Kyoto Entrance Science cheat sheet

Limit definition of derivative: f'(a) = lim_{h→0} [f(a+h) − f(a)]/h; differentiability implies continuity but not conversely (e.g., |x| at 0).

Integration by parts: ∫ u dv = uv − ∫ v du; choose u by LIATE (Log, Inverse trig, Algebraic, Trig, Exponential).

Quadratic roots: x = [−b ± √(b²−4ac)]/(2a); discriminant D = b²−4ac gives two real roots if D>0, one if D=0, complex conjugates if D<0.

Binomial theorem: (a+b)^n = Σ_{k=0}^{n} C(n,k) a^{n−k} b^k; general term T_{k+1} = C(n,k) a^{n−k} b^k.

Arithmetic series S_n = n(a_1+a_n)/2; geometric series S_n = a_1(1−r^n)/(1−r) for r≠1; infinite sum converges iff |r|<1 to a_1/(1−r).

Vector dot product a·b = |a||b|cosθ = a_1b_1+a_2b_2+a_3b_3; perpendicular iff a·b=0. Cross product magnitude |a×b| = |a||b|sinθ gives parallelogram area.

Circle equation (x−a)²+(y−b)²=r²; tangent at (x_1,y_1) on circle is (x_1−a)(x−a)+(y_1−b)(y−b)=r².

Complex numbers: De Moivre (cosθ+i sinθ)^n = cos nθ + i sin nθ; modulus |z|=√(a²+b²), argument arg z = atan2(b,a).

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

Newton's second law: F_net = ma; on an incline, component along slope is mg sinθ, normal force is mg cosθ (no other vertical forces).

Projectile motion: horizontal x = v_0 cosθ·t; vertical y = v_0 sinθ·t − ½gt²; range R = v_0² sin2θ / g on level ground.

Work–energy theorem: W_net = ΔK = ½mv² − ½mv_0²; gravitational potential near Earth U = mgh; spring U = ½kx².

Momentum conservation: Σp_initial = Σp_final when net external force is zero; elastic collision also conserves kinetic energy; perfectly inelastic collision has common final velocity.

Circular motion: centripetal acceleration a = v²/r = rω²; centripetal force F = mv²/r directed toward center.

Simple harmonic motion: x = A sin(ωt+φ), ω = 2π/T; for a spring–mass ω = √(k/m); for a simple pendulum ω = √(g/L).

Ohm's law V = IR; resistors in series R_eq = R_1+R_2+…; in parallel 1/R_eq = 1/R_1+1/R_2+…; electric power P = VI = I²R = V²/R.

Snell's law n_1 sinθ_1 = n_2 sinθ_2; total internal reflection occurs when θ_1 exceeds critical angle sinθ_c = n_2/n_1 (n_1 > n_2).

Photoelectric effect: maximum kinetic energy K_max = hf − W, where W is the work function; threshold frequency f_0 = W/h.

Ideal gas law PV = nRT; at STP (0 °C, 1 atm) one mole occupies 22.4 L; combined gas law P_1V_1/T_1 = P_2V_2/T_2.

Mole concept: n = m/M; number of particles N = n·N_A with N_A ≈ 6.02×10²³ mol⁻¹; molarity M = mol solute / L solution.

pH = −log[H⁺]; pOH = −log[OH⁻]; pH + pOH = 14 at 25 °C; strong acid fully dissociates, weak acid uses K_a.

Le Chatelier's principle: increasing concentration of a reactant shifts equilibrium toward products; increasing pressure shifts toward fewer gas moles; catalyst does not shift equilibrium.

Oxidation numbers: O is usually −2, H is +1; sum equals charge on ion or zero for neutral molecule; oxidation is loss of electrons, reduction is gain (OIL RIG).

Electrochemistry: at anode oxidation occurs, at cathode reduction occurs; standard cell potential E°_cell = E°_cathode − E°_anode; positive E°_cell means spontaneous.

First law of thermodynamics: ΔU = Q + W (or Q − W depending on sign convention); enthalpy change ΔH = Q_p at constant pressure; exothermic ΔH < 0.

Rate law rate = k[A]^m[B]^n; reaction order is sum m+n; half-life of first-order reaction t_{1/2} = ln2/k.

Organic functional groups: alcohol –OH, aldehyde –CHO, ketone –CO–, carboxylic acid –COOH, ester –COO–; esterification of acid plus alcohol produces ester plus water.

Central dogma: DNA → RNA → protein; transcription occurs in the nucleus (eukaryotes), translation on ribosomes in the cytoplasm.

DNA structure: double helix with antiparallel strands; adenine pairs with thymine (2 H-bonds), guanine with cytosine (3 H-bonds); in RNA, uracil replaces thymine.

Mendel's laws: law of segregation (alleles separate during meiosis) and law of independent assortment (genes on different chromosomes assort independently).

Cellular respiration: glycolysis (cytoplasm) yields 2 ATP and 2 pyruvate; Krebs cycle and oxidative phosphorylation in mitochondria yield most ATP (~30–32 total per glucose).

Photosynthesis: light reactions in thylakoid membranes produce ATP and NADPH and release O₂ from water; Calvin cycle in stroma fixes CO₂ using RuBisCO.

Mitosis produces two genetically identical diploid cells; meiosis produces four genetically distinct haploid gametes with crossing over in prophase I.

Enzyme kinetics: enzymes lower activation energy; Michaelis–Menten v = V_max[S]/(K_m+[S]); K_m is substrate concentration at half V_max.

Human circulatory system: blood flows heart → arteries → arterioles → capillaries → venules → veins → heart; left ventricle pumps oxygenated blood to the body via the aorta.

Homeostasis: negative feedback maintains internal stability (e.g., insulin lowers blood glucose, glucagon raises it); osmoregulation via kidneys and ADH.

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

Plate tectonics: divergent boundaries create new crust (mid-ocean ridges), convergent boundaries produce subduction zones and mountain belts, transform boundaries cause strike-slip faults.

Earthquake magnitude: moment magnitude M_w = (2/3) log₁₀ M_0 − 10.7 (M_0 in N·m); each unit increase corresponds to about 32 times more energy release.

Rock cycle: igneous rocks form from cooling magma/lava; sedimentary rocks from compaction and cementation of sediments; metamorphic rocks from heat and pressure.

Atmospheric composition: about 78% N₂, 21% O₂, 0.93% Ar, 0.04% CO₂; ozone layer in the stratosphere absorbs ultraviolet radiation.

Coriolis effect: deflecting force due to Earth's rotation; deflects moving air to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.

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

Geologic time: relative dating uses superposition (older layers below younger) and cross-cutting relationships; radiometric dating uses half-life (e.g., ¹⁴C half-life ≈ 5,730 years).

Water cycle: evaporation, condensation, precipitation, infiltration, runoff; groundwater flows from high hydraulic head to low, governed by Darcy's law Q = K·A·(Δh/L).

Passive voice: be + past participle; use when the agent is unknown or unimportant, e.g., 'The experiment was conducted at 25 °C.'

Relative clauses: defining clauses give essential information and take no commas; non-defining clauses add extra information and are set off by commas, e.g., 'The sample, which was heated, changed color.'

Conditionals: zero (present real), first (future real: if + present, will + base), second (unreal present: if + past, would + base), third (unreal past: if + past perfect, would have + past participle).

Subject–verb agreement: singular subject takes singular verb; ignore prepositional phrases, e.g., 'The number of students is increasing' vs. 'A number of students are present.'

Reported speech: shift tense back one step (present → past, past → past perfect); change pronouns and time expressions (now → then, today → that day).

Common academic collocations: 'conduct research', 'draw a conclusion', 'pose a question', 'play a role', 'account for' - prepositions are fixed and must be memorized.

Linking words: 'however' contrasts, 'therefore' shows result, 'moreover' adds, 'nevertheless' concedes; each should be followed by a comma when starting a sentence.

Reading inference: the answer must be supported by the text, not by outside knowledge; eliminate options that are true in general but not stated in the passage.