1967-01-11 — Page 17

有一察張五第

育敎僑華

中文中學會考試題預習專欄

KEMIE-

日一十月一年七六九一限公竿六十五國民華作

vapour is tilted, the

WAH KIU

期

英文中學會考試題預習專欄

** If the barometer with saturated ether

mercury rises up the

tilted

ether

Vapour

物理科

(十一)·梁海明·

PHYSICS (11) YE«M«Leung]

same level

數學科

(+-)

喬仲强

Solution.

第十次預習試題解答

(1) hot 1 (Real Number)?

* a, b, c, xtil§ ¤ (a+b)x−ab(a+c)x+b+c2=0, 証oc成等比级数,且其公比笃x

(答)在研究根式時負數的偶次方根,本来是没有意義的,不過 一般數學家 為了再開闢一個新的數域,於是假定F= 来代表這個數域的單位,而稱這些數為虛數(Imaginr any Number) 17 x Ą Bk WG PL & IK.

(记)由一元二次方程式之根的公式得方

b(a+c) = {f(a+c) = (a+b)(tre)

ñ b*(a+c)~(a+b2) (b+c®)=a*b*+2abc+bc-(ab+acıb

- zabc-oc- -(b2 zabc+ac)=-(b−ac)*

muzik1⁄2 1⁄2 b-ac=0[1 − (b = ac)*x = = $<]

b=ac & a,b,c£ G.P. a£*£££ b(a+c) - b (a+c) — &

X

a+b

a (a+c)

(2)/2]¶fit 182 (Geometrical Progression)?

arbic, bre-a, crab, a+b-cA G.P. &‡Ʌuğr

(答)在一列

M2, Ms

則此數列箱茑繁何级数 稱為其公比

cta

依和比定理,

r_(c+a_b)+(a+b-c)_za_2

x = (l+c_0)+(c+a_b)=2(c)

λ== (b+c-a)+(a+b=c)

(ature) + (cra-b) 以(1),(2),(3)式代入————

2(eta)

~~(2)

X+X2=λ(2+1)====*== ax ( 1 + 1 ) = 1 + 1 arc

又ツ(2) 代入 リール=ハーフ

H

(三)本題記載之第二部分,田於文值書多(至少有上述= 故選擇時頗感因難,筆者祇就其中最簡單式研討 得上述証域, 讀者於此,當可明白盡量利舉题目中冊给予 之條件,則雖較马業難,亦可迎刃而解矣。

(3)一等比級数え簗わ項第8項等え項各導とする試証

(g-s) logx+(2−p) log y + (p−8) logz

(證)設此等比級數之首項為a公比笃R.依公式l=akt

各取對數

2+ (p-1) Log R

|= loga+ (8—1) logR

log 3 = Rogat (1-1) logR

(3)

19.(6) Some Ilumps of ice are dropped into a highly polished calorimeter two-thirds full of water, A thermometer isuusedi for stirring. As the tem- perature falle, the surface of the metal sud- denly becomes dulled by the formation of a film of minute droplets of water or dew. Record the temperature at which this happens. This temper- ature is called the dew point of the atmosphere! The reason is that the cold metal surface cools the air in its vicinity to a temperature below that for which the water vapour present is suf- ficient to saturate the air. Excess vapour con- densee out, and the highest temperature at whicf this can occur is the dew-point.

(b) (1) In saturated vapour, the vapour pressure

is constant, but the unsaturated vapour pressure is not.

(il) The saturated vapour pressure does not depend on its volume but this is not so in the unsaturated vapour.

[(c) Relative Humidity is defined as the ratio or mass of water vapour present in a givan volume. of air toomass of water vapour necessary to saturate the same volume of air at the same temperature.

But this definition is not convenient to base. experiment on, so we deduce a further relation- ship. Since at a given temperature, the pre- ssure exerted by

vapour in the air

is proportional top te mäsa nf water vapour present,

Pressure of ter vapour in the air

8.V F of water the temp. of, the

but the qua

as the B.V.P

the relative

be found out by vapour, at dew point the room tempe: at different table.

20. (a) We use

mea

(b) Cont

ether are

creases!

of ether

the numeratt

the

dew point he

the S.V.P

S.V P. water vapour at The B.VP water vapour fres can be read from a

shown in the

saturated ether Vapour

urther drope of

rcury level de-

11 a small quantity top of the mercury. At is saturated, and any more ether added will not decrease the

this stage, the eth

mercury level. This shows that the pressure. 18 a maximum |

由此式 消毒 loga 及 lug凡,當可求得x子典九九之根則女亦為其多一報)而此種方程式,精鸾倒數方程式

關係式:

(1)=(2),

(2)−(3),

(4)-(5)

logy-log z = (z-r) logħ

log x_log y = (1-3) log R __(6)

·logz=logy (4-8) byk

logy-logz (8-A) lag R

~ (9-)) (logx_logy) —(†−q) (logy—log 3)

(q-r) log x-(q-r) log y = (p−q) logy-(4-8) log z

• q=r) log X−(q-r) log y_(p−q) logy+(png) logz=0

____ (8~2) log2+(r=f) bgy+(pz) logz=0

ad+c=0 £ a+b+c=2(ab+bc+ca),

a+bac2 2(ab+be+ca)

_b)-2(bc+ca)+

b)=2(a + b) c2 + (C2)-4ab

a+b=c2)-(zab)

並且倒數方程式如為奇次方程式,則或必為草根

(AF) u x2 P4 42 122+112=142 + !!+ 焦項 12(x+1)+1(x+1)-146 設 y=x+2, 9l y=x+2+x

sta 12 (y2− 2) + 11 y −146—0

去括号化箱

分解

W%

24+114-

(3y-10) (4y+17)=0 :: y=10 - 12. 号代入

3x2-10x+3=0

(x-3)(3x-1)=0

(x+4) (4x+1)

X3 B

味粉

-4 或--

+

4ab

•

(6)若x為是數,試求

之極大值典桎小值.

又x之值冬笃何?

(解) 設

Q.E.D.

+b2=c2+zab)(a+b=c2−2ab)

[(a+b)2=c2] [(a−b)_c"]

(a+b+c)(a+b=c) (a−b+c) (a−b−c)

[& a+b+c=

_____ a+b+c" =2(ab+b2c+ca)

(証二)因

a+b+c

a+b=

tube but keeps at

same level. As the volume decreases, the

exɑese vapour con denses back to liquid, and the vapour pressure remaine constant. This shows that the maximum vapour pressure is independent of volume of the vapour..

The boiling point of a liquid is the temper asure at which bubbles of vapour form in the body of the liquid. Every such bubble is held expanded by the pressure of the saturated vap- our inside it. The pressure outside it, which is tending to collapse it, ie the result of the atmospheric pressure acting on the surface of the liquid. If the atmospheric pressure ex- ceeds the S.V.P. the bubble will collapse; if the S.V.P. te equal to or greater than the pressure on the surface of the liquid, the bub he will rise to the surface and escape.

Comparison of evaporation and boiling:

Evaporation.

1. may take place

any temperature. 2. temperature may be

changing

3. vapour escapes only from the surface of -liquidy

Topics for revision this week.

Boiling.

takes place at fired temperature.

2. temperature remaine constant as long as liquid 19 boiling.

3. during boiling,

vapour escapes from the inner of the liquid.

L. Relation between thermal and mechanical energy.

anan afer of heat energy by conduction, convection

and radiation.

Questiona

21.(a) Explain the difference between conduction,

and radiation of heat, conveation, (b) Describe, with a diagram a Thermos": flask

Explain the orlaciples on which the design Le based

(c) What is meant by the statement that heat 18

a form of energy? (a) A calorimeter of water equivalent 10

contains 300 gm, of water at a temperature of 15 C. An automatic stirrer dips into the water and works at the rate of 10 ergs per seo, Calculate the temperature of the calorimeter and its contents after 20 minutes

that all the work done in the liquid 18, con- verted into heat. Take the mechanical equi- valent" of heat - 4.2 x 10". argă per calorie

at a speed of 150. 8. perk se

10 rest by striking arget: How much

the temperature of the let rise if all the kinetic energy is con-

ed into heat in the bullet itself? specific heat of lead - 0.03, mechanical equivalent of heat 4.2 x 10 erge per calorie

A piece of lead falls 3 metres from rest coming to rest again on the ground, Calculate the rise in its temperature, the specific heat of the lead being 0.032. State the assumptions you make in the calculation.

化簡 9x+

答:當

END

(3x+1)=1

204 Ymax = 1; a x=- (註)求之值城,亦可由下表得之,

4+4 (-)-

(س)

(-)

(+)

因D (判别式)不能為負數故之值城為

本次習題尚有加で誰法

2ca 24c+

(ab+be+ca) Q.E.D (一)本題第二証法當較第¬証法,不過第一法乃

移項化簡84

此類題之一個“因子分解”基本問題,故併錄出,以供

(5)何謂倒數方程式?

解方程式 128*411x-1468+

*** A=An a=Qx

則此程式文根;有成倒數之關係者(即如為某一

249 24x+2yx+y

4122 121 (ay−1) x2+ (2y+3)x+(y+3)

因爲宴敷 故判别式須為正數或茑零

·D= (ay+2)=4(2y-1)(y+3)

4y2+8y+4-84 — 20y+12

=-49=124+16

(y2+3y=4).

4(y+4)(y-1). (y+4) (y-1) ≤ 0,亦即此两因子須異號(或其中三 但因子為零)xy+4為较大的因子

42-4

bky z lifta Z3 -41e 12 Pal, Up -4<y<1%

化簡

x+2) ====

-82-4

a+b+c

f+c-a

b+c=a= (a+b+c){

c+a−b= (b+c-a)λ=(a+b+cN

a+b-c— (c+a−b)£= (a+b+c) r3

==the a+b+c=(a+b+c) (1+N*+R2)

第十一次預習試題

此証法較為簡單

)两等园交於A, B.以A为国心小於AB長為半狸作国玄 此二等因於同側之两桌CD試部 BCD三桌共线 (a) A ABCY AB>AC. <Aż 41 BC # D. £ BC ≥ 1§ 3

M. 9| MB: MD= (AB+AC); (AB-AC).

(3)-园典一定回及一已知直线相切,着此定因之直徑垂直

此已知直线`則直括z-端吳典两切桌 同在一直线上,並由 两园内切或相外切情形分别討論之

|(4) 求作一矩形典-已知正方形等情,且其底典高之差等

老定長

(5)右图中:CAD第0因切线cCOA=3d

CD=3A0, 80<E $1¥88- 之長,又其誤差一百分率差何?

(6) A ABC+ 1÷AZTEZUK

5:6-3,試证其餘弦之比!

B. 25:197

آرت