Studienkolleg Entrance Exam Math: Topics, Examples & Practice (2026)

What math is on the Studienkolleg entrance exam? Topics by course type (T/W/M), difficulty levels, practice problems, and a 4-week study plan.

The math section of the Studienkolleg entrance exam (Aufnahmeprüfung) covers secondary school mathematics at the German Oberstufe level — roughly equivalent to grades 11-13 in the German system. For T-Kurs applicants, the exam focuses on algebra, functions, and basic calculus. For W-Kurs applicants, it centers on algebra, percentages, and basic statistics. For M-Kurs applicants, topics overlap heavily with the T-Kurs but at a slightly reduced difficulty. The entire test lasts 45-60 minutes, calculators are not allowed, and all questions are written in German.

This guide breaks down exactly which math topics appear on the entrance exam for each course type, gives you three practice problems with full solutions, provides a German math vocabulary reference, and lays out a focused 4-week study plan.

Which Course Types Include Math in the Entrance Exam?

Not every Studienkolleg course type requires a math test. Here is the breakdown:

Course Type	Math Test Required?	Difficulty Level
T-Kurs (Technical)	Yes — always	High
W-Kurs (Economics)	Yes — always	Medium
M-Kurs (Medicine/Biology)	Yes — at most institutions	Medium-High
G-Kurs (Humanities)	Rarely — some test basic numeracy	Low
S-Kurs (Languages)	No — German only	N/A

If you are applying for the T-Kurs, the math section is the most demanding. W-Kurs math is simpler, focusing on business-relevant calculations. M-Kurs math resembles a lighter version of the T-Kurs.

G-Kurs and S-Kurs applicants can skip this guide — your entrance exam focuses on German language skills only. See our complete entrance exam preparation guide for the German test component.

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T-Kurs Math Topics: What You Need to Know

The T-Kurs entrance exam tests the broadest and most demanding range of math topics. Every Studienkolleg sets its own exam, but the following topics appear consistently across institutions.

Algebra and Equations

This is the foundation. Expect at least 2-3 questions on algebraic manipulation and equation solving.

Core topics:

• Linear equations and systems of linear equations (2 variables, occasionally 3)
• Quadratic equations — solving by factoring, completing the square, and the quadratic formula (Mitternachtsformel / p-q-Formel)
• Polynomial equations of degree 3 and higher (factoring with polynomial division)
• Exponential equations (e.g., 2^x = 16, 3^(2x-1) = 27)
• Logarithmic equations (basic properties of logarithms, change of base)
• Inequalities (linear, quadratic)
• Absolute value equations

Mini-example: At the Studienkolleg in Frankfurt, a recent entrance exam asked students to solve a system of three linear equations with three unknowns — entirely by hand, no calculator. Clean notation and structured work earned partial credit even when the final answer was wrong.

Functions and Analysis

Function analysis is the single most important topic on the T-Kurs entrance exam. You must be confident with:

• Linear functions: slope, intercept, parallel and perpendicular lines
• Quadratic functions: vertex form, standard form, axis of symmetry, roots, sketching parabolas
• Polynomial functions: degree, behavior at infinity, roots, sketching
• Exponential functions: f(x) = a * b^x, growth and decay, graphs
• Logarithmic functions: as inverses of exponential functions, basic graphs
• Trigonometric functions: sin(x), cos(x), tan(x) — unit circle, basic identities, period, amplitude
• Curve sketching (Kurvendiskussion): finding roots (Nullstellen), extrema (Extrempunkte), inflection points (Wendepunkte), monotonicity, symmetry
• Derivatives (Ableitungen): power rule, product rule, chain rule — applied to polynomial and basic trigonometric functions
• Basic integrals: area under a curve, antiderivatives of polynomial functions

The depth of calculus varies. Some Studienkollegs test only derivatives; others include basic integrals as well. Focus your preparation on derivatives first — they appear on virtually every T-Kurs entrance exam.

Geometry

Geometry questions are less frequent than algebra and functions but still appear regularly.

• Pythagorean theorem and its applications
• Area and perimeter of triangles, rectangles, circles, trapezoids
• Volume and surface area of basic 3D shapes (cylinder, cone, sphere, prism)
• Coordinate geometry: distance between points, midpoint, equations of lines and circles
• Basic trigonometry in right triangles (SOH-CAH-TOA)
• Vectors in 2D (addition, subtraction, scalar multiplication, magnitude)

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W-Kurs Math Topics: Business-Focused

The W-Kurs entrance exam is shorter and less demanding than the T-Kurs. The focus shifts from pure mathematics to applied, business-relevant calculations.

Core W-Kurs Topics

Topic	What to Expect
Linear equations and systems	Solve for price, quantity, cost variables
Quadratic equations	Revenue optimization, break-even analysis
Percentages and interest	Simple interest, compound interest, price markup/discount, VAT calculations
Proportional reasoning	Direct and inverse proportion, rule of three (Dreisatz)
Basic statistics	Mean (Mittelwert), median, mode, weighted averages
Functions	Linear and quadratic functions, interpreting graphs in economic contexts
Units and conversions	Currency, measurement units, rates

Mini-example: A W-Kurs entrance exam in Hamburg presented a word problem about a company selling products at a 25% markup on production costs. Students had to calculate the final price including 19% VAT (Mehrwertsteuer), then determine the profit per unit. This requires clean percentage work and careful reading.

What W-Kurs Does NOT Test

You will not encounter calculus, trigonometry, vectors, or advanced geometry on a W-Kurs entrance exam. The math stays firmly within algebra and applied arithmetic.

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M-Kurs Math Topics: Between T and W

The M-Kurs (medicine, biology, pharmacy) entrance exam math overlaps heavily with the T-Kurs but typically leaves out the most advanced topics.

M-Kurs Topic Coverage

• Included: Linear and quadratic equations, polynomial functions, basic curve sketching, derivatives (power rule), exponential functions, basic geometry, trigonometry fundamentals
• Usually excluded: Integrals, advanced trigonometric identities, vectors, complex numbers, polynomial division of degree 4+

Think of the M-Kurs math as T-Kurs minus the hardest 20%. If you prepare thoroughly for the T-Kurs topics listed above, you are more than prepared for the M-Kurs.

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How Hard Is It Compared to Your Home Country?

The difficulty of the entrance exam math depends entirely on your educational background. Here is a realistic comparison:

Country/Region	Typical Experience
China, Vietnam, Iran	Most topics are familiar from high school. The challenge is doing them in German, not the math itself
India	Strong alignment with Class 11-12 CBSE/ISC math. Derivatives and functions will feel familiar
Turkey	Good overlap with Lise (high school) math. Calculus depth may go slightly beyond YKS preparation
Arab countries	Algebra and geometry are well-covered. Calculus (derivatives) may require extra preparation
Latin America	Algebra is familiar, but curve sketching and derivatives may be new. Expect 2-4 weeks of targeted study
Sub-Saharan Africa	Varies widely by country. Students from systems with strong math curricula (Nigeria, Kenya, Ghana) often adapt quickly; others may need 6-8 weeks
South Korea	Strong math education. The Suneung covers all entrance exam topics. Vocabulary is the main challenge

The universal challenge is not the math itself but performing it in German under time pressure without a calculator.

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German Math Vocabulary: Essential Terms

Every math problem on the entrance exam is written in German. If you cannot read the question, you cannot solve it. Memorize this vocabulary before exam day.

Core Mathematical Terms

German	English	Example Context
die Gleichung	equation	”Lösen Sie die Gleichung” = Solve the equation
die Ungleichung	inequality	”Bestimmen Sie die Lösungsmenge der Ungleichung”
die Nullstelle	root / zero	”Berechnen Sie die Nullstellen” = Calculate the roots
die Ableitung	derivative	”Bilden Sie die erste Ableitung” = Find the first derivative
das Integral	integral	”Berechnen Sie das Integral”
der Scheitelpunkt	vertex (of a parabola)	“Bestimmen Sie den Scheitelpunkt”
der Wendepunkt	inflection point	”Berechnen Sie die Wendepunkte”
das Extremum (Pl: Extrema)	extremum	”Bestimmen Sie die lokalen Extrema”
die Steigung	slope / gradient	”Die Steigung der Geraden beträgt 3”
der Achsenabschnitt	y-intercept	”Bestimmen Sie den y-Achsenabschnitt”
die Gerade	straight line	”Die Gerade g hat die Gleichung…“
die Parabel	parabola	”Zeichnen Sie die Parabel”
der Flächeninhalt	area	”Berechnen Sie den Flächeninhalt”
das Volumen	volume	”Berechnen Sie das Volumen des Zylinders”
der Umfang	perimeter / circumference	”Bestimmen Sie den Umfang”
der Bruch	fraction	”Kürzen Sie den Bruch” = Simplify the fraction
der Nenner	denominator	”Erweitern Sie auf den gleichen Nenner”
der Zähler	numerator
die Potenz	power / exponent	”Vereinfachen Sie die Potenz”
die Wurzel	root (square root)	“Berechnen Sie die Quadratwurzel”

Instruction Verbs (Operatoren)

German Verb	Meaning
Berechnen Sie	Calculate
Bestimmen Sie	Determine
Lösen Sie	Solve
Vereinfachen Sie	Simplify
Zeichnen Sie	Draw / Sketch
Zeigen Sie	Show / Prove
Geben Sie an	State / Specify
Untersuchen Sie	Investigate / Analyze
Beschreiben Sie	Describe
Begründen Sie	Justify / Explain why

Tip: The formal “Sie” form is always used in exams. “Berechnen Sie die Nullstellen” is how every math question starts. Get used to this phrasing.

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3 Practice Problems with Solutions

These problems are representative of actual entrance exam questions. Try each one on paper before reading the solution.

Problem 1: Quadratic Function (T-Kurs / M-Kurs Level)

Gegeben ist die Funktion f(x) = x² - 4x + 3.

a) Berechnen Sie die Nullstellen. b) Bestimmen Sie den Scheitelpunkt. c) Skizzieren Sie den Graphen.

Solution:

a) Finding the roots — set f(x) = 0:

x² - 4x + 3 = 0

Using the p-q formula (or factoring): (x - 1)(x - 3) = 0

Nullstellen: x₁ = 1, x₂ = 3

b) Vertex — use completing the square or the formula x_s = -b/(2a):

x_s = 4/(2·1) = 2

y_s = f(2) = 4 - 8 + 3 = -1

Scheitelpunkt: S(2 | -1)

c) The parabola opens upward (a = 1 > 0), passes through (1, 0) and (3, 0), with the lowest point at (2, -1). The y-intercept is f(0) = 3.

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Problem 2: Derivative and Curve Sketching (T-Kurs Level)

Gegeben ist die Funktion f(x) = x³ - 3x.

a) Berechnen Sie die erste Ableitung f’(x). b) Bestimmen Sie die Extrempunkte. c) Bestimmen Sie den Wendepunkt.

Solution:

a) First derivative:

f’(x) = 3x² - 3

b) Set f’(x) = 0 for extrema:

3x² - 3 = 0 → x² = 1 → x₁ = -1, x₂ = 1

Second derivative test: f”(x) = 6x

f”(-1) = -6 < 0 → Maximum at x = -1: f(-1) = -1 + 3 = 2 → Hochpunkt (-1 | 2)

f”(1) = 6 > 0 → Minimum at x = 1: f(1) = 1 - 3 = -2 → Tiefpunkt (1 | -2)

c) Set f”(x) = 0 for inflection:

6x = 0 → x = 0, f(0) = 0

Wendepunkt: W(0 | 0)

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Problem 3: Percentage and Interest Calculation (W-Kurs Level)

Ein Produkt kostet 240 EUR netto. Der Händler gibt 15% Rabatt. Auf den reduzierten Preis wird 19% Mehrwertsteuer aufgeschlagen. Wie hoch ist der Endpreis?

(A product costs 240 EUR net. The dealer gives a 15% discount. 19% VAT is added to the reduced price. What is the final price?)

Solution:

Step 1 — Discount: 240 × 0.85 = 204 EUR

Step 2 — Add VAT: 204 × 1.19 = 242.76 EUR

Endpreis: 242,76 EUR

Note: In German math notation, decimals use a comma (242,76) and thousands use a period or space. Write your answers the German way on the exam.

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Key Formulas to Memorize

You will not receive a formula sheet. Memorize these:

Algebra

• Quadratic formula (p-q-Formel): x = -p/2 ± √((p/2)² - q) for x² + px + q = 0
• Quadratic formula (Mitternachtsformel): x = (-b ± √(b² - 4ac)) / (2a) for ax² + bx + c = 0
• Binomial formulas: (a+b)² = a² + 2ab + b², (a-b)² = a² - 2ab + b², (a+b)(a-b) = a² - b²

Calculus

• Power rule: d/dx [xⁿ] = n · xⁿ⁻¹
• Product rule: (f·g)’ = f’·g + f·g’
• Chain rule: (f(g(x)))’ = f’(g(x)) · g’(x)
• Antiderivative of xⁿ: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ -1)

Geometry

• Circle: A = πr², U = 2πr
• Cylinder: V = πr²h, Surface = 2πr² + 2πrh
• Pythagorean theorem: a² + b² = c²
• Trigonometry: sin(α) = opposite/hypotenuse, cos(α) = adjacent/hypotenuse, tan(α) = sin(α)/cos(α)

Statistics (W-Kurs)

• Arithmetic mean: x̄ = (x₁ + x₂ + … + xₙ) / n
• Compound interest: K_n = K₀ · (1 + p/100)ⁿ
• Percentage: Prozentwert = Grundwert × Prozentsatz / 100

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4-Week Study Plan

This plan assumes you already have a foundation in math from your home country and need to review, fill gaps, and practice under exam conditions. If your math foundation is weak, extend this to 6-8 weeks.

Week 1: Algebra Foundations (2-3 Hours/Day)

Day	Focus	Tasks
1-2	Linear equations, systems	Solve 20 systems of 2 equations; 5 systems of 3 equations
3-4	Quadratic equations	Practice all three solving methods (factoring, completing square, formula); 25 problems
5-6	Powers, roots, logarithms	Simplify expressions, solve exponential equations; 20 problems
7	Review + vocabulary	Redo any mistakes; study German math terms (30 minutes)

Week 2: Functions and Calculus (2-3 Hours/Day)

Day	Focus	Tasks
1-2	Linear and quadratic functions	Sketch graphs, find vertices, convert between forms; 15 problems
3-4	Derivatives	Power rule, product rule, chain rule; differentiate 30 functions
5-6	Curve sketching (Kurvendiskussion)	Full analysis of 5 polynomial functions (roots, extrema, inflection, sketch)
7	Review + vocabulary	Redo weak areas; add 20 more German math terms

Week 3: Geometry + Weak Spots (2-3 Hours/Day)

Day	Focus	Tasks
1-2	Geometry: areas, volumes	Calculate areas and volumes for 15 shapes
3	Coordinate geometry	Distance, midpoint, line equations; 10 problems
4	Trigonometry basics	SOH-CAH-TOA, unit circle; 15 problems
5-6	Fill personal gaps	Focus on your weakest topics from weeks 1-2
7	Review + full vocabulary list	All German math terms; practice reading problems in German

Week 4: Exam Simulation (3 Hours/Day)

Day	Focus	Tasks
1	Practice exam 1	Complete a sample exam under timed conditions (60 min, no calculator)
2	Review exam 1	Analyze every mistake; redo incorrect problems
3	Practice exam 2	Different sample exam, same conditions
4	Review exam 2	Focus on recurring error patterns
5	Practice exam 3	Final timed simulation
6	Light review	Only formulas and vocabulary — no heavy problem-solving
7	Rest	Go into the exam fresh

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Best Resources for Math Preparation

Free Practice Material

• Sample Exams Collection — Download real entrance exam papers from over 20 Studienkollegs
• Math Sample Exams — Sorted by institution and topic
• GeoGebra (geogebra.org) — Free graphing tool for visualizing functions, checking your curve sketches
• Khan Academy (German version) — Video tutorials covering algebra through basic calculus

Textbooks

• “Lambacher Schweizer Mathematik” (Klett) — The standard German high school math textbook. If your Studienkolleg does not provide a specific recommendation, this is the best reference
• “Mathematik für Studienkollegs” — Some institutions publish their own preparation booklets. Ask your target Studienkolleg directly
• “Formelsammlung Mathematik” (any publisher) — A compact formula reference for review. Do not rely on it during study — memorize the formulas — but it helps for checking

Practice Strategy

1. Start with German-language problems from day one. Do not practice in English and hope to translate later. The exam is in German, so your practice must be in German
2. No calculator from day one. Every calculation by hand. Rebuild your mental arithmetic skills
3. Write solutions neatly. German exams award partial credit (Teilpunkte) for clear, structured work. Show every step
4. Time yourself from week 3 onward. You need to solve roughly one problem every 5-7 minutes. Speed matters

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Common Mistakes in the Math Exam

Arithmetic errors under pressure

Without a calculator, simple multiplication and sign errors become the most common source of lost points. Double-check every calculation, especially when working with negative numbers and fractions.

Not reading the question in German carefully

“Bestimmen Sie die Nullstellen” and “Bestimmen Sie die Extrema” require completely different calculations. Misreading one German word can send you down the wrong path and cost 10 minutes.

Skipping steps in written solutions

German math exams expect a clean, logical progression from the given information to the answer. Writing only the final result without showing the method earns zero points at most institutions, even if the answer is correct.

Wrong notation

Use German math notation: commas for decimals (3,14 not 3.14), proper set notation for solution sets (L = {1; 3} not x = 1 or 3), and the German names for points (S(2 | -1) not S(2, -1) — the vertical bar is standard in German math).

Running out of time on the last problem

Many students spend too long on a difficult early problem and then rush or skip the final questions. If you are stuck for more than 3 minutes, mark it and move on. Return to it after finishing the rest.

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Frequently Asked Questions

How long is the math section of the entrance exam?

The math section lasts 45-60 minutes at most Studienkollegs. Some institutions give 90 minutes for a combined German-and-math test. You receive 4-8 problems to solve in that time. Check with your target institution for the exact duration.

Can I use a calculator?

No. The vast majority of Studienkollegs prohibit all calculators, including basic scientific ones. Some institutions in certain federal states allow a simple scientific calculator (wissenschaftlicher Taschenrechner / WTR), but this is the exception. Prepare as if no calculator is allowed.

What tools am I allowed to bring?

Typically: pens, pencils, a ruler, and an eraser. Some Studienkollegs provide blank paper; others expect you to write directly in the exam booklet. A protractor and compass may be permitted but are rarely needed. You may not bring a formula sheet, phone, or smartwatch.

Is the math harder than the German section?

For most international students, the German test is harder to prepare for because language acquisition takes months. The math section rewards focused, short-term preparation — if you know the topics, you can get exam-ready in 4 weeks. Many students from countries with strong math curricula (China, India, Iran, South Korea) find the math manageable but struggle with the German C-Test. Balance your preparation time accordingly.

Do I need to know calculus for the W-Kurs entrance exam?

No. The W-Kurs math section does not include derivatives, integrals, or curve sketching. Focus on algebra, percentages, interest calculations, and basic statistics. If you can solve linear and quadratic equations, calculate compound interest, and interpret simple graphs, you are prepared.

What if I fail the math section but pass the German section?

Most Studienkollegs treat the entrance exam as a combined result. Failing one section typically means failing the entire exam. A very strong German score cannot compensate for a failed math section at most institutions. Some Studienkollegs offer conditional admission with the requirement to improve in a specific area, but this is rare.

Are the math topics the same at every Studienkolleg?

The general topic areas are consistent (algebra, functions, geometry), but the specific difficulty and emphasis vary. Studienkollegs in Bavaria use a centralized exam administered in Munich. Studienkollegs in Hessen coordinate their exams as well. In other states, each institution writes its own test. Download sample exams from your specific target institution to see exactly what to expect.

How is the math score weighted in the overall entrance exam result?

Weighting varies by institution and course type. At many Studienkollegs, the German and math sections carry equal weight (50/50). At others, the German section counts for 60% and math for 40%. For T-Kurs applicants, some institutions weight math more heavily. The exact weighting is usually published on the Studienkolleg’s website or in the exam information documents.

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Your Next Steps

1. Identify your course type (T, W, M, G, or S) and review only the relevant math topics from this guide
2. Download sample exams from the sample exams collection for your target Studienkolleg
3. Print the vocabulary table above and study 5-10 terms per day until you know them all
4. Start the 4-week study plan — or the full entrance exam preparation guide if you also need to prepare for the German test
5. Practice every problem on paper, in German, without a calculator — this is non-negotiable
6. Take at least 2 full timed practice exams in week 4 to build exam stamina

The math section is predictable. The same topic areas appear every semester, the difficulty stays within a known range, and the question formats repeat. Students who prepare systematically for 4 weeks pass. Start today.