The document is the question paper for a Secondary 4 mathematics examination consisting of 11 questions testing topics including algebra, geometry, trigonometry, statistics, and sequences and series. The exam has a maximum score of 100 marks and covers areas such as simplifying expressions, solving equations, calculating lengths, areas, volumes, probabilities, and interpreting graphs. Candidates are instructed to show working, use calculators appropriately, and express answers to a given degree of accuracy.

1. MONTFORT SECONDARY SCHOOL
‘O’ LEVEL PRELIMINARY EXAMINATION 2009
________________________________________________________________
Secondary 4 Express / 5 Normal
MATHEMATICS 4016/02
PAPER 2
Wednesday 2 September 2009 1100 – 1330 2 h 30 min
Additional Materials: Answer Paper
Graph Paper (1 sheet)
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READ THESE INSTRUCTIONS FIRST
Write your name, class and register number 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.
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 number of marks for this paper is 100.
________________________________________________________________
This document consists of 11 printed pages.
Setter: A. Low

2. 2
Mathematical Formulae
Compound interest
n
 r 
Total amount = P1 + 
 100 
Mensuration
Curved surface area of cone = πrl
Surface area of a sphere = 4πr 2
1
Volume of a cone = πr 2 h
3
4 3
Volume of a sphere = πr
3
1
Area of triangle ABC = ab sin C
2
Arc length = r θ , 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 cosA
Statistics
Mean =
∑ fx
∑f
∑ fx  ∑ fx 
2 2
Standard deviation = − 
∑f ∑f 
 

3. 3
Answer all the questions
9 − 6 x + x2
1. (a) Simplify . [2]
2 x 2 − 18
(b) Express as a single fraction in its simplest form,
6 y + 5z
2− .
3 y − 4z
[2]
Express x − 7 x − 9 in the form ( x − a ) − b .
2 2
(c) (i) [1]
(ii) Hence, solve the equation x 2 − 7 x − 9 = 0 giving your answers
correct to two significant figures. [2]
2. The diagram shows a tent. BCFE is the rectangular base of the tent. ABC and
DEF are vertical ends of the tent. BC = EF = 1 m, BE = CF = 2.2 m and
AB = AC = DE = DF = 1.3 m.
.
(a) Show that the height of the tent is 1.2 m. [2]
(b) Calculate the volume of the tent, [2]
(c) Calculate the total surface area of the tent. [2]
(d) The canvas used to make the tent costs 0.02 cents per cm2. Calculate the
cost of the canvas used to make the tent. [2]

4. 4
3. A boy buys a 500 ml packet of mango juice for $1.50 and a 1 litre packet of
orange juice for $2.50.
(a) Mango juice costs x cents more per litre than orange juice. Find the value
of x. [2]
The two packets of juices are geometrically similar.
(b) Given that the height of the packet of orange juice is 25 cm, calculate the
height of the packet of mango juice. Give your answer correct to two
decimal places. [2]
The boy makes a bowl of fruit punch with the juices and water. He mixes water,
mango juice and orange juice in the ratio 1 : 2 : 5.
(c) Find the largest volume, in litres, of fruit punch he can make. [2]
(d) Find the percentage of juice left unused. [2]
4. The diagram shows a rectangle ABCD. M is the midpoint of AD and CM
intersects DN at P. BC = 3 cm, CD = 7 cm and DN = 5 cm.
A N B
5 cm 3 cm
M P
D 7 cm C
(a) Write down the value of sin ∠ BND. [1]
(b) Calculate
(i) ∠ MCD, [2]
(ii) ∠ ADN, [2]
(iii) ∠ MPD. [2]

5. 5
5. A boat travelled 200 m downstream (with the current) from A to B. AB is parallel
to the banks of the river. The speed of a boat in still water is v m/s and the speed
of the current is 2 m/s.
Upstream Downstream
A B
200 m
current
(a) Find an expression, in terms of v, for the time taken, in seconds, for the
boat to travel downstream from A to B. [1]
(b) The boat then travelled upstream (against the current) from B to A.
Find an expression, in terms of v, for the time taken, in seconds, for the
boat to travel from B to A. [1]
(c) The total time taken for the journey downstream and upstream is
4 minutes.
Write down an equation to represent this information, and show that it
simplifies to 3v 2 − 5v − 12 = 0 . [3]
(d) Solve the equation 3v 2 − 5v − 12 = 0 . [2]
(e) Find the time, in seconds, for the boat to travel from B to A. [1]

6. 6
6. The diagram shows a circle ABCE. The tangents at C and E meet at D. Angle
GCD = angle GED = 90° and angle AMB = 54°.
(a) Explain why G is the centre of the circle. [2]
(b) Find
(i) angle ACB, [1]
(ii) angle CAB, [2]
(iii) angle CDE. [3]
(c) Show that triangles ABG and ECG are similar. [2]
(d) State where the centre of the circle CDEG lies. [1]

7. 7
7. The diagram shows a horizontal field PQRS. R is due east of S.
PQ = 245 m, SR = 290 m, angle PSR = 47°, angle SPR = 70° and
angle RPQ = 36°.
(a) Calculate the bearing of P from R. [2]
(b) Show that PR = 225.7 m. [2]
(c) Calculate
(i) how far R is south of P, [2]
(ii) the distance QR, [3]
(iii) the shortest distance from P to QR. [2]
(d) A bird is 50 m vertically above P. Calculate the largest angle of
elevation of the bird when viewed from any point along QR.[2]

8. 8
8. (a) The cumulative frequency curve shows the marks for Mathematics Paper 1
of 80 candidates. The maximum mark of Paper 1 is 80.
(i) Use the graph to find
(a) the median mark, [1]
(b) the upper quartile, [1]
(c) the interquartile range, [1]
(d) the 35th percentile. [1]
(ii) To score a distinction in Paper 1, a candidate has to attain more
than 67 marks. Two candidates are chosen at random. Find the
probability both of them attained distinction. [2]

9. 9
(b) The same 80 candidates sat for Mathematics Paper 2. The box and
whiskers diagram illustrates the marks obtained. The maximum mark for
Paper 2 is 100.
(i) Find the range of marks for Paper 2. [1]
(ii) A candidate is chosen at random. Write down the probability that
this candidate attained at most 40 marks for Paper 2. [1]
(iii) Compare the marks for Papers 1 and 2 in two different ways.
[2]
(iv) Giving a reason, state whether Paper 1 or Paper 2 was more
difficult. [1]

10. 10
9. The diagram shows two circles with radii 6 cm with centres at O and at P. The
circle with centre P passes through O. The two circles intersect at A and B.
2π
(a) Explain why angle AOB = rad. [2]
3
(b) Find the perimeter of the shaded region. [2]
(c) Find the area of the shaded region. [3]
10. The first five terms of a sequence and the difference between successive terms are
shown below.
Sequence 4 13 26 43 64 a
Difference 9 13 17 21 b
(a) Write down the value of a and of b. [2]
(b) Find an expression, in terms of n, for the n th difference. [1]
(c) Find the tenth difference. [1]
(d) Given that the n th term of the sequence is given by pn 2 + qn + r , find the
value of p, of q and of r. [3]

11. 11
1 2
11. The variables x and y are connected by the equation y = x − . Some
2 x
corresponding values of x and y are given in the following table.
x 0.5 1 1.5 2 2.5 3 3.5 4
y −3.75 −1.5 p 0 0.45 0.83 q 1.5
(a) Find the value of p and of q giving your answers correct to two decimal places.
[2]
(b) Using a scale of 4 cm to represent 1unit draw a horizontal x-axis for
0.5 ≤ x ≤ 4.
Using a scale of 2 cm to represent 1 unit on the y−axis, draw a vertical y-axis
for −4 ≤ y ≤ 2.
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 range of values of x for which x 2 − 2 x − 4 ≥ 0 .
[2]
(d) By drawing a tangent, find the gradient of the curve at the point (1, −1.5).
[2]
(e) On the same axes, draw the graph of 4 x + 3 y = 6 . [1]
(f) (i) Write down the x coordinate of the point where the two graphs intersect.
[1]
(ii) The value of x is a solution of the equation Ax + Bx − 12 = 0 . Find the
2
value of A and the value of B. [2]
End of Paper
Answers
x−3 13 z
1. (a) (b)
2( x + 3) 4z − 3 y
( x − 3.5 ) − 21.25 (ii)
2
(c) (i) x = 8.1, -1.1
2. (a) 1.2 m (b) 1.32 m3 (c) 9.12 m2 (d) $18.24
3. x = 50 (b) 19.84 cm (c) 1.6 L (d) 6.67 %
4. (a) 0.6 (b) (i) 12.2° (ii) 53.1° (iii) 49.0°
200 200
5. (a) (b) (d) v = 3 (e) 200 s
v+2 v−2

12. 12
6. (a) Since GE and DE are perpendicular and GC and DC are perpendicular,
GE and GC are radii (rad perpendicular to tan). Radii of the same circle intersect
at the center. Hence G is the center.
(b) (i) 54° (ii) 36° (iii) 72°
(c) AG = EG and BG = CG (radii), angle AGB = angle EGB (vert opp)
By SAS, triangles ABG and ECG are congruent.
(d) Midpoint of DG
7. (a) 333° (c) (i) 201 m (ii) 147 m (iii) 222 m (iv) 12.7°
8. (a) 42 marks (b) 51 marks (c) 15 marks (d)39 marks
3
(ii)
632
(b) (i) 78 marks (ii) 0.25
(iii) P2 median is higher than P1 and it has a larger spread of marks.
(iv) P1 is more difficult as it has a lower median that P2.
2π
9. (a) Since PAO and PBO are equilateral triangles, angle AOB = .
3
(b) 37.7 cm (c) 68.9 cm2
10. (a) a = 89 b = 25 (b) 4n + 5 (c) 45 (d) p = 2, q = 3, r = -1
11. (a) p = -0.58, q = 1.18 (c) x ≥ 3.4 (d) 2.5
(f) (i) x = 1.75 (ii) A = 11, B = -12