Kumamoto Entrance Science flashcards

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

Limit sin x / x

As x approaches 0, sin x / x approaches 1. This fundamental limit is used to derive the derivative of sine and to evaluate related trigonometric limits.

Derivative of cos x

The derivative of cos x is -sin x. This follows from the limit definition and is essential for differentiating trigonometric functions.

Quadratic formula

For ax² + bx + c = 0, x = [-b ± √(b² - 4ac)]/(2a). The discriminant b² - 4ac determines whether roots are real and distinct, repeated, or complex.

Dot product of vectors

a·b = |a||b|cos θ, where θ is the angle between vectors. If a·b = 0, the vectors are perpendicular. It yields a scalar, not a vector.

Geometric series sum

For a geometric sequence with first term a and ratio r (|r| < 1), the infinite sum is a/(1 - r). For finite n terms, Sₙ = a(1 - rⁿ)/(1 - r).

Conditional probability

P(A|B) = P(A∩B)/P(B), provided P(B) > 0. It gives the probability of event A given that event B has occurred.

Newton's second law

The net force on an object equals its mass times acceleration: F = ma. This law relates force, mass, and acceleration in classical mechanics.

Kinematic equation v² = u² + 2as

For constant acceleration, final velocity squared equals initial velocity squared plus twice acceleration times displacement. Useful when time is unknown.

Ohm's law

Voltage V equals current I times resistance R: V = IR. It applies to ohmic conductors at constant temperature.

Snell's law

n₁ sin θ₁ = n₂ sin θ₂ describes refraction at an interface, where n is refractive index and θ is angle from the normal. It predicts bending of light.

Momentum conservation

In an isolated system with no external forces, total momentum before collision equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.

Ideal gas law

PV = nRT relates pressure, volume, moles, and absolute temperature. R is the gas constant, approximately 8.31 J/(mol·K).

Mole concept

One mole contains 6.02×10²³ particles. Moles n = mass / molar mass. At STP, one mole of ideal gas occupies 22.4 L.

pH definition

pH = -log[H⁺], where [H⁺] is hydrogen ion concentration in mol/L. A pH below 7 is acidic, 7 is neutral, above 7 is basic at 25°C.

Le Chatelier's principle

When a system at equilibrium is disturbed, it shifts to counteract the change. Increasing pressure favors the side with fewer gas moles; heating favors endothermic direction.

Molarity

Molarity M = moles of solute / liters of solution. It expresses concentration and is used in dilution calculations: M₁V₁ = M₂V₂.

Oxidation number rules

Oxygen usually has oxidation number -2, hydrogen +1. The sum of oxidation numbers in a neutral compound is zero; in a polyatomic ion, it equals the ion charge.

Faraday's law of electrolysis

The amount of substance deposited or dissolved is proportional to the quantity of electricity passed: moles = It/(nF), where F = 96500 C/mol and n is electrons per ion.

Central dogma

Genetic information flows from DNA to RNA to protein. Transcription copies DNA into mRNA in the nucleus; translation builds protein on ribosomes in the cytoplasm.

Photosynthesis equation

6CO₂ + 6H₂O + light energy → C₆H₁₂O₆ + 6O₂. Light reactions occur in thylakoid membranes; the Calvin cycle fixes carbon in the stroma.

Cellular respiration

C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ATP. Glycolysis occurs in cytoplasm, Krebs cycle in mitochondrial matrix, and electron transport chain on inner membrane.

Mendel's law of segregation

During gamete formation, the two alleles for a gene separate so each gamete receives only one allele. This explains 3:1 phenotypic ratios in monohybrid crosses.

Enzyme competitive inhibition

A competitive inhibitor binds the active site, increasing apparent Km while maximum velocity Vmax remains unchanged. Noncompetitive inhibitors lower Vmax without affecting Km.

DNA base pairing

Adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three hydrogen bonds). In RNA, uracil replaces thymine.

Seismic waves

P-waves are longitudinal and travel through solids and liquids. S-waves are transverse and cannot pass through liquids, revealing Earth's liquid outer core.

Plate boundaries

Divergent boundaries create new crust at mid-ocean ridges. Convergent boundaries cause subduction or mountain building. Transform boundaries produce earthquakes as plates slide past.

Richter scale

A logarithmic scale for earthquake magnitude. An increase of 1.0 corresponds to about 31.6 times more energy release and 10 times greater amplitude.

Rock cycle

Igneous rocks form from cooling magma or lava. Sedimentary rocks form by compaction and cementation of sediments. Metamorphic rocks form under heat and pressure.

Moon phases and tides

Spring tides occur at new and full moons when Sun and Moon align. Neap tides occur at first and third quarters when they are at right angles.

Subject-verb agreement

A singular subject takes a singular verb; a plural subject takes a plural verb. Indefinite pronouns like everyone, each, and neither are singular.

Third conditional

Used for unreal past situations: if + past perfect, would have + past participle. Example: If I had studied, I would have passed.

Comma splice

A writing error joining two independent clauses with only a comma. Correct by using a semicolon, a conjunction, or separate sentences.

敬語 (keigo) types

尊敬語 elevates the listener's actions; 謙譲語 lowers the speaker's actions; 丁寧語 is polite speech using です and ます. Each has specific verb forms.

Japanese particles は and が

は marks the topic of a sentence, while が marks the grammatical subject, often introducing new information or emphasis. Choosing correctly affects nuance.

Japanese conditional forms

と expresses natural consequence; ば is hypothetical; たら means if/when after completion; なら responds to context. Each has different nuance and usage.