26 MATH40003 Practice Paper 6 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: 3fd43c85f493c93f142e4b6ff58b23f5318a218a77acadca86b3f75db218c937
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 (11% down the source page) Historical basis: 2025 Q3, 2026 Q2 and 2022 field-characteristic work. It changes the wording and cycle lengths to test entry recognition, not memory of the current paper. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: a function space plus a permutation with cycles of lengths 2,3,1; ask fixed space, differ- ence image and characteristic exceptions.

English passage 3 (19% down the source page) First key step: Use delta coordinates. Analyse I − T separately on each cycle of τ .

English passage 4 (47% down the source page) Marking points: 4 isomorphism/basis; 4 operator matrix; 4 fixed space; 4 image/rank; 4 character- istic intersection.

English passage 5 (51% down the source page) Common errors: Using τ instead of τ −1 in the matrix; assuming the fixed space has dimension one; dividing by the cycle length without checking characteristic.

English passage 6 (55% down the source page) Final self-check: Check one basis delta function from each cycle; verify the sum constraints; substitute the characteristic intersection vectors.

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 (11% down the source page) Historical basis: 2020 Q4(a), 2021 idempotents and 2026 characteristic exceptions. The polynomial has three distinct roots only when 2 is invertible. 30 minutes.

English passage 2 (14% down the source page) Recognition signal: an operator satisfying A3 = A and the field condition that 2 is invertible.

English passage 3 (17% down the source page) First key step: Factor x3 − x = x(x − 1)(x + 1) and construct the three Lagrange interpolation projectors.

English passage 4 (53% down the source page) Marking points: 4 A2 facts; 10 projectors; 3 direct sum/diagonalisation; 3 characteristic-two exam- ple.

English passage 5 (57% down the source page) Common errors: Using the projectors when 2 is not invertible; proving the images are con- tained in eigenspaces but not the reverse; claiming repeated eigenvalue alone means non-diagonalisable.

English passage 6 (62% down the source page) Final self-check: Expand every projector polynomial using A3 = A; apply P+ to a vector with Av = v; square the characteristic-two matrix.

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 (11% down the source page) Historical basis: 2026 Q3(a), 2021 Q4(b), 2022 Q4(c) and 2025 Q5(a). It mixes three historical entry signals while preserving the 10/10 computation-proof balance. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: symmetric matrix followed by unit-sphere norm and a general QR existence proof.

English passage 3 (19% down the source page) First key step: Use the evident eigenvector (1, 1, 1) and its orthogonal complement; for QR, apply Gram-Schmidt to the columns.

English passage 4 (51% down the source page) Marking points: 6 diagonalisation; 3 square-root factor; 5 norm theorem; 6 QR existence.

English passage 5 (54% down the source page) Common errors: Using non-orthogonal eigenvectors in Q; confusing the matrix square root with entrywise square roots; failing to explain why R is upper triangular.

English passage 6 (57% down the source page) Final self-check: Check ui·uj = δij; check eigenpairs; verify each column of QR is the corresponding original column.

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 (10% down the source page) Historical basis: 2025 Q6 direct A/B coverage, with the added Z2 entry from 2023 Q6. It is the final transfer test for reading group descriptions. 30 minutes.

English passage 2 (15% down the source page) Recognition signal: concrete permutation products, maximum order, cyclicity of four groups, and a simplified subgroup test under a finite-order assumption.

English passage 3 (17% down the source page) First key step: For the product, apply the rightmost cycle first. For cyclicity , use a structural invariant: commutativity, rank, or an explicit generator.

English passage 4 (43% down the source page) Marking points: 4 product/order/sign; 4 maximum order; 5 cyclicity; 7 subgroup theorem.

English passage 5 (45% down the source page) Common errors: Applying permutation factors left-to-right; saying S5 is not cyclic only because it lacks an element of order 120 without justifying possible orders; forgetting integer negative powers in the shear group.

English passage 6 (51% down the source page) Final self-check: Reconstruct the product on all six points; check the least common multiple for the relevant partitions; multiply two shear matrices.