Mid-year Examination 2015/16
Mathematics
Form Three
Name :
Time allowed: 2 hrs. 15 mins.
Class :
(
)
Paper Total: 100 marks
Instructions
1. Answer ALL questions.
2. All answers in Section A should be marked in pencil on the MC answer sheet provided or marks
will be deducted.
3. Answer Sections B and C with blue/black ball pen in the Answer Book.
4. All diagrams and graphs must be drawn neatly in pencil.
5. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant
figures.
6. Anything written on the rough work sheet provided will not be marked.
Section A Multiple choice (30 marks)
1. Factorize a 2 5a b2 5b .
A. (a b)(a b 5)
B. (a b)(a b 5)
C. (a b)(a b 5)
D. (a b)(a 5)(b 5)
2.
3.
( x 1)( x 2
A.
B.
C.
D.
x 1)
3
( x 1) .
x3 1 .
x3 x 2 x 1.
x 3 2 x 2 2 x 1.
( 3x
A.
B.
C.
D.
y ) 2 ( 3x
0.
2 y2 .
6 xy .
12 xy .
y)2
4.
The cost of a television was $3500 last
year. Suppose that its value depreciates
by 40% every year. Find its value next
year.
A. $2 100
B. $1 400
C. $1 260
D. $560
5.
If the price of a book is increased by
60% and then decreased by 40%, find
the percentage change in the price of the
book.
A.
4%
B. 20%
C. 24%
D. 32%
F3_1516_ME.docx
6.
Factorize ac
A. (a b)(c
B. (a b)(c
C. (a b)(c
D. (a b)(c
7.
Which of the following values of x
satisfies the inequality 4 3x 20 ?
A.
5.34
B.
5.333
C.
5.33
D.
5.3
8.
Solve x 3
A.
B.
C.
D.
9.
x
x
x
x
bc ad
d e)
d e)
d e)
d e)
bd
ae
be .
x
.
2
2
2
1
1
If x y and k 0 , which of the
following must be true?
kx ky
I.
x k y k
II.
III.
x
y
2
k
k2
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
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Mid-year Examination 2015/16
Mathematics
10. Which of the following are factors of
x3 x2 6x ?
x
I.
II.
x 2
III. x 3
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
14. Mary has 18 coins. Each of them is either
$2 or $5 coins. The total value of the
coins is not greater than $68. What is the
greatest number of $5 coins Mary can
have?
A.
7
B.
8
C.
9
D.
10
11. If x < y < 0, which of the following must
be true?
x y 0
I.
y x 0
II.
III. x 2 y 2
15. The total surface area of a hemisphere is
72 cm2. Find its radius, correct to 3
significant figures.
A.
2.39 cm
B.
2.76 cm
C.
3.25 cm
D.
3.39 cm
A.
B.
C.
D.
I and II only
I and III only
II and III only
I, II and III
12. A sum of $60 000 is deposited at an
interest rate of 5% p.a. for 4 years,
compounded yearly. Find the amount
correct to the nearest dollar.
A. $72 000
B. $72 930
C. $73 193
D. $73 254
13. Which of the following 2-D pattern(s) can
be folded into a solid?
I.
II.
III.
A.
B.
C.
D.
I only
II only
III only
II and III only
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16. In the figure, VABC is a regular
tetrahedron. The height and volume of
VABC are 9 cm and 486 cm3 respectively.
Find the total surface area of VABC.
A.
B.
C.
D.
54 cm2
162 cm2
216 cm2
648 cm2
17. The running cost of a department store
consists of rent, 50%; labour, 30% and
electricity, 20%. If the rent decreases by
8%, the labour cost increases by 3% and
the electricity cost remains unchanged,
what is the percentage change in the
running cost?
A. 4.9%
B. 3.1%
C. +3.1%
D. +4.9%
18. Mr. Lam wants to let his flat out and earns
an annual income of $128 000 after
paying property tax. How much rent
should he charge each month if the
property tax rate is 16%? (Give the
answer correct to the nearest dollar.)
A. $6667
B. $8333
C. $12 232
D. $12 698
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Mid-year Examination 2015/16
Mathematics
19. The base radius and the slant height of a
right circular cone are 6 cm and 10 cm
respectively. Find the volume of the cone.
A. 60 cm3
B. 80 cm3
C. 96 cm3
D. 120 cm3
20. If x is an even number and 15 2 x
find the least possible value of x.
A. 6
B. 8
C. 10
D. 12
21. Factorize bc
A. (a b)(b
B. (b a)(c
C. (b a)(b
D. (a b)(a
ac
a
a
c
b
2,
(a b) 2 .
c)
b)
a)
c)
23. If the angle and radius of a sector are
decreased by x% and 50% respectively so
that its area is decreased by 85%, find x.
A. 35
B. 40
C. 60
D. 65
The sector in the following figure is
folded to form a right circular cone. Find
the base radius of the cone.
A.
B.
C.
D.
2 cm
3 cm
6 cm
12 cm
A solid sphere of radius 5 cm is cut into
two equal hemispheres. Find the
percentage increase in the total surface
area.
A. 15%
B. 20%
C. 25%
D. 50%
For questions 26 30, refer to the cuboid
ABCDEFGH shown in the following figure.
26.
22. The volume of a square-based pyramid is
150 cm3. If the height of the pyramid is
6 cm, find the length of a side of the base.
A. 5 cm
B. 3 5 cm
C. 5 3 cm
D. 25 cm
24.
25.
How many planes of reflection are there
in the cuboid?
A. 2
B. 3
C. 6
D. 7
27. The projection of D to the plane BCHG is
A. C.
B. E.
C. G.
D. H.
28. The angle between planes ABCD and
ADEF is
A.
BAE .
B.
CDE .
C.
CAF .
D.
BDE .
29. The angle between the line FC and
plane BCHG is
A.
CHF .
B.
CGF .
C.
FCG .
D.
FCH .
30. The angle between planes CDEH and
CDFG is
A.
FDG .
B.
GDE .
C.
GCE .
D.
FDE .
End of Section A
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Mid-year Examination 2015/16
Mathematics
Section B Short questions (30 marks)
1.
(a) Solve the inequality
1
2
x 3 1 x
9
3
4 and represent the solution graphically.
(b) Find the smallest integral value of x satisfying the inequality in (a).
2.
3.
(1 mark)
The population of a city increases by 10% every year.
(a) Find the percentage increase in the population over the next 3 years.
(3 marks)
(b) The population will be 322 102 five years later. Find the population this year.
(3 marks)
(a) Factorize 27 x 3
(2 marks)
(b) Hence simplify
4.
(3 marks)
y3 .
27 x 3
y 3 9 x 2 y 3xy 2
.
3x y
(4 marks)
Figure 1(a) shows a net which can be folded into a right pyramid with a square base
as shown in Figure 1(b), where AB = 12cm.
A
12cm
B
10cm
Figure 1(b)
Figure 1(a)
(a) Find the total surface area of the pyramid.
(b) Find the height of the pyramid.
(c) Find the volume of the pyramid.
5.
See page 7.
6.
See page 8.
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(2 marks)
(2 marks)
(2 marks)
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Mid-year Examination 2015/16
Mathematics
Section C Long questions
(40 marks)
7. Figure 6 shows an inverted right circular conical container. The height and the base radius of the
container are 10 cm and 4 cm respectively. It is filled with hot water to a height of 9 cm.
4 cm
Figure 6
9 cm
10 cm
(a) Find the capacity of the circular conical container in terms of
.
(2 marks)
(b) Find the curved surface area of the circular conical container.
(3 marks)
(c) By finding the radius of the water surface, find the volume of water in terms of . (3 marks)
(d) A sugar block is in a shape formed by a cylinder and two hemispheres of common bases. Its
cross-section is shown in Figure 7. The length of the block is 2.5 cm and the diameter of the
hemisphere is 1 cm.
Figure 7
1 cm
2.5 cm
(i) Find the volume of a sugar block in terms of .
(ii) At most how many sugar blocks can be added to the conical container with hot water as
shown in Figure 6 so that it will not overflow?
(6 marks)
8. Mr. Tsui is planning to rent a car for a trip of 5 days. He has checked online the rates offered by
two car rental companies, A and B, and the rates are shown in the following table.
Company Basic fee ($/day)
Free distance (km/day)
Extra distance ($/km)
A
300
200
1.9
B
500
300
1.6
use up the free distance provided every day.
(a) Which company should he choose if the total distance travelled is 1500 km? Explain your
answer.
(b) Let x km be the total distance travelled by the rental car.
(i) Show that the total cost of renting from Company A is given by $( 1.9 x 400 ).
(3 marks)
(ii) Find, correct to the nearest km, the maximum total distance travelled such that the total cost
of renting from Company A is cheaper than that from Company B.
(4 marks)
(c) Mr Tsui has estimated that his total distance travelled is 2000 km. Company C then offers him
a car rental package of $5400 including fuel. If the cost of the fuel is $1.2/km, Mr Tsui claims
that his total cost spent on fuel and renting a car from Company B is 5% higher than that in
renting from Company C. Do you agree? Explain your answer.
(3 marks)
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Mid-year Examination 2015/16
Mathematics
For questions 9 10, refer to the allowances and tax rates in the current financial year of a city shown
in the following tables.
Allowances:
($)
Basic Allowance
130 000
260 000
Child Allowances:
For each of the 1st to 9th child
80 000
For each child born during the year, the Child Allowance
80 000
will be increased by
Dependent Parent Allowance
Aged 60 or above
30 000
Aged 55 or above but below 60
15 000
Additional Dependent Parent Allowance
Aged 60 or above
30 000
Aged 55 or above but below 60
15 000
Net Chargeable Income ($)
On the first 35 000
On the next 35 000
On the next 35 000
Remainder
Standard tax rate
Rate
5%
10%
16%
23%
20%
9. Mr. and Mrs. Lee are citizens in the city and have a daughter born in this financial year. Mr. Lee
is a policeman while Mrs Lee is a housewife. They live with Mrs. Lee
-year-old father and 54year-old mother. They have to pay $8 450 for the salaries tax this year when assessed on the net
chargeable income.
(a)
Find the total allowances of the Lee family.
(b)
Find their net chargeable income.
(c)
Find Mr Lee
.
10. Let $P be the total income of a citizen, who is single, in the city.
(a) If the citizen has to pay salaries tax at the standard rate,
(i) express the amount of his salaries tax in terms of P;
(ii) show that the total income of the citizen is at least $1 440 000.
(3 marks)
(2 marks)
(2 marks)
(3 marks)
(b) Andy is a citizen in the city and is single. His total income is $1 480 000 this year and thus the
standard rate of tax is used to assess his salaries tax. In order to pay his salaries tax, he begins
to save money 12 months before the due day of paying salaries tax. A deposit of $145 000 is
saved in a bank on the first day of the 1st and the 7th month at an interest rate of 4% per annum,
compounded quarterly.
(i)
Find the amount in his account after 6 months before his second deposit.
(ii)
Find the amount in his account after 12 months, correct to the nearest dollar.
(iii) Will he have enough money to pay his salaries tax on the due day? Explain your
answer.
(6 marks)
End of Paper
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