Back Hub
Edexcel IAL Further Math
Formula Sheet
Ready
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

FP1 - Further Pure 1

Unit WFM01 - Edexcel International A Level Further Mathematics (2018)
Authors: Henrik Gu, Yuchen Wang

1Complex Numbers

Cartesian and Argand diagram, modulus-argument, polar & Euler form.

z = a + bi with i^2 = -1. Real part \Re(z) = a, imaginary part \Im(z) = b. Complex conjugate \bar{z} = a - bi. Modulus |z| = \sqrt{a^2+b^2}; argument \arg z = \tan^{-1}(b/a), range -\pi < \arg z \le \pi.

Arithmetic: add componentwise; multiply as (a+bi)(c+di) = (ac-bd) + (ad+bc)i; divide by multiplying by conjugate of denominator. z \times \bar{z} = a^2 + b^2.

Real-coefficient polynomial rule: complex roots always occur in conjugate pairs. If 3+i is a root, 3-i is also a root; factorise to lower degree.

Polar & Euler form

Polar and Eulerz = r(\cos\theta + i\sin\theta) = re^{i\theta}

Modulus & argument rules

|z_1 z_2| = |z_1||z_2|, \quad \arg(z_1 z_2) = \arg z_1 + \arg z_2
\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}, \quad \arg\!\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2

Trigonometric identities via complex numbers

z + \frac{1}{z} = 2\cos\theta, \qquad z - \frac{1}{z} = 2i\sin\theta

Used to expand powers of \sin/\cos for integration.

Argand Diagram Loci

  • Circle: |z - z_1| = r -> Cartesian: (x-a)^2 + (y-b)^2 = r^2
  • Perpendicular bisector: |z - z_1| = |z - z_2|
  • Half-line: \arg(z - z_1) = \theta
  • Max/min modulus on circle: max |z| = |centre| + radius; min = |centre| − radius

2Roots of Quadratic Equations

Vieta's formulas and common root identities.

If \alpha, \beta are roots of ax^2 + bx + c = 0: \alpha + \beta = -\dfrac{b}{a}, \alpha\beta = \dfrac{c}{a}.

Common root identities

\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta
\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha+\beta}{\alpha\beta}
\alpha^2\beta^2 = (\alpha\beta)^2

Given transformed roots (e.g. \alpha^2, \beta^2 or \tfrac{1}{\alpha}, \tfrac{1}{\beta}), compute their sum and product and reconstruct the new quadratic: x^2 - (\text{sum})x + (\text{product}) = 0.

Common transformations

New rootsSumProduct
\alpha^2, \beta^2(\alpha+\beta)^2 - 2\alpha\beta(\alpha\beta)^2
\frac{1}{\alpha}, \frac{1}{\beta}\frac{\alpha+\beta}{\alpha\beta}\frac{1}{\alpha\beta}
\alpha^2\beta, \alpha\beta^2\alpha\beta(\alpha+\beta)(\alpha\beta)^3

3Numerical Solutions

Change of sign, interval bisection, linear interpolation, Newton-Raphson.

Change of sign

If f continuous and f(a) \cdot f(b) < 0, a root lies in (a, b).

Interval bisection

Take midpoint of interval, discard half with same sign as bound, narrow interval repeatedly.

Linear interpolation

Linear interpolation formulax = a - \frac{f(a)(b-a)}{f(b) - f(a)}

Newton-Raphson

Iteration formulax_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Fast convergence for locating roots. Fails if f'(x_n) = 0.

4Coordinate Systems

Parabola and rectangular hyperbola.

Parabola

Standard form y^2 = 4ax: focus (a, 0), directrix x = -a.

Parametric coordinatesx = at^2, \quad y = 2at

Tangent at point t: ty = x + at^2. Normal at point t: y = -tx + 2at + at^3.

Chord joining t_1 and t_2: y(t_1 + t_2) = 2x + 2at_1 t_2.

Rectangular hyperbola

Standard form xy = c^2. Parametric form: x = ct, y = c/t.

Implicit differentiation\frac{dy}{dx} = -\frac{y}{x}

Tangent at point t: x + t^2 y = 2ct. Normal at point t: t^3 x - ty = c(t^4 - 1).

5Matrix Algebra

Size, multiplication, determinant, inverse.

Matrix multiplication: column count of first = row count of second. Associative but not commutative: (AB)C = A(BC); usually AB \ne BA.

2x2 determinant and inverseM = \begin{pmatrix}a&b\\c&d\end{pmatrix}, \quad \det(M) = ad - bc
M^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix} \quad (\det \ne 0)

Singular matrix (\det = 0): no inverse exists. Solve linear simultaneous equations: form AX = B, solution X = A^{-1}B.

6Transformations

Linear transformations of the plane represented by matrices.

The columns of a 2 \times 2 matrix are the images of \binom{1}{0} and \binom{0}{1}. Area scale factor = |\det(M)|.

Standard transformation matrices\text{Enlargement scale factor } k: \begin{pmatrix}k&0\\0&k\end{pmatrix}
\text{Rotation anticlockwise by } \theta: \begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}

Combined transformations: apply right matrix first, left matrix second. Composition = matrix multiplication in reverse order.

7Series (Sums of Powers)

Closed forms and method of differences.
Standard summation formulas\sum_{r=1}^n r = \frac{n(n+1)}{2}, \quad \sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{r=1}^n r^3 = \left[\frac{n(n+1)}{2}\right]^2

Use these with linearity: \sum (ar^2 + br + c) = a\sum r^2 + b\sum r + cn.

Method of differences (telescoping series)

Rewrite term u_r = f(r) - f(r+1). Sum from 1 to n cancels intermediate terms, only f(1) - f(n+1) left.

8Proof by Induction

Standard four-step technique.

Step 1 - Base case: verify statement holds for n = 1 (or smallest relevant integer).

Step 2 - Assumption: assume true for n = k.

Step 3 - Inductive step: prove statement holds for n = k+1 using the assumption.

Step 4 - Conclusion: since base holds and the step is valid, statement true for all positive integers n.

Common proof types: sum series, divisibility, matrix powers, recurrence relations.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

FP2 - Further Pure 2

Unit WFM02 - Edexcel International A Level Further Mathematics (2018)

1Inequalities

Rational fractional and modulus inequalities.

Rational fractional inequalities

Move all terms to one side to get single fraction = 0. Factor numerator & denominator, find critical values.

Curve sketch rule: odd power roots cross axis, even power roots touch axis (no cross).

Modulus inequalities

Split into piecewise functions to remove modulus sign, solve separate intervals.

General rules|f(x)| < a \;\Rightarrow\; -a < f(x) < a \qquad |f(x)| > a \;\Rightarrow\; f(x) < -a \;\text{ or }\; f(x) > a

2Series - Method of Differences

Telescoping sums.

Rewrite u_r = f(r) - f(r+1) (or similar) so that consecutive terms cancel. The sum \sum_{r=1}^n u_r collapses to endpoints: f(1) - f(n+1). Often used after partial fractions.

3Further Complex Numbers

De Moivre's theorem, roots of unity, exponential form.
Exponential and De Moivree^{i\theta} = \cos\theta + i\sin\theta \qquad (r\, e^{i\theta})^n = r^n\, e^{in\theta}

nth roots of z = re^{i\theta}: r^{1/n}\, e^{i(\theta + 2k\pi)/n} for k = 0, 1, \ldots, n-1. On the Argand diagram they lie on a circle at equal angular spacing.

Applications: expressing \cos n\theta, \sin n\theta in terms of \cos\theta, \sin\theta; simplifying trig sums.

4First-order Differential Equations

Separable variables and integrating factor.

Separable variables

For \dfrac{dy}{dx} = g(x)h(y), rearrange: \displaystyle\int \frac{1}{h(y)}\,dy = \int g(x)\,dx, integrate both sides.

Integrating factor

For \dfrac{dy}{dx} + P(x)y = Q(x): multiply both sides by integrating factor \mu = e^{\int P\,dx}. LHS becomes \dfrac{d}{dx}(\mu y); integrate both sides then divide.

5Second-order Differential Equations

Linear with constant coefficients, homogeneous & non-homogeneous.

Homogeneous: ay'' + by' + cy = 0

Solve auxiliary quadratic am^2 + bm + c = 0:

  • Two distinct real roots m_1, m_2: y_{cf} = Ae^{m_1 x} + Be^{m_2 x}.
  • Repeated root m: y_{cf} = (A + Bx)e^{mx}.
  • Complex conjugate roots m = p \pm qi: y_{cf} = e^{px}(A\cos qx + B\sin qx).

Non-homogeneous: ay'' + by' + cy = f(x)

General solution = Complementary Function (CF) + Particular Integral (PI). Guess PI form matching f(x), substitute to solve constants.

PI guess forms

f(x)PI guessSpecial case
Polynomial of degree nGeneral polynomial of degree nIf c = 0, multiply by x
ke^{mx}\lambda e^{mx}If m is a root of auxiliary, multiply by x (or x^2 if repeated)
k\cos qx or k\sin qx\lambda\cos qx + \mu\sin qxIf p \pm qi are roots, multiply by x

Important: if the PI guess overlaps with the CF, multiply the guess by x (or x^2 if needed) to make it linearly independent.

6Maclaurin & Taylor Series

Power series expansion of a function.

Maclaurin series (expansion at x=0)

Maclaurinf(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots

Standard expansions

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \quad (|x|<\infty)
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots \qquad \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots \quad (|x|<\infty)
\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots \quad (|x|<1)

Taylor series (expansion at x=a)

Taylorf(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots

Composite series

Substitute inner function into standard series. E.g. e^{x^2} = 1 + x^2 + \frac{x^4}{2!} + \ldots

7Polar Coordinates

(r, \theta), conversion, area, tangent.

Conversion

x = r\cos\theta, y = r\sin\theta; r = \sqrt{x^2+y^2}, \theta = \arctan(y/x) (with quadrant care).

Area of a sectorA = \frac{1}{2}\int_\alpha^\beta r^2 \, d\theta

Standard polar curves

Cardioid: r = a(1 + \cos\theta). Rose curves: r = a\sin n\theta.

Tangent to polar curve

Parametric form: x = r(\theta)\cos\theta, y = r(\theta)\sin\theta. Gradient: \dfrac{dy}{dx} = \dfrac{dy/d\theta}{dx/d\theta}.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

FP3 - Further Pure 3

Unit WFM03 - Edexcel International A Level Further Mathematics (2018)

1Hyperbolic Functions

\sinh, \cosh, \tanh and their inverses.
\sinh x = \dfrac{e^x - e^{-x}}{2} \qquad \cosh x = \dfrac{e^x + e^{-x}}{2} \qquad \tanh x = \dfrac{\sinh x}{\cosh x}

Osborn's rule: to translate a trig identity to hyperbolic, replace each \sin with \sinh and each \cos with \cosh, but flip the sign of a product of two \sinh's.

Identity, derivatives\cosh^2 x - \sinh^2 x = 1\dfrac{d}{dx}\sinh x = \cosh x, \; \dfrac{d}{dx}\cosh x = \sinh x, \; \dfrac{d}{dx}\tanh x = \text{sech}^2 x

Inverses in log form: \sinh^{-1}x = \ln(x + \sqrt{x^2+1}), \cosh^{-1}x = \ln(x + \sqrt{x^2-1}) for x \ge 1, \tanh^{-1}x = \tfrac{1}{2}\ln\tfrac{1+x}{1-x} for |x|<1.

2Further Coordinate Systems

Ellipse and hyperbola.

Ellipse \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1: foci (\pm ae, 0) where b^2 = a^2(1-e^2); directrices x = \pm a/e. Parametric (a\cos t, b\sin t).

Hyperbola \tfrac{x^2}{a^2} - \tfrac{y^2}{b^2} = 1: foci (\pm ae, 0) with b^2 = a^2(e^2-1); asymptotes y = \pm \tfrac{b}{a}x. Parametric (a\sec t, b\tan t) or hyperbolic (a\cosh t, b\sinh t).

3Differentiation (further)

Inverse trig and hyperbolic.
\dfrac{d}{dx}(\sinh^{-1}x) = \dfrac{1}{\sqrt{1+x^2}} \qquad \dfrac{d}{dx}(\cosh^{-1}x) = \dfrac{1}{\sqrt{x^2-1}} \qquad \dfrac{d}{dx}(\tanh^{-1}x) = \dfrac{1}{1-x^2}

4Integration (further)

Reduction formulae, standard forms.
Standard forms\int \dfrac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\tfrac{x}{a} + C\int \dfrac{1}{a^2+x^2}\,dx = \tfrac{1}{a}\tan^{-1}\tfrac{x}{a} + C\int \dfrac{1}{\sqrt{x^2+a^2}}\,dx = \sinh^{-1}\tfrac{x}{a} + C\int \dfrac{1}{\sqrt{x^2-a^2}}\,dx = \cosh^{-1}\tfrac{x}{a} + C

Reduction formulae: expressions of the form I_n = \int \ldots recursively in terms of I_{n-1} or I_{n-2}; use integration by parts to derive them, then evaluate iteratively.

5Vectors (3D)

Cross product, vector/scalar equations of a line and plane.
Cross product\mathbf{a} \times \mathbf{b} = \left|\begin{smallmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{smallmatrix}\right|, \qquad |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta

\mathbf{a} \times \mathbf{b} is perpendicular to both \mathbf{a} and \mathbf{b} (right-hand rule). Its magnitude equals the area of the parallelogram they span.

Plane\mathbf{r}\cdot \mathbf{n} = \mathbf{a}\cdot \mathbf{n} \quad \Leftrightarrow \quad n_1 x + n_2 y + n_3 z = d

Distance from point (x_0,y_0,z_0) to plane n_1 x + n_2 y + n_3 z = d: \dfrac{|n_1 x_0 + n_2 y_0 + n_3 z_0 - d|}{|\mathbf{n}|}.

6Further Matrix Algebra

3x3 matrices, eigenvalues, eigenvectors.

Eigenvalue \lambda satisfies \det(A - \lambda I) = 0. Corresponding eigenvector \mathbf{v} solves (A - \lambda I)\mathbf{v} = \mathbf{0}.

For 3x3 the characteristic equation is a cubic; factorise to find the three eigenvalues. Diagonalisation: A = PDP^{-1} where P's columns are eigenvectors and D has eigenvalues on the diagonal.

7Further Trigonometry

Products of trig and hyperbolic; identities via complex exponentials.

Using e^{i\theta} = \cos\theta + i\sin\theta and \sin\theta = \tfrac{e^{i\theta} - e^{-i\theta}}{2i}, \cos\theta = \tfrac{e^{i\theta} + e^{-i\theta}}{2}, derive expressions for \sin^n\theta, \cos^n\theta in terms of multiple angles - useful for integration and summation.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

M2 - Mechanics 2

Unit WME02 - Edexcel International A Level Further Mathematics (2018)

1Kinematics (variable acceleration)

Calculus-based motion; projectiles.

\mathbf{v} = \tfrac{d\mathbf{r}}{dt}, \mathbf{a} = \tfrac{d\mathbf{v}}{dt}. Sometimes a = f(v) or a = f(x) - use a = v\tfrac{dv}{dx} to change variable.

Projectiles: split velocity into horizontal (constant) and vertical (under gravity) components. Time of flight, range, max height derived from SUVAT in each direction.

2Centres of Mass

Systems of particles, laminas, uniform bodies.
Discrete system\bar{x} = \dfrac{\sum m_i x_i}{\sum m_i}, \; \bar{y} = \dfrac{\sum m_i y_i}{\sum m_i}

Composite laminas: treat each piece as a particle at its own centroid. For a body suspended from a point, the centre of mass lies vertically below the point.

3Work, Energy, Power

Energy conservation and non-conservative forces.
W = Fs\cos\theta \qquad \text{KE} = \tfrac{1}{2}mv^2 \qquad \text{GPE} = mgh \qquad P = Fv

Work-energy theorem: total work done = change in KE. When only conservative forces act, KE + GPE is constant.

4Collisions

Impulse, momentum, coefficient of restitution.

Momentum p = mv; impulse I = Ft = \Delta p. In collisions momentum is conserved if no external impulse: m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2.

Coefficient of restitution e = \dfrac{v_2 - v_1}{u_1 - u_2} (0 \le e \le 1). Perfectly elastic: e = 1. Perfectly inelastic: e = 0 (particles stick).

5Statics of Rigid Bodies

Equilibrium under multiple forces and moments.

Rigid body equilibrium: sum of forces = 0 in each direction AND sum of moments about any point = 0. Common exam context: ladder against a wall - resolve horizontally and vertically, take moments about the foot.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

S2 - Statistics 2

Unit WST02 - Edexcel International A Level Further Mathematics (2018)

1Binomial Distribution

Discrete count of successes, cumulative probabilities, and hypothesis testing.

X \sim B(n, p) if there are n independent trials, each with success probability p. P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}. Mean np, variance np(1-p).

Cumulative probabilities

P(X \le x) is found from tables or calculator. Key relations:

P(X > x) = 1 - P(X \le x) \qquad P(X \ge x) = 1 - P(X \le x-1) \qquad P(a \le X \le b) = P(X \le b) - P(X \le a-1)

Hypothesis test for p (binomial test)

Test H_0: p = p_0 against H_1: p < p_0 (one-tail) or H_1: p > p_0 or H_1: p \neq p_0 (two-tail).

  1. State H_0 and H_1 with the significance level \alpha.
  2. Under H_0, X \sim B(n, p_0). Calculate P(X \le x) or P(X \ge x).
  3. For one-tail: if probability \le \alpha, reject H_0.
  4. For two-tail: if probability \le \alpha/2, reject H_0.
  5. State conclusion in context.

Critical region

The set of values of X that lead to rejection of H_0. Find the largest/smallest x such that the tail probability is just \le \alpha.

2Poisson Distribution

Rare events over a fixed interval, cumulative probabilities, and hypothesis testing.
X \sim \text{Po}(\lambda): \; P(X = x) = \dfrac{e^{-\lambda}\lambda^x}{x!}, \; E(X) = \text{Var}(X) = \lambda

Properties

  • Sum of independent Poissons: if X \sim \text{Po}(\lambda_1) and Y \sim \text{Po}(\lambda_2), then X + Y \sim \text{Po}(\lambda_1 + \lambda_2).
  • Approximation to binomial: if n large and p small, B(n, p) \approx \text{Po}(np).

Cumulative probabilities

P(X \le x) from tables. Key relations:

P(X > x) = 1 - P(X \le x) \qquad P(X \ge x) = 1 - P(X \le x-1)

Hypothesis test for \lambda (Poisson test)

Test H_0: \lambda = \lambda_0 against H_1: \lambda < \lambda_0, \lambda > \lambda_0, or \lambda \neq \lambda_0.

  1. State H_0 and H_1 with significance level \alpha.
  2. Under H_0, X \sim \text{Po}(\lambda_0). Calculate P(X \le x) or P(X \ge x).
  3. For one-tail: if probability \le \alpha, reject H_0.
  4. For two-tail: if probability \le \alpha/2, reject H_0.

3Continuous Random Variables

Probability density, cumulative distribution, median, mode, and quartiles.

f(x) is a valid pdf if f(x) \ge 0 and \int_{-\infty}^{\infty} f(x)\,dx = 1. CDF: F(x) = \int_{-\infty}^x f(t)\,dt; P(a < X < b) = F(b) - F(a) = \int_a^b f\,dx.

E(X) = \int x f(x)\,dx, E(X^2) = \int x^2 f(x)\,dx, \text{Var}(X) = E(X^2) - [E(X)]^2.

Median and mode

Median m: F(m) = 0.5, i.e. \int_{-\infty}^m f(x)\,dx = 0.5. Mode: the value of x where f(x) is maximum (may be found by differentiation or inspection).

Quartiles and percentiles

Lower quartile Q_1: F(Q_1) = 0.25. Upper quartile Q_3: F(Q_3) = 0.75. Interquartile range: Q_3 - Q_1.

From CDF to pdf

f(x) = \frac{d}{dx}F(x) at points where F is differentiable.

4Continuous Distributions

Uniform, normal approximation to binomial and Poisson.

Continuous uniform distribution

X \sim U(a, b): f(x) = \tfrac{1}{b-a} for a \le x \le b; mean \tfrac{a+b}{2}, variance \tfrac{(b-a)^2}{12}.

Normal approximation to Binomial

Valid for large n and p not too close to 0 or 1: B(n,p) \approx N(np, np(1-p)). Apply continuity correction (\pm 0.5) because binomial is discrete and normal is continuous.

P(X \le k) \approx P\left(Z \le \frac{k + 0.5 - np}{\sqrt{np(1-p)}}\right)

Normal approximation to Poisson

For large \lambda (typically \lambda > 10): \text{Po}(\lambda) \approx N(\lambda, \lambda). Apply continuity correction.

5Hypothesis Testing

Framework, critical regions, one-tail and two-tail tests.

General procedure

  1. State H_0 (null) and H_1 (alternative); pick a significance level \alpha (e.g. 5%).
  2. Under H_0 compute the probability of observed (or more extreme) result.
  3. If p-value \le \alpha: reject H_0. Else insufficient evidence to reject.
  4. State conclusion in context - never say "prove".

One-tail vs two-tail

H_1TypeRejection rule
p < p_0 or \lambda < \lambda_0Lower one-tailP(X \le x) \le \alpha
p > p_0 or \lambda > \lambda_0Upper one-tailP(X \ge x) \le \alpha
p \neq p_0 or \lambda \neq \lambda_0Two-tailEither tail \le \alpha/2

Critical region

The set of values of the test statistic that leads to rejection of H_0. For a discrete distribution, find the largest (or smallest) value such that the cumulative probability is just \le \alpha.

Types of error

Type I error: reject H_0 when it is true. Probability = \alpha (significance level). Type II error: fail to reject H_0 when it is false. Probability depends on the true parameter value.

These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.

D1 - Decision Mathematics 1

Unit WDM11 - Edexcel International A Level Further Mathematics (2018)

1Algorithms

Sorting and packing.
  • Bubble sort: pass through list swapping adjacent out-of-order pairs; repeat until no swaps. O(n^2).
  • Quick sort: pick a pivot; partition into < pivot and \ge pivot; recurse. Average O(n \log n).
  • Bin packing: first-fit, first-fit decreasing, full-bin - greedy heuristics; not guaranteed optimal.

2Algorithms on Graphs

Minimum spanning tree and shortest path.

Kruskal's (MST): sort edges by weight; add smallest unless it creates a cycle; stop when n-1 edges included.

Prim's (MST): start at any vertex; repeatedly add the smallest edge that connects a visited vertex to an unvisited one.

Dijkstra's (shortest path): initialise source label 0, others \infty; permanently label the smallest unlabelled; update neighbours' temporary labels; repeat.

3Route Inspection

Chinese Postman Problem.

Find the shortest closed route that traverses every edge. If graph is Eulerian (all vertex degrees even), total weight = sum of edges. If some vertices are odd: pair them up; find the pairing whose sum of shortest paths is minimum; add these repeats to the total.

4Critical Path Analysis

Activity networks; earliest/latest event times.

Represent activities on arcs (or nodes). Forward pass: compute earliest event times. Backward pass: compute latest event times. Float of an activity = latest finish − earliest start − duration. Critical path: activities with zero float; their durations sum to the project duration.

5Linear Programming

Graphical and simplex.

Graphical: plot constraints, identify feasible region, evaluate the objective at each vertex - optimum is at a vertex (or along an edge).

Simplex: introduce slack variables; construct initial tableau; pivot on the most negative bottom-row entry to improve; repeat until no negatives left in the objective row.

6Matchings

Bipartite graphs.

A matching pairs elements of one set with elements of another. To improve a partial matching, find an alternating path starting and ending at an unmatched vertex; toggle the edges along it.

Further Mathematics AI Tutor