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Studienkolleg maths test

Tasks covering the Studienkolleg entrance exam topics: expressions, powers and roots, equations and inequalities, linear and quadratic functions. Mark your answers and click “Check” — the test will score you and show the worked solutions.

📐 4 topics · 19 tasks 🏆 35 points
Copyright protectedAll tasks on this page were developed by Studienkolleg Vorbereitung. Copying, downloading, distribution and any commercial use without our written consent is prohibited and will be pursued under the law (§ 106 UrhG).
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Expressions, powers and roots

Simplifying expressions, rules for powers and roots.

11 P
1.11 P
Vereinfachen Sie den Term: 2⋅(320+320+320)2\cdot\left(3^{20}+3^{20}+3^{20}\right)
1.21 P
Berechnen Sie: 813481^{\frac34}
enter a number
1.32 P
Vereinfachen Sie: x⋅x34\sqrt{x}\cdot\sqrt[4]{x^3}
1.43 P
Vereinfachen Sie: (x−9)(x−3)x+3+6x,\dfrac{(x-9)(\sqrt{x}-3)}{\sqrt{x}+3}+6\sqrt{x}, x>0x>0
1.53 P
Vereinfachen Sie den Term: 40x3y4t5⋅(−t310x2y3)2\dfrac{40x^3y^4}{t^5}\cdot\left(\dfrac{-t^3}{10x^2y^3}\right)^2
1.61 P
Berechnen Sie: 12837128^{\frac37}
enter a number
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Equations and inequalities

Absolute values, cubic equations, logarithmic inequalities.

12 P
2.11 P
Lösen Sie die Betragsgleichung: ∣x−5∣=8|x-5|=8
2.23 P
Lösen Sie die Gleichung: x3−2x2−5x+6=0x^3-2x^2-5x+6=0
2.33 P
x+1⋅x−4=6.\sqrt{x+1}\cdot\sqrt{x-4}=6. Geben Sie die Lösung xx an.
enter a number
2.42 P
Lösen Sie die Ungleichung: x2−5x+4≤0x^2-5x+4\le 0
2.53 P
Lösen Sie die Gleichung im Intervall x∈[0; 2π)x\in[0;\,2\pi): cos⁡(2x)=sin⁡(x)\cos(2x)=\sin(x)
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Linear functions

Slope, intercepts, parallel lines.

5 P
3.11 P
Gegeben ist die Gerade 3x+4y=123x+4y=12. Bestimmen Sie ihre Steigung mm und ihren yy-Achsenabschnitt bb.
3.21 P
y=−5x+15.y=-5x+15. Berechnen Sie den xx-Achsenabschnitt.
enter a number
3.32 P
Die Gerade gg verläuft parallel zu y=4x−1y=4x-1 und geht durch den Punkt P(2; 3)P(2;\,3). Bestimmen Sie die Gleichung von gg.
3.41 P
Die Gerade hh ist abgebildet. Bestimmen Sie ihre Gleichung.
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Quadratic functions

Parabola through points, vertex, roots.

7 P
4.13 P
Eine Parabel verläuft durch A(−1; 6)A(-1;\,6), B(1; 0)B(1;\,0) und C(2; 3)C(2;\,3). Bestimmen Sie ihren Scheitelpunkt.
4.21 P
Bestimmen Sie den Scheitelpunkt der Parabel f(x)=x2−6x+5f(x)=x^2-6x+5.
4.31 P
Wie viele Nullstellen hat f(x)=x2+2x+5f(x)=x^2+2x+5 ??
enter a number
4.42 P
Bestimmen Sie die xx-Koordinaten der Schnittstellen: f(x)=x2−4x+3,f(x)=x^2-4x+3, g(x)=2x−5g(x)=2x-5

Want an explanation, not just a score?

In lessons we work through the logic of each solution, the traps the exam typically sets, and how to present your working the way a Studienkolleg expects.

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Four more topics — also free

This is half of it. The other half is in the practice tool

This page has four topics. Four more are waiting in the practice tool — free — an e-mail sign-in is all it takes. It also shows what you have already solved, so you can pick a topic up again instead of starting over.

There is a separate paid tier with the full task bank, but nothing listed below belongs to it — all of it is simply open.

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Four more topics are in the practice tool — free after signing up with your e-mail address. You can see what you have already solved and pick up where you left off.

In total, the trainer has over 14,000 tasks in 38 maths topics – from simplifying expressions to geometry and derivatives, most with a worked solution.

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Maths in the Studienkolleg entrance exam: what is asked and how to prepare

This page is a free online maths test for the Studienkolleg entrance exam (Aufnahmetest, Aufnahmeprüfung): 4 topics and 19 tasks in German, as in the exam. Choose your answers and click “Check” – the test adds up your points per topic and shows the solution to every task. No sign-up, as often as you like.

Who takes the maths test

Applicants for the T-Kurs (engineering, natural sciences, computer science) and the W-Kurs (economics, social sciences) take maths. Some Studienkollegs also have a maths test for the M-Kurs (medicine, biology, pharmacy). German – often including a C-test – is tested for every course.

What the test is like

  • Time: usually 30 to 60 minutes – as stated in most published sample tests.
  • No calculator: almost everywhere “keine Hilfsmittel” – no calculator, no formula sheet, no dictionary. The numbers are chosen so that everything can be done by hand.
  • German only: the tasks are in German, there is no translation.
  • Answers: usually you write down your working and the result; some tests are multiple choice.

Which topics come up

The core is school algebra and functions. The topics that come up most often in sample tests:

  • simplifying expressions: fractions, brackets, powers, roots, logarithms (Terme, Potenzen, Wurzeln, Logarithmen);
  • equations: linear, quadratic, fractional, with roots, exponential and logarithmic (Gleichungen);
  • inequalities and absolute values (Ungleichungen, Beträge), systems of equations (Gleichungssysteme);
  • functions: lines, parabolas, exponential functions – graphs, zeros, vertex (Funktionen);
  • geometry and trigonometry: areas, Pythagoras’ theorem, sine and cosine (Geometrie, Trigonometrie);
  • percentages and word problems (Prozentrechnung, Textaufgaben).

Some tests also include derivatives (Ableitungen). All of these topics are in the online trainer.

German phrasing you should recognise at once

  • Vereinfachen Sie den Term. – simplify the expression;
  • Lösen Sie die Gleichung. / Bestimmen Sie die Lösungsmenge. – solve the equation / find the solution set;
  • Berechnen Sie … – calculate;
  • Geben Sie den Definitionsbereich an. – state the domain;
  • Bestimmen Sie die Nullstellen und den Scheitelpunkt. – find the zeros and the vertex.

How to prepare

Work without a calculator and in German from the start – that trains both your arithmetic and your reading of the task. Practise against the clock too: the trainer has a Challenge mode with 20 tasks in 20 minutes. Unfamiliar words in a task are explained by the trainer’s dictionary of maths terms.

Frequently asked questions

What is asked in the maths entrance test at a Studienkolleg?
School algebra and functions: simplifying expressions, powers and roots, equations and inequalities, linear and quadratic functions; many tests also include logarithms, trigonometry and geometry, and some include derivatives. The tasks are in German.
Is a calculator allowed?
Usually not: most published sample tests say “keine Hilfsmittel” – no calculator, formula sheet or dictionary. So it is worth practising without a calculator, too.
How long is the maths test?
Usually 30 to 60 minutes – the exact time depends on the Studienkolleg and is given in its sample tests.
Is this test free?
Yes. Four topics on this page are open without sign-up, four more are free in the online trainer after signing in by e-mail. The full collection of over 14,000 tasks comes with a subscription.
How do I know I am ready?
Solve the tasks without a calculator and without translating. If you reliably score above 75 % and understand the solutions, you have a solid base; practise the topics with lower scores in the trainer.