Hiroshima Entrance Science cheat sheet

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac > 0 gives two distinct real roots, = 0 gives one repeated root, < 0 gives no real roots.

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius; use it when a side and its opposite angle are both known.

Cosine rule: a² = b² + c² − 2bc·cos A; use it for SAS or SSS triangles, and note it reduces to Pythagoras when A = 90°.

Arithmetic sequence: nth term aₙ = a₁ + (n − 1)d, sum Sₙ = n(a₁ + aₙ)/2. Geometric sequence: aₙ = a₁r^(n−1), sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

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

Definite integral: ∫ₐᵇ f(x) dx equals the signed area between the curve and the x-axis; areas below the axis count as negative, so split the interval at each zero of f.

Counting: permutations nPr = n!/(n − r)! (order matters); combinations nCr = n!/(r!(n − r)!) (order does not); nCr = nC(n − r).

Expected value of a discrete variable: E(X) = Σ xᵢ·P(xᵢ); variance V(X) = E(X²) − (E(X))², and standard deviation is its square root.

Equations of uniformly accelerated motion: v = u + at; s = ut + ½at²; v² = u² + 2as - valid only when acceleration is constant.

Newton's second law: F = ma, with F the net force; weight is W = mg, distinct from mass, which is invariant with location.

Conservation of mechanical energy without friction: ½mv² + mgh = constant; with friction, the lost mechanical energy equals the work done against friction, μN·d.

Momentum p = mv is conserved in any collision or explosion; kinetic energy is conserved only in elastic collisions, and perfectly inelastic collisions share one final velocity.

Ohm's law V = IR; series resistance R = R₁ + R₂ + …; parallel resistance 1/R = 1/R₁ + 1/R₂ + …; power P = VI = I²R = V²/R.

Wave relation v = fλ; for a string or pipe, the fundamental frequency corresponds to the longest standing wave fitting the boundary conditions (nodes at fixed ends).

Refraction: n₁sin θ₁ = n₂sin θ₂; total internal reflection occurs only when light travels from a denser to a less dense medium and the angle exceeds the critical angle sin θc = n₂/n₁.

Ideal gas law PV = nRT; absolute temperature must be used, and at constant volume pressure is proportional to absolute temperature.

Amount of substance: n = m/M; for gases at standard conditions n = V/22.4 L·mol⁻¹ (0 °C, 1 atm); molarity c = n/V in litres.

Avogadro's number N_A ≈ 6.02 × 10²³ particles per mole; number of particles = n × N_A.

pH = −log₁₀[H⁺]; at 25 °C pH + pOH = 14, and neutral solution has pH 7. Strong acid fully dissociates, weak acid only partly.

Neutralisation: acid + base → salt + water; the reacting ratio follows the stoichiometric coefficients, so moles of H⁺ must equal moles of OH⁻ at the equivalence point.

Oxidation is loss of electrons (oxidation number increases); reduction is gain (oxidation number decreases); oxidising agent is itself reduced.

Periodic trends across a period (left to right): atomic radius decreases, ionisation energy increases, electronegativity increases; down a group the reverse holds.

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

Bond types: ionic between metal and non-metal with large electronegativity difference; covalent between non-metals by electron sharing; metallic by delocalised electrons.

Central dogma: DNA → mRNA (transcription, in the nucleus) → protein (translation, at ribosomes); three mRNA bases (a codon) specify one amino acid.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, driven by light energy absorbed by chlorophyll in chloroplasts; light reactions make ATP and NADPH.

Cellular respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + energy; glycolysis in the cytoplasm, then the citric acid cycle and electron transport in mitochondria.

Mendel's law of segregation: the two alleles of a gene separate during gamete formation; a cross of heterozygotes Aa × Aa gives phenotype ratio 3:1 and genotype ratio 1:2:1.

Enzymes are substrate-specific biological catalysts; activity rises with temperature up to an optimum, then falls as the enzyme denatures; pH also has an optimum.

Homeostasis maintains internal conditions near a set point by negative feedback: a deviation triggers a response that opposes the deviation.

DNA structure: double helix of two antiparallel strands; adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three).

Natural selection requires heritable variation, differential reproductive success, and change in allele frequency across generations.

Earth's layered interior: crust, mantle, outer core (liquid, generates the magnetic field by dynamo action), inner core (solid iron–nickel).

Plate tectonics: divergent boundaries create new crust (mid-ocean ridges), convergent boundaries produce subduction and volcanoes, transform boundaries produce earthquakes.

Seismic waves: P waves are longitudinal and travel through solids and liquids; S waves are transverse and cannot pass through liquids, revealing the liquid outer core.

Earthquake magnitude is logarithmic: each whole-number increase in magnitude corresponds to about 31.6 times more released energy.

Rock cycle: igneous rock forms from cooling magma or lava; sedimentary rock forms by compaction and cementation of deposited grains; metamorphic rock forms under heat and pressure.

Greenhouse effect: gases such as CO₂, CH₄ and H₂O absorb outgoing infrared radiation; increasing their concentration raises the equilibrium surface temperature.

Water cycle: evaporation, condensation, precipitation, runoff and infiltration; transpiration from plants is a major return path of water to the atmosphere.

Air pressure decreases with altitude; the troposphere holds most atmospheric mass and is where weather occurs, with temperature falling about 6.5 °C per kilometre.

Independent variable is the one deliberately changed; the dependent variable is measured; controlled variables are held constant to allow a fair comparison.

A control group receives no treatment or a placebo; without it, an observed change cannot be attributed to the treatment.

Correlation does not establish causation: a third confounding variable may drive both observed quantities.

Precision is repeatability of measurements; accuracy is closeness to the true value; random errors scatter around the true value while systematic errors shift all readings in one direction.

Significant figures: in multiplication and division the result keeps the smallest number of significant figures among the inputs; in addition and subtraction, the fewest decimal places.

For repeated measurements, report the mean and the standard deviation; a large standard deviation indicates high variability, not necessarily a wrong result.

On a graph, the slope of a straight-line fit gives the rate of change of y with respect to x; the intercept gives the value of y when x = 0.

A hypothesis must be falsifiable: state a prediction that an experiment could show to be wrong, and distinguish observation from inference.