Graduate Entrance Exam Mathematics Syllabus 2025

This syllabus is compiled from the national graduate entrance examination mathematics outline. It is meant to give candidates a clear and complete guide to what to revise.

Higher Mathematics

I. Functions, Limits, and Continuity

Exam Content

• The concept of a function and the ways of representing one
• Boundedness, monotonicity, periodicity, and parity of functions
• Composite functions, inverse functions, piecewise functions, and implicit functions
• The properties and graphs of the basic elementary functions
• Elementary functions
• Establishing functional relationships
• The definition and properties of limits of sequences and of functions
• Left-hand and right-hand limits of a function
• The concepts of infinitesimal and infinite quantity, and the relation between them
• The properties of infinitesimals and the comparison of infinitesimals
• The four algebraic laws of limits
• The two criteria for the existence of a limit: the monotone bounded criterion and the squeeze criterion
• The two important limits
• The concept of continuity of a function
• The types of discontinuity of a function
• Continuity of elementary functions
• The properties of continuous functions on a closed interval

Exam Requirements

1. Understand the concept of a function, master the ways of representing one, and be able to establish the functional relationship in an applied problem.
2. Understand boundedness, monotonicity, periodicity, and parity of functions.
3. Understand the concepts of composite and piecewise functions, and understand the concepts of inverse and implicit functions.
4. Master the properties and graphs of the basic elementary functions, and understand the concept of an elementary function.
5. Understand the concept of a limit, understand the left-hand and right-hand limits of a function and the relation between the existence of a limit and those two one-sided limits.
6. Master the properties of limits and the four algebraic laws.
7. Master the two criteria for the existence of a limit and use them to evaluate limits; master the method of evaluating limits with the two important limits.
8. Understand the concepts of infinitesimal and infinite quantity, master the method of comparing infinitesimals, and be able to use equivalent infinitesimals to evaluate limits.
9. Understand the concept of continuity of a function (including left-continuity and right-continuity), and be able to classify the type of discontinuity.
10. Understand the properties of continuous functions and the continuity of elementary functions, understand the properties of continuous functions on a closed interval (boundedness, the maximum and minimum value theorems, and the intermediate value theorem), and be able to apply them.

II. Differential Calculus of Functions of One Variable

Exam Content

• The concepts of derivative and differential
• The geometric and physical meaning of the derivative
• The relation between differentiability and continuity
• The tangent and the normal to a plane curve
• The four algebraic laws of derivatives and differentials
• The derivatives of the basic elementary functions
• Differentiation of composite functions, inverse functions, implicit functions, and functions given by parametric equations
• Higher-order derivatives
• The invariance of the form of a first-order differential
• The mean value theorems for differentials
• L’Hospital’s rule
• The criterion for monotonicity of a function
• Extreme values of a function
• Concavity and convexity of a graph, inflection points, and asymptotes
• Sketching the graph of a function
• The maximum and minimum values of a function
• Arc length differential
• The concept of curvature
• The osculating circle and the radius of curvature

Exam Requirements

1. Understand the concepts of derivative and differential, understand the relation between them, understand the geometric meaning of the derivative, be able to find the tangent and normal equations of a plane curve, understand the physical meaning of the derivative and use derivatives to describe certain physical quantities, and understand the relation between differentiability and continuity.
2. Master the four algebraic laws of derivatives and the chain rule for composite functions, and master the derivative formulas of the basic elementary functions. Understand the four algebraic laws of differentials and the invariance of the first-order differential form, and be able to find the differential of a function.
3. Understand the concept of higher-order derivatives and be able to find the higher-order derivatives of simple functions.
4. Be able to find the derivative of a piecewise function, and of implicit functions, functions given by parametric equations, and inverse functions.
5. Understand and use Rolle’s theorem, Lagrange’s mean value theorem and Taylor’s theorem, and understand and use Cauchy’s mean value theorem.
6. Master the method of using L’Hospital’s rule to evaluate indeterminate limits.
7. Understand the concept of extreme values of a function, master the method of using derivatives to decide monotonicity and to find extreme values, and master the method of finding and applying the maximum and minimum values of a function.

III. Integral Calculus of Functions of One Variable

Exam Content

• The concept of an antiderivative; the concepts of indefinite and definite integrals
• The basic formulas of indefinite integration, the properties of indefinite and definite integrals, the mean value theorem for definite integrals, the substitution method, and integration by parts
• Integration of rational functions, rational expressions of trigonometric functions, and simple irrational functions
• Functions with a variable upper limit and their derivatives; the Newton-Leibniz formula
• The concept of improper integrals and their evaluation
• Applications of the definite integral in geometry and physics (the area of a plane figure, the arc length of a plane curve, the volume and surface area of a solid of revolution, the volume of a solid with a known cross-sectional area, work, gravitational force, pressure, centre of mass, centroid, and so on) and the average value of a function

Exam Requirements

1. Understand the concept of an antiderivative, and understand the concepts of indefinite and definite integrals.
2. Master the basic formulas of indefinite integration, master the properties of indefinite and definite integrals and the mean value theorem for definite integrals, and master the substitution method and integration by parts.
3. Be able to integrate rational functions, rational expressions of trigonometric functions, and simple irrational functions.
4. Understand the function of the upper limit of integration and be able to find its derivative; master the Newton-Leibniz formula.
5. Understand the concept of improper integrals and be able to evaluate them.
6. Master the use of the definite integral to express and compute certain geometric and physical quantities.

IV. Vector Algebra and Analytic Geometry of Space

Exam Content

• The concept of a vector and its linear operations
• The dot product, the cross product, and the mixed product of vectors
• The conditions for two vectors to be perpendicular or parallel, and the angle between two vectors
• The coordinate representation of a vector and the operations on it
• Unit vectors, direction numbers, and direction cosines
• The concepts of the equation of a surface and of a space curve
• Equations of planes and of straight lines
• The angles between planes, between a plane and a line, and between two lines, and the conditions for parallelism and perpendicularity
• The distance from a point to a plane and to a line
• Spheres, cylinders, and surfaces of revolution
• The equations of the common quadric surfaces and their graphs
• The parametric and general equations of a space curve
• The equation of the projection curve of a space curve on a coordinate plane

Exam Requirements

1. Understand a rectangular coordinate system in space, and understand the concept of a vector and its representation.
2. Master the operations on vectors (linear operations, dot product, cross product, mixed product), and understand the conditions for two vectors to be perpendicular or parallel.
3. Understand unit vectors, direction numbers and direction cosines, and the coordinate representation of a vector; master the method of doing vector operations with coordinate representations.
4. Master the equations of planes and of straight lines and the methods of finding them.
5. Be able to find the angles between planes, between a plane and a line, and between two lines, and be able to use the relations between planes and lines (parallel, perpendicular, intersecting, and so on) to solve related problems.
6. Be able to find the distance from a point to a line and to a plane.
7. Understand the concepts of the equation of a surface and of a space curve.
8. Know the equations of the common quadric surfaces and their graphs, and be able to find the equations of simple cylinders and surfaces of revolution.
9. Understand the parametric and general equations of a space curve. Understand the projection of a space curve on a coordinate plane, and be able to find the equation of that projection.

V. Differential Calculus of Functions of Several Variables

Exam Content

• The concept of a function of several variables
• The geometric meaning of a function of two variables
• The concepts of limits and continuity for functions of two variables
• The properties of continuous functions of several variables on a bounded closed region
• Partial derivatives and the total differential of a function of several variables
• The necessary and sufficient conditions for the total differential to exist
• Differentiation of composite and implicit functions of several variables
• Second-order partial derivatives
• Directional derivatives and the gradient
• The tangent and the normal plane to a space curve
• The tangent plane and the normal to a surface
• The second-order Taylor formula for a function of two variables
• Extreme values and conditional extreme values of functions of several variables
• The maximum and minimum values of functions of several variables and their simple applications

Exam Requirements

1. Understand the concept of a function of several variables and understand the geometric meaning of a function of two variables.
2. Understand the concepts of limits and continuity for functions of two variables, and the properties of continuous functions on a bounded closed region.
3. Understand the concepts of partial derivatives and the total differential of a function of several variables, be able to find the total differential, understand the necessary and sufficient conditions for the total differential to exist, and understand the invariance of the form of the total differential.
4. Understand the concepts of the directional derivative and the gradient, and master the methods of computing them.
5. Master the methods of finding first- and second-order partial derivatives of composite functions of several variables.
6. Understand the implicit function existence theorem, and be able to find the partial derivatives of implicit functions of several variables.
7. Understand the concepts of the tangent and normal plane to a space curve and of the tangent plane and normal to a surface, and be able to find their equations.
8. Be familiar with the second-order Taylor formula for a function of two variables.
9. Understand the concepts of extreme values and conditional extreme values of functions of several variables, master the necessary condition for the existence of an extreme value, understand the sufficient condition for the existence of an extreme value of a function of two variables, be able to find the extreme values of a function of two variables, be able to use the Lagrange multiplier method to find conditional extreme values, be able to find the maximum and minimum values of simple functions of several variables, and be able to solve some simple application problems.

VI. Integral Calculus of Functions of Several Variables

Exam Content

• The concepts, properties, computation, and applications of double and triple integrals
• The concepts, properties, and computation of the two kinds of line integral, and the relation between them
• Green’s theorem, the condition for a plane curve integral to be path independent, and the antiderivative of the total differential of a function of two variables
• The concepts, properties, and computation of the two kinds of surface integral, and the relation between them
• Gauss’s theorem and Stokes’ theorem
• The concepts of divergence and curl, and their computation
• Applications of line integrals and surface integrals

Exam Requirements

1. Understand the concepts of double and triple integrals, be familiar with the properties of multiple integrals, and be familiar with the mean value theorem for double integrals.
2. Master the methods of computing double integrals (in rectangular and polar coordinates), and be able to compute triple integrals (in rectangular, cylindrical, and spherical coordinates).
3. Understand the concepts of the two kinds of line integral, and be familiar with their properties and with the relation between them.
4. Master the methods of computing the two kinds of line integral.
5. Master Green’s theorem and the condition for a plane curve integral to be path independent, and be able to find the antiderivative of the total differential of a function of two variables.
6. Understand the concepts and properties of the two kinds of surface integral and the relation between them, master the methods of computing the two kinds of surface integral, master the method of computing a surface integral with Gauss’s theorem, and be able to compute a line integral with Stokes’ theorem.
7. Understand the concepts of divergence and curl, and be able to compute them.
8. Be able to use multiple integrals, line integrals, and surface integrals to find certain geometric and physical quantities.

VII. Infinite Series

Exam Content

• The concepts of convergence and divergence of a series with constant terms, and the concept of the sum of a convergent series
• The basic properties of a series and the necessary condition for convergence
• The convergence and divergence conditions of geometric series and p-series
• The comparison test, the ratio test, and the root test for series with positive terms
• The Leibniz test for alternating series
• Absolute convergence and conditional convergence of series with arbitrary terms

Exam Requirements

1. Understand the concepts of convergence and divergence of a series with constant terms and of the sum of a convergent series; master the basic properties of a series and the necessary condition for convergence.
2. Master the convergence and divergence conditions of geometric series and p-series.
3. Master the comparison test and the ratio test for convergence of a series with positive terms, and be able to use the root test.
4. Master the Leibniz test for alternating series.
5. Be familiar with the concepts of absolute convergence and conditional convergence of a series with arbitrary terms, and with the relation between absolute convergence and convergence.
6. Be familiar with the concept of the interval of convergence of a series of functions and of the sum function.
7. Be familiar with the concept of the radius of convergence of a power series, and master the methods of finding the radius of convergence, the interval of convergence, and the region of convergence of a power series.
8. Be familiar with the basic properties of a power series inside its interval of convergence (continuity of the sum function, term-by-term differentiation, and term-by-term integration); be able to find the sum function of some power series inside their interval of convergence, and hence find the sum of some series of terms.
9. Be familiar with the necessary and sufficient conditions for a function to expand into a Taylor series.

Linear Algebra

I. Determinants

Exam Content

• The concept and basic properties of a determinant
• The expansion theorem for a determinant along a row (column)

Exam Requirements

1. Be familiar with the concept of a determinant, and master the properties of determinants.
2. Be able to compute a determinant using its properties and the expansion theorem along a row (column).

II. Matrices

Exam Content

• The concept of a matrix
• The linear operations on matrices, matrix multiplication, and transposition
• Powers of a square matrix, and the determinant of a product of square matrices
• The concept and properties of the inverse matrix, and the necessary and sufficient conditions for a matrix to be invertible
• The adjugate matrix
• Elementary transformations of a matrix, and elementary matrices
• The rank of a matrix, and the equivalence of matrices
• Block matrices and their operations

Exam Requirements

1. Understand the concept of a matrix, and be familiar with the identity matrix, scalar matrix, diagonal matrix, triangular matrix, symmetric matrix and skew-symmetric matrix, together with their properties.
2. Master the linear operations on matrices, matrix multiplication, and transposition, together with their laws; be familiar with the properties of powers of a square matrix and of the determinant of a product of square matrices.
3. Understand the concept of the inverse matrix, master the properties of the inverse matrix and the necessary and sufficient conditions for invertibility, understand the concept of the adjugate matrix, and be able to use the adjugate matrix to find an inverse.
4. Understand the concept of elementary transformations of a matrix, be familiar with the properties of elementary matrices and with the concept of matrix equivalence, understand the concept of the rank of a matrix, and master the methods of finding the rank and the inverse matrix by elementary transformations.
5. Be familiar with block matrices and their operations.

III. Vectors

Exam Content

• The concept of a vector, linear combinations of vectors, and linear representation
• Linear dependence and linear independence of a set of vectors
• A maximal linearly independent subset of a set of vectors, equivalent sets of vectors, and the rank of a set of vectors
• The relation between the rank of a set of vectors and the rank of a matrix
• Vector spaces and the concepts related to them
• The basis, dimension, and coordinates of an n-dimensional vector space
• Changes of basis and of coordinates, and the transition matrix
• The inner product of vectors, and orthogonal normalisation of a linearly independent set of vectors
• Orthonormal bases, orthogonal matrices, and their properties

Exam Requirements

1. Understand the concepts of an n-dimensional vector, of linear combinations of vectors, and of linear representation.
2. Understand the concepts of linear dependence and linear independence of a set of vectors, master the relevant properties and tests for them.
3. Understand the concepts of a maximal linearly independent subset and of the rank of a set of vectors, and be able to find them.
4. Understand the concept of equivalent sets of vectors, and understand the relation between the rank of a matrix and the rank of its row (column) vectors.
5. Be familiar with the concepts of an n-dimensional vector space, subspace, basis, dimension, and coordinates.
6. Be familiar with the formulas for a change of basis and a change of coordinates, and be able to find the transition matrix.
7. Be familiar with the concept of the inner product, and master the Schmidt orthogonal normalisation of a linearly independent set of vectors.
8. Be familiar with the concepts of an orthonormal basis and an orthogonal matrix, and with their properties.

IV. Systems of Linear Equations

Exam Content

• Cramer’s rule for systems of linear equations
• The necessary and sufficient conditions for a homogeneous system of linear equations to have a nonzero solution
• The necessary and sufficient conditions for a non-homogeneous system of linear equations to have a solution
• The properties and the structure of the solutions of a system of linear equations
• A fundamental system of solutions and the general solution of a homogeneous system, and the solution space
• The general solution of a non-homogeneous system of linear equations

Exam Requirements

1. Be able to use Cramer’s rule.
2. Understand the necessary and sufficient conditions for a homogeneous system to have a nonzero solution, and for a non-homogeneous system to have a solution.
3. Understand the concepts of a fundamental system of solutions, of the general solution, and of the solution space of a homogeneous system, and master the methods of finding a fundamental system of solutions and the general solution.
4. Understand the structure of the solutions and the concept of the general solution of a non-homogeneous system.
5. Master the method of solving a system of linear equations by elementary row transformations.

V. Eigenvalues and Eigenvectors of a Matrix

Exam Content

• The concepts and properties of the eigenvalues and eigenvectors of a matrix
• Similarity transformations, similar matrices, and their properties
• The necessary and sufficient conditions for a matrix to be diagonalisable by similarity, and the similar diagonal matrix
• The eigenvalues, eigenvectors, and similar diagonal matrix of a real symmetric matrix

Exam Requirements

1. Understand the concepts and properties of the eigenvalues and eigenvectors of a matrix, and be able to find them.
2. Understand the concept and properties of similar matrices and the necessary and sufficient conditions for a matrix to be diagonalisable by similarity, and master the method of reducing a matrix to a similar diagonal matrix.
3. Master the properties of the eigenvalues and eigenvectors of a real symmetric matrix.

VI. Quadratic Forms

Exam Content

• Quadratic forms and their matrix representation
• Congruent transformations and congruent matrices
• The rank of a quadratic form, and the law of inertia
• The canonical form and the normal form of a quadratic form
• Reducing a quadratic form to canonical form by orthogonal transformation and by completing the square
• Positive definiteness of a quadratic form and of its matrix

Exam Requirements

1. Master quadratic forms and their matrix representation, understand the concept of the rank of a quadratic form, be familiar with the concepts of congruent transformations and congruent matrices, and be familiar with the concepts of the canonical form and the normal form of a quadratic form and with the law of inertia.
2. Master the method of reducing a quadratic form to canonical form by orthogonal transformation, and be able to reduce it by completing the square.
3. Understand the concepts of a positive definite quadratic form and a positive definite matrix, and master the tests for them.

Probability Theory and Mathematical Statistics

I. Random Events and Probability

Exam Content

• Random events and the sample space, and the relations and operations on events
• Complete groups of events
• The concept of probability and its basic properties
• The classical model and the geometric model
• Conditional probability and the basic formulas of probability
• Independence of events
• Repeated independent trials

Exam Requirements

1. Be familiar with the concept of the sample space (the space of elementary events), understand the concept of a random event, and master the relations and operations on events.
2. Understand the concepts of probability and conditional probability, master the basic properties of probability, be able to compute the classical model and the geometric model, and master the addition, subtraction, multiplication, and total probability formulas as well as Bayes’ formula.
3. Understand the concept of independence of events and master probability computation using independence; understand the concept of repeated independent trials and master the methods of computing the probabilities of related events.

IV. Numerical Characteristics of Random Variables

Exam Content

• The mathematical expectation (mean), variance, and standard deviation of a random variable, and their properties
• The mathematical expectation of a function of a random variable
• Moments, covariance, and the correlation coefficient, and their properties

Exam Requirements

1. Understand the concepts of the numerical characteristics of a random variable (mathematical expectation, variance, standard deviation, moments, covariance, and the correlation coefficient), be able to apply the basic properties of these characteristics, and master the numerical characteristics of the common distributions.
2. Be able to find the mathematical expectation of a function of a random variable.

V. The Law of Large Numbers and the Central Limit Theorem

Exam Content

• Chebyshev’s inequality
• Chebyshev’s law of large numbers, Bernoulli’s law of large numbers, and Khinchine’s law of large numbers
• The De Moivre-Laplace theorem, and the Levy-Lindberg theorem

Exam Requirements

1. Be familiar with Chebyshev’s inequality.
2. Be familiar with Chebyshev’s law of large numbers, Bernoulli’s law of large numbers, and Khinchine’s law of large numbers (the law of large numbers for sequences of independent and identically distributed random variables).
3. Be familiar with the De Moivre-Laplace theorem (the binomial distribution converges to the normal distribution) and with the Levy-Lindberg theorem (the central limit theorem for sequences of independent and identically distributed random variables).

VII. Parameter Estimation

Exam Content

• The concept of point estimation, estimators, and estimates
• The method of moments and the method of maximum likelihood
• The criteria for comparing estimators
• The concept of interval estimation
• Interval estimation of the mean and the variance of a single normal population
• Interval estimation of the difference of means and the ratio of variances of two normal populations

Exam Requirements

1. Understand the concepts of point estimation of a parameter, of an estimator, and of an estimate.
2. Master the method of moments (first and second moments) and the method of maximum likelihood.
3. Be familiar with the concepts of unbiasedness, effectiveness (minimum variance), and consistency of estimators, and be able to verify the unbiasedness of an estimator.
4. Be familiar with the concept of interval estimation; be able to find the confidence interval for the mean and the variance of a single normal population, and the confidence interval for the difference of means and the ratio of variances of two normal populations.

VIII. Hypothesis Testing

Exam Content

• Significance testing, hypothesis testing, and the two kinds of error
• Hypothesis tests for the mean and the variance of a single normal population and of two normal populations

Exam Requirements

1. Be familiar with the basic idea of significance testing, master the basic steps of a hypothesis test, and understand the two kinds of error that a hypothesis test may produce.
2. Master hypothesis tests for the mean and the variance of a single normal population and of two normal populations.