University of Montenegro



MOCK EXAM

I Circle one of the given options to best complete the text: (12)

Rational and Irrational numbers

A number r is rational if it can 1. _________________ as a fraction r = p/q where both p and q are integers. To be rational a number 2. _________________ have at least one fractional representation. For example, the number

|  |((√5 + 1)/2)2 + ((√5 - 1)/2)2 |

3. _________________ not at first look rational but it simplifies to 3 which is 3 = 3/1 a rational fraction. 4. _________________ the other hand, the number √5 5. _________________ itself is not rational and 6. _________________ irrational. This is by no means a definition of irrational numbers. In Mathematics, it's not quite true that what is not rational is irrational. Irrationality is a term 7. _________________ for a very special kind of numbers. However, there are numbers which are 8. _________________ rational or irrational. Much of the scope of the theory of rational numbers 9. _________________ by Arithmetic. A major part belongs 10. _________________ Algebra. The theory of irrational numbers belongs to Calculus.

11. _________________ only arithmetic methods it's easy to prove that the number √5 is not rational. Just to remind, √5 stands for the number whose square equals 5. Thus Arithmetic can show that, when squared, no rational number gives 5. It's 12. _________________ to derive in Arithmetic that such a number actually exists.

1. 1. a) write b) being written c) be written d) is writing

2. 2. a) ought b) ought to c) needn’t d) mustn’t

3. 3. a) may b) must c) should d) can

4. 4. a) at b) from c) on d) in

5. 5. a) with b) on c) to d) by

6. 6. a) is called b) calls c) is being called d) call

7. 7. a) reserve b) reserved b) reserves d) reserving

8. 8. a) nor b) neither c) if d) unless

9. 9. a) covers b) covering c) will cover d) is covered

10. 10. a) on b) with c) upon d) to

11. 11. a) used b) having used c) use d) using

12. 12. a) unpossible b) dispossible c) inpossible d)impossible

II Supply the missing articles where needed. (5)

___/___ functions play ___A____ fundamental role in all areas of _/___ mathematics, as well as in other __/____ sciences and engineering. However, the intuition pertaining to ___/__ functions, notation, and even __THE__ very meaning of __THE__ term "function" varies between the fields. __/ (THE)__ more abstract areas of mathematics, such as __/__ set theory, consider very general types of functions, which may not be specified by __A_ concrete rule and are not governed by any familiar principles.

III Complete the text with the appropriate words. (5)

suppose if since then so also

we have shown that by hypothesis arbitrary

___IF_______________ [pic] and [pic] ____THEN_____________ [pic] .

To show that [pic], we need to show that [pic] So we ________SUPPOSE______ [pic]

_____BY HYPOTHESIS____, [pic], ______SO______ [pic] ______ALSO__ by hypothesis [pic], so [pic]

________SINCE______ this was true for any ____ARBITRARY__ [pic] __WE HAVE SHOWN THAT___ [pic]

IV Turn the direct into indirect speech: (5)

1. “We know all the answers,” the students said.

_____The students said that they knew all the answers.__________________________________

2. “You have to give your best if you want to pass,” the teacher said.

_____The teacher said that you had to give your best if you wanted to pass.______________________

3. “What will we study next year?” asked the student.

_____The student asked what they would study the following year._____________________________

4. “You have made great progress,” the lecturer said.

_____The lecturer said you’re you had made great progress.__________________________________

5. “I forgot my homework,” the student said.

_____The student said that he had forgotten his homework.____________________________________

V Complete the text using the correct form of the verbs in brackets: (8)

In 1966 the famed Nobel Prize-winning physicist Richard Feynman, a passionate drummer, _____was asked______ (ask) by a Swedish encyclopedia publisher ___to supply________ (supply) a photograph of himself "beating the drum to give a human approach to a presentation of the difficult matter that theoretical physics represents." Feynman's reply? Dear Sir,

The fact that I beat a drum has nothing to do with the fact that I ___do_____ (do) theoretical physics. Theoretical physics ___is_____ (be) one of the higher developments of human beings, and the perpetual desire to prove that people who do it are human by showing that they do other things that a few other humans do (like playing bongo drums) is insulting to me. I am human enough to tell you to go to hell. Yours, RPF.

II All of the famed philosopher and mathematician Willard Van Orman Quine's work ____was typed_____ (type) on a single 1927 Remington typewriter whose '1,' '!,' and '?' keys he had been removed and replaced with specialized mathematical symbols. Quine was once asked how he managed ____to write________ (write) without using a question mark. "Well, you _______see______ (see)," he __replied_________ (reply), "I deal in certainties."

VI Translate the following text: (10)

I After receiving a sound education in mathematics, classics, and law at La Flèche and Poitiers, René Descartes embarked on a brief career in military service with Prince Maurice in Holland and Bavaria. Unsatisfied with scholastic philosophy and troubled by skepticism of the sort explained by Montaigne, Descartes soon conceived a comprehensive plan for applying mathematical methods in order to achieve perfect certainty in human knowledge. During twenty-year period of secluded life in Holland, he produced a body of work that secured his philosophical reputation. Descartes moved to Sweden in 1649, but did not survive his first winter there.

II Chess is interesting from the mathematical point of view. Many combinatorical and topological problems connected to chess have been known of for hundreds of years. The number of legal positions in chess is estimated to be between 1043 and 1050. Typically an average position has thirty to forty possible moves, but there may be as few as zero (in the case of checkmate or stalemate) or as many as 218.

VII Find synonyms for the following definitions (5).

The English mathematician Charles Babbage, famed for his invention of an early mechanical computer (the so-called "Analytical Engine"), once took issue with one of Tennyson's poems. The poet soon received a letter from the logician:

"In your otherwise beautiful poem," Babbage wrote, "one verse reads,

Every moment dies a man, Every moment one is born.

"If this were true, the population of the world would be at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest:

Every moment dies a man, Every moment 1 1/16 is born.

"Strictly speaking," Babbage added, "the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 is sufficiently accurate for poetry."

argue ____take issue___________ (two words)

more than a particular amount _____excess________

a set of lines that forms one part of a song, poem, or a book ____verse_________

enough ______sufficiently________________

a situation in which there is no movement or activity at all _____standstill______

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