[Turn over

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

The M

aths C

afe

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

The M

aths C

afe

[Turn over

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]

The M

aths C

afe

Page | 4 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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]

The M

aths C

afe

[Turn over

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]

The M

aths C

afe

Page | 6 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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]

The M

aths C

afe

[Turn over

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

The M

aths C

afe

Page | 8 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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

The M

aths C

afe

[Turn over

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]

The M

aths C

afe

Page | 10 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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

The M

aths C

afe

[Turn over

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

The M

aths C

afe

Page | 12 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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

The M

aths C

afe

[Turn over

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

The M

aths C

afe

Page | 14 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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

The M

aths C

afe

[Turn over

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

The M

aths C

afe

Page | 16 AHS Prelim 2010 Sec 4 Mathematics Paper 1

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

The M

aths C

afe