Japan 高校（専門学科）1〜3年 Science and Mathematics (specialist)

Chapters: 7

1. Advanced Mathematics I

Numbers and expressions · Figures and measurement · Quadratic functions · Exponential and logarithmic functions · Data analysis · Counting and probability

Exploring Algebraic Identities – An identity is an equation that is true for every value of the letters. Pictures prove them: a square of side (a + b) splits into a², ab, ab and b², so (a + b)² = a² + 2ab + b². In the same way we get (a − b)², a² − b² = (a + b)(a − b), (x + a)(x + b), (a + b + c)² and (a + b)³. Read backwards, identities help us factorise, calculate fast and simplify rational expressions.
Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.
Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.

2. Advanced Mathematics II

Various expressions · Sequences · Trigonometric functions and the complex plane · Figures and equations · Limits · Differentiation · Integration · Statistical inference

Roots of Polynomials and Polynomial Identities – If a polynomial's roots are known, its coefficients are fixed, and the other way round. For ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a (Vieta's formulas). These let you find expressions in the roots without solving, and build new equations whose roots are changed (transformed roots) by a substitution. A polynomial identity is an equation true for every value of the variable; we prove it by expanding or factorising one side until it equals the other.
Sequences and Series: AP, GP, AM and GM – A sequence is a list of numbers in a definite order: a₁, a₂, a₃, … A series is what we get when we add the terms. In an arithmetic progression (AP) we add the same number d each time; aₙ = a + (n − 1)d and Sₙ = n/2[2a + (n − 1)d]. The arithmetic mean (AM) of a and b is (a + b)/2; n AMs between a and b use d = (b − a)/(n + 1). In a geometric progression (GP) we multiply by the same number r each time; aₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If |r| < 1 the terms shrink and the infinite sum is S∞ = a/(1 − r). The geometric mean (GM) of two positive numbers a and b is √(ab), and n GMs between them use r = (b/a)^(1/(n+1)). For positive a and b, AM ≥ GM, with equality only when a = b.
Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
Statistical Inference – Statistical inference means using a sample to say something about a whole population. A number that describes the population (like the true proportion p or the mean μ) is a parameter. A number worked out from a sample (like p̂ or x̄) is a statistic, and we use it as a point estimate. Different random samples give different answers: this is sampling variability. If we took many samples, their statistics would form the sampling distribution, centred on the true value, with spread called the standard error: SE = √(p(1−p)/n) for a proportion and σ/√n for a mean. Bigger samples give smaller spread. A 95% confidence interval is estimate ± 1.96 × SE; about 95 of every 100 such intervals catch the true value. Simulation helps us check whether a claimed model fits the data. Good inference needs random sampling, and an association in data does not prove cause.

3. Special Topics in Advanced Mathematics

Vectors · Matrices and applications · Discrete graphs · Mathematics in daily life and society

Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
Graph Theory: Dots, Lines and Networks – A graph is a set of vertices (dots) joined by edges (lines). The degree of a vertex is how many edges touch it, and the sum of all degrees is twice the number of edges. An Euler trail uses every edge once and exists only when 0 or 2 vertices have odd degree. A tree is a connected graph with no cycles and n − 1 edges. Weighted graphs model roads and networks; Kruskal’s and Prim’s algorithms find a minimum spanning tree.
Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.

4. Advanced Physics

Force and motion · Waves · Electricity and magnetism · Atoms

Force and Laws of Motion – A force is a push or a pull. Balanced forces (net force zero) do not change motion; an unbalanced force changes speed or direction. Friction opposes sliding. First law: a body keeps its state of rest or uniform motion unless an unbalanced force acts (inertia; heavier bodies have more inertia). Momentum p = mv. Second law: F = ma (rate of change of momentum), 1 N = 1 kg m/s². Third law: forces come in equal and opposite pairs acting on two different bodies. For a system with no outside force, internal forces cancel and total momentum is conserved.
Progressive Waves: Types, Speed and Equation – A wave carries energy from place to place without carrying the matter along. In a transverse wave the particles move at right angles to the wave; in a longitudinal wave they move along it. Speed v = fλ. On a string v = √(T/μ); for sound in a gas v = √(γP/ρ). A wave moving along +x is y = A sin(kx − ωt), with k = 2π/λ and ω = 2πf.
Electricity and Magnetism – Electric charge and magnets are two sides of one idea. Rubbed objects hold static charge; magnets have poles and a field; a current makes a magnetic field; and a changing magnetic field makes a current (electromagnetic induction). Motors, generators and transformers all use these four steps.
Towards Bohr's Model: Light, Photons, Spectra and Bohr's Atom – Light is an electromagnetic wave: c = νλ, and wavenumber ν̄ = 1/λ. But some facts need particles: Planck said energy comes in packets (quanta) E = hν. Einstein used photons to explain the photoelectric effect: an electron comes out only if hν is above the work function W₀ = hν₀, and its kinetic energy is hν − hν₀. Atoms give line spectra, which means electron energies are fixed. Bohr put the electron on fixed orbits with angular momentum nh/2π; energy Eₙ = −2.18 × 10⁻¹⁸ Z²/n² J, radius rₙ = 52.9 n²/Z pm. A jump between orbits gives a photon, and 1/λ = R_H(1/n₁² − 1/n₂²) gives the Lyman, Balmer, Paschen, Brackett and Pfund series. Bohr works only for one-electron species.

5. Advanced Chemistry

Chemistry and daily life · Composition of substances · Chemical change and its uses · States of matter and chemical equilibrium · Properties of inorganic substances · Properties of organic compounds · Role of chemistry

Chemistry in Everyday Life: Medicines, Food, Cleaning and Soil – Chemistry is at work in your home every day. Antacids are weak bases that calm extra stomach acid. Medicines are grouped by what they do: painkillers, antibiotics, antiseptics, antihistamines, tranquilisers. Food chemicals keep food safe and tasty: preservatives, antioxidants and sweeteners. Soap cleans because one end of its molecule loves water and the other loves oil. Farmers add lime to acid soil and fertilisers to feed crops. Some metal ions are needed by the body, others are toxic, which is one reason we recycle.
Chemical Change and Energy – In a chemical change atoms swap partners and new substances form. Five ideas explain most of it: acids and bases (H+ meets OH- and makes water), oxidation and reduction (electrons move), heat (exothermic gives heat out, endothermic takes heat in), rate and equilibrium (how fast, and where it settles), and nuclear energy (a nucleus changes, and far more energy comes out).
Chemical Equilibrium: Kc, Kp, Q and Gibbs Energy – In a closed container a reversible reaction goes both ways. After some time the forward and backward rates become equal, so amounts stop changing, but the reaction does not stop. This is dynamic equilibrium. At equilibrium the ratio of products to reactants (each raised to its coefficient) is a fixed number, the equilibrium constant K. Kc uses concentrations, Kp uses partial pressures, and Kp = Kc(RT)^Δn. Pure solids and liquids are left out of K. The reaction quotient Q tells the direction: Q < K goes forward, Q > K goes backward, Q = K is equilibrium. K and Gibbs energy are linked: ΔG = ΔG° + RT ln Q and ΔG° = −RT ln K.
s- and p-Block Elements: Configuration and Trends – s-block = groups 1 and 2 (outer configuration ns¹ or ns²); p-block = groups 13 to 18 (ns² np¹⁻⁶). Down a group, atoms get bigger and ionisation enthalpy falls; across a period, atoms get smaller and ionisation enthalpy rises overall. Small ions have large hydration enthalpy. The first element of each group behaves differently (small size, high charge density, no d orbitals) and often resembles the element diagonally below it.
Carbon and Its Compounds: Bonding, Hydrocarbons and Naming – Carbon has 4 outer electrons, so it shares electrons (covalent bonds) instead of gaining or losing them. Because it bonds to itself (catenation) and always makes 4 bonds (tetravalency), it forms millions of compounds: chains, branches and rings, saturated or unsaturated. Compounds with the same functional group form a homologous series that differs by –CH₂–, and IUPAC names are built from the number of carbons + a suffix or prefix for the functional group.
Chemistry and Society – Chemistry studies how atoms join and rearrange. By choosing which atoms to join and how, chemists make new materials, medicines, fertilisers and fuels. Atoms are never lost in a reaction, so every product has a cost and a waste that we must plan for.

6. Advanced Biology

Features and evolution of organisms · Life phenomena and substances · Gene expression and development · Responses of organisms to environment · Ecology and environment

Evolution: Origin of Life, Mechanisms and Human Evolution – Life began on the early Earth from simple chemicals: Oparin and Haldane proposed it and Miller made amino acids in a flask. Evidence of evolution comes from fossils, homologous and analogous organs, embryos, molecules and changes we can watch (industrial melanism, drug resistance). Darwin explained it by natural selection acting on variation in populations; the modern synthetic theory adds genes: mutation, recombination, gene flow, genetic drift and natural selection change allele frequencies. If none of these act, frequencies stay constant: Hardy–Weinberg, p² + 2pq + q² = 1. Selection can be stabilising, directional or disruptive. One ancestor spreading into many habitats gives adaptive radiation (Darwin’s finches, Australian marsupials). Humans evolved from Dryopithecus-like apes through Australopithecus, Homo habilis, Homo erectus and Neanderthals to Homo sapiens.
Biochemistry: The Chemistry of Life – Plants build glucose by photosynthesis: 6 CO2 + 6 H2O + light gives C6H12O6 + 6 O2. Four families of big molecules (carbohydrates, proteins, fats, nucleic acids) are built from small units. Cells break nutrients down to release energy, with enzymes speeding every step. Proteins are made by reading DNA, copying it to mRNA and joining amino acids on a ribosome.
Gene Regulation: How Cells Switch Genes On and Off – Every cell carries the same DNA but uses only some genes. Cells control mostly at transcription. In bacteria, the lac operon is a set of genes under one promoter and one operator: a repressor blocks it when there is no lactose; lactose removes the repressor and the genes are read (an inducible operon). In eukaryotes, transcription factors, enhancers and silencers control each gene, and epigenetic marks (DNA methylation, histone packing) can keep genes off without changing the DNA code. Different on/off patterns make different cell types (differentiation).
Stimulus and Response – A stimulus is a change in the surroundings or inside the body. Living things that detect it and respond have a better chance of surviving. The path is always stimulus → receptor → coordinator → effector → response. Simple organisms move by taxis (towards or away) or kinesis (speed changes). Plants bend by tropisms using the growth factor IAA. Animals use receptors such as the Pacinian corpuscle and the rods and cones of the eye, reflex arcs for fast protection, and the brain's medulla to change the heart rate.
Ecology: Ecosystems, Matter Cycles, Energy Sources and New Materials – Ecology is the study of how living things interact with each other and with their surroundings. An ecosystem has producers (plants that make food from sunlight), consumers (animals) and decomposers (fungi and bacteria). Energy enters from the Sun and moves one way along the food chain: only about 10% passes to the next level, the rest is used or lost as heat. Matter, in contrast, is recycled in biogeochemical cycles such as the carbon, water, nitrogen and phosphorus cycles. People need energy, and the main choice is between fossil fuels (limited, add carbon dioxide) and renewable sources such as sun, wind and flowing water. New materials such as composites, smart materials, nanomaterials and plant-based plastics are designed for a job and can lower energy use and waste. Good choices balance strength, weight, safety and recycling.

7. Advanced Earth Science

Overview and structure of the Earth · Earth's activity · History of the Earth · Structure and motion of atmosphere and ocean · Structure and evolution of the universe · Natural environment and human life

Interior of the Earth: Earthquake Waves, Layers and Volcanoes – Nobody can dig to the centre of the Earth, so we learn about the inside from direct sources (mine rocks, deep drilling, lava) and indirect sources (heat and pressure with depth, meteors, gravity, magnetism and, above all, earthquake waves). P-waves travel through solids and liquids; S-waves only through solids. Where the waves do not arrive, we get shadow zones, which prove the outer core is liquid. Earthquakes are measured by magnitude (Richter) and intensity (Mercalli). The Earth has a thin crust, a thick mantle with a soft asthenosphere, a liquid outer core and a solid inner core. Magma that reaches the surface builds volcanoes; magma that cools inside forms intrusive landforms like batholiths, sills and dykes.
Distribution of Oceans and Continents: Drift, Spreading and Plates – In 1912 Alfred Wegener said all continents were once one landmass, Pangaea, surrounded by one ocean, Panthalassa, and that they drifted apart. Matching coastlines, rocks, fossils, glacier deposits and placer gold supported him, but he could not explain the force. Mapping the ocean floor showed ridges, plains and trenches; Harry Hess then proposed sea-floor spreading: new crust forms at mid-ocean ridges and old crust sinks at trenches. This led to plate tectonics: the lithosphere is broken into rigid plates that move on the soft asthenosphere and meet at divergent, convergent and transform boundaries. The Indian plate broke away from the south, moved north, and collided with Asia to raise the Himalayas.
History of the Earth – The universe is about 13.8 billion years old. The Sun and Earth formed from a spinning cloud of gas and dust about 4.6 billion years ago. Earth began hot, then cooled and got oceans. Life started in the sea more than 3.5 billion years ago and filled the air with oxygen. Rock layers and fossils, with the oldest at the bottom, tell us which life and which climate existed at each time. Four big eras (Precambrian, Palaeozoic, Mesozoic, Cenozoic) divide Earth's history, and humans arrived only in the last few seconds of a 24-hour clock.
Atmospheric Circulation and Weather Systems – Air has weight, so it presses down: this is air pressure (about 1013 mb at sea level). Wind blows from high pressure to low pressure, pushed by the pressure gradient force, turned by the Coriolis force (right in the north, left in the south) and slowed by friction near the ground. Uneven heating makes pressure belts (equatorial low, subtropical highs, subpolar lows, polar highs) and three circulation cells, giving planetary winds: trade winds, westerlies and polar easterlies. Belts shift with the seasons, giving seasonal winds like the monsoon; local winds include land and sea breezes and mountain and valley winds. Big bodies of air with the same temperature and moisture are air masses; where two meet is a front. Cyclones are low-pressure storms: tropical cyclones form over warm seas; extratropical ones form along fronts. Thunderstorms and tornadoes are small but violent storms.
Cosmology: The Expanding Universe and the Big Bang – Cosmology is the study of the whole universe: its structure, history and future. Galaxies gather in groups, clusters and filaments around huge voids. Distant galaxies are moving away from us, faster the farther they are (Hubble's law, v = H₀d), because space itself is expanding. Running the expansion backwards leads to a hot, dense beginning about 13.8 billion years ago, the Big Bang. The main evidence is redshift, the cosmic microwave background and the amounts of hydrogen and helium. Most of the universe is dark matter and dark energy.
Natural Hazards – A natural hazard is a natural event that can harm people and property, such as an earthquake, volcano, landslide, flood, drought or cyclone. It becomes a disaster when it hits people who are not ready. Geological hazards come from inside the Earth; meteorological (weather) hazards come from the air and water. Risk = hazard × vulnerability ÷ capacity to cope, so warning systems, strong buildings and trained people cut the damage.