Level 6 – A level further pure maths and statistics full justification
This is an A level further maths proof companion consisting of documents and videos where intuition or proofs are provided for all results in A level further maths pure and statistics modules and A level maths statistics modules used as black boxes, some of which you may not have even realized required justification—when that happens, I explain why justification is needed. Many steps often skipped over in textbooks are justified in this level.
You would benefit from this if one of the following (or equivalent for a non‑British system) describes you:
A level further maths students seeking to go beyond the curriculum
A level further maths students interested in better understanding the material
A level further maths students interested to understand the “why” for results or formulas or methods that are not typically properly justified in A level textbooks or their exercises but still used in A level further maths
Students aiming for an A or an A* in A level further maths
Anyone considering pursuing maths at university
Undergraduate maths students
Maths teachers or prospective maths teachers (A level teachers in particular)
SOURCES: Cambridge notes, math stack exchange, statproofbook, wikipedia
Here are the documents and videos for this level.
Further pure maths:
Table of Contents [Expand]
Page 1 - Polar coordinate integration
Page 1 - Some power series properties
Page 3 - Differential equations solution form justification (Series solutions on page 5)
Page 9 - Vector and matrix stuff (Distance between point and plane, matrix transpose identities, cross product properties, symmetric matrix properties and properties of their eigenvector matrices, cayley hamilton theorem)
Page 13 - A tricky integral problem
Page 13 - Conic section properties
Page 16 - Recurrence relations
Video 1 - Why matrix determinant is volume, matrix inverse formula works
Video 2 - Why l’hopital’s rule works (geometric argument)
Page 7 - Binomial and poisson distribution formulae
Page 8 - Intuition for continuous distribution expectations and integrals
Page 9 - Mean and variance of geometric and negative binomial distributions
Page 11 - Some generating function properties
Page 12 - Sum of normals is normal
Page 14 - Visual proof of unbiased variance estimator
Page 14 - Residual sum of squares formula
After this things get very complicated, as it is difficult the central limit theorem and chi squared tables. Therefore, it is recommended you stop here, read the technical results document which the rest relies on, then come back, as the rest will depend on results from there.
Page 14 - Characteristic functions (main ingredient for CLT proof)
Page 19 - Central limit theorem (we also get normal approximation for free from this)
Page 23 - Chi squared tests
Page 39 - The formula involving s, sigma and chi squared