(Not for sale)

Name: 葉澔謙 ( 24 )

Class: 3C

© Pearson Education Asia Limited 2010

All rights reserved

01 E B 02HK 410 03.i 1 1 2011/5/24 3 34 20

 CONTENTS

1 Rate and Ratio ................................................................................................................. 1

2 Approximation and Errors ........................................................................................... 4

3 Algebraic Fractions and Formulas.............................................................................. 7

4 Identities and Factorization ......................................................................................... 9

5 Linear Equations in Two Unknowns ........................................................................ 13

6 Laws of Integral Indices .............................................................................................. 17

7 Angles related to Rectilinear Figures ....................................................................... 20

8 Rational and Irrational Numbers ............................................................................. 25

9 Pythagoras’ Theorem................................................................................................... 28

10 Areas and Volumes (II) ............................................................................................... 32

11 Basic Concepts of Trigonometric Ratios ................................................................. 37

12 Introduction to Deductive Geometry ...................................................................... 42

13 Statistical Diagrams and Graphs............................................................................... 45

01 E B 02HK 410 03.i 1 1 2011/5/24 3 34 2

 1

1.
(a) 7 35 ( / )
5元1張
2 2
(b) 1100 km 6 600 000 ( / km )
⼈
6000 lkni

(c) 1 75 ( / )
顆1分鐘
1 25

(d) 5 540 ( / )
1.8級1秒

2.
1 2
(a) 3 : 12 = (b) 0.4 : 0.6 = (c) : =
14 23 5 3 310

3. x:y:z

(a) x : y = 1 : 2, y : z = 4 : 1 (b) x : y = 2 : 3, x : z = 2 : 5
241 23 5

4. 1:n

(a) 4 cm : 8 cm = (b) 2 cm : 1 m =
1 2 150

5. 3x = 2y x:y=

A. 1 : 2 B. 2 : 1 C. 2 : 3 D. 3 : 2
r
6. a:b:c=2:3:8 c:a=

A. 1 : 4 B. 2 : 3 C. 3 : 8 D. 4 : 1

01 B 02HK 441 01 01 1 2011/5/24 3 44 02

 7. 4R 15 cm 10 cm

A. 2 : 5 B. 3 : 2 C. 4 : 3 D. 5 : 6 io
8. 6:5 1.65 m

A. 0.9 m B. 1.375 m C. 1.525 m D. 1.98 m

9. 25 3:2

A. 10
ˇ B. 15 C. 18 D. 20
鷽

10. ABC ~ PQR h A

3 cm P
A. 3 B. 3.5 C. 4 ˇ D. 4.5
h cm
B C Q R
6 cm 4 cm

11. 16 20 m 25
30 m
(a)

攀百⼆ǒ 獨⽊橋
染
⼆1 25ms ⼆nmls

(b) 攀⽯

12. $816 9:7

傑⼆8⼼筑
志
⼆357
麗珍 816357
⼆459
傑459357
麗珍志 ⼀
102元
2

01 B 02HK 441 01 02 2 2011/5/24 3 44 04

 1

13. 3:7 45

總數 05 45
步法 150間
3九⼆315
x 105
有⾏為
i雙⼈房
14. 1:2
2:5

ii
5 104
華英法⼆5 104

15. 1 : 16 000 14 cm
(a) ( km )

nooaiǜ
14

器
zitkm
(b) 2 ( km/h )

2.24混
⼆6
下Ikmlh

16. 2 cm : 0.75 m 45 cm
( m )

⺠5⼆步
北上33.75
九⼆16 9m

01 B 02HK 441 01 03 3 2011/5/24 3 44 0

 2

(1 − 4)

1. 2048 = 2000 ( ) 2. 30 217 = 30220 ( )

3. 42.53 = ( ) 4. 0.0258 = 0026 ( )

425

5. 1538
(a) 1450
= (1538 ) - ( 1450 )

=
88
(b) 1600
= (1600 ) - (1538 )

=
62

6. 78 m 2m
( )
(a) = 22 m

= m
1
(b) = m+ 1 m
78
= m
79
(c) = m- m
78
=
In
m

01 B 02HK 441 01 04 4 2011/5/24 3 44 10

 14. 1.6 kg
(a)

i
_o05kg
(b)

5xl00
3.125
15.
(a) 0.1 kg
1 8
1.8 1.9
1.7 2.0

(b) ( ) kg

台 x100

2.8
16. 288 cm 192 cm 288
1 cm
(a)
in
的上限
長度 ⼆ 垃
288
288 5

(b)
鷳的上限 hi
的
⾯積 2881925
上限
以5
5553

17. 2.5 mL 0.04

(a)

者0.04
x 0.1
(b)

上限⼆2.50.1
⼆2.6
下限⼆2.5 a
⼆2 4

01 B 02HK 441 01 0 2011/5/24 3 44 13

 3 Algebraic Fractions and Formulas

Exercise

True / False Questions

1. Determine whether each of the following equality is correct. If it is correct, put a ‘✓’ in the box, otherwise, put a ‘✗’.
a 0 3 4 3× 4
(a) = (b) × =
3a 3 x a b a×b v
5 6 5+6 1 3 1 3
(c) + = (d) + = -
x y x+y x x-2 2- x x-2 x-2

Short Questions

Simplify the following expressions. (2 – 5)

2a b x2 2x
2. × = 3. ÷ =
3b 4a
𨘋6 y y3 型
4 1 3 5
+ = - =
i
4. 5.
2x 2x 是 ab b
ii
In each of the following formulas, find the value of the unknown with the condition given in the bracket. (6 – 7)

V -R
6. A = x2 + 3 [x = 2] 7. P= [V = 4, R = 6]
2
( )-( )
A=( 2 )2 + 3 =
At P= 4 6 =
2 p

Long Questions

Simplify the following expressions. (8 – 10)

2b 4 a2 b3 a+b 15 4a - 2b 2a - b
8. × ÷ 9. × 10. ÷
3a 3
4b 6a 3a - 6b 5a + 5b a + 2b 2a 2 + 4ab

= = =
㵘筋筋 幾叭⽪叔 與
恐
x2day
31
⼭ ⼆
4an
7

01 E B 02HK 410 03.i 2011/5/24 3 34 5

 Junior Secondary Summer Exercise (S2 to S3)

Simplify the following expressions. (11 – 13)

1 1 1 3b 2a - b 5 3
11. + - 12. + 13. +
2a 3a 4a a + 2b 2 a + 4b y-3 6- 2y

= = =
iiiáia Ǜiii Ǚyb
⼆
ù 5btza
izy
2a41
In each of the following, make the letter in brackets the subject of the formula. (14 – 15)

Bx
14. x = L(1 + a) [a] 15. P = A - [x]
3

九⼆ Ltd
䛓 AP
器台 Bi 3 AP
公 xi x 3 A.pl
B
16. The cost price $C of producing n CDs is given by the following formula:
C = 250 + 1.75n
(a) A student union plans to produce 1000 CDs for the school open
day. Find the cost of each CD.

Cisotlt5110001
2000
⼀
thecostofeachCD 器
⼆2
(b) If the budget is $1650, how many CDs can be produced?
1650250 1的n
n 800

17. In a bookstore, each comic book costs $25 and each novel costs $40.
(a) If Amy pays $T for x comic books and y novels, express T in
terms of x and y.

F25x40y
(b) Tony has $250. After buying 4 novels, at most how many comic
books can he buy?

250 25x41401
3.6
8
ihecanbuy3comic
books
atmost
01 E B 02HK 410 03.i 2011/5/24 3 35 03
 4 Identities and Factorization

Exercise

True/ False Questions

1. Determine whether each of the following equations is an identity. If it is an identity, put a ‘✓’ in the box,
otherwise, put a ‘✗’.
(a) 3x = x + 2 x (b) 2( x - 1) = 2 x - 1 x
2 2 2
(c) (2 x + 1) = 4 x + 1 (d) ( x - 5)( x + 5) = x - 25
x

Short Questions

Find the values of A and B in the following identities. (2 – 3)

2. 2x + 1 ≡ Ax + B A= ,B= 3. 4(x - 3) ≡ Ax + B A= ,B= 12

2 1 4
Factorize the following expressions. (4 – 7)

4. 5q + 10 = 5. 3z2 - 6z =
51 多的21
6. a2b - ab2 = abla.by 7. 2c - 4d + 8e =
unite
Expand the following expressions. (8 – 11)
8. (2 x - y )(2 x + y ) 9. 3( x - 3)( x + 3)
= =
4ij 3 in
3x

=
Itfcill5ann.li
10. (5a + 2)2

25⼼20a 4
11. (2b - 7c )2
=
必212117c i
4528bct49c

01 E B 02HK 410 03.i 2011/5/24 3 35 12

 Junior Secondary Summer Exercise (S2 to S3)

Multiple Choice Questions

12. If (2x + 3) - (x - 2) = Px + Q is an identity, find the values of P and Q.

A. P = 1, Q = 1 B. P = 1, Q = 5 C. P = 2, Q = 1 D. P = 2, Q = 5
ˇ

13. Factorize 2(m + n) - m(m + n).

A. (2 + m)(m + n) B. (2 + m)(m - n) C. (2 - m)(m + n) D. (2 - m)(m - n)

ˇ
14. Factorize h2 - 8h + 16.

A. (h + 2)2 B. (h + 4)2 C. (h - 2)2 D. (h - 4)2

ˇ

NF 15. Factorize 8x3 - y3.

A. (2 x 2 - y 2 )(4 x + 2 xy + y ) B. (2 x - y )(4 x 2 + 2 xy + y 2 )
ˇ
C. (4 x 2 + y 2 )(4 x - 2 xy + y ) D. (2 x + y )(4 x 2 - 2 xy + y 2 )

Long Questions

16. Find the values of A and B in the following identities.

(a) 3( x + 2) + A ≡ Bx + 5 (b) (2 x - 1)( x + 4 ) ≡ 2 x 2 + Ax - B

死6 ABx 5 ⼼8x_x4 2itAx

i.tt B 3 BitBi.tt
17. Prove that each of the following equations is an identity.
(a) 3x + 8 = 2( x + 4 ) + x (b) ( x + 1)2 = ( x - 1)2 + 4 x

LHS⼆死 8 L.H.si 2 x 1 12
⼼⼀ itzxtl
3 8
R.H.si 2 x 14x
x2 ti 4x
Yes xizx 1
18. Factorize the following expressions by using grouping terms method.x2 2x 1
(a) m + n + am + an (b) 3 + 3x - 4b - 4bx
i Yes
=mtntalmtnl = 3 1 x 4blltx
⼆lmtn at1

10

01 E B 02HK 410 03.i 10 10 2011/5/24 3 35 22

 4 Identities and Factorization

19. Factorize the following expressions by using the identity of difference of two squares.
(a) 16a 2 - 1 (b) 4 - 25b 2
= =221512
Mai.i
⼆14at1 14a n 1251 251

(c) 49h 2 - k 2 (d) 2m 2 - 18n 2

= =
lnik Ilniqri
⼆ 17htkllitn 2Ini 13啊
2lmtznlm

= 3nlciz.la
20. Factorize the following expressions by using the identities of perfect squares.
(a) a 2 + 6a + 9

a 32
5
(b) 4h 2 - 4h + 1
=
12以2灬灬
⼼
2
i

(c) 9m 2 - 6mn + n 2 (d) 25 x 2 + 40 xy + 16 y 2

= =
13mi213m n i 15如215xlltyyi
13mni 5xyi

21. Factorize the following expressions by using the cross-method.

(a) x 2 - 8 x + 12 x 6 (b) x 2 + 2 x - 15
= x
X 2
=
老妈
比6化⼯ ⼀比⼀2x_x lx 5 x 3 5x_x_

(c) 2 x 2 - 5 x - 12 (d) 3x 2 - 10 x + 8
=
1比 31 x 41YX 4 =
九州⼼
13 名怡
8九 3x_x 比 比⼆ 10九
⼀ ⼀

11

01 E B 02HK 410 03.i 11 11 2011/5/24 3 35 33

 Junior Secondary Summer Exercise (S2 to S3)

22. Factorize the following expressions.

tf
(a) 36h 2 - 64 k 2 (b) 2a 2 + 12ab + 18b 2
= =
lbhi18ki 21àtbab95
⼆lbhtilbhtk 2Icinlapi13啊
213htntkizlzn.tk ⼆21a 3必
413ht4knitki

(c) - a 2 + 10ab - 9b 2 (d) 16 x 2 + 8 xy 2 + y 4

=
ui
a 91 a.by
wii ii.ua =
14x 214x y 1i
14xtji

NF 23. Factorize the following expressions by using the identities of sum and difference of cubes.

(a) a3 + 64 (b) 125h3 - 27 k 3

= 43 = 3
a3 15⼩13k
tatitnà4⼼⼼ 15h3K 25h15hktaki

12

01 E B 02HK 410 03.i 12 12 2011/5/24 3 35 3

 5 Linear Equations in Two Unknowns

Exercise

Short Questions

In each of the following, complete the table for the given equation. (1 - 2)

1. y = 2x - 6 2. 2x + 5y = -6
x -3 0 1 4 x -4 -3 3
2
y y 0
-12 - 6 -
4 2 ⾔ -2 ⼀
号
Solve the following simultaneous linear equations by graphical method. (3 − 4)

x + 2 y = 1 y
3. 
3 x - y = 3 2
3x y 3

x 2y 1

x
1 0 1 2

The solution is x = ( ), y = ( ).
1 0
5 x - 3 y = -10 y
4. 
4 x + 6 y = 13 4x 6y 13

x
2 1 0 1
5x 3y 10

The solution is x = ( -0
5 ), y = ( 2.5 ).

13

01 E B 02HK 410 03.i 13 13 2011/5/24 3 35 43

 Junior Secondary Summer Exercise (S2 to S3)

5. Solve the following simultaneous equations by the 6. Solve the following simultaneous equations by the
method of substitution. method of elimination.
x = 5 y ...... (1) x - 2 y = 9 ...... (1)
 
x - y = 4 ...... (2) x + y = 3 ...... (2)

By substituting (1) into (2), we have (1) - (2):

( )- y = 4 -2 y - ( )= 9-( 3 )
5y
( )y = 4 (
y )y = ( )
4 3 6
y =( ) y =( )
1 2
By substituting y = ( ) into (1), we have By substituting y = (
1 2 ) into (2), we have
x = 5( 1 ) x +( 2 )= 3
=( ) x =( )
5 5
∴ The solution is x = ( ), y = ( ). ∴ The solution is x = ( 5 ), y = ( ).
5 1 2

Multiple Choice Questions

7. Which of the following points does not lie on the straight line y - 2x = 2?
y

A. (0, 2) B. (-1, 0) 5
y 2x = 2
4
C. (1, 4) D. (2, -2)
ˇ 3
2
1
x
0
1 1 2 3
1

8. If P(0, 5) is a solution of the equation x - py + 10 = 0, find the value of p.

A. -10 B. -2 C. 2 D. 10
ˇ
3x + 9 y = -8 y
9. According to the graph on the right, solve  ,
5 x - 7 y = 4 2
correct to 1 decimal place. 5x 7y 4

1
A. (-0.3, -0.8) B. (-0.2, -0.7)
ˇ
C. (-0.3, -0.7) D. No solutions 0
x
1 1 2 3 4

3x 9y 8

14

01 E B 02HK 410 03.i 14 14 2011/5/24 3 35 52

 5 Linear Equations in Two Unknowns

10. In a show, each Class I ticket and Class II ticket cost $12 and $5 respectively. The total number of tickets sold is
90 and the total income from the tickets is $870. If x Class I tickets and y Class II tickets are sold, which of the
following pairs of simultaneous equations can solve for the values of x and y?

 x + y = 870  x + y = 90  x + y = 870  x + y = 90
A.  B.  C.  D. 
12 x + 5 y = 90 ˇ 12 x + 5 y = 870 5 x + 12 y = 90 5 x + 12 y = 870

Long Questions
2 x + 3 y = 4
11. Solve the simultaneous linear equations  by graphical method.
 x - 2 y = -5
2x + 3y = 4 y

x
4 1 0.5 多
名4 4
y
4 2 1 3
x
x - 2y = -5
2
x
x

I i 1
y x
x
-4 -3 -2 -1 0 1 2
The solution is .

 y = 3x
學
12. Solve the following simultaneous equations by the method of substitution.
2 x + y = 5
(a)  (b) 
2 x + 3 y = 11 3 y - 4 x = 10

put 1into2 y 5 Ix
2x313x 11 put03into
Putx intoa
yen ifg 3152x 我 10 2位 g 5
x I 15⼀比⼀比⼆10 y 4
put ⼼00 1比⼆ ⼼ 5
䓬 爪 台 i 瓷
13. Solve the following simultaneous equations by the method of elimination.
2 x + 3 y = 13
o 5 x + 7 y = 8
(a)  (b)  o
2 x + y = 7 o 2 x - 7 y = 13
uirzx 03 i 5xnx⼆8
put into0 zx.it13
Ix 3 t Ix 13 N 21
zxtztx 13 3
x 2 put x 3into
putx 2into 2
2121tyit ⼼態 只
g3 15
i
⾏⾔
i.si
01 E B 02HK 410 03.i 15 15 2011/5/24 3 3 02
 Junior Secondary Summer Exercise (S2 to S3)

14. Solve the following simultaneous equations.

3m - 2n = 3 4u = 3v - 17 o
(a)  o (b) 
4m + 3n = -30 5u + 2v = -4 ⼀
xziqm.cn 9 4u 3iH
o.xziZmt6⼼60
02iAm 51 器 幾台是
3 aoizi 46
putm 3intoa u 2
03
putu z.int
siii 4⼼了 It
3
品 品
正
15. A fast food shop offers hamburger sets and hotdog sets only. A
hamburger set costs $17 and a hotdog set costs $14. If Mr Lee paid
$304 for a total of 20 sets, how many hotdog sets are ordered?
hamburger x x y 20
hotdog
y mnmi3比
x 20 go
put03into20
304
1720yitly304
340 my
y⼝
12into30
puty
九⼆2012
8
i 煰
16. Two years ago, the age of Cindy is
1
of that of her father. The present age of her father is 8 more than 3 times
5
the present age of Cindy. Find the present ages of Cindy and her father.

x zilg.io
vi3xt8
0put
intoQx.iIBx
812了
x 訕台
x計⼆号 2
x 8 20
putx 8into
yi 3⼼ 8
y 32

16

01 E B 02HK 410 03.i 1 1 2011/5/24 3 3 05

 6 Laws of Integral Indices

Exercise

True / False Questions

1. Determine whether each of the following expressions is correct. If it is correct, put a ‘✓’ in the box, otherwise,
put a ‘✗’.
1
(a) 2-3 = (b) ( x 4 )-3 = x 4-3 x
8
(c) (a -2 )3 = (a3 )-2 (d) -b -1 = b
v x
x -3 -3 - ( -7 ) -3 1
(e) =x (f) 2a =
ㄨ
i
x -7 2a 3
-5 NF (h) All binary numbers are even numbers.
(g) 0.000 087 = 8.7 × 10 ㄨ

Short Questions

2. Simplify the following expressions and express your answers with positive indices.

(a) a 2 × a3 × a 4 = (b) (- a 2 )3 =
5 6
d i
b ×b
= (d) (b3 )5 ÷ b7 =
i i
(c)
y
3 4
b ×b

3. Simplify the following expressions and express your answers with positive indices.
3x -2
(a) a -2 × a 4 × a -6 = (b) =
i x5 ㄋㄧㄤ

4
 2a 0 
(c) (-2b -2 )3 = (d)  -3  =
i
 m -2 
-1
b 
i ni
(a0b 2 )4
(e)  3  = (f) =
 n  mi mi (a 2b 4 )0 1 8

4. Use a calculator to find the values of the following expressions, and give your answers in scientific notation.
7.85 × 109 - 2.65 × 1010
(a) = (b) (6.84 × 108 )2 ÷ (3.8 × 10-5 ) =
7.46 × 1016 2.54x⼼ ⼼⼼

17

01 E B 02HK 410 03.i 1 1 2011/5/24 3 3 23

 Junior Secondary Summer Exercise (S2 to S3)

Multiple Choice Questions

5. 5 x-1 =
1 5x
A. 5 x - 5 B. C. D. -5 x
5x 5

6. x7 - x3 =
7
A. x 7-3 B. x 3 C. - x 7+3 D. None of the above

NF 7. In ABCDEF16, what is the place value of A?

i
A. 165 B. 166 C. 10 × 165 D. 10 × 166

NF 8. Arrange the following 3 numbers in descending order.

I. 1010102 II. 1011002 III. 1100002

A. I, II, III B. II, III, I C. III, II, I D. III, I, II

Long Questions

Simplify the following expressions and express your answers with positive indices. (9 – 14)
3
2 3 2 ( - x 3 y )3 2 2a 
9. (a b ) × (ab ) 10. 11. 4a b ÷  3 
( - xy 3 )2 b 

=
dtixib
=
ii =
ybxi
巡8
⼆ oi joy3
i ⼆
是

-3 2
-4 -1 -2 -1
 x  -2 y 0 
12. (3x y ) 2 3
13. (2a b ) 14.   ×  -1 
 y  x 

= = =
nii 主5 x3 4
3X
2ti y
i
yi

18

01 E B 02HK 410 03.i 1 1 2011/5/24 3 3 3

 6 Laws of Integral Indices

Simplify the following expressions and express your answers with positive indices. (15 – 16)
8 2n 27 2n+1
15. 16.
2n- 2 9n-1

= =
ppn
zn2 5
20n 3on3
zn2 了
zcon.cn21
34n5
NF 17. y 如
Convert the following numbers into denary numbers.
(a) 110012 (b) BEAD16

⼆llxni14⼼ ⼼1613
25

48813www
NF 18. Convert the following numbers into binary numbers.

(a) 1810 (b) 4010

Nwwz

NF 19. Convert the following numbers into hexadecimal numbers.

(a) 5910 (b) 201010

313no TDA

20. It is given that heartbeat rate of a human body is about 72 per minute. Suppose there are 365 days in a year,
calculate and express, in scientific notation, the total number of heartbeats a 15-year-old boy experiences since
his birth. (Give your answer correct to 3 significant figures.)

皉424436515
5⼼⼼

19

01 E B 02HK 410 03.i 1 1 2011/5/24 3 3 3

 Junior Secondary Summer Exercise (S2 to S3)

Exercise

True / False Questions

1. In the figure, PRS is a straight line. Determine whether each of the following expressions S
is correct. If the expression is correct, put a ‘✓’ in the box, otherwise, put a ‘✗ ’.
R d

(a) a = b X (b) c = a ˇ b

(c) a + b + c = 180° (d) a + c = d

ˇ ˇ
a c
P Q

Short Questions

Find the unknowns in the following figures. (2 − 12)

2. 3. D 4. H
A
c
b
85°
80° 130°
a 32°
B C 2b 120° K I
E J
F
KJI is a straight line.

a= b= c=
65 20 50
5. P
Q 6. M 7.
4d X
e
46° R
50°
38° f
S 70° Y Z
PQR is a straight line. N P

d= 240 e=
40 f=
ti

20

01 E B 02HK 410 03.i 20 20 2011/5/24 3 3 42

 7 Angles related to Rectilinear Figures

8. 9. D 10.
A H
E 130°
2g

j
h I K
J
B C F G
IJK is a straight line.
DEF is a straight line.

g= h= j = 1200
30 65
11. 12. Z
M
P U 130° m
Y
k
N 42° 105°
130°
68°
X
O
W

k= 145 m= 2600

Find the unknowns in the following figures. Write down the answers on the answer lines and state the suitable
reasons in the brackets provided. (13 − 15)

13. In BDF, A

D
∠BFD + ∠BDF = ∠ (ext. ∠ of )
ABE 80°
132°
∠BFD + 80° = 1320 E
r
C
B F

∠BFD = EBFC is a straight line.

52
r=∠ (corr. ∠s,
BFD AC // DF )

= 520

14. In PSR, L P

∠RPS = ∠RSP = ( ) 44°

280 baseLs A
q
28°
In PQS, Q S
R

∠PQS + ∠QPS + ∠RSP = ( ) QRS is a straight line.

180 Lsumoi⼝
q+( + ) + 28° = 1800
44 28
q= 800

21

01 E B 02HK 410 03.i 21 21 2011/5/24 3 3 44

 Junior Secondary Summer Exercise (S2 to S3)

15. In ABC, E
A
z 112°
∠ABC = ∠BAC = 60 ( )
prop.dk
4
56°
∵ Sum of the interior angles of polygon ABDE = ( - 2) × 180° B D
C
( )
BCD is a straight line.

∴ ∠ABC + ∠BDE + ∠DEA + ∠EAB = 360

+ 56° + 112° + (z + )=
60 60 300
z=
n

Long Questions

16. In the figure, ACD is a straight line. Find x. A C

x D
110°
30°

E
B
ABIDE
岩笑flint.Ls
LBA

70lextLot N
歪恐

17. In the figure, HSK, HTM, KMN and STN are straight lines. Find p. H

46°
S p
T

70° 23°
K N
M

Lksnti230 N Lsumof⼼
LKSNǏ
p4Etilext.Lof⼼
p Hi

22

01 E B 02HK 410 03.i 22 22 2011/5/24 3 3 4

 7 Angles related to Rectilinear Figures

18. In the figure, EADH and FBCG are straight lines. Find n. E A D 60°
35° H
n 110°
⼼⼼⼆3⼼ext.isofpolygon
n356 F
B C
G

n 1550

19. In the figure, ABP is a straight line. AQ and DR intersect at E. Find m. A

R
E 120°

念
m
Q

B
m 7512590120⼆540
130
jni
m P
105° 125°
D
C

20. In the figure, WX = WY. Z is a point on WY such that XY = XZ. W

(a) Find ∠WYX.

40°

⼆LWYxlbase.is的
LWXY
Lwxitnnxtmimǒusumorn
wixio X Y

(b) Find ∠WXZ.

LXIY LZYXlbae.Ls.tl
Lzi 14⼼⼼ Lsunof⼼
Lzito
Lwxziio40
⼆30

23

01 E B 02HK 410 03.i 23 23 2011/5/24 3 3 4

 Junior Secondary Summer Exercise (S2 to S3)

21. Find the value of x in the figure. (x + 1) cm

A B

(2x - 3) cm

C
xtkIx 3
九⼆ 4

22. If each interior angle of a regular polygon is 156°,

(a) find the number of its sides,

In21XIN⼆156in
18On 360 150n
n 15
(b) the size of each exterior angle.

180 15424

23. In the figure, BCD and EDF are straight lines. Find w. A

80° F
w 65°
B
LADCLACDlbase.Ls.AM C r D

LADC65
adj.Lonst.in
LEDC4654800 1800 E

⼭
悲淼 s.AM叫

24

01 E B 02HK 410 03.i 24 24 2011/5/24 3 3 4

 8 Rational and Irrational Numbers

Exercise

True / False Questions

1. Determine whether each of the following is correct. If it is correct, put a ‘✓’ in the box, otherwise, put a ‘✗’.
4 4
(a) 4 is a surd. (b) 2.308 is a rational number.
x x
(c) 7 2 + 2 2 = 9 2 (d) 2 6 = 6 2
v x
45
(e) 12 - 2 = 10 (f) =3
x 5 v

Short Questions

2. Write down the square roots of each of the following numbers.

(a) 25 5 (b) 49
7
(c) 324 (d) 1024
18 32
m
3. Write the following rational numbers in the form (where m, n are integers, and n ≠ 0).
n
(Give your answers in the simplest form.)
1
(a) 8 千 (b) -9
11 毕
⼀

(c) 0.25
卡 (d) -1.32
器
4. Determine whether the following numbers are rational or irrational. Put a ‘✓’ in the appropriate box.
Rational Irrational Rational Irrational

(a) 34.25 (b) 12

ˇ ˇ
1 4 4
(c) - (d) -0.105
2 ˇ ˇ

25

01 E B 02HK 410 03.i 25 25 2011/5/24 3 3 5

 Junior Secondary Summer Exercise (S2 to S3)

NF 5. Express the following surds in their simplest forms.

(a) 32 = ( ______ )2 × ______ (b) 28 = ( ______ )2 × ______

4 2 2 7

= ( ______ )2 × ______ = ( ______ )2 × ______

4 2 27
= ____________ = ____________
zg It

NF 6. Rationalize the denominators of the following expressions.

2 2 ( ) 10 10 ( )
(a) = × 3 (b) = ×
3 3 ( ) 5 5 ( )
3
=
琴 =
V5

NF 7. Simplify the following expressions.

(a) 9 11 + 5 11 = ( ______ + ______ ) 11 (b) 8 7 - 2 7 = ( ______ - ______ ) 7

9 5 8 2
= =
14圿 bf

24 ( )
(c) 6 × 3 5 = 3 × ______ × ______ (d) =
6 5 54 ( )
9
=
3坑 =
⾔

26

01 E B 02HK 410 03.i 2 2 2011/5/24 3 3 04

 8 Rational and Irrational Numbers

Long Questions
NF 8. Simplify the following expressions.
150
(a) 6 × 24 (b) -
5

=
派 =
5g
⼆12
no

NF 9. Rationalize the denominators of the following expressions.

4 3 5
(a) (b)
52 18

= 4 = 的
2巧 wi
⼆
点 器
⼀些
⼀

Simplify theyg
following expressions. (10 − 15)

NF 10. 5 + 20 - 45 NF 13. 2( 8 - 3 )

= =
V5205 3巧 n_n
o 4no
2 + 128
NF 11. 75 - 2 27 + 3 48 NF 14.
3

=
5巧6巧tws =

ill

15
NF 12. 3 × 8 × 10 NF 15.
12 × 10

=
4圬 = 圬⼼5
2圬⼼⼼
忘 27

01 E B 02HK 410 03.i 2 2 2011/5/24 3 3 10

 Junior Secondary Summer Exercise (S2 to S3)

Exercise

Short Questions
Find the unknown in each of the following figures. (1 – 3)

1. 4 2. 3.
y z
41
3
x 12 40
35

x= y= z=
3A 下
3 9
Find the unknown in each of the following figures. (Leave your answers in surd forms.) (4 – 6)

4. 5. 6.
18 12
a
4
b c c
15
5

a = 6.40 b= 9 95 c = 8.49

Determine whether each of the following triangles is a right-angled triangle. If it does, put a ‘✓’ in the box and state
the right angle. Otherwise, put a ‘✗’. (7 – 9)

7. A 8. P
9.
Z

25
14 4 0.9 2 5 4

B C Q R X Y
20 4.1 6

(right angle : ∠ ) (right angle : ∠ ) (right angle : ∠ )

x P v z

28

01 E B 02HK 410 03.i 2 2 2011/5/24 3 3 14

 9 Pythagoras’ Theorem

10. Find the value of x in the figure.

x 2 11
x2 + ( )2 = ( )2 (Pyth. theorem)
in 12
x2 = 144 - 44 12
2
x =
no
x=
no

11. Find the value of y in the figure.

13
y
y2 + ( )2 = ( )2 ( )
2.4y 13
y2 =
6tl 169 2.4y

y2 =
25
y=
5
12. In the figure, determine whether ABC is a right-angled triangle. A
2 2 2 2
∵ AC + BC = ( ) +( ) = 4225
33 56 65
33
2 2
and AB = ( ) = 4225
65
∵ AC + BC ( = / ≠ ) AB 2
2 2 B C
0 56

∴ ABC ( is / is not ) a right-angled triangle. ( )

0

Multiple Choice Questions

13. Find the value of x in the figure, correct to 3 significant figures. 18

6

A. 17.0 B. 18.6 C. 19.0 D. 20.6

ˇ x

14. Which of the following is / are right-angled triangle(s)?

I. II. III.
6 7 73
4 41
16 48

9
20 55

A. I only B. III only C. I and II only D. II and III only

29

01 E B 02HK 410 03.i 2 2 2011/5/24 3 3 1

 Junior Secondary Summer Exercise (S2 to S3)

15. In ABC, BC 2 = AB 2 + AC 2. Which angle is a right angle?

A. ∠A B. ∠B C. ∠C D. none of above

i
16. A ship sails 5.6 km due east and then sails 3.3 km due south. How far is it from the starting point?

A. 6.5 km B. 6.8 km C. 7.5 km D. 7.8 km

Long Questions

17. Referring to the figure, find the length of PR. P

17

Q S
6 R 9
PQ i lotqi
wamintci.no

18. Referring to the figure, find the length of PN. (Leave your answer in surd form.) P N

5 cm
7 cm
Q
NMWE 5 6 cm

it M

in

19. Referring to the figure, find the perimeter of ABC. C

(x 9) cm
45 cm

A x cm B

lxtqiix245
xiox it2025
x 108

30

01 E B 02HK 410 03.i 30 30 2011/5/24 3 3 1

 9 Pythagoras’ Theorem

20. To celebrate Christmas, Christmas lights are hanged from the top of B
building A to the top of building B as shown in the figure. Find the
length of the Christmas lights.
A
60 m

48 m

ABW35122 35 m
37
ithelenght oftheChristmaslightsis

21. At noon, two ships S1 and S2 leave port A at the same time. N

37m.si15B
Ship S1 sails due north at 15 km/h, and ship S2 sails due east at 25 km/h.
15 km/h
Find the distance between two ships at 3 : 00 p.m.
(Give your answer correct to 3 significant figures.)
S1

E
A

45 EN 1452 75 S2
25 km/h

Sz 25 ㄨ 3 Ei 87.5
75

22. In the figure, a ladder AB is placed against a vertical wall. If the top
B
of the ladder slides downwards by 3 m, how far does the foot of the
ladder slide on the ground? (Give your answer correct to 2 decimal 3m

places.) 10 m
9m

D c
Aciiii
Acn
cnwiò
an
AD派via
AD3 64

31

01 E B 02HK 410 03.i 31 31 2011/5/24 3 3 20

 Junior Secondary Summer Exercise (S2 to S3)

1 ( )

Exercise

Short Questions

1. Complete the following table for a circle. (Take π = 3.14.)

Radius Diameter Circumference Area

(a) 2 mm 12 6mm 12.6mni

Hmm
(b) 12 cm
6cm 3ttcm 113cm2

(c) 10m 62.8 m 314ni

20m
(d) 153.86 km2
tkm 14km 44 0km

2. Find the perimeters and the areas of the following figures in terms of π .
(a) (b)

8m

2 mm
6 mm

Perimeter = mm Perimeter =
不 2
4下 8 2
m
Area = mm Area = m
5下 8不
3. In each of the following figures, find AB and the area of sector OAB. (Give your answers in terms of π .)
(a) B (b)
9 cm A 15 m B
O
O 60°
216°

AB = cm AB = m
4 下
5
2
18
下
Area = cm Area = m2
20.3
下 135
下

32

01 E B 02HK 410 03.i 32 32 2011/5/24 3 3 24

 10 Areas and Volumes (II)

4. In each of the following figures, O is the centre of the sector. Find the perimeter and the area of the shaded
region. (Give your answers correct to 3 significant figures.)
(a) A B (b) A

80° 5 cm

O 9 cm

B O

Perimeter = 24.4 cm Perimeter = 26.9 cm

Area = 61.0 cm2 Area = cm2

5. Find the volumes and the total surface areas of the following cylinders. (Give your answers in terms of π .)
(a) 2 cm (b) 3m

7 cm 12 m

Volume = cm3 Volume = m3

28
下 ⼼下
Total surface area = cm2 Total surface area = m2
弧 ⼼
 22 
6. Complete the following table for a cylinder.  Take π = .
 7 

Base radius Height Volume Total surface area

(a) 7 cm 3cm 462 cm3

(b)
Im
3.5 m 44 m3
i
Multiple Choice Questions

7. The figure shows a semi-circle. If the perimeter of the semi-circle is 28 cm, find
the radius of the semi-circle. (Give your answer correct to 2 decimal places.)

A. 3.38 cm B. 4.46 cm C. 5.45 cm D. 8.91 cm O

33

01 E B 02HK 410 03.i 33 33 2011/5/24 3 3 2

 Junior Secondary Summer Exercise (S2 to S3)

8. The figure shows a sector OPQ. If its area is 45π cm2, find the radius of the sector. P

A. 10 cm B. 12 cm C. 15 cm D. 18 cm
ˇ 72°
O

9. As shown in the figure, a piece of cardboard is folded into the 29 cm

shape of a cylinder. Find the volume of the cylinder correct

to the nearest integer. 21 cm 21 cm

A. 1405 cm3 B. 1508 cm3

C. 1789 cm3 D. 1809 cm3

10. The figure shows a cake in the shape of a solid with uniform cross-section.
7 cm
Find the volume of the cake in terms of π .
45o
3 3
A. 3.5π cm B. 6π cm
ˇ 2 cm

C. 9π cm3 D. 12.25π cm3

ˇ

Long Questions

11. The figure is formed by semi-circles. Find its area in terms of π .

lzoixi14冷⼀⼼吃 4 cm 16 cm
200 8128
licni

12. In the figure, the length of PQ is 36π cm.

(a) Find the value of θ . θ
20 cm
P O

Q
2⽢⼼整齊

34

01 E B 02HK 410 03.i 34 34 2011/5/24 3 3 2

 10 Areas and Volumes (II)

(b) Find the area of the sector OPQ. (Give your answers in terms of π .)

Arecilzoix360
恐
40下cni

13. In the figure, ABCD is a square. ABFD and BCDE are sectors with centres A A D

and C respectively. Find the area of the shaded region correct to the nearest E
integer.
10 cm

F
Areāzlimi ⼼⼼主
了
B C
5to

14. As shown in the figure, the metal cube is melted and recast into a
cylinder of the same height. Find the base radius of the cylinder.
(Give your answer correct to 3 significant figures.) 10 cm

10 cm

⼼⼼⼼⼼⼼
的

15. The figure shows a solid which is formed by removing a rectangular wooden 8 cm
64cmzilim.mn1
block from a cylinder. If the wooden block is of the same height as the cylinder,
find the total surface area of the solid. (Give your answer correct to 3 significant 2 cm 5 cm
figures.)

到5
2⽇

35

01 E B 02HK 410 03.i 35 35 2011/5/24 3 3 2

 Junior Secondary Summer Exercise (S2 to S3)

16. As shown in the figure, 80% of the cylindrical container is filled with water. If
some stones, each of volume 64 cm3, are put into the container, at most how many
stones can be put into the container without overflow?
20 cm

zoxnonxi.li
64
7 cm
q.li
overflow
i 9stonescanputintothecontainerwithout

17. The figure shows a cylindrical tank. Water flows from a pipe at a rate of 10π cm3/s.
(a) Find the time required to fill up the tank in minutes.

30 cm
丌118⽂3⼼⼼
60 18 cm
16 2min

(b) If water flows from the pipe into the tank for 9 minutes, find the total surface area that the water is in
contact with the tank. (Give your answer in terms of π .)

llixqx60
玔8 in tioi
qncni

36

01 E B 02HK 410 03.i 3 3 2011/5/24 3 3 2

 11 Basic Concepts of Trigonometric Ratios

11

Exercise

Short Questions

1. Refer to the figures, complete the following.

(a) (b) 24
(c) L
13
5 y 2
1
θ 7 z
25 x
12 M N
2 O 3
MON is a straight line.

( ) ( ) ( )
sin θ = 5 , sin θ = 7 , tan x = 1 ,
( ) ( ) ( )
13 25 2
( ) ( 24 ) ( 1 )
cos θ = 12 , cos θ = , cos y = ,
( 13 ) ( 25 ) ( 2 )
( ) ( ) ( )
tan θ = 5 tan θ = 7 sin z = 1
( ) ( 24 ) ( 2 )
12

2. Find the values of the following expressions correct to 2 decimal places.

(a) sin 30° + cos 40° (b) 2 cos 60.1° - tan 25.9° (c) cos (45° - 20°) i tan 35°

= = =
1It 0 51 0.63

3. In each of the following, find the acute angle θ correct to the nearest degree.
2
(a) sin θ = (b) tan θ = 2 (c) 3 cos θ = 2
5

θ= θ= θ=
24 63 62

37

01 E B 02HK 410 03.i 3 3 2011/5/24 3 3 40

 Junior Secondary Summer Exercise (S2 to S3)

4. Find the value of x in the figure, correct to 5. Find θ in the figure, correct to 1 decimal place.
2 decimal places.
M L
F
8
8 x 15
θ
20° N
D E

EF ( )
∵ tan ∠D = ∵ cos ∠N = MN
( ) ( )
DF
x ( )
∴ tan 20° = cos θ =
( )
(
8 ) 15
x =( )tan 20°
8 ∴ θ =(
5t8 ) (cor. to1 d.p.)
=( ) (cor. to 2 d.p.)
291

Multiple Choice Questions

6. In the figure, tan x - tan y =

x
34
16 16 2 2 3
A. - . B. . C. - . D. .
15 ˇ 15 34 34 y
5

7. If cos x = cos 30° - cos 60°, find the acute angle x correct to the nearest degree.

ˇ A. 69° B. 49° C. 30° D. 21°

8. In the figure, y =
6 75°
y
6 6
A. 6 sin 75°. B. 6 cos 75°. C. . D. .
ˇ sin 75° cos 75°

9. In the figure, BDC is a straight line. Find the length of BD. A

8 cm
8 cos 46° 8 sin 46°
A. cm B. cm
tan 32° ˇ tan 32° B
32° 46°
C
D
8 sin 46°
C. cm D. 8 sin 46° tan 32° cm
sin 32°

38

01 E B 02HK 410 03.i 3 3 2011/5/24 3 3 51

 11 Basic Concepts of Trigonometric Ratios

Long Questions

For questions 10 – 18, give your answers correct to 3 significant figures if necessary.

10. Refer to the figure, find D F

(a) the length of DF, (b) sin x and tan x. 6

x 10

sinx 台 E
DFinto ⼆号
DEN
on
DF8 tanx⼆号
⼆年

11. Refer to the figure, find Y

(a) the length of XY, (b) sin u and cos u. 9 u 以

Z X
40

XYii.ci sinu
炸41
cos u⼆ 年

Find x and θ in each of the following. (12 – 15)

12. A 13. P S
35°
D
x 5
x
10
7 θ
Q 4 R
30°
θ
B C

cos30 洗 cos30 ⼼皓
x 10
0 36ci
x 8lt
⼆
sin35号
九⼆5sin35
sin0洗 x It
姚

39

01 E B 02HK 410 03.i 3 3 2011/5/24 3 3 52

 Junior Secondary Summer Exercise (S2 to S3)

14. N 15. E 6 H

5 50°
5
O θ 60°
x
F x G
I J
11
L M
7 IJHE is a rectangle.
LON is a straight line.

tan5⼼是
xiii to 淺
x 4.195498156
xǖnw
x 4no xzon
⼆
sin04195
些 xnza
co tanan品 513466
⼦
ooto

16. The figure shows a right-angled triangle ABC. A

(a) Find AB and BC. 10 cm

sin35是 35°
AB5 74 B C
cos35
咒
Bczia
5.735764
(b) Find the area of ABC.

Area 此8 mx5 74

D5cni

17. In the figure, Peter is flying a kite. Find the angle θ that the string makes with the
horizontal.
12 m
9m

sin0吾 θ

0 siri1到
1m

0 418

40

01 E B 02HK 410 03.i 40 40 2011/5/24 3 3 54

 11 Basic Concepts of Trigonometric Ratios

18. The figure shows a flagpole HK. Its top H is fastened by two cables HP and HQ. H

(a) Find the height of the flagpole HK.

tan
巒
30° 45°
P
Q 11 m K

(b) Find the distance between P and Q.

烈站0525580
圳
Mi
PQ⼆no5 li
8.05

41

01 E B 02HK 410 03.i 41 41 2011/5/24 3 3 55

 Junior Secondary Summer Exercise (S2 to S3)

12

Exercise

Short Questions

Complete the proof in each of the following. (1 − 2)

1. In the figure, AFB, CGD and EFGH are straight lines. Prove that AB // CD. A
C
Proof
E
∠FGD = ∠CGH (vert.opp.hn ) F
G H
∵ ∠EFB = ∠CGH ( given )
B
∴ ∠EFB = ∠ FGD
D
∴ AB // CD ( )
car.Lequal
2. In the figure, BCD is a straight line. Prove that BAC ∼ BDA. A

Proof 18 cm

∠ABC = ∠DBA ( common )

Ls B
12 cm C 15 cm
D
BC (
12) BA ( ) ( )
= = 2 , = 18 = 2
BA 18 ( ) BD ( ) ( 3 )
3 27
BC BA
∴ =
BA BD
∴ BAC ∼ BDA ( AAA )

3. In the figure, ABC and ADE are straight lines. Name all the isosceles triangles. E

D
LIADBABDEAEBC
45°
A C
B

42

01 E B 02HK 410 03.i 42 42 2011/5/24 3 3 01

 12 Introduction to Deductive Geometry

Long Questions

4. In the figure, EFGH is a straight line. Prove that EH // BD. A C

E H
LAFGuaHlcorr.is F G
AB1lCDIiuGH
LFBDlart.LABncD
i.EHnBDlcorr.Ls.eqeun
B D

5. In the figure, AEB, BDC, AFD and EFC are straight lines. Prove that x = a + b + c. A

E a

F
x
LCDF b c
x ⼼ B C
Ftc.ixuic D

6. In the figure, AEC and BED are straight lines. Prove that ABE ≅ DCE. A D

B C
EBELECBi.EBECLABELDCElgiven.AE

micwertopp.in
三⽇
ABE

7. the figure, AEC and ADB are straight lines. Prove that EBC is an
InDCEIAMLDEBELEBclalt.L.DE A
isosceles triangle.
D E

B C

lBaLAED
LEcBlcorrL.DEnBc
LEBELECB
inEBCisanisoscelestriangle

43

01 E B 02HK 410 03.i 43 43 2011/5/24 3 3 02

 Junior Secondary Summer Exercise (S2 to S3)

8. The figure shows a regular hexagon ABCDEF. Prove that FD ⊥ CD. A F

LFED 162YIN 吣
B E

LEDF180no
302
⼼肛 ⼼30 C D

qo
iFDUD
9. In the figure, ABC and CDE are equilateral triangles. A

(a) Prove that ADC ≅ BEC. D

E
哭兒prop.ci创
LACE 42
35° 42°
LDcnoo B C
no.itncipilpni.in
(b) Find ∠ADC.
ADCEABEccsMLDAHEBciziiai.IN

351801上sumoim
LADC 103

10. In the figure, ADC is a straight line. A

3 cm
(a) Prove that ABD ∼ ACB. D
6 cm
x cm
y cm
LABRLACBgiven
LBAc cABlcomm.nu
LADB
ABcizsumoini.AABD
B
14 cm
C
AACBu.AM

(b) Find the values of x and y.

㗊㗊 您㗊
i 知⾔
it x it

44

01 E B 02HK 410 03.i 44 44 2011/5/24 3 3 03

 13

13

1. 5 –7
(a)
20
x
(b) 15

10
x
1416 x
(c) 10.5 5 X
x
0
3 6 9 12 15 18 2
名
10

2. 40
20
(a)
15
x
x

xxffnf
(b) 10

129.5cm 5
x
0
129.5 139.5 149.5 159.5 169.5 179.5 189.5
(cm)

3. 1 2
1

(cm) 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70
2 4 7 11 6

45

01 B 02HK 441 01 045 45 2011/5/24 3 4 4

 2

(cm) 20.5 30.5 405 50 5 60.5 to5

2 6 13 24 30 30

4.
(a) 2 50

24 40

30
(b) 60
20

10
= %×
w
0
= 60 % × 30 1 1.5 2 2.5 3 3.5 4 4.5

=
孔
∴ 60 = 30

5. 20

9 10 11 12 13
2 4 5 8 1

(a)

(b)
圖
圓形

6.

2 2

60
60 50 i
40
30
50
20
10
40 0
A B A B

(a) (b)

2
46

01 B 02HK 441 01 04 4 2011/5/24 3 4 4

 13

7.
I.
20

15
II. $20
10
III. $25
5
A. I B. II
0
10 15 20 25 30 35 40 45
C. I II ˇ D. I III
($)

8. 80

80
A. 48 B. 50
60
C. 55 D. 59
ˇ 40

20

0
40 45 50 55 60 65 70

9.
I. II. III.

0 0 0

A. I B. III C. I II D. II III
ˇ
10. 50
I. II. III.

0 0 0
50 50 50

ˇ A. I B. II C. I III D. II III

47

01 B 02HK 441 01 04 4 2011/5/24 3 4 51

 11. /

A. B.

ˇ C. D.

12.
( )
(a) 5-9 4
10 - 14 8
15
15 - 19 11
10 20 - 24 8

5 25 - 29 6
30 - 34 3
0
2 7 12 17 22 27 32 37

(b) 9.5 – 29.5

名
25

13. 90

(a) 80
40

30

⼦ 23
六⽉ intnati 20

12
以
⼆7.8 10 a
4
胡⼆niiiinxno 0
35tn 35 45 55 65 75 85 95 105

(b) ⼆⽉
48

01 B 02HK 441 01 04 4 2011/5/24 3 4 53

 13

14.
(a) 17 0
100 22 45
100
27 65
75 32 90
x 37 100
50
x
25

0
x
17 22 27 32 37

(b)

分
下四位數 if
25

(c)
分
上四位數 if
in
27

㖚xwo
65

15. 2008
2009
(a) 2008 2009

⾔ 12
⼆zi3
8

(b) 2008 2009

4
(a)
0
相同
不 2008 2009

(c)

49

01 B 02HK 441 01 04 4 2011/5/24 3 4 55

 Answers
1 Rate and Ratio 19. (a) (4a - 1)(4a + 1) (b) (2 - 5b)(2 + 5b)
(c) (7h - k)(7h + k) (d) 2(m - 3n)(m + 3n)
1. (a) $5 / sticker (b) 6000 people / km2
20. (a) (a + 3)2 (b) (2h - 1)2
(c) 1.25 lucky stars / min (d) 1.8 stairs / s
(c) (3m - n)2 (d) (5x + 4y)2
2. (a) 1 : 4 (b) 2 : 3 (c) 3 : 10
21. (a) (x - 2)(x - 6) (b) (x + 5)(x - 3)
3. (a) 2 : 4 : 1 (b) 2 : 3 : 5 4. (a) 1 : 2 (b) 1 : 50
(c) (2x + 3)(x - 4) (d) (3x - 4)(x - 2)
5. C 6. D 7. B 8. B 9. B 10. D
22. (a) 4(3h - 4k)(3h + 4k) (b) 2(a + 3b)2
11. (a) 1.25 m/s, 1.2 m/s (b) rock climbing
(c) -(a - 9b)(a - b) (d) (4x + y2)2
12. $102 13. 150 14. 5 : 10 : 4
23. (a) (a + 4)(a2 - 4a + 16) (b) (5h - 3k)(25h2 + 15hk + 9k2)
15. (a) 2.24 km (b) 67.2 km/h 16. 16.875 m

5 Linear Equations in Two Unknowns

2 Approximation and Errors 1. x −3 0 1 4
1. 2000 2. 30 220 3. 42.5 4. 0.026
y −12 −6 −4 2
5. (a) (1538) - (1450) = 88 (b) (1600) - (1538) = 62
( 2)
2. x −4 −3 2 3
6. (a) m =1m (b) 78 m + 1 m = 79 m
2 y 0.4 0 −2 −2.4
(c) 78 m - 1 m = 77 m 3. x = (1), y = (0) 4. x = (-0.5), y = (2.5)
7. C 8. C 9. B 10. A 11. B 12. C 5. By substituting (1) into (2), we have (5 y ) - y = 4
1
13. (a) 0.25°C (b) 14. (a) 0.05 kg (b) 3.125% (4 ) y = 4
110
y = (1)
15. (a) 1.8 kg (b) 2.8%
16. (a) Upper limit of the length = 288.5 cm, By substituting y = (1) into (1), we have x = 5(1)
upper limit of the width = 192.5 cm = (5)
The solution is x = (5), y = (1).
(b) Upper limit of the area = 55 536.25 cm2
6. (1) - (2): -2 y - ( y ) = 9 - (3)
17. (a) 0.1 mL (b) Upper limit = 2.6 mL, lower limit = 2.4 mL
(-3) y = (6)
y = (-2)
3 Algebraic Fractions and Formulas
By substituting y = (-2) into (2), we have x + (-2) = 3
1. (a) ✗ (b) ✓ (c) ✗ (d) ✓
2 x =5
1 xy 5 3 - 5a
2. 3. 4. 5. The solution is x = (5), y = (-2).
6 2 2x ab
7. D 8. C 9. A 10. B
2 ( 4 ) - (6 )
6. A = ( 2) + 3 = 7 7. P= = -1 11. 2x + 3y = 4 x - 2y = -5
2
1 7 x -4 -1 2 x -3 -1 1
8. 1 9. 10. 4a 11.
a - 2b 12a y 4 2 0 y 1 2 3
2a + 5b 7 x 3
12. 13. 14. a = -1 15. x = ( A - P ) y
2(a + 2b ) 2( y - 3) L B
16. (a) $2 (b) 800 17. (a) T = 25x + 40y (b) 3 2x + 3y = 4 4

x - 2y = -5
4 Identities and Factorization 3

1. (a) ✓ (b) ✗ (c) ✗ (d) ✓ 2

2. A = 2, B = 1 3. A = 4, B = -12 4. 5(q + 2) 1
5. 3z(z - 2) 6. ab(a - b) 7. 2(c - 2d + 4e)
x
8. 4x2 - y2 9. 3x2 - 27 -4 -3 -2 -1 0 1 2
10. 25a2 + 20a + 4 11. 4b2 - 28bc + 49c2
The solution is x = -1, y = 2.
12. B 13. C 14. D 15. B
12. (a) x = 1, y = 3 (b) x = 0.5, y = 4
16. (a) A = -1, B = 3 (b) A = 7, B = 4
13. (a) x = 2, y = 3 (b) x = 3, y = -1
18. (a) (m + n)(1 + a) (b) (1 + x)(3 - 4b)

01 E B 02HK 410 03.i 50 50 2011/5/24 3 3 31

 14. (a) m = -3, n = -6 (b) u = -2, v = 3 5. (a) 32 = (4 )2 × 2 (b) 28 = (2)2 × 7
15. 12 16. Cindy: 8 years old, her father: 32 years old
= (4 )2 × 2 = (2)2 × 7
=4 2 =2 7
6 Laws of Integral Indices
1. (a) ✓ (b) ✗ (c) ✓ (d) ✗ 2 2 ( 3) 10 10 (5)
6. (a) = × (b) = ×
3 3 ( 3) 5 5 (5)
(e) ✓ (f) ✗ (g) ✓ (h) ✗
2. (a) a9 (b) -a6 (c) b4 (d) b8 2 3 =2 5
=
1 3 8 3
12
3. (a) (b) (c) - (d) 16b
a
4
x
7
b
6
7. (a) 9 11 + 5 11 = (9 + 5) 11 (b) 8 7 - 2 7 = (8 - 2) 7
2 3 8
(e) m n (f) b = 14 11 =6 7
4. (a) -2.5 × 10-7 (b) 1.2312 × 1022 24 ( 24 )
x
7 (c) 6 × 3 5 = 3× 6 × 5 (d) =
5. C 6. D 7. A 8. C 9. a8b5 10. - 54 (54 )
y
3
= 3 30
2
b
10
27 y
6
ab
2
4y
3
=
11. 12. 13. 14. 15. 25n + 2 16. 34n + 5 3
2a x
12
2 x
2 13 10
17. (a) 2510 (b) 4881310 18. (a) 100102 (b) 1010002 8. (a) 12 (b) - 6 9. (a) (b)
13 2
19. (a) 3B16 (b) 7DA16 20. 5.68 × 108
10. 0 11. 11 3 12. 4 15
2
7 Angles related to Rectilinear Figures 13. 4 - 6 14. 3 6 15.
4
1. (a) ✗ (b) ✓ (c) ✓ (d) ✓
2. 63° 3. 20° 4. 50° 5. 24° 6. 40° 7. 71° 9 Pythagoras’ Theorem
8. 30° 9. 65° 10. 120° 11. 145° 12. 260° 1. 5 2. 37 3. 9
13. In BDF, ∠BFD + ∠BDF = ∠DBE (ext. ∠ of ) 4. 41 5. 99 (or 3 11) 6. 72 (or 6 2)
∠BFD + 80° = 132° 7. ✗ 8. ✓, P 9. ✓, Z
∠BFD = 52° 10. x 2 + (2 11 )2 = (12)2 (Pyth. theorem)
r = ∠BFD (corr. ∠s, AC // DF) x 2 = 144 - 44
= 52° x 2 = 100
14. In PSR, ∠RPS = ∠RSP = 28° (base ∠s, isos. ) x = 10
In PQS, ∠PQS + ∠QPS + ∠RSP = 180° (∠ sum of ) 11. y + (2.4 y ) = (13)2 (Pyth. theorem)
2 2

q + (44° + 28°) + 28° = 180° 6.76y 2 = 169

q = 80° y 2 = 25
15. In ABC, ∠ABC = ∠BAC = 60° (prop. of equil. ) y =5
∵ Sum of the interior angles of polygon ABDE 12. ∵ AC 2 + BC 2 = (33)2 + (56)2 = 4225 and AB 2 = (65)2 = 4225
= (4 - 2) × 180° (∠ sum of polygon) ∵ AC 2 + BC 2 (= / ≠ )AB 2
∴ ∠ABC + ∠BDE + ∠DEA + ∠EAB = 360° ∴ ABC (is / is not ) a right-angled triangle.
60° + 56° + 112° + (z + 60°) = 360° (converse of Pyth. theorem)
z = 72° 13. C 14. D 15. A 16. A
16. 100° 17. 41° 18. 155° 19. 130° 20. (a) 70° (b) 30° 17. 10 18. 12 cm (or 2 3 cm)
21. 4 22. (a) 15 (b) 24° 23. 35° 19. 270 cm 20. 37 m 21. 87.5 km 22. 3.64 m

8 Rational and Irrational Numbers 10 Areas and Volumes (II)

1. (a) ✗ (b) ✓ (c) ✓ (d) ✗ (e) ✗ (f) ✓ 1. Radius Diameter Circumference Area
2. (a) 5, -5 (b) 7, -7 (c) 18, -18 (d) 32, -32 (a) 2 mm 4 mm 12.56 mm 12.56 mm2
8 100 1 33 (b) 6 cm 12 cm 37.68 cm 113.04 cm2
3. (a) (b) - (c) (d) -
1 11 4 25
(c) 10 m 20 m 62.8 m 314 m2
4. (a) Rational (b) Irrational 153.86 km2
(d) 7 km 14 km 43.96 km
(c) Irrational (d) Rational

01 E B 02HK 410 03.i 51 51 2011/5/24 3 3 00

 2. (a) Perimeter = 8π mm, Area = 8π mm2 12 Introduction to Deductive Geometry
(b) Perimeter = (4π + 8) m, Area = 8π m2 1. ∠FGD = ∠CGH (vert. opp. ∠s)
3. (a) 
AB = 3π cm, Area = 13.5π cm 2 ∵ ∠EFB = ∠CGH (given)
∴ ∠EFB = ∠FGD
(b) 
AB = 18π m, Area = 135π m 2
∴ AB // CD (corr. ∠s equal)
4. (a) Perimeter = 34.4 cm, Area = 61.1 cm2
2. ∠ABC = ∠DBA (common angle)
(b) Perimeter = 26.9 cm, Area = 23.1 cm2 BC 12 ( 2) BA (18) ( 2)
= = , = =
5. (a) Volume = 28π cm3, Total surface area = 36π cm 2 BA 18 (3) BD ( 27 ) ( 3)
3
(b) Volume = 108π m , Total surface area = 108π m 2 BC BA
∴ =
BA BD
6. Basic radius Height Volume Total surface area
∴ BAC ∼ BDA (ratio of 2 sides, inc. ∠)
(a) 7 cm 3 cm 462 cm3 440 cm2
3. AEC, BEC, ABE, BDE, ADB
1
(b) 2m 3.5 m 44 m3 69 m 2 9. (b) 103° 10. (b) x = 9, y = 7
7

7. C 8. C 9. A 10. D
13 Statistical Diagrams and Graphs
11. 16π cm2 12. (a) 324° (b) 360π cm2
1. (a) Time spent by a group of teenagers
13. 57 cm2 14. 5.64 cm 15. 258 cm on surfing the Internet in a week
20
16. 9 17. (a) 16.2 minutes (b) 924π cm2
15

Frequency
11 Basic Concepts of Trigonometric Ratios 10

(5) (12) (5)

1. (a) sin θ = , cos θ = , tan θ = 5
(13) (13) (12)
(7 ) (24) (7) 0
(b) sin θ = , cos θ = , tan θ = 3 6 9 12 15 18 21
( 25) (25) (24) Time (h)
(1) (1) (1) (b) 14 hours - 16 hours (c) 10
(c) tan x = , cos y = , sin z =
( 2) ( 2) ( 2) 2. (a) Heights of 40 students in a class
2. (a) 1.27 (b) 0.51 (c) 0.63 20

3. (a) 24° (b) 63° (c) 62°

15
EF
4. ∵ tan ∠D =
Frequency

( DF ) 10
x
∴ tan20° =
(8 ) 5

x = (8) tan20°
0
= (2.91) (cor. to 2 d.p.) 129.5 139.5 149.5 159.5 169.5 179.5 189.5
Height (cm)
( MN )
5. ∵ cos ∠N = (b) 134.5 cm
( LN )
3. Length less than (cm) 20.5 30.5 40.5 50.5 60.5 70.5
(8 )
cos θ = Cumulative frequency 0 2 6 13 24 30
(15)
θ = (57.8°) (cor. to1d.p.) 4. (a) 24
6. B 7. A 8. D 9. B (b) ∵ Corresponding cumulative frequency
4 4 = 60% × total frequency
10. (a) 8 (b) sin x = , tan x =
5 3 = 60% × 50
40 9 = 30
11. (a) 41 (b) sin u = , cos u=
41 41
∴ From the graph, 60th percentile = 2.5 hours
12. x = 8.66, θ = 44.4° 13. x = 2.87, θ = 36.9°
5. (a) bar chart (b) pie chart 6. Figure (b)
14. x = 5.96, θ = 31.7° 15. x = 2.89, θ = 67.1°
7. D 8. C 9. B 10. A 11. C
16. (a) AB = 5.74 cm, BC = 8.19 cm (b) 23.5 cm2
17. 41.8° 18. (a) 11 m (b) 8.05 m

01 E B 02HK 410 03.i 52 52 2011/5/24 3 3 20

 12. (a) Time required for a group of students
to solve a mathematics problem
15
Frequency
10

0
2 7 12 17 22 27 32 37
Time (min)
(b) 33
7 2
13. (a) In June: 7 %, in July: 26 % (b) July
9 3
14. (a) BMI of 100 citizens
100
Cumulative frequency

75

50

25

0
17 22 27 32 37
BMI
(b) Lower quartile = 20, upper quartile = 29 (c) 35%
15. (a) 2 : 3 (b) No
(c) Yes. The diagram gives an impression that the number
of sales of bottle of water in 2009 grew significantly
compared with the sales in 2008.

01 E B 02HK 410 03.i 53 53 2011/5/24 3 3 23