Entrance examinations - Mathematics

Mathematics entrance examination program is developed on basis of the exemplary program recommended by the Ministry of Education and Science of the Russian Federation.

MATHEMATICS

This program consists of three sections.
In the first section there are listed the main mathematical concepts every applicant has to know for passing both written and oral examinations.
The second section represents the list of questions for the theoretical part of an oral examination.
The third section specifies the knowledge and skills requirements of an applicant for passing written and oral examinations.
Extent of knowledge of the material described in the program corresponds to a high school mathematics course. Applicant may use all means from this course, including elements of calculus. But to solve examination problems it is enough to have solid knowledge of the concepts and their properties listed in the present program. Objects and facts studied at general education school also can be used by applicants, but provided that they are capable to explain them and to prove.

 

Major mathematical concepts and facts

Arithmetic, Algebra and Pre-Calculus
Natural numbers (N). Prime and composite numbers. Devisor, multiple. The greatest common divisor, The least common multiple.
Divisibility rules for 2, 3, 5, 9, 10.
Whole numbers (Z). Rational numbers (Q), their addition, subtraction, multiplication and division. Comparison of rational numbers.
Real numbers (D), their representation as decimal fractions.
Number line. Modulus of a real number, its geometrical meaning.
Numerical expressions. Variable expressions. Formulas of abridged multiplication.
Power with natural and rational exponent. Arithmetical root.
Logarithms, their properties.
Monomial and polynomial.
Polynomial in one variable. Root of polynomial by example of quadratic trinomial.
Concept of function. Representations of functions. Range of definition. Range of function.
Graph of function. Function increase and decrease; periodicity, even and odd functions.
Sufficient condition for function increase (decrease) in the interval. Concept of function extremum. Necessary condition for function extremum (Fermat’s theorem). Sufficient condition for extremum. Function largest and least value in the interval.
Definition and main properties of functions: linear, quadratic  , power  , exponential , logarithmic, trigonometric functions

(), arithmetical root .
Equation. Roots of an equation. Concept of equivalent equations.
Inequations. Inequation solution. Concept of equivalent inequalities.
Set of equations of inequalities. System solution.
Arithmetic and geometric progression. Formula of the nth term and sum of first n terms of arithmetic progression.
Sine and cosine of sum and of difference of two arguments (formulas).
Sum-to-product formula .
Differentiation. Its physical and geometrical meaning.
Derived functions      .

Geometry
Line, ray, segment, polyline; length of line segment. Angle, angle value. Vertical and adjacent angles. Circumference, circle. Parallel lines.
Examples of transformation, types of symmetry. Similarity transformation and its properties.
Vectors. Vector operations.
Polygon, its apexes, sides, diagonals.
Triangle. Its median, bisector, altitude. Types of triangles. Relationships between sides and angles of a right triangle.
Quadrangle: parallelogram, rectangle, rhombus, square, trapezoid.
Circumference, circle. Centre, span, diameter, radius. Tangent to circle. Arc of circle. Sector.
Central and inscribed angles.
Formulas for area of: triangle, rectangle, parallelogram, rhombus, square, trapezoid.
Length of circumference. Arc length. Radian measure. Area of a circle and of a sector.
Similarity. Similar figures. Similar figures area ratio.
Plane. Parallel and intersecting planes.
Parallelism of line and plane.
Angle between line and plane. Perpendicular to plane.
Dihedral angle. Linear angle of dihedral angle. Perpendicularity of two planes.
Polyhedrons. Their apexes, faces, diagonals. Straight and oblique prisms; pyramids. Regular prism and regular pyramid. Parallelepipeds, their types.
Solids of revolution: cylinder, cone, sphere, ball. Centre, diameter, radius of a sphere and a ball. Plane, tangent to sphere.
Surface area and volume formulas for a prism.
Surface area and volume formulas for a pyramid.
Surface area and volume formulas for a cylinder.
Surface area and volume formulas for a cone.
Volume formula of a ball.
Surface area formula of a sphere.

 

Primary formulas and theorems

Algebra and Pre-Calculus
Properties of function and its graph.
Properties of function and its graph.
Properties of function and its graph.
Quadratic formula.
Factorization of quadratic trinomials.
Properties of numerical inequalities.
Logarithm of product, exponent, quotient.
Definition and properties of function and , their graphs.
Definition and properties of function  и , their graphs.
Solution of equations type .
Reduction formulas.
Dependencies between trigonometric functions with identical argument.
Trigonometric functions of double argument.
Derivative of the sum of two functions.

Geometry
Properties of an isosceles triangle. Properties of points equidistant from the segment endpoints. Characteristics of parallel lines.
Angle sum of a triangle. Sum of exterior angles of a convex polygon.
Characteristics of parallelogram, its properties.
Circumscribed circle of a triangle.
inscribed circle of a triangle.
Tangent to circle and its properties.
Measure of an inscribed angle in a circle.
Characteristics of similarity of triangles.
Pythagoras' theorem.
Formula of area of a parallelogram, triangle, trapezoid.
Distance formula between two points of plane. Circle equation.
Line-plane parallelism characteristic.
Planes parallelism characteristic.
Line-plane perpendicularity theorem.
Perpendicularity of two planes.
Theorems of parallelism and perpendicularity of planes.
Theorem of three perpendiculars.

 

Primary knowledge and skills

Applicant has to know:
To perform arithmetic operations with numbers in the form of standard and decimal fractions, with desired precision to round off these numbers and computational results; to use calculators or reckoners.
To carry out identity transformations of polynomials, fractions containing variables, expressions containing exponents, indicative, logarithmic and trigonometrical functions.
To make graphs of linear, quadratic, exponential, indicative, logarithmic and trigonometrical functions.
To solve equations and inequalities of the first and second degree, equations and inequalities leading to them; to solve systems of equations and inequalities of the first and second degree and those leading them. Here, in particular, are related the elementary equations and inequalities containing exponential, indicative, logarithmic and trigonometrical functions.
To solve problems with equations and systems of equations.
To draw geometric shapes and to make elementary constructions in the plane.
To use geometrical representations while solving algebra problems, to use algebra and trigonometry methods while solving geometric problems.
To perform vector operations in the plane (adding and subtracting vectors, vector-number multiplication) and to use properties of these operations.
To use concept of a derivative while examining functions on increase (decrease), extrema and while making graphs of functions.