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Exemplary program of the entry examinations on Math


Numeracy, algebra and pre-calculus.

Natural numbers (N). Simple and composite numbers. Divider, multiple, least common multiple.
Test for divisibility by 2, 3, 5, 9, 10.
Whole numbers (Z). Rational numbers (Q), addition, subtraction, multiplication and division. Comparison of rational numbers. Real numbers (R), representation thereof in form of decimal numbers. Plotting of numbers on a straight line. Module of real number, and its geometric meaning.
Numerical expressions. Expressions with variables. Short multiplication formulae.
Powers with natural and rational exponents. Arithmetical root. Logarithms and their features. Monomial and polynomial. Polynomial with one variable. Root of polynomial by the example of quadratic trinomial.
The notion of functional. Function specification methods. Range of definition. Range of function.
Function graph. Function rise and fall; periodicity, evenness, oddness.
Sufficient condition for rise (fall) of function in the interval. Notion of the extremum of function. Essential condition of the extremum of function (Fermat problem). Sufficient condition of the extremum. Maximum and minimum value of function on the interval.
Definition and main properties of functions: linear, quadrant y = ax2 + bx + c, power y = axn (n ? N), y = k/x, indicative y = ax, a > 0, logarithmic, trigonometric functions (y = sin x, y = cos x; y = tg x, y = ctg x), arithmetical root y =??х
Equation, Equation roots, Notion of the equivalent equation. Inequations. Solutions of inequations. Notion of equivalent inequations. System of equations and inequations. System solutions.
Arithmetic and geometric progression. Formulae of the n-th order and sum of the first n-th orders of the arithmetic progression. Formulae of the n-th grade order and sum of the first n-th orders of geometric progression.
Sinus and Co sinus of the sum and difference of the two arguments (formulae).
Transformation into product of a sum sin ??± cos ?; cos ??± cos ?
Definition of the derivative. Its physical and geometrical meaning.
Derivatives of functions y = sin x, y = cos x; y = tg x; y = ax; y = axn (n ? N), y = ln x

Geometry. Line, beam, interval, broken; length of interval. Angle, angle value. Vertical and adjacent angles. Circular curve, circle. Parallel lines. Transformation examples of figures, types of symmetry. Similarity transformation and its properties.
Vectors. Operations with vectors.
Polygon, its cortex, sides, diagonals.
Triangle. Its median, mean line, and height. Types of triangles. The relations between sides and angles of the rectangular triangle.
Quadrangle: parallelogram, rectangle, rhomb, square, trapezium.
Circular curve, circle. Centre, bisecant, diameter, radius, tangent to circle. Ark of circle, Sector.
Central and inscribed angles. Formulas of squares: triangle, rectangle, parallelogram, rhomb, square, trapezium.
Length of circular curve and length of circular arc.
Radian length of angle. Square of circle and sector.
Similarity. Similar figures. Relation of squares of similar figures. Subspace. Parallel and crossing subspaces.
Parallelism of line and subspace. Angle between the line and subspace. Perpendicular line to subspace.
Dihedral angles. Linear angle of dihedral angle. Perpendicularity of two subspaces.
Polyhedrons. Heights, faces, and diagonals. Right-angle and oblique prisms; pyramids. Regular prism and regular pyramid. Parallelepipeds and their types.
Lathe objects: cylinder, cone, globe, full sphere. Centre, diameter, radius of sphere and full sphere. Subspace, line tangent to sphere.
Formula of surface area and prism volume
Formula of surface area and pyramid volume
Formula of surface area and cylinder volume
Formula of surface area and cone volume
Formula of full sphere volume
Formula of sphere area


Algebra and Pre-calculus

Properties of function y = kx + b and graph
Properties of function y = k/x and graph
Properties of function y = ax2 + bx + c and graph
Properties of the roots of the quadratic trinominal on linear factors
Properties of numerical inequations
Logarithm of the product, power, partial
Definition, properties, and graphs of functions y = sin x; y = cos x
Definition, properties, and graph of function y = tg x
Definition, properties, and graph of function y = ctg x
Solution of equations type sin x = a, cos x = a, tg x = a
Reduction formulas
Dependencies between trigonometric functions of one and the same argument
Trigonometric functions of double argument
Derivative sum of two functions


Properties of the oblique triangle
Properties of points equally distant from the ends of segment
Signs of parallelism of straight lines
Total of triangle angles. Total of external angles of convex polygon
Signs and properties of parallelogram.
Escribed circle. Inscribed circle.
Tangent to a circle and its properties
Value of angle escribed in a circle
Signs of triangle similarity
The Pythagorean theorem
Formulas of parallelogram, triangle, trapezium squares
Formula of the distance between two points of subspace. Circular curve equation
Parallelism sign of the straight line and subspace
Parallelism sign of subspaces
Theorem of perpendicularity of the straight line and subspace
Perpendicularity of two subspaces
Theorems on parallelism and perpendicularity of subspaces
Theorem on three perpendiculars