Chiba Entrance Science cheat sheet

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a; the discriminant D = b² − 4ac gives two real roots if D > 0, one if D = 0, none (real) if D < 0.

Arithmetic and geometric sequences: aₙ = a + (n − 1)d with sum Sₙ = n(a + l)/2; geometric aₙ = ar^(n−1) with Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1.

Sine and cosine rules: a/sin A = b/sin B = c/sin C = 2R; a² = b² + c² − 2bc·cos A. Triangle area = ½ab·sin C.

Differentiation: (xⁿ)′ = nx^(n−1), (sin x)′ = cos x, (cos x)′ = −sin x, (eˣ)′ = eˣ, (ln x)′ = 1/x. Product rule (fg)′ = f′g + fg′; quotient rule (f/g)′ = (f′g − fg′)/g².

Integration: ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C. Definite integral equals the signed area under the curve between the limits.

Vectors in the plane: dot product a·b = |a||b|cos θ = a₁b₁ + a₂b₂; perpendicular when a·b = 0. Position vector of the point dividing AB internally in ratio m:n is (na + mb)/(m + n).

Probability and counting: P(A∪B) = P(A) + P(B) − P(A∩B); permutations nPr = n!/(n−r)!; combinations nCr = n!/(r!(n−r)!).

Complex numbers: i² = −1; modulus |a + bi| = √(a² + b²); conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b².

Limits and the derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; a function is continuous at x = a when lim(x→a) f(x) = f(a).

Kinematics under constant acceleration: v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as. These apply only while acceleration is constant.

Newton's second law: F = ma; net force causes acceleration in the same direction. Friction limit: f_max = μN, where N is the normal force.

Work, energy and power: W = Fs·cos θ, kinetic energy = ½mv², gravitational potential energy near Earth = mgh, power P = W/t = Fv.

Momentum and impulse: p = mv; impulse = FΔt = Δp. In an isolated system total momentum is conserved in every collision; kinetic energy is conserved only in elastic collisions.

Circular motion: centripetal acceleration a = v²/r = rω²; centripetal force F = mv²/r. Period T = 2πr/v.

Waves: v = fλ; for a string fixed at both ends the fundamental frequency is f₁ = v/(2L), and harmonics are integer multiples. Sound speed in air ≈ 340 m/s at room temperature.

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

Refraction and lenses: Snell's law n₁sin θ₁ = n₂sin θ₂; total internal reflection occurs when the angle exceeds the critical angle sin θc = n₂/n₁ (n₁ > n₂). Thin lens: 1/f = 1/a + 1/b.

Thermodynamics and gas law: ideal gas PV = nRT; first law ΔU = Q − W (work done by the gas). Absolute temperature must be used in gas calculations.

Amount of substance: n = m/M = N/N_A = V/22.4 L at standard conditions (0 °C, 1 atm) for gases. Avogadro's number N_A ≈ 6.02 × 10²³ mol⁻¹.

Molar concentration: c = n/V (mol/L). Dilution keeps moles constant: c₁V₁ = c₂V₂.

Acid–base neutralization: for a monoprotic acid and base, c_a V_a = c_b V_b at the equivalence point; pH = −log₁₀[H⁺], and at 25 °C pH + pOH = 14.

Oxidation numbers change in redox reactions: oxidation is loss of electrons (increase in oxidation number), reduction is gain. Balance by conserving atoms and charge.

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

Gas laws: Boyle PV = constant at fixed T; Charles V/T = constant at fixed P; combined PV/T = constant; ideal gas PV = nRT with T in kelvin.

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

Organic functional groups: alcohols –OH, aldehydes –CHO, carboxylic acids –COOH, esters –COO–. Esterification of a carboxylic acid with an alcohol is catalyzed by acid.

Solubility rules and precipitation: most nitrates and alkali-metal salts are soluble; AgCl, BaSO₄ and CaCO₃ are sparingly soluble and form precipitates in double replacement reactions.

Cell theory: all organisms are made of cells, the cell is the basic unit of life, and new cells arise from existing cells by division.

Prokaryotic cells lack a membrane-bound nucleus and organelles; eukaryotic cells have a nucleus, mitochondria, and (in plants) chloroplasts and a cellulose cell wall.

Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, occurring in chloroplasts; light reactions make ATP and NADPH, the Calvin cycle fixes CO₂ into sugar.

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

DNA structure: a double helix of two antiparallel strands; adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three). Replication is semi-conservative.

Transcription and translation: DNA is transcribed to mRNA in the nucleus, then ribosomes translate mRNA codons into amino acids using tRNA; three bases code for one amino acid.

Mendelian genetics: a monohybrid cross of heterozygotes gives a 3:1 phenotype ratio; a test cross reveals an unknown genotype; sex-linked traits on the X chromosome appear more often in males.

Homeostasis and the nervous system: negative feedback keeps internal conditions stable; a neuron transmits an action potential, and synaptic transmission uses chemical neurotransmitters across a gap.

Ecology: energy flows through food chains with roughly 10% transferred per trophic level; decomposers recycle nutrients; photosynthesis and respiration drive the carbon cycle.

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

Plate tectonics: divergent boundaries create new crust at mid-ocean ridges; convergent boundaries produce subduction zones, trenches, and volcanic arcs; transform boundaries cause strike-slip earthquakes.

Earthquake magnitude is logarithmic (Richter scale): each whole-number increase represents about 31.6 times more energy release.

Rock types: igneous rocks form from cooling magma or lava, sedimentary rocks from compacted deposits, metamorphic rocks from heat and pressure. The rock cycle links all three.

Earth's rotation on its axis (about 24 hours) causes day and night; revolution around the Sun (about 365.25 days) with a 23.4° axial tilt causes the seasons.

Moon phases arise from the changing angle between Earth, Moon and Sun; a solar eclipse requires the Moon between Earth and Sun at new moon.

Weather: air moves from high to low pressure; low-pressure centers bring rising air and precipitation, high-pressure centers bring descending air and clear skies.

The greenhouse effect: atmospheric CO₂, water vapor and methane absorb outgoing infrared radiation; increasing CO₂ raises the average surface temperature.

The water cycle: evaporation, condensation, precipitation, and runoff; groundwater recharges slowly compared with surface flow, so aquifer depletion is a long-term risk.

Subject–verb agreement: a singular subject takes a singular verb; intervening phrases do not change it ("The box of tools is heavy").

Tense consistency: keep a passage in one tense unless a time shift is clearly signaled by an adverb or clause.

Pronoun reference must be unambiguous: a pronoun agrees with its antecedent in number and gender, and every "it" or "they" must point to exactly one noun.

Dangling and misplaced modifiers: an introductory participial phrase must describe the subject of the main clause ("Walking home, she saw…" not "Walking home, the rain…").

Comma rules: use commas for items in a series, after introductory elements, and around non-restrictive clauses; do not join two independent clauses with only a comma (comma splice).

Semicolons join two independent clauses or separate complex list items; colons introduce an explanation, list, or quotation after a complete clause.

Reading strategy: identify the main idea, then locate supporting details; for inference questions, choose the answer that is supported by the text rather than merely plausible.

Vocabulary in context: use surrounding sentences and contrast or cause–effect signal words (however, therefore, although) to infer an unfamiliar word's meaning.

Parallel structure: items joined by and, or, or a series must share grammatical form ("reading, writing, and speaking" not "reading, writing, and to speak").