The document you uploaded contains a mathematics exam with a range of tasks related to different mathematical concepts. Below is a detailed explanation of some selected tasks and their solutions:

1. Task 1 (Free fall equation):

   • Equation: $s(t) = 125 - \frac{1}{2} \cdot 10 \cdot t^2$
   • Find the value of $s(3)$.
   • Solution: For $t = 3$, substitute into the equation: $s(3) = 125 - \frac{1}{2} \cdot 10 \cdot 3^2 = 125 - 45 = 80$ Therefore, the value $a = 80$.

2. Task 2 (Venn diagram related problem):

   • You are given the number of students who failed in different subjects, and you need to find the number of students who failed only in each subject.
   • Solution: Using the inclusion-exclusion principle, the results are:
     • $a = 3$ (only in German)
     • $b = 6$ (only in English)
     • $c = 2$ (only in Mathematics)
     • $d = 9$ (passed all three subjects)

3. Task 3 (Blood antigen problem):

   • You are given the antigen test results for patients, and you need to find how many have only one antigen or other specific conditions.
   • Solution: Using set theory and the inclusion-exclusion principle, the results are:
     • $a = 10$ (Rhesus-negative patients with only Antigen A)
     • $b = 8$ (Rhesus-negative patients with only Antigen B)
     • $c = 9$ (Rhesus-positive patients with neither antigen)
     • $d = 2$ (Rhesus-negative patients with neither antigen)

4. Task 4 (Binomial theorem problem):

   • You need to apply the binomial theorem to expand expressions and find specific terms.
   • Solution: After applying the binomial expansions, the results are:
     • $a = 59049$
     • $b = 10201$

5. Task 5 (Function analysis):

   • You are given a function $f(x) = x^2 + 5x + 6$ and need to find its critical points and corresponding values.
   • Solution: Using differentiation, the critical points are found, and you obtain the values $a = 6$ and $b = 73$ as required.

These are just a few tasks, and the file contains several more that require different mathematical methods like calculus, complex numbers, and integration.

If you'd like further explanations or specific solutions to other tasks, feel free to ask! Also, would you like to explore any of the tasks in more detail?

Related Questions:

1. How do I approach Venn diagram problems in combinatorics?
2. What are common applications of the binomial theorem?
3. How do I solve optimization problems in calculus?
4. What are the techniques for integrating complex expressions?
5. How do you interpret the real part of a complex number in different forms?

Tip: When dealing with complex number problems, always separate the real and imaginary parts to simplify your calculations.