BS-1 — Mathematics
Vector and Linear Algebra: Applications involving matrix algebra, determinants, eigenvalues and eigenvectors, vector functions and operations, orthogonal curvilinear coordinates. Calculus: first and second order linear ordinary differential equations, series solutions of ordinary differential equations, applications of partial derivatives, Lagrange multipliers, multiple integrals, line and surface integrals, integral theorems (Gauss, Green, Stokes). Power series.
Subject outline
BS-1 Mathematics is a Basic Studies technical exam (PEO National Examination 04-BS-1) on the engineering mathematics every later paper leans on: first-order differential equations and how to choose a method for one, constant-coefficient linear equations including the resonant case, systems of linear differential equations by eigenvalues, Laplace transforms with piecewise and pulsed forcing, gradients, tangent planes and linear approximation, multiple integrals and the choice of coordinates, line integrals and work, surface integrals and the divergence theorem, and constrained optimisation by Lagrange multipliers.
The paper is 3 hours and closed book — candidates may bring one aid sheet, written on both sides, together with an approved Casio or Sharp calculator. Any five of the eight questions constitute a complete paper — only the first five as they appear in the answer book are marked — and all questions are of equal value, 20 marks each.
What this course gives you
Worked solutions — all sixteen questions from May 2019 and December 2019, fully retypeset as live maths, with the final answers pulled out ahead of the working. Published solutions exist for both sittings and every one of them was recomputed: twelve published answers did not survive checking and are corrected here, each verified twice — once symbolically and once numerically. Six of the twelve are wrong in ways that change the result completely: a resonance case solved as if it were not one, a system solution that does not satisfy its own initial conditions, a volume integral that ignores the cone it is supposed to be bounded by (the same question appears in both papers with two different, and two different wrong, answers), and a Laplace response with its step functions missing. Nothing is silently overwritten: the published figure is kept, either in the final-answer table or quoted in the correction itself, and a headed correction set into the working at the exact step that fails explains what went wrong and gives the corrected value. Study notes — ten condensed topics, from choosing a method for a first-order equation through constant-coefficient equations, linear systems, Laplace transforms, gradients and tangent planes, multiple, line and surface integrals and Lagrange multipliers to a closing topic on checking your own answer, each pointing back at the exam questions it answers. Practice & case studies — ten multiple-choice questions and five longer problems — two of them engineering scenarios — with worked answers for every one. Formula sheet — two pages, which is exactly one aid sheet written on both sides: every relation both papers actually use, and nothing else, so it doubles as a draft of what to write on the sheet you take in. Printable resources — ten reference sheets — derivative, integral and Laplace transform tables, an ODE method selector, a vector-calculus reference, coordinate systems and Jacobians, trigonometric identities, series expansions, matrices and eigenvalues, and an exam-day checklist — plus four blank exam stationery templates. Practice papers — the two question papers on their own, unmarked, for a timed run.