M101: Eignungsprüfung im Fach Mathematik der Universitäten Cambridge und Oxford
M101: Mathematics Entrance Examinations for Cambridge and Oxford
Dieser Kurs wird nur auf Englisch unterrichtet.

This course is taught only in English.

Course M101 prepares students for the Cambridge STEP (Sixth Term Examination Paper) and the Oxford MAT (Maths Admissions Test). Please note that the syllabus for M101 covers more material than is actually required for STEP. The Cambridge STEP is a mathematics examination designed to test candidates on questions that are similar in style to the Mathematical Tripos. It requires substantial mathematical maturity. Candidates preparing for the Abitur or the International Baccalaureate must complete a bridging course before they can commence with M101.

Cambridge University:

Each of Mathematics I, II and III will be a 3-hour paper divided into three sections as follows:

Section A: Pure Mathematics – eight questions
Section B: Mechanics – three question
Section C: Probability and Statistics – two questions

Please note that calculators are not permitted.

Oxford University:

Candidates applying to Oxford cover only parts of the pure maths syllabus of M101. The Oxford MAT aims to test the depth of mathematical understanding, as opposed to the breadth of knowledge. Candidates preparing for the Abitur or the International Baccalaureate may need to complete a bridging course before they can commence with M101.

Section A: Pure Mathematics

  1. Algebra
    • Groups and Symmetries
    • Integers Modulo n
    • Permutations; Permutation Groups
    • Isometry and Matrix Groups
    • Subgroups, Direct Groups and Isomorphism
    • Cayley's Theorem; Lagrange's Theorem
    • Isomorphism Classes
    • Internal Automorphisms
    • Quotient Groups
    • Homomorphisms
    • Soluble Groups
  2. Functions
    • Domain and Range of a Function
    • Injective, Surjective and Bijective Functions
    • Even and Odd Functions; Symmetry
    • The Elementary Functions: Polynomial Functions, Rational Functions, Trigonometric Functions and the Modulus Function Composition of Functions
    • Transformation of Functions
    • Curve Sketching
    • Trigonometry: Radian Measure, Arc Length of a Circle and Area of a Segment
    • Trigonometric Functions and Trigonometric Identities
    • Inverse Trigonometric Functions
  3. Linear Algebra
    • Vector Product – Vector Triple Product (Lagrange's Formula); Jacobi Identity
    • Basic Dyadic Algebra
    • Vector Equations of Lines and Planes
    • Hyperplanes; Affine Sets
    • Distance between Skew Lines
    • Matrices
    • The Transpose of a Matrix
    • Symmetric and Skew-Symmetric Matrices
    • Transformations in Two Dimensions
    • Further Transformations in the x-y Plane
    • Compound Transformations
    • Determinants (Wronskian and Pfaffian)
    • Transformations in Three-Dimensional Space
    • Inverse Transformations
    • Systems of Linear Equations
    • Vector Space
    • Image and Kernel of a Linear Transformation
    • Subspaces; Bases and Linear Independence
    • Rank-Nullity Theorem
    • Orthogonality and Least Squares
    • Euclidean n-Space
    • Cauchy-Schwarz Inequality
    • Inner Product Spaces
    • Gram-Schmidt Process and QR Factorization
    • Orthogonal Transformations and Orthogonal Matrices
    • Geometrical Interpretation of Determinants Eigenvectors and Eigenvalues
    • Characteristic Polynomial
    • Geometric and Algebraic Multiplicity of Eigenvalues
    • Complex Numbers and Complex Eigenvalues
    • Diagonalisation and Similarity
    • Symmetric Matrices
    • Quadratic Forms
    • Singular Value Decomposition
    • Complex Inner Product Spaces
  4. Geometry
    • Euler's Theorem
    • Similar Figures
      • Homothetic Figures
      • Homologous and Antihomologous Points
      • Center of Similitude; Circle of Similitude
      • Inversely Similar Figures
    • Coaxial Circles and Inversions
      • Power of a Point
      • The Radical Axis
      • Conjugate Coaxial Systems
      • The Ratio Locus
      • Inversion
      • Peaucellier Inversor
    • Triangles and Polygons
      • Anharmonic Ratio
      • Theorem of Ptolemy
      • Distance Relations
      • The Cyclic Quadrangle
    • Geometry of Circles
      • The Power Theorem of Casey
      • Poncelet's Theorem
      • Circles of Antisimilitude
      • The Self-Conjugate Triangle
      • Stereographic Projection
    • Tangent Circles
      • Steiner Chains
      • The Arbelos
      • The Problem of Apollonius
      • Casey's Theorem
      • Theorem of Hart
      • Circles Intersecting at Given Angles
    • Triangles
      • The Theorem of Miquel
      • The Pedal Triangle
      • Simson Line
      • Theorems of Ceva and Menelaus
      • Isogonal Conjugates
      • Isotomic Conjugates
      • Circumcenter and Orthocenter
      • Theorem of Gauss and Bodenmiller
      • Theorems of Hagge
    • Inscribed and Escribed Circles
      • The Incenter and Excenters
      • The Inscribed Circle
    • The Nine Point Circle
      • The Theorem of Feuerbach
      • Simson Lines
    • Further Theory
      • Exsymmedians
      • The Gergonne Point of a Triangle
      • The Isogonic Centers
      • Antipedal Triangles
      • The Nagel Point
      • The Spieker Circle
      • The Fuhrmann Circle
      • Theorems of Desargues, Pascal and Brianchon
      • Pedal Triangles and Circles
      • Theorems of Fontene
      • Circles of Droz-Farny
      • The Brocard Configuration
      • Circles: Tucker, Lemoine, McCay and Taylor
      • The Steiner Point and the Tarry Point
      • The Neuberg Circles
      • The Circles of Apollonius
      • The Theorem of Schoute
  5. Number Theory
    • Sets and Proofs
    • Number Systems and Decimals
    • Inequalities
    • nth Roots and Rational Powers
    • Complex Numbers and Polynomial Equations
    • Proof by Contradiction and Disproof by Counterexample
    • Proof by Induction
      • Principle of Mathematical Induction I
      • Principle of Mathematical Induction II
      • Principle of Strong Mathematical Induction
    • The Euclidean Algorithm
    • Prime Factorization
    • Congruence of Integers
    • Fermat's Little Theorem
    • kth Roots Modulo m
    • Multiplication Principle
    • Binomial Coefficients and the Binomial Theorem
    • Multinomial Coefficients
    • Further Set Theory
    • Equivalence Relations
    • Permutations
  6. Calculus I
    • Limit Theorems
    • Continuity at a Point and Continuity on Intervals
    • The Pinching Theorem
    • The Intermediate-Value Theorem
    • Extreme-Value Theorem
  7. Calculus II
    • Differentiability and Continuity
    • Derivatives
    • Implicit Differentiation
    • Differentials
    • Differentiating the Trigonometric Functions
    • Newton-Raphson Method
  8. Calculus III
    • The Mean-Value Theorem
    • Rolle's Theorem
    • Increasing and Decreasing Functions
    • Local and Absolute Extreme Values
    • Max-Min Problems
    • Concavity and Points of Inflection
    • Vertical and Horizontal Asymptotes
    • Vertical tangents and Vertical Cusps
  9. Calculus IV
    • The Fundamental Theorem of Integral Calculus
    • Change of Variables
    • Mean-Value Theorem for Integrals
    • Riemann Sums
    • Integration with Respect to y
    • Volume by Parallel Cross Sections; Discs and Washers
    • The Shell Method
    • The Centroid of a Region
    • Pappus's Theorem on Volumes
  10. Calculus V
    • Inverse Functions
    • The Logarithm Function
    • The Exponential Function
    • Integration of Trigonometric Functions
    • Inverse Trigonometric Functions
    • The Hyperbolic Sine and Cosine
    • The Hyperbolic Cotangent, Hyperbolic Secant and Hyperbolic Cosecant
    • The Hyperbolic Inverses
    • Integration by Parts
    • Powers and Products of Sine and Cosine
    • Other Trigonometric Powers
    • Integrals Involving Trigonometric Substitutions
    • Partial Fractions
    • Numerical Integration
  11. Calculus VI
    • The Conic Sections in Polar Coordinates
    • Intersection of Polar Curves
    • Area in Polar Coordinates
    • Arc Length
    • Area of a Surface of Revolution
    • The Centroid of a Curve
    • The Cycloid
  12. Calculus VII
    • The Least Upper Bound Axiom
    • Sequences of Real Numbers
    • Limit of a Sequence
    • Convergent and Divergent Sequences
    • The Pinching Theorem for Sequences
    • Important Limits
    • Indeterminate Forms
    • L'Hospital's Rule
    • The Cauchy Mean-Value Theorem
    • Other Indeterminate Forms
    • Improper Integrals
    • Stirling's Formula
    • The Gamma Function
  13. Calculus VIII
    • Infinite Series
    • Arithmetic, Geometric and Harmonic series
    • Divergence Test
    • The Integral test – The Harmonic Series, the p-Series
    • The Basic Comparison Test, The Limit Comparison Test
    • The Root Test; The Ratio Test
    • Absolute and Conditional Convergence; Alternating Series
    • Shanks Transformation
    • Taylor Polynomials in x and Taylor Series in x
    • Lagrange Formula for the Remainder
    • Power Series; Maclaurin Series; Standard Expansions
    • Differentiation and Integration of Power Series
    • Abel's theorem
  14. Calculus IX
    • Differential Equations
    • Exact Differential Equations
    • The Integrating Factor
    • Second Order Linear Differential Equations
    • The Particular Integral
    • The Failure Case
  15. Complex Numbers
    • De Moivre's Theorem
    • Cube Roots of Unity
    • The nth Roots of Unity
    • The Exponential Form for a Complex Number
    • Osborn's Rule
    • Loci
    • Transformations of the Argand diagram
    • Fourier Coefficients
    • Dirichlet Conditions
    • Complex Form of Fourier Series

Section B: Mechanics

  1. Elastic Strings and Springs
    • Hooke's Law
    • The Modulus of Elasticity
    • Springs
  2. Motion in a Horizontal Circle
    • Angular Velocity
    • The Magnitude of the Radial Acceleration
    • The Conical Pendulum
    • Banked Tracks
  3. Work and Energy
    • Elastic Potential Energy
    • Elastic Springs
  4. Motion in a Vertical Circle
    • Motion Restricted to a Circular Path
    • Motion Not Restricted to a Circular Path
  5. Collisions
    • Elastic Impact
    • Newton's Law of Restitution, Coefficient of Restitution
    • Multiple Impacts
  6. Centre of Mass
    • Centre of Mass of a Rigid Body
    • Finding a Centre of Mass by Integration
  7. Equilibrium of Rigid Bodies
    • General Conditions
    • Equilibrium on a Horizontal Plane; Equilibrium on an Inclined Plane
  8. Variable Acceleration
    • Motion with Constant Acceleration
    • Acceleration and Velocity as a Function of Displacement
    • Simple Harmonic Motion
  9. Vectors
    • Forces – Vector Representation
    • Power of a Variable Force; Work Done by a Variable Force
  10. Vector Equation of a Curve
    • The Helix
    • Motion of a Particle on a Curve
    • Vector Calculus
    • Motion of a Particle in a Plane Using Polar Coordinates
  11. Vector Product and its Application
    • Moment of a Force About a Point
    • Vector Moment of a Couple
    • Systems of Forces in Three Dimensions
  12. Impact
    • Direct and Oblique Impact
    • Impact – Fixed Surface and Moving Objects
  13. Projectiles
    • Inclined Axes
    • Time of Flight
    • Projection Down an Inclined Plane
  14. Motion
    • Basic Equation of Motion
    • Harmonic Motion
      • Damped Harmonic Motion; Properties of Damped Harmonic Motion
      • Forced Harmonic Motion
    • Motion of a Particle with Varying Mass
  15. Centre of Mass
    • Centre of Mass and Centre of Gravity
    • Work Done and Change in Kinetic Energy
    • Motion of the Centre of Mass in a Uniform Gravitational Field
    • Centroid
    • Stability of Equilibrium
  16. Rotation
    • Rotation of a Rigid Body about a Fixed Axis
    • Properties of a Rotating body
    • Kinetic Energy of Rotation
  17. Moment of Inertia
    • Moment of Inertia of a Set of Particles
    • Moment of Inertia of a Ring and a Disc
    • Moment of Inertia of a Solid of Revolution
    • Moment of Inertia of a Surface of Revolution
    • Moment of Inertia of a Rod
    • Non-Uniform Bodies
    • Radius of Gyration
    • Second Moment of Area and Volume
    • The Parallel Axis Theorem
    • The Perpendicular Axis Theorem
    • Compound Bodies
  18. Rotation About a Fixed Axis
    • Work Done by an External Couple
    • Angular Acceleration
    • Small Oscillations – Compound pendulum; Equivalent Simple Pendulum
    • Force Exerted by Axis
    • The Impulse of a Torque
    • Instantaneous Impulse of a Torque
    • Conservation of Angular Momentum
  19. Rotation and Translation
    • Motion of a Lamina in its Own Plane
    • Rotation and Translation of a Rigid Body
    • Rolling Without Slipping
    • Rolling and Slipping
    • Impulsive Motion
  20. Shearing Force and Bending Moment
    • Internal Stresses in a Rigid body
    • Concentrated and Distributed Forces
    • Bending Moment and Shearing Force

Section C: Probability and Statistics

  1. Introduction to Probability
    • Set Theory
    • Counting Methods
    • Permutations and Combinations
    • Multinomial Coefficients
  2. Conditional Probabilities
    • Conditional Probability
    • Bayes' Theorem
    • Independent Events
  3. Random Variables and Distributions
    • Random Variables and Discrete Distributions
    • Continuous Distributions
    • Functions of a Random Variable
    • Markov Chains
  4. Expectation
    • Expectation Algebra
    • Covariance and Correlation
    • Conditional Expectation
  5. Distributions
    • The Bernoulli and Binomial Distributions
    • The Hypergeometric Distributions
    • The Poisson Distributions
    • Joint Distribution Functions
    • Independent Random Variables
    • The Normal Distributions
    • The Multinomial Distributions
    • The Bivariate Normal Distributions
  6. Large Random Samples
    • Markov's Inequality
    • Chebyshev's Inequality
    • The Weak Law of Large Numbers
    • The Central Limit Theorem
    • The Strong Law of Large Numbers
    • Jensen's Inequality
  7. Estimation
    • Statistical Inference
    • Bayes Estimators
    • Maximum Likelihood Estimators
  8. Sampling Distributions of Estimators
    • The Sampling Distribution of a Statistic
    • The Chi-Square Distributions
    • The t Distributions
    • Confidence Intervals
    • Unbiased Estimators
  9. Significance Testing
    • Null and Alternative Hypothesis
    • Critical Region and Critical Values
    • One-Tailed and Two-Tailed Tests
  10. Linear Statistical Models
    • The Method of Least Squares
    • Regression
    • Statistical Inference in Simple Linear Regression
    • Bayesian Inference in Simple Linear Regression
    • The General Linear Model and Multiple Regression