National Year 12 Further Mathematics
Chapters: 11
1. B Complex numbers (part 1)
Complex number basics
- 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.
2. C Matrices (part 1)
Matrix algebra and transformations
- 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.
3. D Further algebra and functions (part 1)
Roots of polynomials · Summation of series · Graphs and inequalities · Standard Maclaurin series
- 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.
- Rational Graphs, Conics and Inequalities – To sketch y = (ax + b)/(cx + d), find where the bottom is zero (vertical asymptote), the value of a/c far away (horizontal asymptote) and the axis crossings. For a quadratic over a quadratic, set y = k and use the discriminant to find which values y can take. The conics y² = 4ax, x²/a² + y²/b² = 1 and x²/a² − y²/b² = 1 move and stretch by simple swaps (x → x − p, x → x/s). Rational and polynomial inequalities are solved with critical values and a sign check, never by multiplying by an unknown sign.
- 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.
4. E Further calculus (part 1)
Volumes and mean values
- Solids of Revolution – A solid of revolution is the 3D shape you get when a flat shape turns a full 360° around a straight line (the axis). A rectangle makes a cylinder, a right triangle makes a cone, a half-circle makes a sphere and a trapezium makes a frustum. Their volumes are V = πr²h, V = ⅓πr²h, V = ⁴⁄₃πr³ and V = ⅓πh(R² + Rr + r²). In calculus, any curve y = f(x) turned about the x-axis gives V = π∫y² dx.
5. F Further vectors (part 1)
Lines in 3D
- 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 1)
Polar basics
- 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 1)
Hyperbolic basics
- 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. A Proof
Proof by induction
- Proof by Mathematical Induction – Mathematical induction proves that a statement P(n) is true for every natural number n. Step 1 (base case): show P(1) is true. Step 2 (inductive step): assume P(k) is true for some k, and use it to show P(k + 1) is true. Then, like a line of dominoes, P(1) makes P(2) true, P(2) makes P(3) true, and so on for ever.
9. Optional application 1 Mechanics (part 1)
MA Dimensional analysis · MB Momentum and collisions · MC Work, energy and power · MD Circular motion (constant speed)
- Dimensions and Dimensional Analysis – The dimensions of a quantity show how it is built from base quantities: mass [M], length [L], time [T] (and current [A], temperature [K], amount [mol], luminous intensity [cd]). Force = [M L T⁻²]. In a correct equation every term has the same dimensions (principle of homogeneity). We use this to check equations, to convert units from one system to another, and to find how one quantity depends on others. It cannot give number constants like 2π, and it cannot handle sums or trig and log functions.
- Elastic and Inelastic Collisions in 1D and 2D – In every collision, total momentum is conserved (no outside force during the short hit). In an elastic collision kinetic energy is also conserved. In an inelastic collision some kinetic energy becomes heat, sound or dent energy; if the bodies stick together it is perfectly inelastic. In 1D, elastic collision gives v₁ = (m₁ − m₂)u₁/(m₁ + m₂) and v₂ = 2m₁u₁/(m₁ + m₂) when body 2 starts at rest. In 2D, momentum is conserved separately along x and y.
- Work, Kinetic Energy, Work–Energy Theorem and Power – Work is done when a force moves something along its direction: W = F·s = F s cos θ. For a changing force, work is the area under the F–x graph. A moving body has kinetic energy K = ½mv². The work–energy theorem says: net work done on a body = change in its kinetic energy. Power is how fast work is done: P = W/t = F·v.
- Centripetal Force, Car on a Level Road and on a Banked Road – A body moving in a circle is always changing direction, so it needs a net force towards the centre: the centripetal force F = mv²/r. It is not a new kind of force; tension, gravity, friction or a part of the normal force supplies it. On a level road only friction supplies it, so vmax = √(μs r g). On a road banked at θ, a part of the normal force helps: with no friction the ideal speed is v₀ = √(r g tanθ), and with friction vmax = √[r g (μs + tanθ)/(1 − μs tanθ)].
10. Optional application 2 Statistics (part 1)
SA Discrete random variables · SB Poisson distribution · SC Type I and II errors (part 1) · SD Continuous random variables (part 1) · SE Chi-squared tests for association · SH Confidence intervals (normal)
- 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.
- 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.
- 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.
- 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.
- 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.
11. Optional application 3 Discrete (part 1)
DA Graphs · DB Networks · DC Network flows (part 1) · DD Linear programming · DE Critical path analysis (part 1) · DF Game theory · DG Binary operations (part 1)
- 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.
- Linear Programming (Class 12): find the best answer with a graph – Linear programming finds the biggest profit or the smallest cost when you must obey some rules. The rules are straight-line inequalities (constraints). Together they cut out a region of allowed points (the feasible region). The goal, Z = ax + by (the objective function), is always best at a corner of that region. So: draw the lines, shade, find the corners, put each corner in Z, pick the largest or smallest. If the region is open (unbounded), check once more that the answer really holds.
- 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.