This document contains instructions and questions for a mathematics exam. It provides information about the exam such as the date, time allowed, materials permitted, and instructions for completing and submitting the exam. The exam contains 7 multi-part questions testing a variety of mathematics concepts including algebra, geometry, trigonometry, statistics, and matrix operations.

1. Name Reg. No. Class 4 ( )
DUNMAN HIGH SCHOOL
PRELIMINARY EXAMINATION 2009
SECONDARY FOUR
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4016/02 MATHEMATICS
PAPER 2 100 MARKS
TUESDAY 15 SEPTEMBER 2009 2 hours 30 minutes
Additional Materials: Writing Paper
Graph paper (1 sheet)
String
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INSTRUCTIONS TO CANDIDATES
Do not open this booklet until you are told to do so.
Write your name, Register Number and class on all the work you hand in.
Write in dark blue or black pen on both sides of the paper.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid / tape.
Answer all questions.
If working is needed for any question it must be shown with the answer.
Omission of essential working will result in loss of marks.
Calculators should be used where appropriate.
If the degree of accuracy is not specified in the question, and if the answer is not exact, give
the answer to three significant figures. Give answers in degrees to one decimal place.
For π , use either your calculator value or 3.142, unless the question requires the answer in
terms of π .
At the end of the examination, fasten all you work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total of the marks for this paper is 100.
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This question paper consists of 14 printed pages.
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2. 2
Mathematical Formulae
Compound interest
n
⎛ r ⎞
Total amount = P ⎜1 + ⎟
⎝ 100 ⎠
Mensuration
Curved surface area of a cone = πrl
Surface area of a sphere = 4πr 2
1 2
Volume of a cone = πr h
3
4 3
Volume of a sphere = πr
3
1
Area of triangle ABC = ab sin C
2
Arc length = rθ , where θ is in radians
1 2
Sector area = r θ , where θ is in radians
2
Trigonometry
a b c
= =
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A.
Statistics
Mean =
∑ fx
∑f
∑ fx ⎛ ∑ fx ⎞
2
2
−⎜
⎜∑f ⎟
Standard deviation =
∑f ⎝
⎟
⎠

3. 3
p2 − q p
1 (a) Given that = , express q in terms of p. [3]
q 2
(b) Express as a fraction in its lowest terms,
3 − 2x x
− . [3]
x − 5x + 6 3 − x
2
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2
1st 2nd 3rd
pattern pattern pattern
In the diagram above, each pattern is made up of dots, lines and small triangles. In the
1st pattern, there are 9 dots, 15 lines and 7 small triangles.
(a) How many small triangles are there in the
(i) 4th pattern,
(ii) n th pattern? [2]
(b) How many lines are there in the n th pattern? [1]
(c) If there are d dots, l lines and T triangles in one of these patterns, write down
an equation connecting d, l and T. [2]
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4. 4
3 A cylindrical container which has an internal diameter of 60 cm and an internal height
22
of 1.05 m weighs 7 kg when empty. (You may assume that π = .)
7
(a) Find the weight of the container when it is full of oil, if the density of oil is
7
g/cm3 .
9
(b) How many times will the oil in the container fill a hemispherical bowl of
internal diameter of 7 cm? [5]
(c) Find the internal surface area of the hemispherical bowl in contact with the
oil. [2]
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4 In May 2007, the Credit Bureau Singapore released the following data on
Singaporeans’ home loans/ mortgages for the period from March 2005 to March
2007.
No of Singaporeans with: March 2005 March 2006 March 2007
2 or more home loans 19901 25977 41078
2 or more home loans valued at 1416 1962 2925
a total of more than S$1 million
More than S$1 million 2381 2381 4291
in home loans
The information for those Singaporeans with 2 or more home loans over this period of
⎛ 19901 ⎞
comparison can be represented by the matrix P = ⎜ 25977 ⎟ .
⎜ ⎟
⎜ 41078 ⎟
⎝ ⎠
The information for those Singaporeans with 2 or more home loans valued at a total
of more than S$1 million over this period of comparison is represented by a matrix Q.
(i) Write down the matrix Q. [1]
(ii) Calculate the matrix ( P − Q ) . [1]

5. 5
(iii) Describe what is represented by the elements of ( P − Q ) . [1]
The information for those Singaporeans with home loans in 2005 is represented by the
matrix A = (19901 1416 2381) .
Information for those Singaporeans with home loans in 2007 is represented by the
matrix B.
(iv) Write down the matrix B. [1]
(v) Show that the matrix C, in terms of A and/ or B, which has its elements
showing the increase of each category over the period of 2005 to 2007 is
( 21177 1509 1910 ) . [1]
⎛ 1 ⎞
⎜ 19901 0 0 ⎟
⎜ ⎟
1
(vi) A matrix D is given by ⎜ 0 0 ⎟ . Evaluate (100 CD ) , rounding
⎜ 1416 ⎟
⎜ ⎟
⎜ 0 1 ⎟
⎜ 0 ⎟
⎝ 2381 ⎠
off each element to the nearest whole number. [1]
(vii) Describe what is represented by the elements of the matrix (100 CD ) . [2]
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6. 6
5 In Singapore, the rate for the usage of water for the month of July in 2009 is as
follows:
Water used : $1.17 per m3
Water borne fee : $0.28 per m3
Sanitary Appliance fee : $2.80 per fitting
Water Conservation tax : 30% of the amount payable for water used (before
GST)
Goods and Services tax (GST): 7% of all the above fees/ tax
(i) In July, the GST payable for water used only by a Pasir Ris 5-room household
is $3.11.
Calculate the amount, excluding GST, paid for water used in July by this
household. [2]
(ii) Show that the amount of water used by this household in July, is
approximately 38.0 m3. [1]
(iii) Hence, by using the result found in (ii), find the overall water bill if this
household has 2 sanitary fittings. [2]
(iv) If the national average of water usage per month for a typical 5-room HDB flat
in Singapore is 19.1 m3,
(a) how many percent above average is the water usage for this
household? [2]
(b) what is the average water usage per day for a typical 5-room HDB flat
in Singapore for the month of July? [1]
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7. 7
6
C
P
D
B
42°
R A
H
The points D, H, R and P lie on the circumference of a circle. DR is a diameter of the
circle, DA is a tangent to the circle at D, RDC, RHA and CBH are straight lines and
ˆ
DRH = 42° .
(a) Find, with reason,
(i) ˆ
DHR , (ii) ˆ
RDH ,
(iii) ˆ
DAR , (iv) ˆ
RPH . [4]
(b) ˆ
Given also that DBH = 107° , find
(i) ˆ
RCH , (ii) ˆ
DHC . [2]
(c) Show that the triangles DHR and AHD are similar. [2]
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8. 8
Q
7
P
R S
A B C
18 cm
The diagram shows three semicircles each of radius 18 cm with centres at A, B and C
in a straight lines shown above. A fourth circle centre at P and with radius r cm is
drawn to touch the other three semicircles. Given that BPQ is a straight line which is
tangential to the two semicircles with centres A and C at point B,
(a) show that r = 4.5 cm. [3]
(b) ˆ
Find the value of PAC in radians. [2]
(c) Calculate the area of the shaded region. [3]
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9. 9
8
P Q
108.3° 0.874 km
1.3 km
R
North
T
26.3°
S
In the diagram, ST represents the northward-bound MRT line. The quadrilateral PQRS
formed the fence that boarded a carnival for the F1 Night Race in September. The
point P is due west of S and PS is parallel to QR. Given that PRT is a straight line,
ˆ ˆ ˆ
QR = 0.874 km, PS = 1.3 km, PQR = 108.3° RST = 26.3° and SRT = 90° . Find
(i) the bearing of R from T, [1]
(ii) the length of PR, [1]
Hence, show that PQ = 0.54 km, [2]
(iii) ˆ
QPR . [1]
(iv) The base of the Singapore Flyer is at point Q. If the angle of depression of P
from the highest point of the flyer is 8° , find the height, in metres, of the
entire flyer. [1]
(v) A man walked from P along PS and reached a point X such that the angle of
elevation of the highest point of the flyer is a maximum. Find this maximum
angle of elevation. (You may ignore the height of the man.) [3]
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10. 10
9 According to the Straits Times, a check on a random selection of basic goods at
several supermarkets in Singapore revealed an increase in the prices since the
beginning of the year. In particular, a pack of fresh chicken (between 1 to 1.3 kg) now
cost 70 cents more than its original cost at the beginning of the year.
In 2008, Yusof budgeted $234 for fresh chicken to be used during his wedding
reception in January 2009.
(i) If x represents the number of packs of fresh chicken (between 1 to 1.3 kg)
which Yusof could buy at the beginning of 2009, write down an expression, in
terms of x, for the original cost of a pack of fresh chicken (between 1 to 1.3
kg). [1]
(ii) Yusof found that he would get 7 packs of fresh chicken (between 1 to 1.3 kg)
less than that at the beginning of the year if he decided to delay the wedding
reception till September 2009.
Write down an expression, in terms of x, for the current cost of a pack of fresh
chicken (between 1 to 1.3 kg). [1]
(iii) Write down an equation in x, and show that it reduces to x 2 − 7 x − 2340 = 0 . [3]
(iv) Solve the equation x 2 − 7 x − 2340 = 0 . [2]
(v) Calculate the percentage increase in the price of a pack of fresh chicken
(between 1 to 1.3 kg). [2]
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11. 11
10 Answer the whole of this question on a sheet of graph paper.
In a recent Olympic diving event, a male participant stood on a platform and
performed a dive into the water.
During the dive, the horizontal distance of the participant away from the platform,
x m, and the corresponding vertical distance of the participant above the platform,
y m, are related by the equation
13 x2
y= x− .
10 2
Some corresponding values of x and y are given in the table below.
x 0 1 2 3 4 5 6
y 0 0.8 0.6 −0.6 −2.8 −6 p
(a) Find the value of p. [1]
(b) Using a scale of 2 cm to 1 unit, draw a horizontal x-axis for 0 ≤ x ≤ 6 .
Using a scale of 2 cm to 1 unit, draw a vertical y-axis for − 11 ≤ y ≤ 1 .
On your axes, plot the points given in the table and join them with a smooth
curve. [3]
(c) Use your graph to find the distance(s) the participant was from the platform
when he was 0.5 m above the platform. [2]
(d) Use your graph to find the maximum height above the platform reached by the
participant. [1]
(e) By drawing a tangent, find the gradient of the curve at the point (3, −0.6).
What can be said about the movement of the participant at this instant? [3]
(f) The participant entered the water when he was 4.4 m away from the platform
horizontally. Use your graph to determine the height of the platform above the
water for this participant. [1]
(g) Is the graph useful in finding the position of the participant beyond a
horizontal distance of 4.4 m? Justify your answer. [1]
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12. 12
11 A bag holds some coloured balls. There are 15 red, 3 blue and 2 white balls. Two
balls are picked from the bag at random and the colours are noted. The tree diagram
below shows the possible outcomes and some of their probabilities.
Second Pick
b Red
First Pick 3
Red 19 Blue
3
4 2
White
19
15
19 Red
3 2
20 19
Blue Blue
c
White
15
a Red
19
White d Blue
1
White
19
(a) State, leaving your answers as fractions in lowest terms, the values of a, b, c
and d. [2]
(b) Expressing your answers as a fraction in its lowest terms, find the probability
that
(i) both balls are white, [1]
(ii) at least one ball is red. [2]
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13. 13
12 In a bid to make our society more environmentally friendly, a survey was conducted
and the cumulative frequency curve shown on page 14 illustrates the number of plastic
bags used, by 200 Singaporeans in a week.
(a) Use the graph to find
(i) the median number of plastic bags used, [1]
(ii) the lower quartile, [1]
(iii) the interquartile range, [1]
(b) A person is considered to be a ‘reddie’ if he uses more than 18 plastic bags in
a week. A Singaporean is chosen at random. Calculate, leaving your answer
as a fraction in its lowest term, the probability of getting a ‘reddie’. [2]
(c) Given that 19.5% of Singaporean surveyed are ‘green crusaders’, use the
graph to find the minimum number of plastic bags used by a Singaporean who
is not a green crusader. [2]
(d) The frequency table for this set of data is given below. Showing your method
clearly, prove that the values are as shown in the table. [2]
Number of plastic Number of Singaporeans
bags used per week surveyed
0< x≤4 10
4< x≤8 29
8 < x ≤ 12 52
12 < x ≤ 16 75
16 < x ≤ 20 30
20 < x ≤ 24 4
(e) Calculate,
(i) the mean, [3]
(ii) the standard deviation. [2]
(f) A similar survey was also conducted in Hong Kong and the table below shows
the results of the processed data.
Mean 11.96 Compare, briefly, the results for the
Standard Deviation 2.90 two countries. [1]
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14. Cumulative Frequency 14
200
Cumulative Frequency
Curve showing the
190
distribution of number of
plastic bags used by 200
180 Singaporeans in a week
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0 5 10 15 20 25
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Number of plastic bags used in a week
200
--- End of Paper 2 ---