MUNSANG COLLEGE
2013-2014 Second Term Examination
F. 5 Mathematics Compulsory Part
Paper 1
Class : _______
Name : _____________________
Class Number : _____
Subject teacher: CHF / CYL / LPS / MKL / MKW / WFL (Please circle as appropriate)
Time allowed :
2 hours 15 minutes
Full mark
105
:
This question-answer book consists of 24 printed pages.
Instructions to candidates:
1. This paper must be answered in English with a blue / black ball pen.
2. Write your name, class and class number in the space provided on this cover and circle the initial of your
subject teacher.
3. This paper consists of THREE sections, A(1), A(2) and B.
Section A(1) carries 35 marks, Section A(2) carries 35 marks and Section B carries 35 marks.
4. Answer ALL questions in this paper. Write your answers in the spaces provided in this Question-Answer
Book. Do not write in the margins. Answers written in the margins will not be marked.
5. Graph paper and supplementary answer sheets will be supplied on request. Write your name, class and
class number on each sheet, and fasten them INSIDE this book.
6. Unless otherwise specified, all working must be clearly shown.
7. The diagrams in this paper are not necessarily drawn to scale.
8. Unless otherwise specified, numerical answers must be exact or correct to 3 significant figures.
9. Calculator pad printed with the “HKEA Approved” / “HKEAA Approved” label is allowed.
Remove the calculator cover / jacket.
1
Section A(1) (35 marks)
m 2 mn 
3
1.
Simplify
m n
2
2
and express your answer with positive indices.
2.
Make a the subject of the formula
1 b
1

1 .
2a  b 2b
(3 marks)
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2
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(3 marks)
3.
Factorize
(a) x 2  25 y 2 ,
(b) x 2  25 y 2  2 x  10 y .
4.
Angel spent $206 buying a total of 48 apples from the supermarket, including Fuji apples and Gala
apples. The selling prices of a Fuji apple and a Gala apple are $5 and $3 respectively. Find the
numbers of Fuji apples and Gala apples purchased by Angel.
(4 marks)
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3
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(3 marks)
6.
In a polar coordinate system, O is the pole. The polar coordinates of the points A and B are 2, 20
and 4, 40 respectively. Let D be the image after reflecting B about the line OC, where
C  10, 0 .
(a) Find the polar coordinates of D.
(b) Find the length of AD in surd form.
(c) What kind of triangle is OAD ?
(5 marks)
A  {x : x is a multiple of 2 and 21  x  1051 } and
B  {x : x is a multiple of 3 and 53  x  2110 } .
(a) Write down the smallest and the biggest elements of A  B .
(b) Find the number of elements of A  B .
Let
(4 marks)
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4
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5.
7.
Find the equation of the circle passing through the points A 8, 0 , B 0,  4  and C 2, 0  .
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(4 marks)
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5
8.
A fair die is rolled twice. A two-digit integer is then formed with the two numbers obtained, the
first number being the tens digit and the second being the units digit.
(a) Write down all the two-digit integers formed in the table below.
(b) Find the probability that the two-digit integer formed is prime.
(4 marks)
Second Number (Units Digit)
1
2
3
4
5
6
2
3
4
5
6
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First Number
(Tens Digit)
1
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6
9.
Figure 1 shows the bar chart of the scores of F.5H students in an I.Q. Test.
Distribution of the scores of the I.Q. Test
18
Number of students
16
14
12
10
8
6
4
2
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1
2
3
4
5
Score
(a)
(b)
(c)
6
7
8
Figure 1
How many students are there in the class?
Find the mean score and standard deviation of the distribution.
It is known that one of the scores of the distribution is inaccurately recorded. Does it affect the
median score of the class? Explain your answer.
(5 marks)
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7
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0
Section A(2) (35 marks)
10. The following stem-and-leaf diagram shows the scores of Hennes in 31 Mathematics tests where a, b,
c and d are integers such that 0  a , b , c , d  9 . It is given that the inter-quartile range is half of the
range. The mean score equals the median score.
Stem (tens)
1
2
3
4
5
6
0
1
3
3
1
2
2
3
5
9
a
b
5
d
5
b
6
7
b
6
7
c
9
9
9
Find the values of d and the inter-quartile range.
(1 mark)
Hence find the values of a and c .
(3 marks)
Hence find value of b.
(2 marks)
A score is chosen at random from the diagram. Find the probability the units digit of the score
is greater than its tens digit.
(1 mark)
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(a)
(b)
(c)
(d)
Leaf (units)
5
0
1
1
1
0
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8
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9
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11. Let f ( x)  2 x 3  x 2  ax  b , where a and b are real constants.
It is given that f (x) is divisible by ( x  2) . When f (x) is divided by ( x  1) , the remainder
is 10 .
(a) Find the values of a and b.
(3 marks)
(b) Susan claims that f (n)  f (n  1) is even for any integer n . Do you agree? Explain your
answer.
(2 marks)
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10
12. In Figure 2, A, B, C and D are concyclic points. AC
and BD meet at point E. F and G are points on ED
and AD respectively such that AF//BC and FG//EA .
It is given that AD  4, BC  8, BE  4 and CE  6 .
(a) Write down three triangles which are similar to
EBC .
(2 marks)
(b) Find the lengths of EF and FD .
(3 marks)
(c) Find the ratio of the area of AFG to the area of
DFG.
(2 marks)
A
4
B
G
4
E F
D
6
8
Figure 2
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C
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11
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13. Phoebe owns a fish farm with an area of 300 m2 for keeping marine fish. From past experience,
breeding of fish is impossible in an artificial environment like fish farms. The growth rate of marine
fish may vary according to the population density. Also, 10% of the stock may die at the most due
to water pollution or diseases etc. In June 2014, Phoebe imports 130 fry (i.e. very young fish) for
free and stocks all of them in the farm which is initially empty. After one year, she sells all the live
fish remaining in the farm with a profit of $10 per kg. Suppose the total weight of the remaining
live fish, W kg, can be modelled by W  y  800 . In the model, y partly varies directly as x and
partly varies directly as x2, where x is the number of remaining live fish in the farm. It is given that
y  2475 when x  110 and y  2400 when x  120.
(a) Express y in terms of x.
(3 marks)
(b) Name the graph of the equation in (a) and state the equation of the axis of symmetry of the
graph.
(2 marks)
(c) Would Phoebe make a total profit of $32510 ? Explain your answer.
(3 marks)
(d) The partner of Pheobe claims that they had better not to stock 130 fry? Do you agree?
Explain your answer.
(1 mark)
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12
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13
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14. Figure 3 shows a right cone of base radius x cm and
height y cm. A right cylinder of base radius r cm
and height 2r cm is enclosed in the cone with the
upper rim touching the inner curved surface of the
cone. The cone and cylinder both stand on a
horizontal plane.
xy
(a) Show that r 
.
(3 marks)
2x  y
(b) Suppose x and y are measured to be 24 and 32
correct to the nearest integer.
Find the
minimum volume of the cylinder correct to the
nearest cm3. Explain your answer.
(4 marks)
y cm
2r cm
r cm
x cm
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Figure 3
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14
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15
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Section B (35 marks)
15. It is given that
a  9i
 3  bi , where i 2  1 and a, b are real constants.
2i
(4 marks)
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b.
Find the values of a and
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16
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16. Six letters ‘A’, ‘B’, ‘C’, ‘a’, ‘b’ and ‘c’ are arranged randomly in a row.
(a) Find the number of possible arrangements such that all the small letters are placed together.
(2 marks)
(b) If all the small letters are placed together, find the probability that ‘A’ and ‘a’ are next to each
other.
(2 marks)
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17
17. In Figure 4, P is the centre of the circle
S : x 2  y 2  12 x  18 y  k  0 . L1 is the tangent to
the circle at A(3, 5). L1 meets the x-axis at the
point B. L2 is the tangent from B to the circle at
another point C.
(a) Find the value of k, and the centre and radius
of S.
(3 marks)
(b) Find the equation of L1 .
(2 marks)
(c) Find the coordinates of B.
(1 mark)
(d) Find the length of AC in surd form without
calculating the coordinates of C.
(4 marks)
y
S
L2
P
L1
C
A
O
x
B
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Figure 4
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18
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19
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18. Franco is the manager of an international company. He joins a savings plan in June 2014 by
depositing an amount $20,000 at Munsang Bank every month, compounded at r% per month. Let
$Sn be the total amount of the savings just before (n1)th deposit where n is a positive integer. It is
given that S6 : S3 = 2.015 .
(a)
n

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(b)
(c)

20000 1  r %  1  r %   1
(2 marks)
r%
Find the value of r correct to 1 decimal place.
(3 marks)
Using the result of (b), find the least value of n such that Sn first exceeds 1,000,000.
(3 marks)
Show that S n 
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20
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21
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19. In Figure 5, VABCD is a right pyramid with a square base of side 20 m. The height of VABCD is
10 7 m.
(a) Find the length of VB and the value of cosBVA .
(3 marks)
(b) Find the angle between VCD and the base ABCD.
(2 marks)
The pyramid VABCD is then placed on a horizontal plane such that VCD is in contact with the
horizontal plane as shown in Figure 6.
(c) Find the height of B above the horizontal plane in Figure 6.
(2 marks)
(d) Two ants P and Q, each of negligible size, are initially at B and V respectively. Ants P and Q
start to crawl directly to V and A respectively at the same time along the slant edges, moving at a
constant speed of 1 m/min. Let X and Y be the positions of P and Q after t minutes. Find the
time elapsed when XY is perpendicular to VB.
(2 marks)
V
A
D
A
D
C
B
20 m
C
Figure 6
Figure 5
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22
V
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B
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23
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END OF PAPER
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24
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