MATH40004 Practice Paper 6 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: 08599668e5a59df237604f70bf26d9a36555e6a635f97535d66715a774209f5a
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 Subtract the two logarithm series; divide only after the odd series has been established; prove iteration convergence before solving the limit equation.

English passage 2 (64% down the source page) Common errors Treating endpoint divergence as conditional convergence; using a first omitted term as an error bound for a positive series without bounding the full tail; solving L = tanh L first.

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 For a wire, use line mass aρ0dθ and moments. For the differentiated Fourier series, first find the contin- uous even function’s coefficients.

English passage 2 (58% down the source page) Common errors Counting the base arc twice in the net moment; expanding sin θ instead of the optimum equation; differentiating a discontinuous Fourier series without midpoint values.

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 The homogeneous system is a harmonic oscillator. For resonant forcing, multiply the usual sinusoidal ansatz by t. For an integrating factor depending on y, solve the exactness equation after multiplication.

English passage 2 (89% down the source page) Common errors Using a nonresonant ansatz; differentiating xp incorrectly when forming yp; guessing e−y instead of solving the exactness condition.

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 A singular branch is part of the domain, not a fixed-point branch. In the transform problem, the shift x −2 becomes a phase translation. For the sphere, use an order-by-order implicit expansion.

English passage 2 (88% down the source page) Common errors Calling the singular line a fixed branch; treating the collision as transcritical; losing the −2ω phase; omitting ℓ2