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This document is an entrance exam for 12th-grade math students from the 2024–2025 academic year. It contains three parts: multiple-choice questions, true/false questions, and short-answer questions. Here is a breakdown:

1. **Multiple Choice (Part I)**:  
   - There are 12 questions, each with four options. Topics include trigonometry, functions, limits, series, geometry, probability, and calculus. For example:
     - Solve for all solutions of \(2\sin x = 3\).
     - Maximum value of the function \(y = 3\cos x + 1\).
     - A geometric series calculation.
     - Limit calculations and basic probability questions.

2. **True/False (Part II)**:  
   - Contains 4 questions with sub-parts a), b), c), and d), where students decide whether the statements are true or false. Topics range from calculus, geometry, combinatorics, to logarithmic and exponential functions.

3. **Short Answer (Part III)**:  
   - This section requires more detailed solutions, covering problems like:
     - Finding minimum velocity in a motion problem.
     - A question involving exponential growth of a forest area.
     - Geometric problems involving volumes of prisms and pyramids.
     - Probability calculations.

Would you like to solve any specific questions from the exam?

Here are 5 possible directions for exploration:
1. Solving the trigonometric equation in Question 1.
2. Calculating the maximum of a trigonometric function in Question 2.
3. Exploring limits and calculus in Question 4.
4. Working on the probability of students' selection in Question 12.
5. Solving the volume problem of pyramids or prisms in geometry.

**Tip**: For problems involving geometric volumes, always visualize the shape and use the correct formula for each solid.

This document is an entrance exam for 12th-grade math students from the 2024–2025 academic year. It contains three parts: multiple-choice questions, true/false questions, and short-answer questions.

Explore a variety of challenging math problems from the 2024–2025 12th-grade entrance exam. Topics include trigonometric equations, geometric series, limits, and probability. Learn how to solve them step-by-step, ideal for Grade 12 students.

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