National Year 13 Further Mathematics
Chapters: 12
1. B Complex numbers (part 2)
De Moivre and roots
- De Moivre's Theorem and Roots of Unity – A complex number can be written by its length r and angle θ: z = r(cos θ + i sin θ) = re^{iθ}. When you multiply, the lengths multiply and the angles add. So zⁿ = rⁿ(cos nθ + i sin nθ): this is De Moivre's theorem. It gives quick powers, formulas for cos nθ and sin nθ, and the n roots of any number. The n roots of 1 sit evenly on the unit circle like the corners of a regular polygon.
2. C Matrices (part 2)
3×3 systems and eigen-theory
- Determinants: Minors, Cofactors, Adjoint, Inverse and Linear Systems – A determinant is one number made from a square matrix, written |A|. For a 2 × 2 matrix it is ad − bc, and it equals the (signed) area made by the columns. For a 3 × 3 matrix we expand along a row using minors and cofactors. |A| = 0 means A is singular and has no inverse. Half of a determinant gives the area of a triangle. The adjoint (transpose of the cofactor matrix) gives A⁻¹ = (adj A)/|A|, and then a system AX = B is solved by X = A⁻¹B. The value of |A| and (adj A)B tell us if a system is consistent.
3. D Further algebra and functions (part 2)
Maclaurin series and limits · Modulus and reciprocal graphs
- Maclaurin and Taylor Series – A Maclaurin series writes a function as an endless polynomial: f(x) = f(0) + f′(0)x + f″(0)x²/2! + … . Near x = 0 a few terms copy the curve very well. The standard series for eˣ, sin x, cos x, ln(1+x) and (1+x)ⁿ must be known with their ranges of validity. A Taylor series does the same around any point a. Series also make hard limits easy: replace each function by its first terms.
- Rational Functions: Graphs, Asymptotes and Sketching – A rational function is one polynomial divided by another, like y = (2x + 1)/(x − 1). It is not defined where the bottom is zero. Near those x-values the graph shoots up or down along a vertical asymptote. Far out, it settles near a horizontal (or slant) asymptote. Find domain, asymptotes, intercepts and holes, then sketch.
4. E Further calculus (part 2)
Advanced integration
- 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.
5. F Further vectors (part 2)
Planes and vector product
- Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.
6. G Polar coordinates (part 2)
Polar curves
- Polar Coordinates – Polar coordinates give the position of a point by its distance r from a fixed point (the pole) and the angle θ turned anticlockwise from a fixed ray (the initial line). Convert with x = r cosθ, y = r sinθ, and back with r² = x² + y², tanθ = y/x (check the quadrant). Equations r = f(θ) draw circles, cardioids, limaçons, roses and spirals. Calculus: slope dy/dx = (dy/dθ)/(dx/dθ), area = ½∫r²dθ.
7. H Hyperbolic functions (part 2)
Hyperbolic functions
- Hyperbolic Functions – Hyperbolic functions are built from eˣ and e⁻ˣ: cosh x = (eˣ + e⁻ˣ)/2, sinh x = (eˣ − e⁻ˣ)/2, tanh x = sinh x/cosh x. The point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, so cosh²x − sinh²x = 1. Their inverses have log forms, e.g. arsinh x = ln(x + √(x² + 1)). They differentiate neatly (d/dx sinh x = cosh x, d/dx cosh x = sinh x) and give standard integrals such as ∫ 1/√(x² + 1) dx = arsinh x + c.
8. I Differential equations
First and second order ODEs
- Differential Equations – A differential equation connects a function y with its derivatives. Its order is the highest derivative present and its degree is the power of that derivative (when the equation is a polynomial in derivatives). A general solution has arbitrary constants; a condition like y(0) = 1 fixes them to give a particular solution. Class 12 solves first-order equations of three kinds: variables separable, homogeneous (put y = vx) and linear dy/dx + Py = Q (multiply by the integrating factor e^∫P dx).
9. J Numerical methods
Numerical integration and ODEs
- Numerical Methods: Finding Roots and Areas Step by Step – Some equations and areas cannot be found with a neat formula. Numerical methods get as close as we like with simple repeated steps: check a change of sign, halve the interval (bisection), slide down tangents (Newton-Raphson), repeat x = g(x) (fixed-point iteration), add up trapeziums for an area, and walk along a slope in small steps (Euler).
10. Optional application 1 Mechanics (part 2)
MD Circular motion · ME Centres of mass and moments
- Motion in a Vertical Circle – A ball on a string moving in a vertical circle speeds up at the bottom and slows down at the top, because gravity does work on it. At every point the net force towards the centre must be mv²/r. At the bottom T = mg + mv²/r (largest); at the top T = mv²/r − mg (smallest). The string stays tight at the top only if v_top ≥ √(gr). Using energy conservation, this needs u ≥ √(5gr) at the bottom.
- Centre of Mass: Two Particles, Rigid Body and Uniform Rod – The centre of mass is the one point that moves as if all the mass of a system were packed there. For two particles on a line, x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂). It lies on the line joining them, closer to the heavier one. For many particles, x_cm = Σmx/Σm (same for y and z). For a uniform rod of length L, the centre of mass is at L/2, its middle. Internal forces cannot move the centre of mass; only an outside force can: M·a_cm = F_ext.
11. Optional application 2 Statistics (part 2)
SC Type I and II errors · SD Continuous random variables · SE Yates correction · SF Exponential distribution · SG-SH t-tests and confidence intervals
- Hypothesis Testing – A hypothesis test checks a claim about a population using a sample. Start with the null hypothesis H₀ (no change, e.g. p = 0.5) and the alternative H₁ (what we suspect, e.g. p > 0.5). Choose a significance level such as 5%. Work out how likely the sample result (or more extreme) is if H₀ were true: the p-value. If the p-value is below the level, or the result falls in the critical region, reject H₀. Otherwise there is not enough evidence to reject it. Type I error = rejecting a true H₀; Type II error = not rejecting a false H₀.
- Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
- Chi-Square Test for Independence and Homogeneity – A chi-square (χ²) test checks if counts in a table are too far from what we would expect by chance. For a contingency table: E = row total × column total ÷ grand total, χ² = Σ (O − E)² ÷ E, df = (r − 1)(c − 1). If χ² is bigger than the critical value (or p < significance level), reject H0 of no association. For 2×2 tables, Yates' correction uses (|O − E| − 0.5)².
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.
- Confidence Intervals – A confidence interval is a range of believable values for an unknown population number (a mean μ or a proportion p), worked out from one sample. It has the shape estimate ± margin of error, where margin of error = critical value × standard error. A 95% level means the method catches the true value in about 95% of samples. Higher confidence gives a wider interval; a bigger sample gives a narrower one. If a claimed value lies outside the interval, the data give evidence against the claim.
12. Optional application 3 Discrete (part 2)
DA Planarity and isomorphism · DC Network flows · DD Simplex algorithm · DE Critical path analysis · DF Game theory (larger games) · DG Binary operations and groups
- 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.
- Graph Algorithms – A graph is a set of vertices joined by edges, which can carry weights. Breadth-first search (BFS) explores in layers using a queue and finds the fewest-edge path. Depth-first search (DFS) goes deep using a stack or recursion and backtracks. Trees can be traversed pre-order, in-order and post-order. Dijkstra's algorithm finds shortest paths from one vertex when weights are non-negative. Kruskal's and Prim's algorithms build a minimum spanning tree. Route inspection finds the shortest closed route using every edge; the travelling salesperson problem asks for the shortest tour of every vertex. In a flow network, the maximum flow equals the capacity of the minimum cut.
- The Simplex Algorithm: Solving Linear Programs with Tableaux – The graphical method works for two variables, but real problems have many. The simplex algorithm solves them with a table (tableau). Add a slack variable to each ≤ constraint, start at the origin, and pivot: choose the most negative number in the objective row, use the ratio test to choose the row, and clear the column. Each pivot moves to a better corner of the feasible region. When the objective row has no negative numbers, the tableau is optimal and you read the answer. To minimise C, maximise −C.
- Critical Path Analysis – A big project is made of many activities. Some must wait for others. We draw them as an activity-on-node network. A forward pass gives each activity its earliest start; a backward pass gives its latest finish. Float = latest start − earliest start tells how much an activity can slip. Activities with zero float form the critical path: the longest route, which fixes the shortest possible project time. A Gantt (cascade) chart turns the network into bars on a time line, and a resource histogram shows workers needed per day. Moving activities inside their float to smooth that histogram is resource levelling.
- Game Theory: Making the Best Choice When Others Choose Too – Game theory studies decisions where your result depends on what others choose. A pay-off matrix lists each player's gain for every pair of choices. A dominated strategy is always worse and can be removed. A Nash equilibrium is a pair of choices where no player gains by changing alone; the prisoner's dilemma shows it can be worse for everyone than cooperating. In a zero-sum game, the play-safe (maximin/minimax) strategies meet at a saddle point when the game is stable; otherwise players use a mixed strategy, found by drawing expected-pay-off lines and taking the highest point of the lower edge.
- Group Theory: Binary Operations, Groups, Rings and Fields – A group is a set with one operation that is closed, associative, has an identity and gives every element an inverse. Clock arithmetic Z6 shows all of it: tables, subgroups, the order of an element and Lagrange's theorem. Rings and fields add a second operation.