Course detail

Přípravný kurz z matematiky

FSI-K-MATAcad. year: 2024/2025

The course is designed for high school students preparing for studies at a university with a technical focus. The content covers high school material, making it suitable for use as preparation for final exams as well.

The course concludes with a test that is equivalent in difficulty to the mathematics entrance exam for the Faculty of Mechanical Engineering at Brno University of Technology.

Course Content:

  1. Polynomial Operations: Binomial theorem, simplifying algebraic expressions, powers, roots, and rationalizing fractions.
  2. Linear Equations and Inequalities: Single-variable linear equations and inequalities, equivalent transformations. Systems of two linear equations with two unknowns. Solving linear equations or inequalities involving absolute values.
  3. Quadratic Equations and Inequalities: Single-variable quadratic equations and inequalities, relationships between roots and coefficients, factorization of quadratic trinomials, graphical solutions of quadratic equations. Equations and inequalities with variables in the denominator, basic irrational equations.
  4. Applied Problems: Direct and inverse proportionality, rule of three, percentages.
  5. Functions of a Single Real Variable: Domain, range, and graph of a function. Linear functions, quadratic functions, rational functions, and functions with absolute values – including graphs.
  6. Exponential and Logarithmic Functions: Graphs of exponential and logarithmic functions, solving exponential and logarithmic equations.
  7. Trigonometric Functions: Degree and radian measure, formulas, graphs, and solving trigonometric equations and inequalities.
  8. Sequences: Arithmetic and geometric sequences.
  9. Plane Geometry: Focus on triangles, solving right and general triangles. Pythagorean theorem, Thales' theorem, Euclidean theorems, sine and cosine rules. Circle and circular arc, inscribed and central angles.
  10. Geometric Calculations: Perimeter, area, surface area, and volume of basic shapes in the plane and space.
  11. Analytic Geometry in the Plane: Distance between two points, vectors, lines in the plane, and conic sections.
  12. Complex Numbers: Basic operations, algebraic and trigonometric forms, De Moivre’s theorem, solving quadratic equations.
  13. Combinatorics: Variations, combinations, permutations, Pascal’s triangle, factorials, and binomial coefficients.
  14. Final Test.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

Mgr. Jana Hoderová, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

High school mathematics knowledge is assumed. The course builds upon elementary principles, which are always reviewed at the beginning of each lesson. The aim is to standardize the participants' level of knowledge.

The course emphasizes solidifying fundamental knowledge, demonstrating the interconnections between concepts and formulas, and providing thorough practice.

Rules for evaluation and completion of the course

The course spans 26 hours over 13 weeks, with 2 hours per week.

Attendance is not monitored, and participants can attend in person, watch a live online stream, or view recorded lessons, which are made available on the e-learning platform afterwards.

In the 13th week, participants take a final test, which is conducted exclusively in person. The test is equivalent in difficulty to the entrance exam for the Faculty of Mechanical Engineering at Brno University of Technology.

To pass the final mathematics test, participants must score at least 50%. Successfully passing this test (or the physics test in the K-FYZ Preparatory Physics Course) allows participants to request an exemption from the mathematics and physics entrance exams for the Faculty of Mechanical Engineering at Brno University of Technology.

Aims

The aim of the course is to prepare applicants for studies at the Faculty of Mechanical Engineering at Brno University of Technology for successfully completing their first semester.

A solid foundation in mathematics is essential for all technical subjects, including physics, statics, mechanics, strength of materials, elasticity, and more.

Study aids

Participants have access to the course’s e-learning platform, where thematic materials for 12 lessons (the 13th lesson being the final test) are published.

Each lesson includes a set of exercises recommended for practice.

The textbook Martišek, Faltusová: Mathematics - A Handbook for Preparing for Entrance Exams is recommended for the course and is available in the Digital Library at https://hdl.handle.net/11012/249728

Prerequisites and corequisites

Not applicable.

Basic literature

MARTIŠEK, Dalibor a Milana FALTUSOVÁ. Matematika: příručka pro přípravu k přijímacím zkouškám. Brno: Akademické nakladatelství CERM, 2004. ISBN 80-214-2578-4 (CS)

Recommended reading

Not applicable.

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme C-PRI-H Preparatory course for study applicants

    specialization PMF , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

Mgr. Jana Hoderová, Ph.D.

Syllabus

  1. Počítání s mnohočleny, binomická věta, úpravy algebraických výrazů, mocniny, odmocniny, usměrňování zlomků.
  2. Lineární rovnice a nerovnice o jedné neznámé, ekvivalentní úpravy. Soustava dvou lineárních rovnic o dvou neznámých. Řešení lineární rovnice, resp. nerovnice s absolutními hodnotami.
  3. Kvadratické rovnice a nerovnice o jedné neznámé, vztahy mezi kořeny a koeficienty kvadratické rovnice, rozklad kvadratického trojčlenu. Grafické řešení kvadratické rovnice. Rovnice a nerovnice obsahující neznámou ve jmenovateli, jednoduché iracionální rovnice.
  4. Úsudkové příklady - přímá a nepřímá úměrnost, trojčlenka, procenta
  5. Funkce jedné reálné proměnné: definiční obor, obor funkčních hodnot, graf funkce. Lineární funkce, kvadratická funkce, lineární lomená funkce, funkce s absolutní hodnotou – grafy.
  6. Exponenciální a logaritmické funkce - grafy. Exponenciální a logaritmické rovnice. Goniometrické funkce, stupňová a oblouková míra, vzorce, grafy. Goniometrické rovnice a nerovnice.
  7. Posloupnosti - aritmetická a geometrická.
  8. Planimetrie - zaměřeno na trojúhelník. Řešení pravoúhlého a obecného trojúhelníku. Věta Pythagorova a Thaletova, věty Eukleidovy, věta sinová a kosinová. Kruh a kružnice, obvodový a středový úhel.
  9. Výpočty obvodu, obsahu, povrchu a objemu základních útvarů v rovině a v prostoru.
  10. Analytická geometrie v rovině (vzdálenost dvou bodů, vektory, přímka v rovině, kuželosečky).
  11. Komplexní čísla (základní operace, algebraický a goniometrický tvar, Moivreova věta, řešení kvadratických rovnic).
  12. Kombinatorika (variace, kombinace, permutace, Pascalův trojúhelník, faktoriál, kombinační číslo).
  13. Závěrečný test. 

Elearning

eLearning: currently opened course