第 4 页

1．（a+b）n=？

2．ai= ？（i=）

3．表通常十进数345为二进数

4．加法及乘法之交换律、结合律、分配律如何？

五、证明自椭圆二焦点至其任意切线之垂直距离，其乘积为一常数。

国立山东大学

一九四七年

一、如方程式ax3+3bx2+3cx+d=0有二相等之根，则其系数间之关系为（bc﹣ad）2=4（ac﹣b2）（bd﹣c2），试证之。

二、试解

三、试求椭圆中诸平行弦之中点轨迹的方程式。

四、已知二圆C1:x2+y﹣6x=0、C2:x2+y2﹣4=0，求通过C1、C2之两交点及另一点（2，﹣2）之圆的方程式。

五、1．求cos40°cos60°cos80°之值。

2．试∠A+∠B+∠C=90°，则tanAtanB+tanBtanC+tanCtanA=1，试证之。

国立厦门大学

一九四七年

一、等积各三角形，证明其等腰者周界为极小，当其底相等时。

二、ABC三角形中，自A、B两点各作对边垂线，垂足为DE，设MN为DE及AB之两中点，证明MN⊥DE。

三、三角形ABC中，其边为a、b、c，内接圆半径为r，试证：。

四、证明。

五、求中之x45之系数。

上海圣约翰大学

一九四七年

Algebra

1. Determine the graph of y=x2+x+1. Find the coordinates of its vertex and the equation of its axis.

2. (1) There are two groups of boys on the street corner. One boy leaves the first group and joins the second. The groups are then equal in size. If one boy had left the second group and joined the first, the first group would then have been three times as large as the second. Find the original size of each group.

(2) Simplify

3. Solve and verify x2+3x﹣6(x2+3x﹣3)+2=0

4. (1) Determine the real value of k, so that the roots of equation, (k﹣1) x2﹣kx﹣2x+4=0, may be real and equal.

(2) Simplify

5. Evaluate to four significant figure by logarithm,

Geometry

1. Show that the line joining the mid-points of the diagonals of a trapezoid is half the difference of its bases.

2. With BC, one leg of ΔABC, as diameter, a circle is described intersecting the hypotenuse AC at P. From P draw a tangent intersecting AB at D. Show that AD=BD.

3. Transform the difference of two squares into a rectangle, the ratio of two sides being 2:3.

4. Given:∠1=∠2, a, b, c, find d.

Trigonometry

1. (1) Prove that:

(2) Prove that:

2. (1) Solve 1+

(2) Solve secx﹣cotx=cscx﹣tanx

3. The base of a right circular cone has a diameter of 25 feet and its slant height is 40 feet. The surface of the cone is cut along a straight line from its vertex to a point on the base, and the surface is then spread out flat to form a sector of a circle. Find the angle of its sector in degrees.

4. A tower of 20. 7 feet high stands at the edge of the water on a bank of a river. From a point directly opposite to the tower on the other side of the river above the water, the angle of elevation of the top of the tower is 27°17' and the angle of depression of the image of its top in the water is 38° 12' . Find the width of the river.

5. The sides of a triangle are 149, 163 and 222. To find

(1) The area of the triangle.

(2) The radius of the inscribed circle.

(3) The angles of the triangle.

6. Two towers, A and B, on the shore of a lake can be observed from only one point C on the opposite shore. The lines joining the bases of two towers subtend an angle of 63°42' at C. The heights of the towers are 132 feet and 89 feet, and the angle of elevation of the tops as seen from C are 8°13' and 7°21' respectively. Find the distance AB.

Analytical Geometry

1. Find the distance from the point (﹣1,﹣4) to the straight line which is drawn through (﹣2, 6) and the perpendicular to the line joining (3, 6) and (﹣5,﹣1) .

2. Show that the tangent to a hypobola makes equal angles with the focal radii drawn to the point of tangency.

3. Show that the tangents to the parabola y2=4px at the extremities of the latus return are perpendicular and meet at the intersection of the directrix with x-axis.

4. Find the locus of point which moves so that the sum of its distance from the points (﹣7,4) and (9,4) is always 20 units. Determine the center, foci, vertices, axis and eccentricity. Locate all points.

A figute is required for each problem

* * *

[1]　即因式分解。

[2]　1946年中央政治学校与中央干部学校合并，定名为国立政治大学。

英文之部

国立北平师范大学

一九三三年

Ⅰ.1. Give the plural of the following nouns.

fish, child, alumnus, datum, mercy, thief, calf, tooth, son-in-law.

2. Give the comparatives and the superlatives of the following adjectives.

good, ugly, interesting, bad.

3. Write down the past tense and past participle of each of following verbs.

blow, drink, wring, teach, know, forget, read, steal, swear, ride.

4. Define the following and give examples.

metaphor, simile, copula, declension, dative case.

5. Correct or justify the following sentences.

(1) He said that the earth is round.

(2) He I am she. She me till death and beyond it.

(3) One of the greatest historical wonders are the Great Wall of China.

(4) Being very old the work is too heavy for me.

(5) Will I come to see you tomorrow?

(6) He is the most diligent of any student in this school.

(7) The father, together with the son, go to the Park.

(8) Gulliver's Travels was written by Swift.

(9) You are younger than him.

(10) I wanted to have talked with you when I met you yesterday.

Ⅱ. Write a Composition on one of the following subjects.

Autumn Reveries

A Farewell to Middle School

An Ideal University

Science and National Defense

Ⅲ. Translate the following into idiomatic Chinese.

"Nobody." he says," could possibly doubt that the chief concern of the law-giver must be the education of the young". His whole theory of ethics demands a process of "habituation" or training in moral habits, as a necessary propaedeutic to virtue; and he regards the city state and its authorities as alone competent to furnish this training…… Pericles could claim for Athens: " Our city as a whole is an education to Greece" and the city which was a school for other cities was also a school of taste and conduct and character for its own members. It taught its citizens by a spontaneous didactic " to be lovers of beauty without extravagance and lovers of wisdom without unmanliness"; it provided, above all other cities," recreations from toil for the spirit—contests (in tragedy and in comedy and in lyric poetry) all the year round, and beauty in public buildings to delight the eye day by day." The Athens of Pericles, after all, was an educational State.

1．不准请求监试委员解释题目

2．只需要抄写第几级第几节，无须抄写全文

国立北洋工学院

一九三三年

N. B. Candidates are requested to devote one hour to the translation and two hours to the composition.

Ⅰ. Write a composition of not less than 300 words on any one of the following subjects.

The Greatest Need in China at Present

Why I Want to Study Engineering Courses

Ⅱ. Translate the following into English.

1．今日国家大势如此危急任何部分事业皆有重新改造之必要凡有一技之长必须尽其所能为事业谋进步为人类谋幸福不幸我国人缺创造之精神无论所学者是何专门而结果大多数趋于政治之一途驯至政界中人供过于求而有用之材皆变成社会寄生物不能表现其所学习之技能欲救此弊只有希望多数学生不向政界谋生活各专其所能致力于各种事业庶几人人皆有职业而社会亦可发达矣。

2．江西省之玉山萍乡铁路经行政院于本年六月十三日决议由铁道部积极筹办据关系方面负责人声言此路在商业上国防上均关重要东端与浙江省直杭江路接轨即直通东方大港货物由乍浦进出极形便利该路由萍乡至玉山尚待实测全线工程至少必须四千万元如此巨款非政府目前所能筹集庚款亦多作他用尚望全国金融界予以协助如款项集齐工程方面思念可告成功云。

国立交通大学（上海本部）

一九三三年

Ⅰ. Punctuate the following paragraph and supply all necessary capitals.

soul and body body and soul how mysterious they were there was animalism in the soul and the body had its moments of spirituality, the senses could refine and the intellect could degrade who could say where the fleshly impulse ceased or the physical impulse began how shallow ware the arbitrary definitions of ordinary psychologists and yet how difficult to decide between the claims of the various schools was the soul a shadow seated in the house of sin or was the body really in the soul as giordano bruno thought the separation of spirit from matter was a mystery and the union of spirit with matter was a mystery also oscar wilde the picture of dorian gray.

Ⅱ. Correct the following sentences and state a reason for each correction.

1. Having been reared in the country, the city often oppresses me.

2. Mother told sister that the doctor had advised te get more sleep.

3. Neither the doctor nor the nurse were able to give him any relief.

4. Taking the child in her arms, she set down and sung it to sleep.

5. Arson is when a person maliciously sets fire to a house.

Ⅲ. Define the following terms as used in rhetoric: exposition, unity, clause, smile, usage.

Ⅳ. Write a composition on one of the following subjects.

How Aeroplanes Will Benefit China

The Farm Problem in China

南开大学

一九三三年

Ⅰ. What do you mean by simple, compound, complex, loose, periodic and balanced sentences? Give an example of each.

Ⅱ. Point out all the noun clauses, adjective clauses and adverbial clauses in the following sentences. Also point out the phrases and classify them according to their form (prepositional, participial, infinitive) and use (noun, adjective, adverbial, independent) .

1. It is difficult to teach a cat, which is naturally active, any kind of athletic tricks.

2. Did you learn whether he passed in his examination?

4. We can enjoy even unpleasant surroundings, provided we look for something interesting in them.

5. First of all, a dog wearing a silver collar attracted the attention of my friend.

6. All I can say is that I remember seeing him leave the room.

Ⅲ. Correct the following sentences.

1. The governor of Jehol was ill treated the troops and heavy taxed on the people.

2. The real heroes of China are the soldiers whom used their blood to fought against Japan.

3. I do not know what will they do?

4. Thomas A. Edison is one of the greatest inventors who has ever lived.

5. Being a guest, we were surprised at his rudeness.

Ⅳ. Indicate the accents of following words by underlining the syllables accented.

1. atmosphere 4. contract (verb) 7. expert (adjective)

2. contribute 5. Japanese 8. absolutely

3. excellent 6. economist 9. comment

10. record (verb)

Ⅴ. Translate the following passage into Chinese.

Since the Commercial Revolution of the sixteenth century, the people of all continents have been brought into communication with one another. The horizon of the civilized world is no longer limited to a single continent, but includes the entire globe. As a result of exploration, of travel, of oceanic steamships and world-commerce, of world-politics, and of the intermingling of races by migration. Our mental outlook has become incomparably broader. Moreover, we have built up an economic structure that is world-wide, with the result that each continent is dependent upon the others for everyday articles of food and commerce. No nation, no continent, can now shut itself off from the rest of the world as in bygone ages.

Ⅵ. Write a composition of not less than 200 words on the following subject.

The Evils of Civil War

焦作工学院

一九三三年

Ⅰ. Write the following composition.

A Description of This Room

Ⅱ. Translate the following into Chinese.

Engineering in its early uses referred specially to the operations of these who constructed engines of war and executed works intended to serve military purposes. Such military engineers were long the only ones to whom the title was applied. But about the middle of 18th century there began to rise a new class of engineers who concerned themselves with works which, though they might be in some cases, as in the making of roads, of the same character as those undertaken by military engineers, were neither exclusively, and those men by, way of distinction came to be known as civil engineers.

Ⅲ. Correct the mistakes which you find in the following sentences.

1. We want him to quickly to go away.

2. I am very amused by this words.

3. we do not like to hear news which tell us about the lost battle.

4. He got that what he had been longing for.

5. the house which I live needs repairing.

6. I went so late, I missed the train.

7. Many of the people who I know were there.

8. Why you went to that school yesterday?

9. Please you hand me your book.

10. As a rule, the health of the people in towns is poorer than the people in the country.

11. It is importance that one should write ligibly.

12. I make myself to take exercise early in the morning.

13. He would have took this away himself but they stopped him.

14. These books are not to sell under fifty cents a copy.

15. Were I there, I will have prevented that accident.

Ⅳ. Comprehension Test

It was a very black and very dirty street down which I made my way that November morning at half-past five. There was no paving, there was no sidewalk, there was no lights. Rain had been following for several days, and I waded through seas of mud and sloshed through lakes of water. There were men in front of me and men behind me, all plodding along, through the muck and mire, just as I was plodding, their tin lunch-pails rattling as mines was rattling. Some of us were going to work, some of us were going to look for work. The steel-mill lay somewhere in the darkness ahead of us.

Hardly out of sight of the smoking factory chimneys and scarcely out of hearing of the noise and bustle of Barcelona, busiest and most restless city of Spain, a medieval Benedictine monastery clings to the face of a fantastic stone peak that rises boldly from the brown foothills of Catalonia. It is Montserrat, the nation's holiest shrine, to which thousands of the Spanish faithful make pilgrimage each year to pay homage to what is called the Black Virgin.

Directions:

Answer the following sentences correctly, you will find after each sentence the word yes or no. If the sentence is a corrected-statement of some words or ideas from the two paragraphs, you must underline the word yes and then write down, in the blank space that follows, such words from above passage as will justify you answer. On the contrary, if the sentence disagrees with anything in the above quotation, you must underline the word no and write the word with which the sentence disagrees.

1. I was walking alone down the street that morning.

yes, no——

2. I had a tin lunch-pail with me.

yes, no——

3. None of the people who walked down the street were without work.

yes, no——

4. Much water and mud were on the street.

yes, no——

5. We could see the steel-mill as we walked along.

yes, no——

6. The monastery is far from Barcelona.

yes, no——

7. Barcelona is an industrial town.

yes, no——

8. The monastery is on the side of a mountain.

yes, no——

9. Montserrat is the name of the monastery.

yes, no——

10. Thousands of Spanish faithful, that is, religious believers go to visit the monastery.

yes, no——

厦门大学

一九三三年

第一次

Ⅰ. Distinguish a phrase from and a clause, a subordinate clause from a principal clause.

Ⅱ. Illustrate with examples the use of the relative pronouns who, which, that, and what.

Ⅲ. Write one simple sentence in the present tense, two complex sentences in the present perfect, and three compound sentences in the future.

Ⅳ. Diagram the following sentences.