999 is RM12.00.
Price Index on year 2000, where 1999 as a base year is
1.3 Finding Q or Q given relevant information 105. Find the price for one tine of biscuit in year 2000.]
01 [ Answer RM12.60 ]

Example 3
a) The shoe price in year 2000 was RM 50.00 and price
index in year 2004 was 126 with year 2000 as a base

year. Find the price in year 2004. [Answer : RM96.60]

Exercise 3 b Index price for one piece of ‘roti canai’ at year 2005 was
a) Price index for one bottle of syrup in year 2000 is 130.
Take year 1998 as the base year, find the price for one equal to 150, using year 1995 as the base year. If the price
bottle of syrup in year 2000, if the price in year 1998 is for one piece of ‘roti canai’ was RM0.60 at year 2005,
RM5.00 [ Answer RM6.50 ]
calculate its price at year 1995.

154

Example 4

a The table below shows the items price and index number b Tables below shows price index for petrol, gasoline and

Items Price in Price in Price Index in gas. Find the value of x, y and z.
1990 1995 1995 based on
Items 1986 1990 1990
year 1990
(1975 = 100) (1975 = 100) (1986 = 100)
P x RM 0.70 175
Petrol 250 280 z
Q RM 2.00 RM 2.50 125
Gasoline x 180 120
R RM 2.50 y 120
Gas 175 y 140

Find the value of : [ Answer x = 150 y = 80 , z = 112 ]
(a) x [ answer 4 ]
(b) y [ answer 3 ]

Exercise 4 3 Table bellow shows the pries index for flour ,milk
Find the value of x , y and z
and sugar .Find the value of a, b and c
Price in Price in Price Index
Items 1999 2000 Items 1986 1990 1990
( 1999=100) (1975=100) (1975=100) ( 1986=100)
P RM 2.00 RM 2.50 x flour
Q RM 3.00 y milk 250 280 c
R 110 sugar a 180 120
z RM 5.00 125 140
175 b

(Answer : a = 150 , b = 245 , c = 112 )

Homework Text Book Exercise 11.1.3 page 264

Students will be able to:

2. Understand and use the concept of composite index to solve problems.

2.1 Calculate composite index.
2.2 Find index number or weightage given relevant information.

2.3 Solve problems involving index number and composite index.

2. Calculating composite index

The composite index is the weighted mean for all the items in a certain situation.
If I1 , I2, I3,….IN represent the index numbers for N items respectively weightages w1, w2,w3,….wN then the
composite index is

I   w1I1
 w1

= I1w1  I 2w2  I3w3  ..............  I N wN
w1  w2  w3  ........  wN

155

Example 5 Solution
a The following table indicates the price index and the
weightages for several types of fruits in year 2005 with
2000 as the base year.

Fruit Index Weightage

Banana 120 4
Pineapple 132 3
Mango 150 1
Guava 160 3

Calculate the composite index for year 2005 with 2000

as the base year.[ Answer 136.91 ]

Example 6 Solution
b) The following table indicates price index in year 2005
with 2000 as the base year for the material that needed
for prawn noodle and the weightages respectively.

Material Price Weightage
Index
Noodle 3
Prawn 113 2
Vegetable 104 4
118 1
Gas 102

Calculate the composite index for year 2005 with 2000 as Solution
the base year. [ Answer 112.1 ]

Exercise 7
1. The following table shows several items for Abu’s

family monthly expenses in year 2005 with the price
index and weightages using 2000 as the base year.

Material Price Index Weightage

Food 116 9

Rental 110 5

Electric and gas 112 2

Clothes 99 1

Other 115 3

Calculate the composite index for year 2005 with 2000

as the base year. [ answer 113.1 ]

Exercise 8

2. The following table indicates the price index and the Solution

weightages for several types of foods in year 2005

using 2000 as the base year.

Item Index Weight

Rice 120 5
Vegetable 136 3
125 6
Meat 104 2
Daily 115 4
Others

Calculate the composite index for year 2005 with 2000
as the base year [ Answer 121.1 ]
Homework Text Book Exercise 11.2.1 page 266

156

Finding index number or weightage given relevant information

Example 9

a) The above table shows the change in prices of four Solution

models of goods from year 2000 to year 2002 with

respective weightages.

Model Index Weight

A 110 1

B 100 2

C 98 4

Dp 3

Given that the composite index is 104.7 and year 2000
as base year, find the change in price index of model D.
[ Answer p = 115 ]

b) Table below indicates the index numbers for several goods in Solution
year 1999 and year 1990 is taken as the base year.

Goods Index Number Weight

P 110 k

Q 140 6
R 150 8
S 180 2

Given that the composite index is 142, find the value of k.
[ Answer : 4 ]

Exercise 9
1. Table above shows index numbers of certain goods in year Solution

2001 where year 2000 as is taken the base year.

Goods Index Number Weight

E 130 5
F 150 7
G m 8
H 120 2

Given that the composite index is 141, find the value of m. Solution

[ Answer m = 145.25 ]

2. Table 1 shows the price index of several type of cloths which
are sold in a shop in year 2001 and year 1995 is taken as the
base year.

Cloths Price Index, I Weight, w

Dress 130 5
Trouser 120 x

Jacket 110 4

Gown 150 3

TABLE 1
Given that the composite index is 125 at year 2001 where year
1995 is taken as the base year, find the value of x.
[ Answer x = 8 ]

Homework Text Book Exercise 11.2.2 page 267

2.3 Solving problems involving index number and composite index

157

1.Table 3 shows the price index for the monthly expenses in the year DIAGRAM 1

2005 based on the year 200 [ Ans. a RM550 b)RM 109.17 c) RM1832 Others Food (b) Based on the
information in Table 3 and
Monthly Expenses (RM) Price Index 50o Diagram 1, calculate the
composite index for the monthly
Food 110 Utility 60o 100o expenses in the year 2005
House Rental 105 based on the year 2000.
Car Instalment 100 80o 70o
120 Car House (c) Given that the monthly
Utility 115 Installment Rental expenses for the year 2005
Others is RM2000. Calculate the monthly
expenses for the year 2000
TABLE 3
(a) Given that the monthly expenses on food in the year 2000 was
RM500. Calculate the monthly expenses on food in the year 2005.

Homework Text Book Exercise 11.2.3 page 269

SPM QUESTIONS

PAPER 2

YEAR 2003
1. Diagram 1 is a bar chart indicating the weekly cost of the items P , Q , R , S and T for the

year 1990 . Table 1 shows the prices and the price indices for the items.

Weekly cost ( RM )

33
30

24
P
15
12

0

PQ R S T Items
P
DIAGRAM 1
Price Index in 1995 based
Items Price in 1900 Price in 1995 on 1990
P x RM 0.70 175
Q RM 2.50 125
R RM 2.00 RM 5.50 y
S RM 4.00 RM 9.00 150
T RM 6.00 z 120
RM 2.50

TABLE 1

158

(a) Find the value of [ 3 marks ]
(i) x
(ii) y
(iii) z

(b) Calculate the composite index for items in the year 1995 based on the year 1990 .
[ 2 marks ]

(c) The total monthly cost of the items in the year 1990 is RM 456 . Calculate the
corresponding total monthly cost for the year 1995 .
[ 2 marks ]

(d) The cost of the items increases by 20 % from the year 1995 to the year 2000 .
Find the composite index for the year 2000 based on the year 1990.
[ 3 marks ]

YEAR 2004
2. Table 2 shows the price indices and percentage of usage of four items , P , Q , R and S ,

which are the main ingredients in the production of a type of biscuit.

Price index for the year 1995 Percentage of usage

Item based on the year 1993 (%)

P 135 40

Qx 30

R 105 10

S 130 20

TABLE 2

159

(a) Calculate
(i) the price of S in the year 1993 if its price in the year 1995 is RM 37.70 ,

(ii) the price index of P in the year 1995 based on the year 1991 if its price index
in the year 1993 based on the year 1991 is 120.
[ 5 marks ]

(b) The composite index number of the cost of biscuit production for the year 1995
based on the year 1993 is 128.

Calculate
(i) the value of x ,

(ii) the price of a box of biscuit in the year 1993 if the corresponding price in the
year 1995 is RM 32 .
[ 5 marks ]

YEAR 2005
3. Table 3 shows the prices and the price indices for the four ingredients , P , Q , R and S ,

used in making biscuits of a particular kind . Diagram 2 is a pie chart which represents the
relative amount of the ingredients P , Q , R and S , used in making biscuits .

Price per kg

Ingredients ( RM ) Price index for the
year 2004 based on
P Year Year
Q the year 2001
R 2001 2004 x
S
0.80 1.00 140
150
2.00 y 80

0.40 0.60

z 0.40

TABLE 3

160

P

Q 60o
120o

S

100o

R

DIAGRAM 2 [ 3 marks ]
(a) Find the value of x , y and z .

(b) (i) Calculate the composite index for cost of making these biscuits in the year
2004 based on the year 2001 .

(ii) Hence , calculate the corresponding cost of making these biscuits in the year
2001 if the cost in the year 2004 was RM 2985 .
[ 5 marks ]

(c) The cost of making these biscuits is expected to increase by 50 % from the year 2004
to the year 2007 .
Find the expected composite index for the year 2007 based on the year 2001.
[ 2 marks ]

161

YEAR 2006
4. A particular kind of cake is made by using four ingredients , P , Q , R and S . Table 4 shows

the prices of the ingredients .

Price per kilogram ( RM )

Ingredient Year 2004 Year 2005
P
Q 5.00 w
R
S 2.50 4.00

xy

4.00 4.40

TABLE 4

(a) The index number of ingredient P in the year 2005 based on the year 2004 is 120 .

Calculate the value of w. [ 2 marks ]

(b) The index number of ingredient R in the year 2005 based on the year 2004 is 125 .
The price per kilogram of ingredient R in the year 2005 is RM 2.00 more than its
corresponding price in the year 2004 .

Calculate the value of x and of y . [ 3 marks ]

(c ) The composite index for the cost of making the cake in the year 2005 based on the
year 2004 is 127.5 .

162

YEAR 2007
5. Table 4 shows the prices and the price indices of five components , P , Q , R , S and T,

used to produce a kind of toy .
Diagram 6 shows a pie chart which represents the relative quantity of components used.

Price ( RM ) for the

Component year Price index for the
year 2006 based on
P Year Year
Q the year 2004
R 2004 2006 125
S 110
T 1.20 1.50 150
y
x 2.20 1.40

4.00 6.00

3.00 2.70

2.00 2.80

TABLE 4

SR [ 3 marks ]
72o 90o

T 36o Q
144o
P

DIAGRAM 6
(a) Find the value of x and y .

(b) (i) Calculate the composite index for the production cost of the toys in the year
2006 based 2004 .

[ 3 marks ]

(c) The price of each component increase by 20 % from the year 2006 to the year
2008 .
Given that the production cost of one toy in the year 2004 is RM 55 , calculate the
corresponding cost in the year 2008.
[ 4 marks ]

163

PAPER 2 ANSWERS (INDEX NUMBER)
1. a)
i) x = 0.40
b) ii) y  137.5
c) iii) z = 3.00
d)
2. a) I = 140.9
RM 642.5
b)
169.10
i) P93  RM 29.00
ii) I = 162
i) x = 125
ii) P93  RM 25

3. a) x 125, y = 2.80, z = 0.50

b) i) I 129.4
ii) P01  2306.80

c) Expected composite index = 194.1

4. a) w = 6.00

b) x = 8.00
y = 10.00

c) i) P04  24.00

ii) m = 4

164