Japan 高校2年 Mathematics II
Chapters: 5
1. Various expressions
Cubic expansion and factorisation; binomial theorem · Polynomial division and rational expressions · Complex numbers · Discriminant; relations between roots and coefficients · Factor theorem and higher-degree equations · Proof of equalities and inequalities
- Binomial Theorem for Positive Integers – A binomial is a two-term expression like a + b. The binomial theorem tells us how to expand (a + b)ⁿ for any positive integer n without multiplying again and again: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. There are n + 1 terms. The power of a goes down by 1 and the power of b goes up by 1 in each term, and the two powers always add up to n. The coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle, where each number is the sum of the two above it. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The pattern was known long ago in India (Pingala's Meru Prastara) and in Persia and China, and Pascal later studied the triangle in detail. The theorem is proved by mathematical induction using ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.
- Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
2. Figures and equations
Division of segments with coordinates · Equations of lines and circles · Loci · Regions defined by inequalities
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
- 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).
- Linear Inequalities in One Variable – An inequality compares two expressions with <, >, ≤ or ≥. A linear inequality in one variable looks like ax + b < c. Its answer is usually a whole range of numbers, not one number. We solve it like an equation: we may add or subtract the same number on both sides, and multiply or divide by the same positive number. If we multiply or divide by a negative number, the sign must flip. We show the answer on a number line: a hollow dot for < or > (end not included) and a filled dot for ≤ or ≥ (end included), with the shaded part showing all solutions. Double inequalities like −1 ≤ x < 3 give a piece of the line. If x must be a natural number or an integer, only the whole numbers in that range count.
3. Exponential and logarithmic functions
Rational exponents · Exponential functions and graphs · Logarithms and properties · Logarithmic functions and graphs
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- 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.
4. Trigonometric functions
General angles and radians · Graphs of trigonometric functions · Basic identities · Addition and double-angle formulas
- 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.
5. Ideas of calculus
Derivative and derived function · Increase/decrease and extrema · Indefinite and definite integrals; area
- Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
- Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
- 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.