Kumamoto Entrance Science cheat sheet

Limit of sin x / x as x→0 equals 1; limit of (1 - cos x)/x² as x→0 equals 1/2.

Derivative rules: (xⁿ)' = n xⁿ⁻¹, (sin x)' = cos x, (cos x)' = -sin x, (eˣ)' = eˣ, (ln x)' = 1/x.

Integration by parts: ∫u dv = uv - ∫v du; choose u by LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

Arithmetic sequence: aₙ = a₁ + (n-1)d; sum Sₙ = n(a₁ + aₙ)/2. Geometric: aₙ = a₁ rⁿ⁻¹; Sₙ = a₁(1 - rⁿ)/(1 - r).

Probability of independent events: P(A∩B) = P(A)P(B); conditional probability P(A|B) = P(A∩B)/P(B).

Quadratic formula: x = [-b ± √(b² - 4ac)]/(2a); discriminant D = b² - 4ac determines number of real roots (D>0 two, D=0 one, D<0 none).

Vector dot product: a·b = |a||b|cos θ; if a·b = 0 then vectors are perpendicular.

Complex number modulus: |a + bi| = √(a² + b²); conjugate of a+bi is a-bi; (a+bi)(a-bi) = a² + b².

Equations of motion (constant acceleration): v = u + at, s = ut + ½at², v² = u² + 2as.

Newton's second law: F = ma; weight W = mg; friction f = μN.

Work-energy theorem: net work W = ΔK = ½mv² - ½mu²; gravitational potential energy U = mgh.

Momentum conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ for isolated systems; impulse J = FΔt = Δp.

Ohm's law V = IR; electrical power P = VI = I²R = V²/R; resistors in series add, in parallel 1/R = Σ1/Rᵢ.

Wave equation: v = fλ; for a string, v = √(T/μ) where T is tension and μ is linear density.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; critical angle sin θc = n₂/n₁ (n₁ > n₂).

Ideal gas law: PV = nRT; pressure from kinetic theory P = (1/3)ρ⟨v²⟩; absolute temperature relates to average kinetic energy (3/2)kBT per molecule.

Mole concept: n = m/M = N/Nₐ = V/22.4 L at STP (for gases); Nₐ = 6.02×10²³ mol⁻¹.

Molarity M = moles of solute / liters of solution; dilution M₁V₁ = M₂V₂.

Ideal gas law PV = nRT; R = 8.31 J/(mol·K) or 0.0821 L·atm/(mol·K).

Acid-base neutralization: moles H⁺ = moles OH⁻; pH = -log[H⁺]; pOH = -log[OH⁻]; pH + pOH = 14 at 25°C.

Le Chatelier's principle: increasing pressure shifts equilibrium toward fewer gas moles; increasing temperature favors endothermic direction.

Oxidation numbers: O usually -2, H usually +1; sum of oxidation numbers in a neutral compound is zero.

Electrolysis: moles of substance = (It)/(nF), where F = 96500 C/mol and n is electrons per ion.

Rate law: rate = k[A]^m[B]^n; reaction order determined experimentally, not from stoichiometry.

Central dogma: DNA → RNA → protein; transcription in nucleus, translation on ribosomes.

Photosynthesis equation: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂; occurs in chloroplasts (light reactions in thylakoid, Calvin cycle in stroma).

Cellular respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ATP; glycolysis in cytoplasm, Krebs cycle in mitochondrial matrix, electron transport chain on inner membrane.

Mendel's law of segregation: alleles separate during gamete formation; law of independent assortment applies to genes on different chromosomes.

DNA base pairing: A-T, G-C; RNA has uracil instead of thymine; DNA replication is semiconservative.

Enzyme activity: increases with temperature up to optimum, then denatures; pH also has an optimum; competitive inhibitors raise Km, noncompetitive lower Vmax.

Homeostasis: negative feedback maintains internal stability (e.g., insulin lowers blood glucose, glucagon raises it).

Ecological pyramid: energy decreases ~90% at each trophic level (10% rule); producers have the most biomass/energy.

Earth's interior layers: crust, mantle, outer core (liquid), inner core (solid); seismic S-waves cannot pass through liquid outer core.

Plate tectonics: divergent boundaries create new crust (mid-ocean ridges), convergent boundaries cause subduction or mountain building, transform boundaries cause earthquakes.

Earthquake magnitude: Richter scale is logarithmic; an increase of 1.0 corresponds to ~31.6 times more energy release.

Rock cycle: igneous (from magma/lava), sedimentary (compaction/cementation of sediments), metamorphic (heat/pressure).

Weather: warm fronts bring gradual precipitation; cold fronts bring rapid storms; Coriolis effect deflects winds right in Northern Hemisphere.

Greenhouse effect: CO₂, CH₄, H₂O absorb infrared radiation; increasing greenhouse gases raise average global temperature.

Moon phases: synodic month ~29.5 days; spring tides occur at new and full moon, neap tides at first and third quarter.

Solar system: terrestrial planets (Mercury, Venus, Earth, Mars) are rocky; Jovian planets (Jupiter, Saturn, Uranus, Neptune) are gas/ice giants.

Subject-verb agreement: singular subject takes singular verb; indefinite pronouns (everyone, each, neither) are singular.

Verb tense consistency: use past perfect for an action completed before another past action (had + past participle).

Passive voice: be + past participle; use when the doer is unknown or unimportant.

Relative clauses: use 'who' for people, 'which' for things, 'that' for defining clauses (not after comma).

Conditionals: zero (present), first (will + base), second (would + base for unreal present), third (would have + past participle for unreal past).

Punctuation: comma splice is joining two independent clauses with only a comma; use semicolon or conjunction instead.

Reading strategy: identify main idea, supporting details, inference, and author's tone; scan for keywords before reading closely.

Composition structure: clear thesis statement, topic sentences per paragraph, supporting evidence, and conclusion that restates thesis.

敬語 (keigo): 尊敬語 (sonkeigo) elevates the listener's actions; 謙譲語 (kenjōgo) lowers the speaker's actions; 丁寧語 (teineigo) is polite form (です・ます).

助詞 (particles): は marks topic, が marks subject, を marks direct object, に marks direction/time, で marks location of action/means.

動詞の活用 (verb conjugation): 五段 (godan), 一段 (ichidan), 不規則 (irregular: する, 来る); て-form used for requests, ongoing actions, and sequencing.

条件形 (conditionals): と (natural consequence), ば (hypothetical), たら (if/when), なら (contextual).

受身形 (passive): 五段 verbs change -u to -areru; 一段 verbs add -rareru; する becomes される; 来る becomes 来られる.

漢字 (kanji) readings: 音読み (on'yomi) from Chinese, 訓読み (kun'yomi) native Japanese; compound words often use on'yomi.

文章読解 (reading comprehension): identify 主語 (subject), 述語 (predicate), and 接続詞 (conjunctions) to follow logical flow.

作文 (writing): use consistent 常体 (plain) or 敬体 (polite) style; avoid ambiguity by placing modifiers directly before the modified noun.