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.