Geometry

1. Study of affine space by analytical method

(subspaces - linear varieties, their parametric and general equations, intersections and mutual positions)

2. Study of Euclidean space by analytical method

(scalar product of vectors and metrics, perpendicularity of subspaces, distances of subspaces, angles)

3. Affine representations of spaces

(analytical expression of affine mapping, invariants of affine transformations, group of similarities of Euclidean space)

4. Axiomatic construction of geometry: incidental and ordered plane

(axioms of incidence and arrangement and their consequences, models of incident and ordered plane.)

5. Axiomatic construction of geometry: Hilbert's and Euclidean planes

(axioms of similarity and their consequences: triangles of similarity of triangles, properties of a triangle, construction of perpendiculars and parallels; axioms of parallelism and axioms of continuity)

Combinatorics, probability and statistics

1. Mathematical induction (principle of mathematical induction; connection with good arrangement of natural numbers; examples of use).

2. Pigeon/Dirichlet principle (formulation and some applications).

3. Combinatorial principles (addition principle, multiplication principle, bijection principle, counting in two ways).

4. Binomial coefficients and binomial theorem (definition and formula for binomial coefficients and some of their properties; binomial theorem formulation).

5. Principle of inclusion and exclusion (formulation and examples of use).

6. Probability, its basic properties. Conditional probability and independence of events. Complete Probability Theorem, Bayes Theorem.

7. Probability distributions, their properties and characteristics (distribution function, density, mean value, dispersion). Special types of distributions (alternative, binomial, geometric, exponential, normal). Central limit theorem.

8. Descriptive statistics (location and variability characteristics). Point estimates (random selection; estimates of mean and dispersion and their properties).

9. Confidence intervals for the mean value. Hypothesis testing, one-choice tests on the mean value.

Algebra and theoretical arithmetic

1. Linear representations and their matrices, product of matrices, inverse matrices.

2. Vector spaces and subspaces, linear combinations of vectors, linear representations.

3. Finite-dimensional vector spaces, base and dimension of finite-dimensional vector space.

4. Systems of linear equations, the existence of a solution of an inhomogeneous system of linear equations, the structure of the set of solutions of a homogeneous system of linear equations.

5. Divisibility in the field of integers. Theorem on division with the rest. The largest common divisor and the smallest common multiple of two integers. Euclidean algorithm for calculating the greatest common divisor.

6. Prime numbers, their properties, theorem about the decomposition of a natural number into the product of prime numbers. Number systems.

7. Congruences, divisibility criteria of natural numbers expressed in the decimal system, Euler's theorem, small Fermat's theorem.

Mathematical analysis

1. Limits of sequence and function, basic theorems about limits.

2. Continuity, properties of continuous functions on intervals, optimization - search for global extrema of continuous functions on closed intervals, relationship between continuity and differentiability of a function.

3. Derivation of a function, Lagrange's theorem on mean value and its use in investigating the monotonicity of functions, necessary and sufficient conditions for the existence of local extrema of differentiable functions.

4. Approximation of differentiable function by polynomials, equation of tangent, equation of Taylor polynomial of n-th degree.

5. Indefinite integral and primitive function, basic integration formulas, per partes method and substitutions.

6. Riemann integral, definition and calculation, heuristic derivation of formulas for calculation of area content, length of curve, volume of rotating body and surface of rotating body.

The course 1-UMA-951/15 Fundamentals of Mathematics has two parts:

A) School mathematics test

The test uses the types of tasks from mathematics tests for the external part of the Matura exam and from mathematics tests at the entrance exams at FMFI UK, a total of 20 short-answer tasks or with a choice of several options.

B) Oral exam

The student draws an assignment, which has 3 parts - three different circuits

1. geometry, 2. combinatorics, probability and statistics, 3. algebra and theoretical arithmetic, 4. mathematical analysis.

Each part contains

- the task from the relevant area, the solution of which (including the justification of individual steps) the student will demonstrate during the oral answer,

- definition of the area of ​​the relevant heading, which is related to the solved task; in the oral answer the student will state the basic concepts and statements of this area, or their relationship to the problem.

Maximum points:

• 20 points from the school mathematics test (1 point for each correct answer),

• 25 points for each of the three parts of the assignment (10 for solving the problem, 15 for the theoretical part),

thus a maximum of 20 + 3.25 = 95 points in total.

A student completes the course if he/she obtains at least 5 points for each of the three parts of the assignment and a total of at least 46 points.

slovak, enghlish

0/100

State exam from selected areas of the core subjects of the program.