1. ANDERSON SECONDARY SCHOOL
2009 Preliminary Examination
Secondary Four Express / Four Normal / Five Normal
CANDIDATE
NAME
CENTRE INDEX
S
NUMBER NUMBER
MATHEMATICS 4016/02
Paper 2 03 September 2009
2 hours 30 minutes
Additional Materials: Writing paper (10 sheets)
Graph paper (2 sheets)
Geometrical instruments
READ THESE INSTRUCTIONS FIRST
Write your name, centre number and index number on all the work you hand in.
Write in dark blue or black pen 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.
Answer all the questions.
If working is needed for any question it must be neatly and clearly shown in the
space below the question.
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 100.
This document consists of 13 printed pages.
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

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
a 2 = b 2 + c 2 − 2bc cos A
Statistics
Σfx
Mean =
Σf
2
Σfx 2 ⎛ Σfx ⎞
Standard deviation = −⎜
⎜ Σf ⎟⎟
Σf ⎝ ⎠
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

3. 3
1 (a) (i) Factorise 2a 3 − 18a − 9 + a 2 completely. [2]
2a 3 − 18a − 9 + a 2
(ii) Hence simplify . [2]
2a 2 + 5a − 3
2x + 3 2 1
(b) Solve the equation − = . [3]
x −1 x −1 3
2
(c) Express 1 + 4 x − 2 x 2 in the form a + b( x + c) 2 , where a, b, and c are constants.
Hence, solve the equation 1 + 4 x − 2 x 2 = 0 , leaving your answers to
2 decimal places. . [3]
2
Diagram 1 Diagram 2 Diagram 3 Diagram 4
In each diagram, unit equilateral triangles shaped are arranged inside a larger
equilateral shaped triangle. The unit triangles are shaded if they have one or two of
their vertices on the edges of the larger triangle. Those with none or three of their
vertices on the larger triangle are not shaded.
The table below shows some of the patterns.
Diagram Number 1 2 3 4 5 … n
Number of shaded unit triangles 0 3 6 9 a … p
Total number of unit triangles 4 9 16 25 b … q
Number of unshaded unit triangles 4 6 10 16 c … r
(a) By considering the number patterns in the above table,
(i) state the value of a, of b and of c, [3]
(ii) find, in terms of n, an expression for p and for q. [2]
(b) By considering the relation among p, q and r, show that r = n 2 − n + 4 . [2]
(c) Form an equation relating p and q. [1]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

4. 4
3 In the triangle KLM, KL = LM and ∠LKM = ∠NLM .
K
N
L M
(a) Prove that LN = NM . [2]
(b) Prove that ΔKLM and ΔLNM are similar. [2]
(c) Given that KL = 8 cm and KM = 12 cm, find
(i) the length of LN,
area of ΔKLN
(ii) the ratio . [4]
area of ΔKLM
4 A P B
Q
D C
ABCD is a rectangle. Points P and Q lying on the lines AB and BC respectively
such that AP = AD = x cm , PB = BQ and AB = y cm .
(a) Find an expression, in terms of x and/or y, for the area of
(i) ΔAPD,
(ii) ΔBPQ. [2]
(b) If the area of ΔAPD : area of ΔBPQ = 4 : 1, form an equation in x and y, and
show that it can be simplified to 3x 2 − 8 xy + 4 y 2 = 0 . [2]
(c) Solve this equation 3x 2 − 8 xy + 4 y 2 = 0 , expressing x in terms of y. [2]
4+3 2
(d) Show that the perimeter of the quadrilateral PQCD is x cm. [3]
2
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

5. 5
5 T
North
A
35
B
52
C
A, B and C are points on horizontal ground where B is due south of A,
AB = 35 m, BC = 52 m and area of triangle ABC is 468.7 m2.
(a) Triangle ABC is represented on a map with a scale of 1 : 250 .
Find the area on the map, giving your answer correct to nearest whole. [2]
(b) Show that the bearing of C from B is 329°. [3]
(c) Find the length of AC. [2]
A vertical mast TA stands at A and the angle of elevation of T from B is 22°.
Find
(d) the height of TA, [2]
(e) the angle of depression of C from T. [1]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

6. 6
6 (a) The exchange rate between Singapore dollars (S$) and United State dollars
(US$) was S$1 = US$0.65 .
For every transaction, the bank charges a 1.5 % commission.
A traveller bought US$3500 from the bank.
Calculate the total amount, in Singapore dollars (S$), the traveller paid to
the bank. [2]
(b) A bank offers its customers different saving and investment schemes to grow
their money:
Option A: Deposit a sum of money in a special saving account to earn
an interest of 0.4 % per annum compounded monthly.
Option B: Deposit a sum of money in a fixed deposit account at a simple
interest of r % per annum.
Option C: Invest in a financial investment product which guarantees
yearly 3.2 % interest return plus a special bonus of $200 for
investing for a full 5 years.
For Option C, the following charges also apply:
An initial sales charge of 3% on the amount of money invested and
a yearly administrative fee of 1.5%.
A retiree has $ 30 000 to invest for a duration of 5 years.
(i) Calculate the amount of money he will receive at the end of 5 years if
he takes up Option A. [2]
(ii) Find the value of r, to three decimal places, if the interest received
from Option B is three times more than that obtained from Option A
at the end of 5 years. [2]
(iii) If Option C is chosen, find the percentage return to his investment
at the end of 5 years. [2]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

7. 7
7
1.4
0.6
Figure 1 Figure 2
A goldsmith made a solid gold pendant which comprises of a right cone and
a hemisphere of radius 0.6 cm. The height of the pendant is 1.4 cm.
(a) Show that the length of the slant edge of the cone is 1 cm. [1]
(b) Find, in terms of π, the total surface area of the pendant. [2]
(c) Show that the volume of pendant is 0.24π cm3. [2]
The goldsmith decided to make another solid pendant with the same amount of
material. The new pendant was in the shape of prism of length 2x cm and
the cross-section of the prism is a regular hexagon with side x cm.
(d) Find
(i) the area of hexagon, in terms of x,
(ii) the value of x. [4]
8 y
A(−1, 3)
x
B (2, −1)
The coordinates of the points A and B are (−1, 3) and (2, −1) respectively.
1
(a) The point A is reflected in the line x = .
2
State the coordinates of C, the image of A under this reflection. [1]
(b) Find the value of AB 2 . [1]
(c) Find the area of triangle ABC. [1]
(d) Hence, find the shortest distance from the point C to the line AB. [1]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

8. 8
9 (a)
In the diagram, BD is a diameter of a circle centre O.
∠APD = 22° and ∠ADB = 35° .
Find, stating the circle properties as you use them,
(i) ∠BDC,
(ii) ∠BAC,
(iii) ∠AOB. [4]
(b)
ABCD is a cyclic quadrilateral.
The tangents to the circle, centre O, at A and B meets at the point E.
BD bisects ∠CBO and ∠CAD = 18° .
Stating the circle properties as you use them, prove that
(i) the points A, E, B and O lie on the circumference of a circle. [2]
(ii) CB is parallel to DO. [3]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

9. 9
10 Answer the whole of this question on a sheet of graph paper.
A parts manufacturer makes a profit of y thousand dollars for x thousand pieces of
a certain component produced and y = 5 x − x 2 − 2 .
The table below gives some of the corresponding values of x and y.
x 0 0.5 1.0 1.5 2.0 2.5 3.0 4.0
y −2.0 0.25 2.0 3.25 4.0 4.25 4.0 2.0
(a) Using a scale of 4 cm to represent one unit on the horizontal axis and
a scale of 2 cm to represent one unit on the vertical axis, draw the graph of
y = 5 x − x 2 − 2 for 0 ≤ x ≤ 4 by joining the points with a smooth curve. [3]
(b) Use your graph to find
(i) the number of pieces of the component the company must produce
in order to obtain the maximum profit, [1]
(ii) the minimum number of pieces the company must produce in order
to cover the cost of production, [1]
(iii) the range of values of x for which the profit is more than $2850. [1]
y
(c) (i) On the same axes, draw the graph of = 1 for 0 ≤ x ≤ 3.0 . [1]
x
(ii) Write down the x-coordinate of the point where the two graphs
intersect. [1]
(iii) State briefly what the value of this x-coordinate represents. [1]
(iv) The value of x in (c)(ii) is the solution of the equation
x 2 + Ax + B = 0 . Find the value of A and of B. [2]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

10. 10
11 In an international beauty pageant, each contestant’s height was measured and
their results were displayed on the cumulative frequency curve.
60
50
40
Cumulative Frequency
30
20
10
0
1.5 1.6 1.7 1.8 1.9 2.0 2.1
Height of contestant (in metres)
(a) Using the above cumulative frequency curve, find
(i) the median height,
(ii) the interquartile range,
(iii) the minimum height of the top 10 tallest contestants. [3]
(b) On a sheet of graph paper, draw a box-and-whisker diagram to illustrate
the above information. [2]
ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

11. 11
(c) The information from the cumulative curve above is tabulated in the frequency
distribution table below.
Height
1.5 < x ≤ 1.6 1.6 < x ≤ 1.7 1.7 < x ≤ 1.8 1.8 < x ≤ 1.9 1.9 < x ≤ 2.0 2.0 < x ≤ 2.1
(x m)
Number of
p 11 21 q 5 1
contestants
Find the value of p and of q. [2]
(d) (i) Show that the mean height of the distribution is 1.77 m.
(ii) Calculate the standard deviation of the distribution. [3]
(e) If two contestants are randomly chosen, find the probability that
(i) both contestants’ heights are 1.8 m or less, [1]
(ii) at least one of the contestants is taller than 1.8 m. [1]
(f) During the pageant, all the contestants were required to wear 10-cm
high heel shoes.
Without further calculation, explain what effect the wearing of the high heel
shoes has on the mean and standard deviation. [1]
(g) A new contestant’s height, 1.77 m, is included in the computation.
Without further calculation, explain clearly the effect the new height has
on the standard deviation. [1]
ANDSS 4E5N Prelim 2009 Math (4016/02) [End of paper

12. 12
ANDERSON SECONDARY SCHOOL
Secondary Four Express / Five Normal / Four Normal
Preliminary Examination 2009
MATHEMATICS Paper 2 4016/02
1 (a) (i) (2a + 1)(a + 3)(a − 3) 9 (a) (i) ∠BDC = 33°
(2a + 1)(a − 3)
(ii) (ii) ∠BAC = 33°
2a − 1
(b) x = ±2 (iii) ∠AOB = 70°
(c) x = 2.22 or − 0.22 (to 2 d.p.) 10 (a)
2 (a) (i) a = 12 , b = 36 , c = 24 (c)(i)
(ii) p = 3(n − 1)
q = (n + 1) 2
2
⎛p ⎞
(c) q = ⎜ + 2⎟
⎝3 ⎠
16
3 (c) (i) LN = cm
3
area of ΔKLN 5
(ii) =
area of ΔKLM 9
1 2 (b) (i) 2500 pieces
4 (a) (i) x cm 2 (ii) 450 pieces
2 (iii) 1.3 < x < 3.7
1
(ii) ( y − x) 2 cm 2 (c) (ii) x = 0 .6
2
2 (iii) The number of thousands of
(c) x= y
3 pieces of the components to
5 (a) 75 cm2 be produced such that the
(b) 28.4 m profit made per piece is $1.
(c) 14.1 m (iv) A = −4 , B = 2
(d) 26.4° 11 (a) (i) 1.775 m
6 (a) S$5465.38 (to 2 d.p.) (ii) 0.15 m
(b) (i) $30605.94 (to 2 d.p.) (iii) 1.875 m
(ii) r = 0.135 (to 3 s.f.) (b)
1
(iii) 2 %
3
7 (b) 1.32π cm 2
3 3 2 p = 4 , q = 18
(d) (i) x cm 2 (c)
2
(d) Mean = 1.77
(ii) x = 0.525 (to 3 s.f.)
Standard deviation = 0.109 (to 3 s.f.)
21
8 (a) C is (2, 3) (e) (i)
59
38
(b) AB 2 = 25 (ii)
59
(f) Mean is increased by 0.1 m.
(c) 6 units2
Standard deviation remains the same.
ANDSS 4E5N Prelim 2009 Math (4016/02) – Answer Key

13. 13
12
(d) units (g) Standard deviation decreases.
5
ANDSS 4E5N Prelim 2009 Math (4016/02) – Answer Key