Studienkolleg · Maths

Entrance test maths: every topic

Most Studienkollegs test school maths for the T, M and W courses: algebra, equations, geometry and, for the T-Kurs, often trigonometry. Each page below explains one topic with the rules you need, the typical mistakes and free practice questions with worked solutions.

Topics marked "München" are named by Studienkolleg München for its maths test. Studienkolleg München

Arithmetic and terms

FractionsMünchenFractions are in almost every Studienkolleg maths entrance test, usually without a calculator.Powers and rootsPower rules let you simplify long products in a few steps.LogarithmsA logarithm answers one question: to which power must I raise the base to get this number? log₂(8) = 3 because 2³ = 8.Binomial formulasThe three binomial formulas save time when you expand brackets and are the key to factorising and completing the square.Terms and bracketsMünchenSimplifying terms is the basic skill behind every equation.

Equations and inequalities

Linear equationsMünchenLinear equations are the most common task type in entrance tests.Quadratic equationsMünchenQuadratic equations are a core topic of the T-Kurs entrance test at München and many other Kollegs.Higher-degree equationsMünchenEquations with x³ or x⁴ look hard but almost always reduce to a quadratic.Systems of equationsA system of equations asks for values that make all equations true at once.InequalitiesInequalities are solved almost like equations.Polynomial divisionMünchenPolynomial division works like long division with numbers.Root equationsRoot equations need one extra step that equations without roots do not: squaring can create solutions that are not real solutions, so you must check every result..Exponential and log equationsWhen x sits in the exponent, you either write both sides with the same base or take the logarithm.Absolute value equationsThe absolute value |x| is the distance of x from zero, so it is never negative.

Geometry

Pythagorean theoremMünchenIn a right triangle, the two shorter sides (legs) and the longest side (hypotenuse) are linked by one equation.Area and perimeterMünchenArea tasks test formulas and unit conversion.Volume and surface areaSolid geometry asks for volume, surface area or a missing length.Intercept theorems and similarityMünchenWhen two lines through one point are cut by parallels, the pieces are in the same ratio.CirclesMünchenCircle tasks combine area formulas with angle rules.AnglesAngle tasks look simple but often hide a small equation.

Trigonometry

Sine, cosine and tangentIn a right triangle, sine, cosine and tangent link an angle with the ratio of two sides.RadiansRadians measure an angle by the length of its arc on a circle of radius 1.Trigonometric equationsMünchenTrigonometric equations usually have more than one solution in an interval.

Functions

Quadratic functionsA quadratic function f(x) = ax² + bx + c draws a parabola.Exponential functionsExponential functions describe processes that grow or shrink by the same factor in each step: populations, interest, radioactive decay.

Probability and statistics

Probability basicsProbability questions ask how likely an event is.Counting and combinatoricsCombinatorics counts possibilities without listing them.Mean, median and spreadDescriptive statistics summarise a list of data with a few numbers.

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