Back to coursesMATH40004 Practice Paper 2 REVISED 2
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: b9d123fac534efbd06d9ceaa6cf109321ebf300c5d0ae4788f842a92944471ea
Source date: 2026-08-06Source page 1
Source page 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (13% down the source page)
Recognition signal
Three independent convergence tests, followed by a binomial-to-antiderivative method and an iterated
chain rule.
English passage 2 (59% down the source page)
Common errors
Using a derivative-heavy route for the binomial series; giving six decimals without controlling the positive tail;
solving L = sin L before proving convergence.
Source page 3
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (13% down the source page)
Recognition signal
Exploit evenness first; for the shifted function use the Fourier shift identity rather than re-integrating.
English passage 2 (62% down the source page)
Common errors
Writing a0 as the constant term instead of a0/2; failing to expand cos(n(x −π/2)); losing the factor 2π in the
convolution theorem.
Source page 4
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (13% down the source page)
Recognition signal
Euler–Cauchy means use z = log x; a repeated eigenvalue with one eigenvector means a Jordan chain.
English passage 2 (82% down the source page)
Common errors
Missing the −Y ′ term in x2y′′; failing to multiply a resonant ansatz by z2; using a second eigenvector when
none exists.
Source page 5
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (13% down the source page)
Recognition signal
Factor the autonomous vector field. At branch intersections, compare stability on both sides before
naming the bifurcation.
Source page 6
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (38% down the source page)
Common errors
Treating every branch meeting as pitchfork; interchanging the coordinates of the saddle points; classifying by
trace alone when the determinant is negative.