Kagawa Entrance Science cheat sheet

Quadratic formula: roots of ax²+bx+c=0 are x = (−b ± √(b²−4ac))/(2a); discriminant D=b²−4ac gives D>0 two real roots, D=0 double root, D<0 complex conjugates.

Trig identities: sin²θ+cos²θ=1, tanθ=sinθ/cosθ; addition formulas sin(α±β)=sinαcosβ±cosαsinβ, cos(α±β)=cosαcosβ∓sinαsinβ.

Derivative rules: (xⁿ)′=nxⁿ⁻¹, (sin x)′=cos x, (cos x)′=−sin x, (eˣ)′=eˣ, (ln x)′=1/x; product rule (fg)′=f′g+fg′.

Integration: ∫xⁿdx=xⁿ⁺¹/(n+1)+C (n≠−1), ∫(1/x)dx=ln|x|+C; area between curves = ∫|f−g|dx over the interval.

Arithmetic/geometric sequences: arithmetic aₙ=a₁+(n−1)d, Sₙ=n(a₁+aₙ)/2; geometric aₙ=a₁rⁿ⁻¹, Sₙ=a₁(1−rⁿ)/(1−r).

Vectors: dot product a·b=|a||b|cosθ; perpendicular ⇔ a·b=0; component form gives a·b=a₁b₁+a₂b₂.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); independent ⇔ P(A∩B)=P(A)P(B).

Complex numbers: i²=−1; modulus |a+bi|=√(a²+b²); De Moivre (cosθ+isinθ)ⁿ=cos nθ+isin nθ.

Number theory: Euclidean algorithm for GCD; integers a,b coprime ⇔ ∃ integers x,y with ax+by=1.

Kinematics: v=v₀+at, x=v₀t+½at², v²=v₀²+2ax for constant acceleration.

Newton's second law F=ma; friction f=μN; on an incline, component along slope = mg sinθ.

Energy and momentum: KE=½mv², PE=mgh; momentum p=mv conserved in isolated systems (elastic: KE conserved too).

Circular motion: centripetal acceleration a=v²/r; period T=2πr/v.

Waves: v=fλ; for a string v=√(T/μ); refraction obeys Snell's law n₁sinθ₁=n₂sinθ₂.

Electricity: Ohm's law V=IR; power P=VI=I²R; series resistors add, parallel 1/R=1/R₁+1/R₂.

Thermodynamics: Q=mcΔT; ideal gas PV=nRT; first law ΔU=Q−W.

Modern physics: photon energy E=hf=hc/λ; photoelectric threshold hf₀=W (work function).

Mole concept: n=m/M; one mole contains 6.02×10²³ particles; molar volume of gas at STP ≈ 22.4 L.

Concentration: molarity c=n/V (mol/L); dilution c₁V₁=c₂V₂.

Acid-base: pH=−log[H⁺]; strong acid/strong base neutralization; Kw=[H⁺][OH⁻]=1.0×10⁻¹⁴ at 25°C.

Redox: oxidation = loss of electrons (increase in oxidation number); reduction = gain; balance by conserving electrons.

Periodic trends: atomic radius decreases across a period, increases down a group; ionization energy generally increases across a period.

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

Organic functional groups: alcohol −OH, aldehyde −CHO, carboxylic acid −COOH, ester −COO−; esterification is acid-catalyzed with water elimination.

Electrochemistry: at the anode oxidation occurs; at the cathode reduction; standard cell potential E°cell=E°cathode−E°anode.

Central dogma: DNA→RNA (transcription)→protein (translation); codons are triplet base sequences on mRNA.

Cell division: mitosis produces identical diploid cells; meiosis halves chromosome number and generates genetic variation via crossing over.

Mendel's laws: segregation of alleles and independent assortment; test cross reveals genotype.

Photosynthesis: 6CO₂+6H₂O→C₆H₁₂O₆+6O₂ in chloroplasts; light reactions make ATP/NADPH, Calvin cycle fixes CO₂.

Cellular respiration: glycolysis→citric acid cycle→electron transport; net ATP yield far higher with O₂ than anaerobic fermentation.

Enzymes lower activation energy; activity depends on temperature and pH, denaturing at extremes.

Homeostasis: negative feedback maintains internal conditions (e.g., blood glucose via insulin and glucagon).

Ecosystems: energy flows one way and decreases ~90% per trophic level; matter cycles (carbon, nitrogen).

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

Plate tectonics: divergent boundaries create new crust, convergent boundaries produce subduction and volcanoes, transform boundaries cause earthquakes.

Rock types: igneous (from magma), sedimentary (deposition/compaction), metamorphic (heat and pressure).

Earthquake magnitude uses the logarithmic Richter scale; epicenter is the surface point above the focus.

Weather: air moves from high to low pressure; Coriolis effect deflects winds; fronts mark air-mass boundaries.

Rock cycle links igneous, sedimentary, and metamorphic rocks through melting, weathering, and metamorphism.

Solar system: Kepler's third law T²∝a³; inner planets are rocky, outer planets gas/ice giants.

Geologic time: relative dating uses superposition (older layers below); radiometric dating gives absolute ages.

Subject-verb agreement: singular subjects take singular verbs; indefinite pronouns like 'everyone' are singular.

Tense consistency: keep tenses coherent within a sentence; use past perfect for an action completed before another past action.

Relative clauses: 'who' for people, 'which' for things, 'that' for restrictive clauses; commas mark non-restrictive clauses.

Conditionals: first (if + present, will), second (if + past, would), third (if + past perfect, would have + past participle).

Reading strategy: identify main idea, supporting details, and the author's purpose; distinguish fact from opinion.

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

Comparatives: use '-er/more' for two items, '-est/most' for three or more; avoid double comparatives.

Composition: clear thesis, topic sentences, transitions (however, therefore), and a concluding statement.

Particles: は marks topic, が marks subject, を marks direct object, に marks direction/time, で marks location of action or means.

Verb forms: ます (polite), dictionary form (plain), て-form for requests/sequence; た-form marks past.

Transitive vs intransitive pairs: 開ける/開く, 始める/始まる; transitive takes を, intransitive takes が.

Keigo: 尊敬語 (respectful, e.g., いらっしゃる), 謙譲語 (humble, e.g., 伺う), 丁寧語 (polite です/ます).

Kanji reading: distinguish 音読み (Chinese-derived) from 訓読み (native); context determines correct reading.

Conditional forms: と (natural consequence), ば (hypothetical), たら (if/when), なら (given that).

Passive and causative: られる for passive, させる for causative; combine as させられる (causative-passive).

Reading comprehension: track subject omission (frequent in Japanese) and identify referents of こ/そ/あ pronouns.

Identify the independent variable (manipulated), dependent variable (measured), and controlled variables in an experiment.

Distinguish correlation from causation; a controlled experiment with a control group is needed to establish causation.

Significant figures: results should not exceed the precision of the least precise measurement.

Graphs: slope represents rate of change; area under a curve can represent accumulated quantity; linear fit implies constant rate.

Error types: random error reduces with repeated trials; systematic error shifts all readings consistently.

Hypothesis testing: a hypothesis must be falsifiable; predictions are derived and compared with data.

Unit analysis: check dimensional consistency; cancel units to verify a formula's correctness.

Data tables: compute means and note outliers; report uncertainty (e.g., ± range or standard deviation).