Studienkolleg Math: What the Entrance Exam Tests and What the T-Kurs Teaches

Many applicants prepare for the Studienkolleg entrance exam by opening a calculus book. That is usually the wrong starting point. The math part of the entrance exam (the Aufnahmeprüfung or Aufnahmetest) mostly tests solid school algebra and geometry. Calculus, vectors and analytic geometry are what you learn inside the Studienkolleg, and they are tested at the end in the Feststellungsprüfung.

This guide separates the two clearly. First, what the math entrance test covers for the T, M and W courses, with worked examples. Then, the full T-Kurs syllabus you will study once you are admitted, so you can get a head start.

Two different exams, two different syllabuses

	Entrance exam (Aufnahmeprüfung)	Final exam (Feststellungsprüfung)
When	Before admission	After about two semesters
Purpose	Decides whether you get a place	Gives you your university entrance qualification
Math level	School algebra, equations, geometry	Full course syllabus, including calculus for T-Kurs
Also tested	German, usually at about B1+ to B2 level	German and the other subjects of your course

Each Kolleg writes its own entrance exam, so details vary. Below we use Studienkolleg München as a concrete, officially documented example, because it publishes exactly what its tests cover. For the other Kollegs, see our entrance exams by city.

The math entrance test for the T-Kurs

At Studienkolleg München the T-Kurs math test takes 45 minutes and covers algebra and geometry, in particular:

• simplifying algebraic fractions (Bruchterme),
• quadratic equations,
• polynomial division,
• equations with trigonometric functions,
• geometry, especially the Pythagorean theorem and related results, the intercept theorems (Strahlensätze) and circle calculations.

Other Kollegs test a similar range, sometimes adding linear equations and inequalities, powers and roots, or simple functions.

Worked examples

These are our own practice examples at the level described above.

1. Simplify an algebraic fraction. Simplify (x² − 9) / (x² + 3x).

Factor the top and bottom: x² − 9 = (x − 3)(x + 3) and x² + 3x = x(x + 3). Cancel the common factor (x + 3). The result is (x − 3) / x, valid for x ≠ 0 and x ≠ −3.

2. Solve a quadratic equation. Solve x² − 5x + 6 = 0.

Look for two numbers that multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3) = 0 and x = 2 or x = 3. The quadratic formula gives the same result.

3. Polynomial division. Divide x³ − 6x² + 11x − 6 by (x − 1).

The result is x² − 5x + 6 with no remainder. Combined with example 2, this means the cubic has the roots 1, 2 and 3. Exams often chain steps like this.

4. A trigonometric equation. Solve 2 sin x = 1 for 0 ≤ x < 2π.

sin x = 1/2, so x = π/6 or x = 5π/6. Remember the second solution: sine is positive in the first and second quadrants.

5. Pythagoras. A right triangle has legs of 6 cm and 8 cm. How long is the hypotenuse?

c² = 6² + 8² = 36 + 64 = 100, so c = 10 cm.

If these felt easy, you are in good shape for the entrance test. If one of them felt shaky, that topic is where your preparation should start.

The math entrance test for the M-Kurs and W-Kurs

At Studienkolleg München the math test for the M and W courses takes 30 minutes and focuses on:

• basic arithmetic with whole numbers, fractions and real numbers,
• fractions and percentages,
• simple equations,
• quantitative word problems.

Worked examples

1. Fractions. Calculate 3/4 − 2/3. The common denominator is 12: 9/12 − 8/12 = 1/12.

2. Percentages. A price of 80 euros rises by 15% and then falls by 15%. What is the new price? 80 × 1.15 = 92, then 92 × 0.85 = 78.20 euros. Not 80: the second percentage is taken of a larger amount.

3. A simple equation. Solve 3(x − 2) = x + 8. 3x − 6 = x + 8, so 2x = 14 and x = 7.

The percentage example is a classic trap. Many tests include at least one item where the intuitive answer is wrong.

The German part: equally important

Every applicant takes a German test, whatever the course. At Studienkolleg München it takes 50 minutes, is set at about B1+ to B2 level and includes a C-Test (you complete the second half of every second word in short texts), grammar and vocabulary tasks, reading comprehension and a short writing task.

At Kollegs that test both German and math, you usually need to pass both. A strong math result does not make up for a weak German result. Plan your German preparation as a parallel track from day one.

The T-Kurs syllabus you will study after admission

Once you are admitted to a T-Kurs, the math syllabus becomes much broader. Across Kollegs it typically includes:

• numbers, terms and equations
• functions and their properties
• sequences and series
• trigonometry
• geometry
• differential calculus
• integral calculus
• vectors and analytic geometry
• statistics and probability
• complex numbers (at some Kollegs)

This is the material behind the Feststellungsprüfung at the end of the course. The M-Kurs covers a narrower version and places more weight on statistics and probability. You can see full topic breakdowns on our T-Kurs and M-Kurs pages.

You do not need to master this list before the entrance exam. But if you have time left after preparing for the entrance test, starting on functions and basic differential calculus makes the first semester noticeably easier.

How to prepare, step by step

1. Find your target Kolleg's own information. Many Kollegs publish topic lists and sample papers (Musterprüfungen). These are the best guide to your actual exam.
2. Secure the algebra foundation. Fractions, powers, factoring and equations appear in almost every task. Weakness here spreads into every other topic.
3. Practise without a calculator unless your Kolleg allows one. Many tasks are designed to be solved by hand.
4. Work under time pressure. 30 to 45 minutes is short. Practise full sample papers with a timer.
5. Keep German running in parallel. Daily C-Test practice and reading are the most efficient habits.

To see where you stand across the full topic list, use our free Studienkolleg readiness check. If you are not sure which course you need, read T-Kurs vs M-Kurs.

Summary

The Studienkolleg entrance exam tests whether your school math is solid, not whether you already know calculus. For the T-Kurs that means algebraic fractions, quadratic equations, polynomial division, basic trigonometry and geometry. For the M and W courses it means arithmetic, fractions, percentages and simple equations. Pass both math and German, and the broader syllabus, calculus included, comes after admission.

Sources: Studienkolleg München: math entrance exam, Studienkolleg München: German entrance exam, DAAD: Studienkollegs.