Closed-Ended Questions (Zadania Zamknięte)

This section presents five multiple-choice questions covering various mathematical concepts relevant to the matura rozszerzona matematyka.

The first question deals with comparing the values of different mathematical expressions, including logarithms and trigonometric functions. Students must determine the correct order of these values.

Question 2 involves geometry, specifically a square inscribed in a circle. Students need to calculate the side length of the square given the radius of the circle.

Definition: Inscribed square - A square that fits inside a circle with all four vertices touching the circle's circumference.

The third question focuses on sequences and limits, asking students to determine the limit of a given sequence as n approaches infinity.

Question 4 deals with function analysis, requiring students to identify properties of a given rational function, such as its behavior and extrema.

The final closed-ended question is a probability problem involving the selection of three-digit numbers. Students must calculate the probability of selecting a number divisible by either 3 or 7.

Highlight: This section tests students' ability to quickly analyze and solve diverse mathematical problems, a crucial skill for the matura matematyka rozszerzona.

Open-Ended Questions (Zadania Otwarte)

This section contains eight open-ended questions that require detailed solutions and explanations, mimicking the format of the matura rozszerzona matematyka 2024.

Question 6 asks students to prove a logarithmic identity involving multiple variables. This problem tests students' understanding of logarithm properties and algebraic manipulation.

Example: The question requires proving that log(ab) = (x+1)y / (y+1), given certain conditions for a, b, c, x, and y.

Question 7 involves geometry, specifically an equilateral triangle with parallel lines drawn inside it. Students must prove a relationship between the area of a smaller triangle formed within the larger one and the given dimensions.

Question 8 presents a cubic equation where the coefficients form a geometric sequence. Students need to solve this equation using their knowledge of polynomial factoring and geometric sequences.

Vocabulary: Geometric sequence - A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Complex Geometry and Algebra Problems

This section continues with more challenging open-ended questions, focusing on complex geometry and algebra problems relevant to the matura rozszerzona matematyka.

Question 9 requires students to find the equations of tangent lines to a circle that are perpendicular to a given line. This problem combines knowledge of circle geometry, line equations, and perpendicularity conditions.

Question 10 involves coordinate geometry, asking students to find the coordinates of a triangle's vertex given the coordinates of two other vertices and the centroid. Students must also calculate the area of a smaller triangle formed within the original one.

Definition: Centroid - The point where the medians of a triangle intersect, also known as the center of mass or center of gravity of the triangle.

Question 11 is a parametric function problem. Students need to determine the values of a parameter for which a quadratic function has two distinct roots with a specific difference between them. This question tests students' understanding of quadratic functions and the relationships between roots and coefficients.

Advanced Trigonometry and Geometry

The final section of the document covers advanced trigonometry and geometry problems, providing excellent preparation for the matura rozszerzona matematyka 2024.

Question 12 presents a complex trigonometric equation involving sine and cosine functions. Students must solve this equation for a specific interval, demonstrating their ability to manipulate trigonometric expressions and use trigonometric identities.

Highlight: This problem is particularly challenging and requires a deep understanding of trigonometric functions and their relationships.

The final question, number 13, involves calculating the area of a parallelogram given specific conditions about its diagonals. This problem combines knowledge of parallelogram properties, trigonometry, and algebraic manipulation.

Vocabulary: Parallelogram - A quadrilateral with opposite sides parallel and equal in length.

These final problems showcase the level of difficulty and complexity students can expect in the matura matematyka rozszerzona, emphasizing the importance of thorough preparation and practice with arkusze maturalne matematyka rozszerzona PDF materials.

Zadania otwarte - część piąta

Ta strona zawiera zadanie trygonometryczne.

Highlight: Zadanie 12 wymaga rozwiązania równania trygonometrycznego z funkcjami sinus i cosinus w przedziale (0, 2π).

Example: Przekształcenie równania do postaci (sin x - cos x)(2 cos² x + 1) = 0 i rozwiązanie go metodą rozkładu na czynniki.

Vocabulary: Tożsamość trygonometryczna - równość zawierająca funkcje trygonometryczne, prawdziwa dla wszystkich dopuszczalnych wartości zmiennych.

Zadania otwarte - część szósta

Ostatnia strona zawiera zadanie z geometrii płaskiej dotyczące równoległoboku.

Highlight: Zadanie 13 wymaga obliczenia pola równoległoboku, znając długości jego boków i kąt między przekątnymi.

Example: Wykorzystanie wzoru na pole równoległoboku P = ab sin α, gdzie α jest kątem między bokami.

Vocabulary: Przekątna równoległoboku - odcinek łączący przeciwległe wierzchołki równoległoboku.

Page 8: Trigonometric Equations

The section deals with complex trigonometric equations involving sine and cosine functions over specific intervals.

Definition: The solution requires understanding of trigonometric identities and solving equations in the interval (0,2π).

Page 9: Parallelogram Properties

This final page focuses on parallelogram area calculations using diagonal properties and trigonometric functions.

Example: The solution demonstrates how to calculate the area of a parallelogram given the lengths of its sides and the angle between its diagonals.

Overview of Mathematical Problems

This document presents a series of mathematical problems designed to prepare students for the matura rozszerzona matematyka. The problems cover a wide range of topics and difficulty levels, providing comprehensive practice for the exam.

Highlight: The problems are structured to mimic the format of the matura matematyka rozszerzona, including both closed and open-ended questions.

The first section contains closed-ended questions (zadania zamknięte) covering topics such as logarithms, trigonometry, and limits. These questions test the student's ability to quickly analyze and solve mathematical problems.

Example: One question asks students to determine the order of three given numbers, including a logarithm, a trigonometric function, and an exponential expression.

The second part of the document focuses on open-ended questions (zadania otwarte), which require students to show their work and provide detailed solutions. These problems cover more complex topics and often involve multiple steps to reach the final answer.

Vocabulary: Zadania otwarte - Open-ended questions that require detailed solutions and explanations.