Kyoto Entrance Science common mistakes
Kyoto Entrance Science · Understand the traps and how to avoid them.
In Mathematics III, when computing an improper integral such as ∫₀¹ x^(−1/2) dx, the candidate substitutes the endpoint 0 directly into the antiderivative 2√x, writes 2√0 = 0, and reports the finite value without ever writing a limit as the lower bound approaches 0.
The integrand has a singularity at the endpoint, so the fundamental theorem cannot be applied directly; candidates pattern-match to the routine definite-integral procedure and treat the unbounded endpoint as an ordinary number, often because the antiderivative looks harmless and finite.
What to do instead
Before integrating, classify the integral: check whether the integrand is continuous and bounded on the closed interval. If it is not, write the limit explicitly (lim_{t→0⁺} ∫_t¹ ...) and evaluate the limit, and state the convergence or divergence as the final answer. Drill a mixed set of convergent and divergent improper integrals so the classification step becomes automatic.
In Mathematics B (vectors and complex plane), the candidate solves a geometry problem in the complex plane by treating |z − a| as if it were the signed distance z − a, or by applying the triangle inequality |z₁ + z₂| ≤ |z₁| + |z₂| as an equality without checking when equality holds.
Students carry over real-number habits (squaring, dropping absolute values, assuming linearity) into the complex setting, and the modulus is introduced as a formula rather than a geometric distance, so the locus interpretation is missing.
What to do instead
For every complex-plane problem, first translate the condition into a locus: |z − a| = r is a circle centred at a, |z − a| = |z − b| is the perpendicular bisector. Then solve algebraically with z = x + iy and confirm the result matches the geometric picture. When using inequalities, write the equality condition (same argument) alongside.
In Mathematics A (probability and counting), the candidate computes the number of outcomes in a sample space without deciding whether order matters and whether sampling is with or without replacement, then divides one count by another count built under a different convention, producing a probability greater than 1 or a wrong denominator.
Counting is taught as a menu of formulas (permutations, combinations) and candidates pick one by keyword rather than by modelling the experiment; the numerator and denominator are often counted under inconsistent conventions.
What to do instead
Write one sentence defining the sample space before counting: 'I list ordered pairs of distinct cards, so the denominator is ₅₂P₂.' Count the numerator under exactly the same convention, or switch both to unordered. Sanity-check that the answer lies in [0,1] and that the denominator equals the total number of equally likely outcomes.
In Mathematics II (calculus and trigonometric/exponential functions), the candidate differentiates or integrates a composite function by acting on the outer function only, e.g. writing d/dx sin(2x) = cos(2x) or ∫ cos(2x) dx = sin(2x), omitting the inner derivative or the factor 1/2.
The chain rule is understood conceptually but not yet automatised, and under time pressure the candidate's working memory drops the inner factor; the error is invisible because the result still 'looks like' a trigonometric answer.
What to do instead
Adopt an explicit substitution step: let u = 2x, du = 2 dx, so dx = du/2, and rewrite the integral in u before integrating. For derivatives, write the inner function and its derivative on a separate line. Practise ten composite derivatives and integrals daily until the inner factor appears without prompting, and verify by differentiating the answer.
In Physics, the candidate writes Newton's second law as F = ma with F meaning a single applied force, ignoring friction, normal force, tension, or the component of gravity along an incline, and then reports an acceleration that is too large.
Textbook problems often present one force, and candidates memorise 'F = ma' as a recipe rather than as a statement about the net force; free-body diagrams are skipped because they feel like extra work.
What to do instead
Make a labelled free-body diagram mandatory before any equation: draw every force acting on the body, choose axes (often along the incline), and write ΣF = ma component by component. Never write F = ma until the diagram is complete. Check the limiting cases (frictionless, horizontal) to confirm the expression reduces sensibly.
In Physics electricity and magnetism, the candidate confuses the direction of the magnetic force on a moving charge with the direction of the magnetic field or of the velocity, using Fleming's left-hand rule and the right-hand rule interchangeably and misplacing the result by 90°.
Three vectors (v, B, F) are involved and the rules are memorised as hand gestures without the underlying cross product F = qv × B, so sign conventions and the distinction between positive and negative charge are lost.
What to do instead
Anchor everything to F = qv × B: compute the cross product componentwise or by the determinant, then multiply by the sign of q. Use the hand rule only as a check on the computed direction. Work problems with both positive and negative charges and with v anti-parallel to B (zero force) to test the rule.
In Chemistry, the candidate balances an oxidation–reduction equation by adjusting coefficients on the molecular equation without separating it into half-reactions, so charge is not conserved and the electron count does not match between oxidation and reduction.
Trial-and-error balancing works for simple combustion equations and candidates generalise it to redox, where atom balance alone is insufficient; the electron-transfer bookkeeping is not made explicit.
What to do instead
Always split into oxidation and reduction half-reactions, balance atoms other than O and H, balance O with H₂O and H with H⁺ (or OH⁻ in basic solution), then balance charge with electrons. Multiply each half-reaction so the electrons cancel, add, and verify both atom and charge balance on each side. Practise one acidic and one basic example per session.
In Chemistry (organic), the candidate identifies a reaction product by name-matching the reagent (e.g. 'KMnO₄ means oxidation') without checking the substrate's functional group, the conditions (heat, acid, dilute vs concentrated), and the regiochemistry, and so predicts the wrong product or ignores a possible side reaction.
Organic reactions are learned as a list of reagent–product pairs rather than as mechanisms; candidates do not ask what the substrate can actually do or how conditions change the outcome.
What to do instead
For each reaction, write three things before naming a product: the functional groups present, the role of the reagent (electrophile/nucleophile/oxidant), and the conditions. Draw the curly-arrow mechanism for the key step. Build a comparison table for reagents that behave differently under different conditions (dilute vs concentrated acid, cold vs hot permanganate).
In Biology, the candidate answers a genetics question by writing a Punnett square for the phenotype ratio without first establishing the parental genotypes from the described phenotypes, and reports 3:1 for a cross that is actually a test cross or involves linkage.
The 3:1 and 9:3:3:1 ratios are memorised as answers rather than as consequences of specific parental genotypes; the candidate skips the step of inferring genotypes from the information given.
What to do instead
Begin every genetics problem by listing the alleles, assigning symbols, and writing the genotype of each individual described. Only then construct the Punnett square. Check whether the question implies a test cross (crossing with the recessive homozygote) or independent assortment, and state the assumption explicitly. Verify that the ratio you report follows from the genotypes you wrote.
In Earth Science, the candidate reads a geological cross-section or a description of rock layers and determines the relative age of units by their vertical position alone, ignoring cross-cutting intrusions, faults, and unconformities that truncate the sequence.
The principle of superposition is the first and most memorable rule, and candidates apply it mechanically without considering the other relative-dating principles that override it in deformed sequences.
What to do instead
Apply the principles in a fixed order: superposition, original horizontality, cross-cutting relationships (an intrusion or fault is younger than what it cuts), inclusions (a fragment is older than its host), and unconformities. For each boundary in the section, write which principle decides it. Practise on diagrams containing at least one fault and one unconformity.