Kyoto Entrance Mathematics study guide
Kyoto Entrance Mathematics · Work through your exam, one topic at a time.
Kyoto University's general entrance examination in mathematics is the university-set written paper that follows the National Center for University Entrance Examinations (Daigaku Nyūshi Center Shiken) stage. It is intended for high-school graduates and equivalent applicants seeking undergraduate admission to Kyoto University, and it is a constructed-response paper: candidates write full solutions, including reasoning and proofs, rather than selecting answers. The Kyoto Entrance Mathematics study guide has 13 sections and about 4,110 words.
01Numbers and Expressions
Numbers and Expressions rewards exact manipulation: factoring polynomials, handling radicals and absolute values, and simplifying rational expressions without changing the domain. Kyoto entrance problems often combine these skills inside a short proof or a parameter question.
262 words, about 1 min02Equations and Inequalities
Equations and Inequalities tests equivalent transformations: solving linear and quadratic equations, systems, polynomial and rational inequalities, and absolute-value statements. Kyoto problems often ask for a parameter range or for the exact set of solutions, so endpoint behavior matters.
341 words, about 2 min03Sets and Logic
Sets and Logic tests precise language: set operations, De Morgan's laws, quantifiers, converse and contrapositive, and direct or contradiction proofs. Kyoto entrance questions use short statements about integers or real numbers and ask whether a condition is necessary, sufficient, or both.
312 words, about 2 min04Quadratic Functions and Graphs
Quadratic Functions and Graphs tests the connection between algebra and geometry: vertex form, axes of symmetry, discriminants, ranges, and maxima or minima on restricted domains. Kyoto problems often place a quadratic inside a parameter or inequality question.
345 words, about 2 min05Trigonometric Functions and Trigonometric Ratios
Trigonometric Functions and Trigonometric Ratios tests radian measure, the sine and cosine graphs, identities, general solutions, and triangle ratios. Kyoto questions often combine an equation with a restricted interval or ask for exact values without a calculator.
327 words, about 2 min06Exponential and Logarithmic Functions
Exponential and Logarithmic Functions tests laws of exponents and logarithms, domains, monotonicity, and equations that require checking for extraneous solutions. Kyoto problems often use a substitution such as t = 2^x or ask for the range of a log expression.
356 words, about 2 min07Differentiation and Its Applications
Differentiation and Its Applications tests limits, derivative rules, tangent lines, monotonicity, and extrema. Kyoto entrance problems often require a sign chart and a clear distinction between local and global behavior on a stated interval.
312 words, about 2 min08Integration and Its Applications
Integration and Its Applications tests antiderivatives, the fundamental theorem, substitution, areas between curves, and volumes of revolution. Kyoto problems often ask for a region whose boundaries change, so splitting the integral is part of the work.
299 words, about 1 min09Sequences and Series
Sequences and Series tests arithmetic and geometric patterns, recurrence relations, summation, and mathematical induction. Kyoto problems often define a sequence recursively and ask for a closed form or a limit.
342 words, about 2 min10Vectors in the Plane and in Space
Vectors in the Plane and in Space tests components, dot and cross products, angles, lines, and planes. Kyoto problems often combine a perpendicularity condition with a distance or area computation.
334 words, about 2 min11Complex Numbers and Complex Plane
Complex Numbers and Complex Plane tests arithmetic, conjugates, modulus, polar form, De Moivre's theorem, and roots of polynomial equations. Kyoto problems often ask for a locus or for all complex solutions in a specified form.
309 words, about 2 min12Probability and Statistics
Probability and Statistics tests counting, probability rules, conditional probability, expectation, and basic data analysis. Kyoto questions often require choosing between permutations and combinations before any probability is computed.
251 words, about 1 min13Number Theory and Integers
Number Theory and Integers tests divisibility, the Euclidean algorithm, congruences, and integer solutions. Kyoto problems often ask for the remainder of a power, the gcd of large integers, or all solutions to a linear congruence.
320 words, about 2 min