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ANGLICAN HIGH SCHOOL
Preliminary Examination 2010 Secondary Four
NAME
Class / Index Number 4 /
MATHEMATICS 4016 / 01
Paper 1 17 August 2010
2 hours
Candidates answer on the Question Paper.
READ THESE INSTRUCTIONS FIRST Write in dark blue or black pen in the spaces provided on the Question Paper. Do not use staples, paper clips, highlighters, glue or correction fluid. 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 your 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 80.
For Examiner’s Use
This Question Paper consists of 14 printed pages.
Write down the model of your calculator used:
Signature of Parent/Guardian & Date
Name of Parent/Guardian 80
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Page | 2 AHS Prelim 2010 Sec 4 Mathematics Paper 1
Mathematical Formulae
Compound interest
Total amount = 𝑃 1 +𝑟
100 𝑛
Mensuration
Curved surface area of a cone = πrl
Surface area of a sphere = 4πr2
Volume of a cone = 1
3𝜋𝑟2
Volume of a sphere = 4
3𝜋𝑟3
Area of triangle ABC = 1
2 ab sinC
Arc length = 𝑟𝜃, where 𝜃 is in radians
Sector area = 1
2𝑟2𝜃, where 𝜃 is in radians
Trigonometry
𝑎
sin𝐴 =
𝑏
𝑠𝑖𝑛𝐵 =
𝑐
𝑠𝑖𝑛𝐶
𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos𝐴
Statistics
Mean =
𝑓𝑥
𝑓
Standard deviation = 𝑓𝑥2
𝑓−
𝑓𝑥
𝑓
2
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Answer all the questions.
1 Expressed as the product of prime factors,
360 = 2
3 3
2 5 and
84 = 22 3 7
Use these results to find
(a) the higher common factor of 360 and 84.
(b) the smallest positive integer k, such that 360k is a cube number.
Answer (a) [1]
(b) k = [1]
2 (a) Write the following numbers in order of size, starting with the largest.
648.2 ,13
112 ,648.2 ,684.2
(b) Calculate the simple interest when $ 12500 is invested for 9 months at 0.2 % per
annum.
Answer (a) [1]
(b) $ [1]
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3 Given that 94 p and 16 q , find
(a) the greatest possible value of p q ,
(b) the smallest possible value of 𝑝 + 𝑞
𝑞
(c) the smallest possible value of )( 22 qp
Answer (a) [1]
(b) [1]
(c) [1]
4 The rate of exchange between Japanese yen (¥) and Singapore dollars ($) was
¥ 64.1 = $ 1.
(a) Joanne bought a handbag for ¥ 23 000 in Japan. How much is the cost of the
handbag in SGD?
(b) A similar handbag costs $ 414.40 in Singapore. What is the difference in the
price between the two countries in SGD?
[Correct your answers to the nearest dollars]
Answer (a) $ [1]
(b) $ [1]
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5 (a) Write the following numbers correct to four significant figures,
(i) 385.047
(ii) 0.00192287.
Answer (a) (i) [1]
(ii) [1]
(b) Express 9
16 as a percentage.
Answer (b) % [1]
6
(a)
(i) Calculate
2.51− 1.23−7.836 3
20.7 ×96.45+18.01 , showing all the figures on your
calculator display.
(ii) Give your answer correct to 3 decimal places.
Answer (a) (i) [1]
(ii) [1]
(b) Evaluate (2.77 10 –3
) (5.911 10 –7
) ,give your answer in standard form.
Answer (b) [1]
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7 Factorise the following expressions completely
(a) ac + 2bc – 3 ad – 6 bd
(b) 2x2 – 32
Answer (a) [1]
(b) [1]
8 A map is drawn to a scale of 1: 250 000. Find
(a) the actual distance, in kilometres, of a highway which is 28 cm in length on the
map.
(b) the area, in cm2, of a garden on the map which has an actual area of 37.5 km
2.
Answer (a) km [1]
(b) cm2
[1]
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9 Liquid is poured into the empty container, as shown in the diagram, at a constant rate.
It filled the container up to a height of d1 in 5 minutes and takes another 5 minutes to
fill up to d2.
On the axes in the answer space, sketch the graph showing the depth (d) of the liquid
varies with time (t).
[2]
Answer
10 The line 𝑥
2−
𝑦
3 = 6 cuts the y–axis at the point P. Find
(a) The coordinates of the point P,
(b) The gradient of the line.
Answer (a) (……. , …….) [1]
(b) [1]
Liquid in
d2
d1
d1
d2
Depth of
liquid (d)
Time (minutes)
0 5 10
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11 In the diagram, points A, B and C are on level ground. 86ABC . C is due north
of A and B is in the direction 050 from A. Find
(a) the bearing of B from C.
(b) Find the area of ∆ 𝐴𝐵𝐶 when AC = 5 m and AB = 4 m.
Answer (a) o
[1]
(b) m2
[2]
12 Given that 13
25
3
2
x
xyy
(a) Calculate the value of y when x = −1
8
(b) Express y in terms of x.
Answer (a) y = [2]
(b) [2]
N
B
A
C
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13 The temperature at 0500 is – 12o C
The temperature at 1330 is 24o C
(a) Find the difference between the two temperatures.
(b) Assuming that the temperature rises at a steady rate, find
(i) the temperature at 0800, correct your answer to one decimal place.
(ii) the time when the temperature is 0o C
Answer (a) o
[1]
(b) (i) o
[2]
(ii) [1]
14 If = {pupils of a school}, F = {female pupils} and S = {Singaporeans}.
(a) Draw a Venn diagram to represent the above information.
[2]
(b) Shade the area that represents male Singaporean pupils.
Answer (a) & (b)
(c) Express each of the following statements in set notation.
(i) Female pupils who are Singaporeans.
(ii) Male pupils who are non Singaporeans.
Answer (c) (i) [1]
(ii) [1]
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15 The scale drawing in the answer space below shows a circle with diameter AB and Q is
a point on the circumference of the circle.
(a) Construct the bisector of angle QAB.
(b) Construct a right–angled triangle ABC such that C lie on the bisector of angle
QAB and ABC = 90o. Label the point C.
(c) Measure and write down the size of angle ACB.
Answer
[2]
Answer (c) ACB = o
[1]
A
B
Q
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16 In the diagram, AFDB and CGEB are straight lines. BDE and DEF are two isosceles
triangles and FDE = 36o.
(a) Find FEC.
Answer (a) FEC = o
[2]
(b) By construction, determine the maximum number of isosceles triangles that can
be added in the given diagram such that the vertices of the triangle(s) are on the
two given lines.
Answer (b) [2]
(c) Justify your answer in (b) with reason. [1]
Answer
C
A
36
F
E
B
D
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17 It is known that a certain parabola cuts the x–axis at –3 and 4.
(a) Write down the equations of two quadratic curves that fit this description, and
sketch them on the axes in the answer space.
Answer (a) [1]
[1]
[3]
(b) Write down the equation(s) of the line of symmetry of the curves that you have
drawn.
(b) [1]
x
y
O
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18 The points A, B, C and D lie on a circle. A smaller circle, centre C, passes through
points B, E and F and AFC is straight line. ADC = 79o and BAC = 21
o.
Giving your reasons, find
(a) angle CBA,
(b) angle FCB,
(c) angle CFE, if CF is parallel to EB.
Answer (a) CBA = ° [1]
(b) FCB = ° [1]
(c) CFE = ° [2]
79
21
F
C
A
D
B
E
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19 ABCD is a parallelogram. 𝐴𝐵 = 𝒂, 𝐵𝐶 = 𝒃, 3BX = 2XC and AE = 3
5𝐴𝑋.
(a) Given that 𝒂 = 25 and 𝒃 =
50 , find
`
(i) 𝐴𝑋
(ii) 𝐵𝐸
(iii) 𝐸𝐷
Answer (a) (i) units [2]
(ii) [1]
(iii) [1]
(iv) Hence, show that B, E and D are collinear.
Answer
[1]
(b) Write down, as a fraction in its simplest form, the value of
𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐸
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 𝐴𝐵𝐶𝐷
(b) [2]
A
B C
D
X
E a
b
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20 The cumulative frequency shows the time in minutes taken by a group of 32
competitors to complete a Sudoku puzzle.
Using the graph, to estimate
(a) the median,
(b) the interquartile range,
(c) the number of competitors that took 14 minutes or less to complete the game,
(d) the percentage of competitors that took 19.5 minutes or more to complete the
game.
Answer (a) min [1]
(b) min [2]
(c) [1]
(d) % [2]
Cumulative Frequency Curve of Time Taken To Solve Sudoko Puzzle
Number of Competitors
Time (min)
4
0
8
12
16
20
24
28
32
8 10 12 14 16 18 20 22 24 26
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21 The first four terms in a sequence of numbers, p1, p2, p3, p4, …, are given below.
p1 = 12 + 2
2 + 2
2 = 9
p2 = 22 + 3
2 + 6
2 = 49
p3 = 32 + 4
2 + 12
2 = 169
p4 = 42 + 5
2 + 20
2 = 441
(a) Write down an expression for p5 and show that p5 = 961.
Answer (a)
[1]
(b) Write down an expression for p6 and evaluate it.
Answer (b)
[1]
(c) Show that pn = n4 + 2n
3 + 3n
2 + 2n + 1.
Answer (c)
[3]
(d) Given that p10 = 102 + 11
2 + r
2 = k, express k as a perfect square in terms of r.
Answer (d) k = [1]
(e) Given that pw = w2 + (w + 1)
2 + r
2 = 5257
2 , find the value of r and of w.
Answer (e) r = [1]
w = [1]
END OF PAPER
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