MATH40004 Practice Paper 1 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: 96aeead49f49540eb812a1702f41cc4f9731501781202ab823909cd6d20036ef
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 Taylor theorem + a quotient whose numerator has a known odd power series. The first decisive step is to expand arctan x, not to differentiate the quotient repeatedly.

English passage 2 (77% down the source page) Common errors Forgetting endpoint convergence of the arctangent series; claiming decimal accuracy without an error theorem; differentiating g repeatedly instead of reading coefficients.

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 A geometric loss law plus a bulk first-order loss. The key substitution is the geometric length scale u = V 1/3.

English passage 2 (74% down the source page) Common errors Using V ′ = 4πR2R′ and then losing a factor; cubing a negative u after extinction; using the point value rather than the midpoint value at a Fourier jump.

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 A constant 2 × 2 system: calculate eigenvalues first. For the forcing, split the ansatz into an et part and a constant part.

English passage 2 (89% down the source page) Common errors Calling the spiral unstable because it rotates; using one ansatz for both forcing frequencies; differentiating xmynM without the product rule.

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 One-dimensional autonomous dynamics: factor the right-hand side, make a sign line for each parameter range, then place the branches in the bifurcation diagram.

English passage 2 (86% down the source page) Common errors Confusing stable/unstable branch line style; forgetting the transform rule xy ↔iby′; solving for zx before checking Fz̸ = 0.