MATH40004 Practice Paper 3 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: 44af619b8aee498701c61d7aa1bd7ee20048b88f357b99d03570e39528fdfa28
Source date: 2026-08-06

Source 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 This question mixes global graph information, nonzero-centre Taylor expansion, a Riemann sum and a parametric arc-length calculation.

English passage 2 (89% down the source page) Common errors Calling the stationary inflection at zero a maximum; using width 1/n instead of π/(4n); forgetting that an infinite number of turns can have finite length.

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 Volumes use disc/shell formulas; the rod uses mass times centre = moment; Laplace inversion uses partial fractions.

English passage 2 (59% down the source page) Common errors Using the surface-area formula instead of volume; leaving ρ0 in the centre of mass; confusing 1/(s2 + 9) with the transform of sin 3t without the factor 3.

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 Use oddness for the Fourier coefficients; use periodic boundary terms twice before applying Parseval.

English passage 2 (50% down the source page) Common errors Using the function value at the periodic jump; retaining an a0 term despite zero mean; saying E > 0 at ε = 2 without checking the n = 1 mode.

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 For the system, eigenvectors give the separatrices. For the transform integral, compute the cosine trans- form first. For Euler–Cauchy, stay in z = log x.

English passage 2 (88% down the source page) Common errors Reversing stable and unstable eigenlines; using the cosine transform directly in Plancherel without the factor two; choosing A = log z instead of integrating twice.