1977-02-21 — Page 19

育教僑華頁三第張五第 日四初月正年巳丁麼夏 WAH KIU YAT. PO

1977中學會考試題預習專欄

明德社主編

附加數學 (十五)

岑俊彦•

Additional Mathematics(15).

Complex number (11)

Worked Examples:

3. Find the roots of the

equation

iz2 + 32 -

21 = i

Solution

-3+32

-

4(1)(-21)

21

-3+

1. If z =

coserisine and

z = cose-isine,

find (1) z + Z

(ii) z

(iii) zz, and (z-ž)2

and by using (z+ž)2, prove

that cos20 = 2cos2-1

and cos20 =. 1-2sin20

respectively. Solution:

1) z + Z = 2cdso

ii) z - ž = 2ising

iii) zz = (cosorisino) (cose.

-işine)

cos-0-

= COS + sin

>

z 2 + 2 z z + z 2

NOW, On substitution, we have (2cose)?

=(cos +igine)2+2(1)+(cose-

isine)

=(cos20+isin20)+2+(cos20-1

sin29)by De Mrivre's theo-

rem

2

4cos 9 2cos20 + 2

co $20 = 2c0s 01

2

(z - z.)?

= 22-227

On substitution, we have

(2isine)?

=(cose+igine) -2(1)+(co50-

-4sin cos20+1sin20-2+cos

29-isin29%

= 2cos29 2

cos29 1-2sin

2. Find the cube roots of

unity and represent them

in an are and Diagram. Solution: The given expres- sion is

1+oi cis(360°)

= ((cis(360°)} Cis (360°)

where n = 0, 1

3

cis(120 When n = 0;

when n =

x

when n 2 ZĄ

where 2.

글.

- cis 0° =1

=

14 01 cis120°

=ci s240°

and Z2 are

the 3 re uired roots..

1977中學入學試試題預習專欄

愛智慧社主編

數學科 (十五)

正負數,代數符號、代数式及簡易方程式

* BEK——— ( -10) (−10 ) - ( -10) x ( -10)

第10選一注意不同文字不能相加。

第11刪一除號機的「K」可與前一「K」相瀉,期

FKKжK×× 第12期一注意滿辦。

́ 13———— ——5ƒX › 3x2 - 3× ( -5)× C

-5)

第14到一文字與數字不能相加。

#15-(-x)3 − ( − × ) × ( - )× (-

第21期

X)

14X應移項。

· 1922 ——— Y K10MM »

· 15 25 BUY +14-21+7.

2

寬29個一注您所凍爲轟媽現年的歲數。

第30題兒所有貴X,依通常搶所有爲X-200

1. (-8)+(-8) - ?

A 16.

D-8

B 8

E -16

ABC DE

1. -----

.or

i or zi

the required roots are: 121 Ans.

Exercise 10.

1. Given that z

cos0+isino,

z=coso-isino, write down, by expansion, the expres- sions

a) (z + ž

b) (z - ż

c) (z + z)

in term of z and Z. Hence D prove that

.:

cos30 = 4cos30 3cöse

3

sin30 = 3sino-4sin'e and cos50 = 5cos0-20cos38+16cos?@ 2. In each of the Following cases, find the indicated roots and represent them in an argand Diagram.

a) the fifth roots of unity. b) the cube roots of -81 c) the fourth roots of −2-2/31

13. Find the roots of the

forlowing equations in the form a + ib:

(a) 2z2 + 31

2

B

122

+52

iz + 5z

0

英文 (十六)

English (16) Comprehension

魯榮光

Comprehension Exercise (3)

Read the passage carefully and answer the questions that follow it

And there is another material, named concrete, which gives the men of today fresh chances of expressing beauty in architecture. It is a very old material, used by the ancient Romans and known long before. The Roman made great concrete roofs supported only by the walls on which they rested; but in the Middle Ages this art was lost, and roofs were made with a multitude of small stones

stones

for the arched ribs of the yault, and more stones for arching across those ribs. And now in the course of ages concrete has re- turned to play its part in the history of architecture.

2. (-7) ( − 2) = ?

A -9

D-5-

5. (+12) - ( -3)=7

日一廿月二年七七九一层公年六十六國民懒中

報日僑華

一期星

C. it can be used to build.

an arch

What is this new mater- ial which is helping to form a new style? Concrete is a mixture of a powder. called cement, made from clay and chalk, and water, sand, and broken brick or gravel. This pudding, mixed and left for a time, be- comes as hard as stone. So you may say, you have only made stone instead of quar- rying one: why should that alter the style of archite- cture? The answer is, in part, that this stone can be of any size we wish. We do not take it to the build- ing in a lorry, nor it into position with ist crane. We carry our great stone to the building in bucketsful of liquid pudding. Wherever we want this con- crete stone we fix up a wooden trough, into which the mixture can be poured. When it hardens, the wooden trough is removed and the great artificial stone is in position. Now a concrete beam, like stone, would crack if heavily weighted, but we.. now know a way. ----unknown to the ancient "omans of strengthening concrete. Before pouring in the mixture, long steel rods can be placed in the trough. When the mix- ture is poured in and hardens, the steel is embedded in the concrete and gives it great additional strength.

Now this trough oan be made in any shape. You know how a jelly is shaped: the cook does not make a stiff jelly, then carve it into shape: she pours it while liquid into a mould, and when it has set it stands up shaped like the mould, Jellies: are always the same shape. We can shape concrete like

jelly, and we can get unusual shapes far more easily than

in stone and we can model a building into queer shapes as we can plasticine. Questions

5. It is suggested that

A. the use of concrete in.

"architecture allows of

auty.

B. the use of concrete in

making roofs plays an important part in the history of architecture C. The "omana first named the material 'concrete',

D. In the Middle Ages

use of concrete was. "considered to be an art.

the

E. Only the Romans knew how

to use concrete in archit ecture.

6. The use of concrete alters

the style of architecture because

concrete has now return- ed to play its part

B. it is mixture of cement, clay and chalk, water and sand.

11. KXKXK+K×K×K-?

A K D`1

12. A--2 B-3, 2A-8-7

B.4K E.0

CK.

11.

A B C D E

EX

A 9

3 7

C

1

D -7

E -9

A B C D E

12. ☐☐☐☐☐

B.75 E-225

C 25

ABCDE 13. ☐☐☐

C X-4

14.

ABCDE

13. X--5-3x2 -?

A 225

D-75

B-7

E 9

A B C D E

A 25

B 15

4

D -15

E. -25

4. (-4)

ABCDE 3. COLOU

A -12

B -8

C-4

D 4

E 12

A B C D E

4.

B -9 E 36

B -10

E 40

C 10

ABCDE 6. ☐☐☐☐☐

15. (-X)3-

B -6 E 6

C -2

A X3

B

A B C D E

7. 0000

D'~3X

-x3 E. 3x3

C 3X

8 20 E-100

.CO

A -15

D 15

6. (-15)-(-25)-?

A. -40 25

7..(+2)-(+4)=?

A -8

D 2

8 (-10)( -10) -7?

A 100

D-20

9. X+2X-4X-?

A B C D E

5. 0000

14. 6x-8-5X+-4-9.

A 11X-12 B 11X-4 D X+4

EX-12

D. we know the way of

strengthening concrete

E. it can be moulded easily. 7. Which of the following word is not related to the other four?

A. mould

B. crane

C. trough

D. pudding

E. arching

8. Which of the following

suggested title best sums up the passage?

A. The use of concrete in modern times. B. The place of concrete

in the history of architecture.

C. Concrete and Beauty

D. The use of concrete

E. Now concrete is made

and used.

news agencies, Chinese boa- rding houses and hotels.

6. It launches community campaigns such as "Keep Ho ng Kong. Clean" and "Fight Crime" campaigns.

7. It manages the trust funds of the community.

8. It helps to improve the private housing mana- gement by assisting the setup of Mutual 1d Commit- tees.

9. It carries out commun- ity and youth projects.

10. It gives advices and assistance in stautory dec- laration, verification of marital status, customary adoptions of orphans and application for travel documents, etc.

11. It provides translat- ion for all government dep- artments.

12. Officers of the depa- rtment maintain personal relationships and direct

KABAREE) contacts with leading Chin-

孔繁盛·

Economic and Public Affairs

(15)

Government Secretariat

1. It is the centre of public administration in Hong Kong

2. It is headed by The Chief Secretary, who is responisible for the over- all direction and supervis- ion of the activities of the various administrative departments.

3. The Chief Secretary is the Governor's principal adviser on policy.

4. He is also the chief government spokesman.

5. There are 40 or so go- vernment departments under the authority of the Gover- nment Secretariat. They are headed by either a director. commissioner or registrar.

The two government de- partments most closely con- cerned with the reactions of the people to government ·· policies and plans are the Home Affairs Department and the New Territories Admini- stration.

Home Affairs Department

1. It acts as a bridge between government and peo-

It explains the gover- nment policies and provides a channel for people's opi- nion and complaints.

3. It acts as a co-ordin ator for other government departments providing publ- ic services.

4. It offers a service for mediation or reconcill- ation of many and varied disputes, particularly family and tenancy disputes 15. It is responsible for

the registry of newspapers,

18. M÷M-M-M-?

ΑΟ

D 2M

19. X —4—20 • X −7·

A 80 D.5

8.1

E M

CM

B. 24 E-24

A B C D E 18. ☐☐☐☐☐

C 16

ABCDE

19.

Во E 40

C 10

ABCDE 20. ☐☐☐O

B 16

E -16

C 4

21.

A B C D E

20. 2x+10= -10 · X - ?

A-10

D 10

21. 24-4X-40 = X=?

A 64

22, -10, Y-?

ese residents and with law- ful Chinese Societies, and traditional organisations: in Hong Kong.

13. There are 10 City: District Offices located in the major urban areas of Hong Kong Island and Kowlo- on. They are under the con- trol of the department. They serves as local-ag- ents of the government.

New Territories Administra tion

1. It also sounds out pu- blic reaction from local organisations in the New Territories.

2 District officers sta- tioned in the seven geogra phical districts (Islands,

· Tsuen Wan, Tuen Mun, Yuen Long, Tai Po, Sha Tin, and Sai Kung) Liaise closely. with rural committees elec- ted by and from village representatives.

3. It maintains close co- ntact with the Heung Tee Kuk--the Rural Consultati- ve Council!--which is form- ed by the leaders of the New Territories.

4. It co-ordinates the administration and public services in the New Terri- tories.

5. It explains government policies and actions to N.T people.

6. It acts as the channel. for opinions and complaints. from the N.T. people.

7. It controls the Crown land in the New Territories 8. It advises the govern- ment on the needs and cust- ons of the N.T. people.

D

A 0

D 9.

2

ABCDE

26.

C-3

A B C D E 27. ☐☐☐

ABCDE

27. 9X2-81 X-7

28.哥哥有X元,弟弟所有是哥哥的4憎少2元,下列那

一個代數式可以表示弟弟所有的欸项?

A 4X-2 B 4X+2 C 4(x-2)

04(X-2) E ×(x+2)

28. L

29,四年截,媽媽的年歲是妹妹的3倍,若當時姊姊是

X讚,如如現在幾歲?

C· 3(X+4)

EX(3+4).

B 3CX-4). D. 3X-+-4

29.

30.兄弟二人共有郵票80枚,已知兄比弟每20枚,若兒

所有爲X,解下列那一方模式可以算出兄所有郵票

A.3X-4

ABCDE

.::

B X+X+20-30

ABCDE

C 80-X-20

D X-20-80

30..

D

2

*

A 20

D 0.5

B 10 E 0.2

C. S

22.

ABCDE

的數目?

23. X+Y-X-Y +Y=↑

A 2X+20-80

2X

B X

E-1

C 1

23.

C.12

24.

ABCDE

E 2X-20-80

A B C D E

(1) E

(2) C

(3) 8.

(4) A

(5)D

(600

C 70 C

(8) A

C 1

(9) C

(10) A

(11) A

(12) D

ABCDE

(13) B

(14) C

(18) B

(16) B

(17) C

( 18 ) D

(19) B

(20) A

(21) D

( 22 ) E

(23) D

(24) E

(25) E

(26) A

(27) C

(28), A

( 29 ) D

(30) E

ABCDE

8. DODOU

16.下列各數式中,那一個的個最大?

A: (-2) XB

C (-2)+(-8)

E (-2) (+8)

ABCDE 15. DOUJ

D (-2)-(-8)

17.下列各款式中,那一個的僅是質數?

A 8(-2) (-1)

24. 36-2y-4Y+12 » Y=7

48

B 24 E. 4

25. 7(Y+14) -28–7 · Y- 2.

A 11

7

A 7X

B 3X

C -X

D -3X

E -7X

ABCDE

10. 31 -4x+2y=7

A 5Y-4X BY 4X

9. Q☐☐DO

G (-2)x5

C 3Y-2X

D 4X+SY E 4X-5Y

ABCDE

10.

B (-2) (-8)

ABC DE 16. J

B 7

B 6-8-+4

6 -11

0 (372

26. Yж4-Y+Y=?

4

A B C D E

A

B

17.

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