Topics

1. Equations and inequalities
2. Quadratic functions
3. Figures and measurements
4. Plane figures
5. Set theory and logic
6. The number of possible outcomes and probability
7. Expressions and proofs/Equations of higher degree
8. Figures and equations
9. Various functions
10. The concept of differentiation/integration
11. Sequences (Progressions) of numbers
12. Vectors
13. Limits
14. Differential calculus
15. Integral calculus
16. Matrices and their applications
17. Expressions and curves


Purpose of the Examination
The purpose of this examination is to test whether students from other countries have the basic scholastic ability in math considered necessary for studying at the undergraduate level at a Japanese university.
There are two courses. Course 1 is for undergraduate faculties and departments for which a basic knowledge of math is considered sufficient. Course 2 is for undergraduate faculties and departments for which math is very important.

At the time of taking the examination the examinee must choose whether to take Course 1 or Course 2; the examinee should follow the instructions given by the university or the department to which he is applying.

The topics covered by the examination are listed below. The symbols are the ones used in Japanese high school text books; the English version of the test uses standard English terms, and the Japanese version of the test uses terms used in Japanese high school text books.

The Course 1 examination covers only topics 1 to 6.
The Course 2 examination covers all 17 topics.

Subject Topics
1. Equations and inequalities Mathematics l
(1) Numbers and expressions
1) Real numbers
2) Expansion and factorization of a polynomial
(2) Linear inequalities
(3) Quadratic equations

2. Quadratic functions Mathematics ll
(1) Quadratic functions and its graphs
(2) Variation in values of quadratic functions
1) Maximum value and minimum value of a quadratic function
2) Quadratic inequalities

3. Figures and measurements Mathematics ll
(1) Trigonometric ratios
1) Sine, cosine, tangent
2) Relations between trigonometric ratios
(2) Trigonometric ratios and figures
1) Sine formulas, cosine formulas
2) Measurement of figures

4. Plane figures Mathematics A
(1) Properties of triangles
(2) Properties of circles

5. Set theory and logic Mathematics A
(1) Sets and the number of elements
(2) Propositions and proofs

6. The number of possible outcomes and probability Mathematics A
(1) Permutations, Combinations
(2) Probability and its fundamental laws
(3) Independent trials and probability

7. Expressions and proofs/Equations of higher degree Mathematics ll
(1) Expressions and proofs
1) Division of polynomials, fractional expressions
2) Proofs of equalities and inequalities
(2) Equations of higher degree
1) Complex numbers and quadratic equations
2) Equations of higher degree

8. Figures and equations Mathematics ll
(1) Points and lines
1) Coordinates of a point
2) Equation of a line
(2) Circles
1) Equation of a circle
2) Relative position of a circle and a line

9. Various functions Mathematics ll
(1) Trigonometric functions
1) General angles
2) Trigonometric functions and their basic properties
3) Addition theorems for trigonometric functions
(2) Exponential and logarithmic functions
1) Expansion of exponents
2) Exponential functions
3) Logarithmic functions

10. The concept of differentiation/integration Mathematics ll
(1) The concept of differentiation
1) Differential coefficients and derivatives
2) Applications of the derivative
Tangent lines, increase/decrease in function value
(2) The concept of integration
1) Indefinite integrals and definite integrals
2) Areas of figures

11. Sequences (Progressions) of numbers Mathematics B
(1) Sequences and their sums
1) Arithmetic sequences (Arithmetical progressions) and geometric sequences (geometrical progressions)
2) Various sequences
(2) Recurrence formulae and mathematical induction
1) Recurrence formulae and sequences
2) Mathematical induction

12. Vectors Mathematics B
(1) Vectors in a plane
1) Vectors and their operations
2) Scalar product (Inner product) of vectors
(2) Space coordinates and vectors
Space coordinates and vectors in a space

13. Limits Mathematics lll
(1) Limits of sequences
1) Limit of {r n}
2) Sum of an infinite geometric series
(2) Functions and their limits
1) Composite functions and inverse functions
2) Limit of value of a function

14. Differential calculus Mathematics lll
(1) Derivatives
1) Derivatives of the sum/difference/product/quotient of two functions
2) Derivatives of composite functions
3) Derivatives of trigonometric functions, exponential functions, logarithmic functions
(2) Applications of the derivative
Tangent lines, increase/decrease in value of functions, velocity, acceleration

15. Integral calculus Mathematics lll
(1) Indefinite integrals and definite integrals
1) Integrals and their basic properties
2) Integration by substitution, integration by parts
3) Integrals of various functions
(2) Applications of integration
Area, volume

16. Matrices and their applications Mathematics C
(1) Matrices
1) Matrices and operations on them
Sum, difference, and scalar multiple
2) Product of matrices, inverse matrices
(2) Applications of matrices
1) Systems of linear equations
2) Translation of a point

17. Expressions and curves Mathematics C
(1) Quadratic curves
1) Parabolas
2) Ellipses and hyperbolas
(2) Parametric representations
Parametric representation of a curve