25 MATH40003 Practice Paper 5 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: a32fc09c7102215740b74eeafac3d31150f85361044601b6edaedd8708eb837d
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: 2021 Q5, 2020 Q5 and 2025 Q4(b). It directly covers the previously under-trained high-mark determinant/minor ability chain. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: sparse matrices, eigenspace dimensions and the largest nonzero minor.

English passage 3 (17% down the source page) First key step: For the monomial matrix, permute rows to a diagonal matrix. For rank, prove “rank at least k iff some k-minor is non-zero”.

English passage 4 (46% down the source page) Marking points: 4 monomial determinant; 4 added entry; 5 characteristic factor; 7 rank theorem.

English passage 5 (49% down the source page) Common errors: Expanding a huge determinant instead of using the structure; assuming geomet- ric multiplicity automatically implies algebraic multiplicity without the basis argument; proving only one direction of the rank theorem.

English passage 6 (54% down the source page) Final self-check :Test part (b) on a 2 × 2 example; verify the block matrix has µI2 in the first two columns; identify the selected rows and columns in the rank proof.

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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: 2024 Q4(b). Directly restores the tridiagonal determinant recurrence and the 2x2 diagonalisation method. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: a tridiagonal matrix determinant dn and request for a closed form.

English passage 3 (17% down the source page) First key step: Expand along the last row, then the last column of the remaining minor, to obtain a second-order recurrence.

English passage 4 (52% down the source page) Marking points: 5 recurrence; 3 matrix form; 8 diagonalisation/closed form; 4 numerical check.

English passage 5 (54% down the source page) Common errors :Using b + c as both the diagonal and an eigenvalue; omitting d0 = 1; shifting the power by one.

English passage 6 (58% down the source page) Final self-check: Check n = 0 and n = 1 in the closed formula; substitute it into the recurrence algebraically.

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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: 2025 Q5 direct A/B coverage and 2020 Q1(f). It combines the 2025 QR/integers question with a concise O2 classification, while retaining a two-hour workload. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: apply Gram-Schmidt to three given columns, then classify a matrix group using the integer constraints on orthogonal columns.

English passage 3 (17% down the source page) first key step :Compute q1, subtract its projection from v2, then subtract both projections from v3; the R entries are qi · vj.

English passage 4 (59% down the source page) Marking points: 8 QR; 5 O2 classification; 7 signed permutation classification/group/order.

English passage 5 (61% down the source page) Common errors: Forgetting to square the norm in projection denominators; allowing a zero or two non-zero integer entries in a unit column; proving only that permutation matrices are included, not classifying all of On(Z).

English passage 6 (66% down the source page) Final self-check: Check QT Q = I and QR = A; count choices column by column; verify the two O2 forms have determinant ±1.

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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 Q6, 2023 Q4(d), 2022 Q6(d) and the finite-group counting result. This gap paper restores several rotation abilities but keeps each proof short and independent. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: the set of squares, power maps on roots of unity, permutation cycle types, counting elements of prime order, and the Cayley action.

English passage 3 (17% down the source page) First key step: Use the subgroup test for squares in an abelian group; use element orders for the power map; count order-p elements by their unique cyclic subgroups.

English passage 4 (49% down the source page) Marking points: 6 squares; 4 unit roots; 4 cycle count; 3 divisibility; 3 Cayley.

English passage 5 (51% down the source page) Common errors: Assuming (gh)2 = g2h2 in a non-abelian group; failing to divide by 2! when counting two 3-cycles; not proving distinct order-p subgroups meet trivially.

English passage 6 (55% down the source page) Final self-check: For the power map, check kernel via orders; sum the two cycle-shape counts; evaluate the Cayley embedding at the identity.