This document contains instructions and questions for a mathematics preliminary examination. It consists of 7 questions testing skills in algebra, trigonometry, geometry, statistics, and problem solving. Students are instructed to show their working, use formulas provided, and give answers to a specified degree of accuracy. A total of 100 marks are available across the exam.

1. BUKIT PANJANG GOVERNMENT HIGH SCHOOL
PRELIMINARY EXAMINATIONS 2008
SECONDARY FOUR EXPRESS / FOUR N(A)’O’
MATHEMATICS 4016/2
Paper 2
18 August 2008 (Mon) 08 00 - 10 30
Additional materials:
Answer paper
Graph paper
2 hours 30 minutes
INSTRUCTION TO CANDIDATES
Write your answers and working on the separate answer paper provided.
Write your name, class and index number in the spaces provided on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use paper clips, highlighters, glue or correction fluid.
Answer all the 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 100.
________________________________________________________________________
This paper consists of 7 pages.
Setter : Mr Ng Yong Heng

2. Mathematical Formulae
Compound Interest
r n
Total amount = P (1  )
100
Mensuration
Curved surface area of a cone =  rl
Surface area of a sphere = 4 r 2
1 2
Volume of a cone = r h
3
4 3
Volume of a sphere = r
3
1
Area of triangle ABC  ab sin C
2
Arc length = r , where  is in radians
1 2
Sector area = r  , where  is in radians
2
Trigonometry
a b c
 
sin A sin B sin C
a 2  b 2  c 2  2bc cos A
Statistics
Mean =
 fx
f
 fx  fx 
2
2

Standard deviation = 
f 
 f  
Page 2

3. Answer all the questions.
1. The diagram shows a point A which lies 10 km south of a point B. The points C and D
are both 7km from B and the bearing of C from A is 035o . The points A, C, D and E all
lie on a straight line. Calculate
(a)  ACB, [2]
(b) the bearing of C from B, [2]
(c) the length of CD, [2]
(d) the shortest distance from B
to the line ACD. [2]
2. (a) Factorise completely
(i) 2 x 4  32 [3]
(ii) 9a 2  12ab  4b 2  12ac  8bc [2]
(b) A hemispherical bowl with outer radius R is made of metal with thickness t .
Show that the volume, V , of metal needed to make the bowl is given by the
2
formula V   [ R  ( R  t ) ] .
3 3
[2]
3
(i) Given a bowl in which R  10cm and t  0.7cm , evaluate V , correct to 4
significant figures. [1]
(ii) Rearrange the formula to express t in terms of  , V and R. [3]
3. Mr Bean travels by car from Town A to Town B, a total distance of 100 km, at a
constant speed of x km/h.
(a) Write down an expression, in terms of x , for the number of hours he takes to drive
from Town A to Town B.
[1]
(b) He finds that the time for the return journey is shorter by 25 minutes if he
increases his speed by 12 km/h. Write down an expression, in terms of x ,
for the number of hours he needs. [1]
(c) Form an equation in x and show that it simplifies to x 2  12 x  2880  0 [4]
(d) Solve the equation x 2  12 x  2880  0 . [2]
(e) Find the time taken for the return journey (in hour and minutes). [1]
Page 3

4. 4. (a) Given that   {x : x is an integer, 1  x  19} , A  {x : x is a prime number } and
B  {x : x is a multiple of 5},
(i) find n( A  B ), [2]
(ii) list the members of the set A  B '. [2]
(iii) find n( A  B ) ' [2]
(b) The table shows the results of four football teams in a competition.
Played Won Drawn Lost
Alberto 6 3 2 1
Bebeto 6 2 3 1
Chocolato 6 1 4 1
Danieleno 6 0 1 5
(i) The points system is: A win is awarded 3 points, a draw 1 point and a loss
gets 0 points. Write down two matrices, which, on multiplication, will display
the total number of points gained by each team. Evaluate this product.
[2]
(ii) The organizer has a reward system for teams competing in the tournament.
For every game played, each team is awarded $2 000. A winning team gets
another $3 000, whereas both teams in a draw get $1 000. No extra
monetary award is given for a game lost. Set up three matrices such that
their product will give the total amount of money paid by the organizer. [2]
5. The table below shows the exchange rates of some of the currencies displayed by a
money exchanger in Lot 1 shopping mall, Choa Chu Kang on 18 July 2008.
Currency Buying Selling
Singapore dollar (S$) Chinese Renminbi (RMB) 0.1980 0.2004
to one units of foreign
currency Malaysian Ringgit (RM) 0.4198 0.4212
(a) Mr Li from The People’s Republic of China exchanged RMB 24 950 for Singapore
dollars (S$). How much, in S$ (correct to the nearest dollar), did he receive?
[2]
(b) If Mr Li were to exchange that sum of Singapore currency back into Renminbi, how
much, in RMB (correct to the nearest Renminbi), would he lose? [3]
(c) After spending S$ 2 440 in Singapore, Mr Li divided the remainder in the ratio 3 : 2
to exchange for Malaysian Ringgit and Chinese Renminbi respectively. How much, in
RM (correct to the nearest Ringgit) and RMB (correct to the nearest Renminbi), would
he receive respectively, assuming that all the exchange rates remained unchanged)?
[4]
(d) Mr Li spent RM 2 600 in Kuala Lumpur, Malaysia. Again, he exchanged the
balance of Malaysian Ringgit for Chinese Renminbi (RMB) in Lot 1 shopping mall,
Page 4

5. Choa Chu Kang. How much, in RMB (correct to the nearest Renminbi), would he
receive? (Exchange process: RM  S$  RMB)
[2]
6. (a) In the diagram, BT is the diameter of the circle and PT is the tangent to the circle
at T . Given that ACB  68o and ABC  66o , calculate
(i) ABT [2] (ii) CAT [1] (iii) APT [2]
(b) Calculate the volume of the frustum ABCDEFGH shown in the diagram, in which
AB  DC  8cm, BC  AD  6cm, EF  HG  4cm, FG  EH  3cm and the height of the
frustum is 5cm. [3]
7. ABC is a triangle lying in a horizontal plane with BAC  90o and BC  22.5m . V is a
point vertically above A , and the angles of elevation of V from B and C are 46.6o
and 40o respectively.
(a) Show that the height of VA is
approximately 14.8m . [4]
(b) Calculate BVC . [3]
(c) Show also that the volume of
VABC is approximately 609m 3 . [3]
Page 5

6. 8. Answer the whole of this question on a sheet of graph paper.
10
The following is an incomplete table of values for the graph of y  x  8
2
x
1 1
x 1
2 1 2 2 3 4
1 11 1 1
y 12 4 10
4 a 12 b 3 2
(a) Find the values of a and b. [2]
(b) Using scales of 4 cm to represent 1 unit on the x  axis and 1 cm to represent 1
10 1
unit on the y  axis, draw the graph of y  x   8 for values of x between
2
x 2
and 4.
[3]
(c) Use your graph to find
(i) the minimum value of y ; [1]
(ii) the two values of x for which y  2 . [2]
1
(d) By drawing a tangent, estimate the gradient of the curve when x  2 . [2]
2
(e) By drawing a suitable straight line, find the solution to the equation
10
x 2   16  2 x [2]
x
9. In the diagram, AXB is an arc of a circle centre O and radius 10 cm with
 AOB = 0.9 radians. AYB is an arc of a circle centre P and radius 5 cm with
 APB =  radians.
Calculate
(a) the length of AB. [2]
(b) the value of  . [2]
(c) the length of the arc AXB. [1]
(d) the area of the segment AYB. [3]
(correct to 2 decimal places)
(e) the area of the segment AXB. [3]
Page 6

7. ( f) the area of the shaded region
bounded by the arcs AYB and
AXB. [1]
10. The diagram shows the cumulative frequency curve for the marks of 100 students in a
Science test.
Cumulative Frequency
(a) Using the graph to estimate
(i) the median, [1]
(ii) the upper quartile, [1]
(iii) the interquartile range. [1]
(iv) the fortieth percentile mark. [1]
(b) Given that 45% of the students passed the test, use the graph to find the passing
mark. [2]
(c) The same 100 students took an English test. The box-and-whisker plot below
shows the marks obtained. The maximum mark was again 80.
Page 7

8. (i) Find the interquartile range. [1]
(ii) Albert commented that the Science test was more difficult than the English
test. Do you agree? Give two reasons to support yourself. [2]
End of Paper
ANSWERS:
7. (b) 62.1o
o o
1. (a) 125.0 (b) 160
(c) 8.03cm (d) 5.73cm
2. (a) (i) 2( x 2  4)( x  2)( x  2)
(ii) (3a  2b)(3a  2b  4c) 8. (a) a = 3, b = 1
(b) Scales [1]
(b) (i) 2094 cm3
1
Coordinates [1]
Smoothness [1]
(ii) t  R   R 3  3V 
3
 2 
(c) (i) 0.7  0.9
100 100 (ii) 1.1  1.3, 2.3  2.5
3. (a) (b)
x x  12 (d) 3.10  3.70
100 100 5 (e) 0.6  0.8, 2.5  2.8
(c)  
x x  12 12
(d) x  48 or  60( NA) 9. (a) 8.70cm (b) 2.11 radians
(e) 1 hour 40 min (c) 9cm (d) 15.65 cm 2
(e) 5.83 cm 2 (f) 9.82 cm 2
4. (a) (i) 10 10. (a) (i) 39.5  41.5 marks
(ii) {2, 3, 7, 11, 13, 17, 19} (ii) 49  51 marks
(iii) 18 (iii) 21  23 marks
(iv) 35  37 marks
 3 2 1  11 (b) 43 marks
  3   
2 3 1    9 (c) (i) 28 marks
(b) (i)  1 
 1 4 1    7 (ii) I don't think so. Although the
  0    upper quartile of the English
 0 1 5  
 1  test is higher than that of the
Science test, the median of
(ii) the English test is lower than
 6 3 2 the Science test. Secondly,
  2000  the maximum mark of the
6 2 3  
 1 1 1 1   3000    76000  English test is lower than the
6 1 4 
  1000   Science test. Thirdly, the
 6 0 1  interquartile range of the
English test is also lower.
5. (a) S$ 4 940 (b) RMB 299
(c) RM 3 561; RMB 4 990 (d) RMB 2
013
6. (a) (i) 22o (ii) 44o (iii) 22o
(b) 140cm3
Page 8