Vocabulary and grammar exercises

I. Find the English equivalents of the following Russian word combinations:

1). совокупность точек; 2). весь класс; 3). как необходимое так и доста- точное условие; 3). известная плоская кривая, 5). обязательно удовлет- ворять; 6). точно говоря; 7). единственное условие; 8). предварительное изучение; 9). тесно связанный; 10). находиться на графике; 11). запи- сывать кратко; 12). две основные задачи; 13). проходить через данную точку; 14). смысл утверждения.

a. both necessary and sufficient condition; b. strictly speaking; c. implication of a statement; d. a single condition; e. to lie on a locus; f. closely related; g. the entire class; h. the totality of points; i. to path through a given point; j. two fundamental problems; k. a preliminary study; l. a familiar plane curve; m. to write briefly; n. necessarily satisfy.

III. Give the corresponding plural form of the following nouns and their Russian equivalents.

1. focus → foci

Calculus, genius, locus, modulus, nucleus, radius;

2. axis → axes

Thesis, emphasis, analysis, basis, hypothesis, crisis, phasis, parenthesis;

3. datum→ data

Continuum, medium, spectrum, minimum, maximum, phenomenon, criterion, polyhedron;

4. vertex → vertices

Matrix, directrix, bisectrix, index

5. formula → formulae (formulas)

Abscissa, huperbola, lacuna, corona

III. Insert the suitable words: (a locus, preliminary, interpretation, convenience, closely, plane, procedure, implication, constitutes, determine).

1. Analytic geometry considers the entire class of all … curves. 2. Every student must understand the … of the equation of a locus. 3. … is the totality of points whose coordinates satisfy the equation f (x, y) = 0. 4. To draw the graph of a function we must do a … study of the function. 5. The equation of a locus has an analytic … . 6. If we know the coordinates of a point we can … its position in a plane. 7. For … we shall focus attention on this characteristic of the locus. 8. All the foci, possessing one property, are … related. 9. This theorem … a common property of fields. 10. The … of obtaining the equation of a locus consists of four steps.

IV. Read and translate the following sentences paying attention to the translation of “it”.

1. It is interesting to solve this problem. 2. It must be emphasized that division by zero is impossible. 3. It is convenient to use the method of elimination for solving simultaneous equations. 4. It follows from the theorem that the obtained relation is true. 5. To solve this equation it is necessary to simplify it. 6. It will be proved that this point lies on the curve. 7. It is an equation in two unknowns. 8. It is easier to ask a question than to answer it. 9. I found that article and began translating it at once. 10. Take this book, it is on the shelf. 11.It often snows in winter. 12. He wants to make it himself. 13. It is clear that all the numbers p₂, m₂, n₂ are different from zero.

V. Read and translate the following sentences paying attention to the translation of the modal verb “should”.

1. Any point on the curve should possess the unique property. 2. We should apply this rule for solving the problem. 3. A circle is a plane curve which should satisfy a unique property. 4. We should have a proper description of the problem if we want to receive a solution. 5. This locus should be investigated by the students. 6. We should determine the equation of this geometric figure.

VI. Read and translate the sentences with different functions of the Infinitive.

a) 1. To solve the equation is to find the numerical values of the unknowns. 2. To check the addition of a column of figures is to add in the reverse order. 3. To define which of the numbers is less is not difficult. 4. To evaluate an expression means to substitute the numerical equivalents for the letters. 5. To find the logarithm of the given number means to find the exponent of the power according to the given power and the given base. 6. To prove a theorem in a deductive system is to snow that it is a necessary logical consequence of some previous propositions.

b) 1. To change a mixed number to an improper fraction we multiply the denominator of the fraction by the integer, add the numerators and place the sum over the denominator. 2. To understand some formulas she used the text-book of mathematics. 3. To raise a power to a power it is sufficient to multiply their exponents. 4. To simplify an expression in two unknowns one transforms it in the same way as equations in one unknown. 5. To keep the number unchanged in value we must multiply it by the same power of ten.

c) 1. All operations to perform in succession will give the expected result. 2.The choice of methods as a rule depends on the problem to be solved. 3. Euler was the first to prove this law. 4. There will be six independent elements to be determined. 5. The proof to be tested concerns the conditions of convergence of function series.

d) 1. The professor told him to prove the theorem. 2. Our teachers asked us to do all necessary calculations. 3. This force caused the body to move. 4. The students wanted to investigate the properties of square matrices. 5. The word “smooth” is used to suggest that the motion of a point has no abrupt changes of direction.

g) 1. A similar device may be used in triple integrals. 2. Mathematics is changing constantly and algebra must reflect these changes if it wants to stay alive. 3. A more general treatment of the whole subject is to be found in the monograph.

VII. State the functions of the Infinitives and translate the sentences.

1. It is often required to find quantities from the given derivatives. 2. The methods to be described above were widely used. 3. To find the square of a number means to multiply it by itself. 4. In order to check the accuracy of the root you must raise the number to the power. 5. It is impossible to draw more than one line through two points. 6. Apollonius was the first to introduce the terms “ellipse”, “parabola”, “hyperbola”. 7. To say that a = b means that these letters are different symbols representing the same natural numbers. 8. To find out the truth is necessary. 9. By the use of algebra we can reduce complex problems to simple formulas. 10. The methods to be followed are based on some peculiar properties of these equations.

VIII. Translate into English:

1.Мы изучим две основные проблемы аналитической геометрии. 2. Это определение выражает необходимое и достаточное условие. 3.Урав- нения линейного преобразования рассматривались в предыдущем тексте. 4. Каждая точка на кривой должна удовлетворять конкретному (данному) правилу для кривой. 5. Сейчас мы определяем простую кривую типа С. 6. В алгебре студенты строят графики любого типа. 7.Обе задачи составляют основную проблему аналитической геометрии. 8. Мы можем составить уравнение геометрического места точек. 9. Умножение, по существу, обратно делению. 10. Геометрическое место точек имеет важное свойство.

IX. Read the sentences and translate them paying attention to the verbs “to have to”, “to be to”:

1. How the solution is to be performed the author has explained above 2 Further we are to give some theorems. 3. To avoid memorizing everything we are to study how different facts are related and how the more complicated ones depend upon other simpler ones. 4. The Greeks treated some numerical problems, but in doing so their method was to give geometrical interpretation to numbers. 5. To improve your knowledge of literature you have to read much. 6. It is to be proved that there are at least 2 points invariant under this transformation. 7. We have finally to consider the case in which x is negative.

X. Answer the following questions:

1. What are the two fundamental problems of analytic geometry? 2. How many pairs of values of x and y satisfy the equation f (x, y) = 0? 3. What is the locus or graph of the equation? 4. What point lies on the locus of the equation F (x, y) = 0? 5. What are the coordinates of the point of a locus restricted by? 6. By what is a geometric figure generally given? 7. How may a circle be defined? 8. How many conditions may a locus satisfy? 9. What is the difference between the first and the second problems?