24 MATH40003 Practice Paper 4 ANSWERS

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: 8081cc3c65f0ffdc30e1649ca8d12fe60553750a58b3df76bf8c7f2efadd9658
Source date: 2026-08-06

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Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (11% down the source page) Historical basis :2020 Q2, 2021 Q1 and 2022 Q1. It directly trains one row reduction across different fields and the affine-coset theorem. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: solve the same system over several fields, then explain the general solution-set structure .

English passage 3 (17% down the source page) first key step :Perform row operations over the integers as far as possible before reducing modulo the field characteristic.

English passage 4 (46% down the source page) Marking points :8 field solutions; 6 fibre/coset theorem; 6 rank criterion and dimension.

English passage 5 (48% down the source page) Common errors :Dividing by 6 in F2 or F3; treating a finite-field free parameter as infinitely many real values; asserting the coset form without first choosing a particular solution.

English passage 6 (52% down the source page) Final self-check :Substitute each solution family into the original equations in the appropriate field; verify the affine dimension equals the number of free variables.

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Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (11% down the source page) Historical basis :2021 Q3 and 2022 Q3. The main methods are direct linearity/idempotence checks, image-kernel decomposition, the independent-chain proof for nilpotent degree, and finite geo- metric series. 30 minutes.

English passage 2 (16% down the source page) Recognition signal: the appearance of T 2 = T or Nm = 0.

English passage 3 (18% down the source page) first key step :For idempotents, write v = Tv + (v −Tv). For nilpotents, use v, Nv, . . . , N m−1v and apply powers of N successively.

English passage 4 (49% down the source page) Marking points :6 classification; 5 idempotent decomposition; 5 nilpotent degree; 4 explicit in- verses.

English passage 5 (53% down the source page) Common errors :Calling a non-linear map idempotent under the course definition; claiming im T = ker T for a non-zero idempotent; applying only one power of N and concluding all coeffi- cients vanish.

English passage 6 (58% down the source page) Final self-check :Check every map sends 0 correctly; multiply the finite series; test the shift on bn.

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Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (11% down the source page) Historical basis :2021 Q4. Course method: diagonalise the 2 × 2 recurrence matrix and use the or- thonormal eigenbasis of a symmetric matrix for norm estimates. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: a two-dimensional recurrence plus a general real symmetric norm statement.

English passage 3 (17% down the source page) first key step :Write (an, bn)T = Mn−1(a1, b1)T . For the norm theorem, expand an arbitrary unit vector in an orthonormal eigenbasis.

English passage 4 (59% down the source page) Marking points :10 recurrence; 2 Pythagoras; 8 norm extrema.

English passage 5 (61% down the source page) Common errors :Using Mn instead of Mn−1; forgetting to normalise eigenvectors in the spectral theorem; proving only inequalities and not attainment.

English passage 6 (65% down the source page) Final self-check :Check the closed formula at n = 1; run one recurrence step; plug a unit eigen- vector into the extrema formula.

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Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (11% down the source page) Historical basis :2020 Q6 and 2024 Q6(c). The methods are cyclic-group arithmetic, Lagrange, the kernel/coset criterion, and standard explicit homomorphisms. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: finite cyclic groups, element orders, coprime-order subgroups, homomorphism kernels and cosets .

English passage 3 (17% down the source page) first key step :Use ord(gk) = 12/ gcd(12, k), and for the fibre statement rewrite equality as x−1y ∈ker ϕ.

English passage 4 (53% down the source page) Marking points :5 order distribution; 4 subgroup product; 5 kernel/fibres; 3 prime-order injection; 3 examples.

English passage 5 (57% down the source page) Common errors :Listing subsets that are not subgroups; assuming AB = G from orders without proving injectivity or using the product formula; using inverse or transpose as a homomor- phism on GL2 without checking order reversal.

English passage 6 (62% down the source page) Final self-check :Sum the element counts to 12; check the intersection order; multiply the shear matrices explicitly.