Kyoto Entrance Science flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Derivative definition

The derivative at a point a is the limit as h approaches 0 of [f(a+h) − f(a)]/h. It gives the instantaneous rate of change and the slope of the tangent line at that point.

Discriminant

For a quadratic ax²+bx+c=0, the discriminant D=b²−4ac. If D>0 there are two distinct real roots; if D=0 one repeated real root; if D<0 two complex conjugate roots.

Integration by parts

The formula ∫u dv = uv − ∫v du. Choose u by the LIATE order (logarithmic, inverse trig, algebraic, trig, exponential) to make the remaining integral simpler.

Dot product

For vectors a and b, a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃. It is zero exactly when the vectors are perpendicular, and it equals the product of magnitudes times the cosine of the angle between them.

De Moivre's theorem

For a complex number in polar form, (cosθ + i sinθ)^n = cos nθ + i sin nθ. This makes powers and roots of complex numbers easy to compute.

Inclined plane forces

On a frictionless incline of angle θ, the component of gravity along the slope is mg sinθ and the normal force is mg cosθ. These determine acceleration a = g sinθ down the slope.

Projectile range

For a projectile launched at speed v₀ and angle θ on level ground, the horizontal range is R = v₀² sin2θ / g. Maximum range occurs at θ = 45°.

Centripetal force

An object moving in a circle of radius r at speed v has centripetal acceleration v²/r directed toward the center. The required net force is F = mv²/r.

Simple harmonic motion

Motion described by x = A sin(ωt+φ) with angular frequency ω = 2π/T. For a spring–mass system ω = √(k/m); for a simple pendulum ω = √(g/L).

Snell's law

When light crosses a boundary between media, n₁ sinθ₁ = n₂ sinθ₂, where n is the refractive index and θ is measured from the normal. It governs refraction and total internal reflection.

Photoelectric effect

Light of frequency f ejects electrons from a metal only if f exceeds the threshold f₀ = W/h. The maximum kinetic energy is K_max = hf − W, where W is the work function.

Ideal gas law

PV = nRT relates pressure, volume, amount in moles, and absolute temperature. At STP (0 °C, 1 atm) one mole of ideal gas occupies about 22.4 L.

pH definition

pH = −log[H⁺], where [H⁺] is hydrogen ion concentration in mol/L. At 25 °C, pH + pOH = 14; a pH below 7 is acidic, above 7 is basic.

Le Chatelier's principle

When a system at equilibrium is disturbed by a change in concentration, pressure, or temperature, it shifts in the direction that partially counteracts the change. A catalyst speeds attainment but does not shift equilibrium.

Electrochemical cell potentials

For a galvanic cell, E°_cell = E°_cathode − E°_anode using standard reduction potentials. Oxidation occurs at the anode, reduction at the cathode; a positive E°_cell indicates a spontaneous reaction.

First-order half-life

For a first-order reaction, concentration decays as [A] = [A]₀e^{−kt}. The half-life is t₁/₂ = ln2/k ≈ 0.693/k, independent of initial concentration.

DNA base pairing

In double-stranded DNA, adenine pairs with thymine via two hydrogen bonds, and guanine pairs with cytosine via three. The two strands are antiparallel and run in opposite directions.

Mendel's law of segregation

During meiosis, the two alleles of a gene separate so that each gamete receives only one allele. This explains the 3:1 phenotypic ratio in a monohybrid cross of heterozygotes.

Cellular respiration ATP yield

Complete oxidation of one glucose molecule yields about 30–32 ATP: 2 from glycolysis, 2 from the Krebs cycle, and the rest from oxidative phosphorylation in the mitochondria.

Enzyme Michaelis–Menten equation

The rate is v = V_max[S]/(K_m+[S]). K_m is the substrate concentration at half the maximum rate and indicates enzyme affinity: a lower K_m means higher affinity.

Mitosis versus meiosis

Mitosis produces two genetically identical diploid cells for growth and repair. Meiosis produces four genetically distinct haploid gametes, with crossing over and independent assortment increasing variation.

Earth's seismic waves

P-waves are longitudinal and travel through solids and liquids; S-waves are transverse and cannot travel through liquids. The S-wave shadow zone reveals Earth's liquid outer core.

Plate boundary types

Divergent boundaries create new crust at mid-ocean ridges; convergent boundaries produce subduction zones and mountains; transform boundaries cause strike-slip earthquakes. Each type has distinct geological features.

Rock cycle

Igneous rocks form from cooling magma or lava; sedimentary rocks form by compaction and cementation of sediments; metamorphic rocks form from heat and pressure. Any type can transform into another.

Coriolis effect

Earth's rotation deflects moving air and water: to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. The deflection is zero at the equator and strongest at the poles.

Radiometric dating

Radioactive isotopes decay with a fixed half-life, so the ratio of parent to daughter isotopes gives the age of a sample. Carbon-14 (half-life ≈ 5,730 years) dates organic remains up to about 50,000 years.

Passive voice formation

The passive is formed with the correct tense of 'be' plus the past participle, e.g., 'The solution was heated.' Use it when the agent is unknown or unimportant, or to focus on the action.

Third conditional

Used for unreal past situations: if + past perfect, would have + past participle. Example: 'If the sample had been heated, it would have decomposed.' It describes a counterfactual outcome.

Non-defining relative clause

A clause that adds extra, non-essential information about a noun and is set off by commas, e.g., 'The sample, which was heated, changed color.' It cannot use 'that' and cannot be omitted without losing meaning.

Subject–verb agreement

The verb agrees with the grammatical subject, ignoring prepositional phrases. 'The number of students is increasing' (singular subject) differs from 'A number of students are present' (plural).

Reported speech tense shift

When reporting speech, shift the tense one step back: present becomes past, past becomes past perfect. Pronouns and time expressions also change, e.g., 'now' becomes 'then'.

Linking words

Conjunctions that show logical relationships: 'however' contrasts, 'therefore' shows result, 'moreover' adds information, 'nevertheless' concedes. When starting a sentence, they are followed by a comma.