MATH40007 · Practice Paper

MATH40007 Revised Historical Coverage Paper 5

Revision questions and worked solutions, presented read-only. This review interface was prepared after the recorded study period.

Status: Completed
Questions completed: 4 / 4
Suggested time: 120 minutes
Source date: 2026-08-11

Question 120 marks

Wheel-graph Laplacian and eigenbasis

Question 1 The wheel graph 𝑊5 has a central node 0 joined to four outer nodes 1–4, and the outer nodes form a cycle. 0 1 2 3 4 (a) Give the dimensions and ranks/nullities of an incidence matrix (edges by vertices). (4 marks) (b) Write the graph Laplacian and find the dimension of its null space. (4 marks) (c) Find a real orthonormal eigenbasis and all eigenvalues. (12 marks) (Total: 20 marks) 17 MATH40007 Revised Historical-Coverage Practice Pack Questions

Worked solution and marking guidance

Question 1 There are 8 edges and 5 vertices. An edge-by-vertex incidence matrix is 8 × 5, rank 4, right-null dimension 1 and left-null dimension 4. The Laplacian in node order 0 , 1, 2, 3, 4 is 𝐿 = [ 4 −1 −1 −1 −1 −1 3 −1 0 −1 −1 −1 3 −1 0 −1 0 −1 3 −1 −1 −1 0 −1 3 ] . An orthogonal eigenbasis before normalisation is (1, 1, 1, 1, 1), (0, 1, 0, −1, 0), (0, 0, 1, 0, −1), (0, 1, −1, 1, −1), (−4, 1, 1, 1, 1), with eigenvalues 0, 3, 3, 5, 5. Normalise by √ 5, √ 2, √ 2, 2, √ 20 respectively. Question difficulty allocation: 𝐴 = 8, 𝐵 = 4, 𝐶 = 4, 𝐷 = 4.

Question 220 marks

Random-walk hitting probabilities

Question 2 Consider the graph below. 𝑈 and 𝑉 are target stations, 𝑇 is a stop station and all choices are uniform. 𝐴 𝐵 𝐶 𝑈 𝑉 𝑋 𝑇 (a) Give the dimensions and nullities of an incidence matrix. (4 marks) (b) Among starting nodes 𝐴, 𝐵, 𝐶, which maximises the probability of reaching 𝑈 or 𝑉 before 𝑇? Give all three probabilities. (6 marks) (c) Starting at 𝑇, find the probability of reaching 𝑈 or 𝑉 before returning to 𝑇. (6 marks) (d) Close station 𝑉 and remove its incident edges. Starting at 𝐴, find the probability of reaching 𝐶 before returning to 𝐴. (4 marks) (Total: 20 marks) 18 MATH40007 Revised Historical-Coverage Practice Pack Questions

Worked solution and marking guidance

Question 2 There are 10 edges and 7 vertices, so the incidence matrix is 10 × 7, rank 6, right-null dimension 1 and left-null dimension 4. For hitting 𝑈 or 𝑉 before 𝑇, harmonic equations give 𝑃 𝐴 = 32 81 , 𝑃 𝐵 = 58 81 , 𝑃 𝐶 = 38 81 , 𝑃 𝑋 = 19 81 . Thus 𝐵 is best. Starting at 𝑇 and conditioning on the first step, 𝑃𝑇 = 𝑃 𝐴 + 𝑃𝐶 + 𝑃𝑋 3 = 89 243 . After closing 𝑉, solve with boundary values 𝑃 𝐴 = 0, 𝑃𝐶 = 1. The internal values are 𝑃𝐵 = 𝑃𝑈 = 1/2, 𝑃𝑇 = 3/5, 𝑃𝑋 = 4/5. After leaving 𝐴, 𝑃(hit 𝐶 before return 𝐴) = 1/2 + 1 + 3/5 3 = 7 10 . Question difficulty allocation: 𝐴 = 4, 𝐵 = 6, 𝐶 = 6, 𝐷 = 4.

Question 320 marks

Mass-scaled square-cycle vibrations

Question 3 Four masses form a square cycle of unit springs. Masses 1 and 3 have mass 1; masses 2 and 4 have mass 1/4. 1 2 34 Let 𝐾 be the cycle Laplacian, 𝑀 = diag(1, 1/4, 1, 1/4) and K̃ = 𝑀 −1/2𝐾 𝑀−1/2. (a) Write 𝐾 and K̃. (4 marks) (b) Let 𝑆 exchange nodes 1 ↔ 3 and 2 ↔ 4. Explain why 𝑆2 = 𝐼 and 𝑆K̃ = K̃ 𝑆. (4 marks) (c) Use the 𝑆 = +1 and 𝑆 = −1 subspaces to find all eigenvalues and eigenvectors ofK̃. (8 marks) (d) Give all natural frequencies and identify the rigid mode. (4 marks) (Total: 20 marks) 19 MATH40007 Revised Historical-Coverage Practice Pack Questions

Worked solution and marking guidance

Question 3 The cycle Laplacian is 𝐾 = [ 2 −1 0 −1 −1 2 −1 0 0 −1 2 −1 −1 0 −1 2 ] , and K̃ = [ 2 −2 0 −2 −2 8 −2 0 0 −2 2 −2 −2 0 −2 8 ] . The half-turn is an involution and preserves all masses and springs, so it commutes with K̃. The eigenpairs are 0 : (2, 1, 2, 1), 2 : (−1, 0, 1, 0), 8 : (0, −1, 0, 1), 10 : (−1, 2, −1, 2). Therefore the natural frequencies are 0, √ 2, 2 √ 2, √ 10 . The zero-frequency vector is the rigid translation in mass-scaled coordinates. Question difficulty allocation: 𝐴 = 8, 𝐵 = 6, 𝐶 = 2, 𝐷 = 4. 9 MATH40007 Revised Historical-Coverage Practice Pack Course-method solutions

Question 420 marks

Half-plane conductor and complex potential

Question 4 The upper half-plane is a unit conductor. The interval −1 < 𝑥 < 1 on the boundary is a grounded electrode, while the rest of the boundary is insulated. Let ℎ(𝑧) = − 𝑚 2𝜋 log ( 𝑧 + √︁ 𝑧2 − 1 ) , where the branch is chosen so that √ 𝑧2 − 1 ∼ 𝑧 at infinity. (a) Show that the electrode is grounded. (4 marks) (b) Identify the source represented by the far-field behaviour. (4 marks) (c) Find the complex current density. (4 marks) (d) Find the normal current density on the electrode and its total flux. (4 marks) (e) Find the current on the insulated boundary 𝑥 > 1 and its leading singular behaviour as 𝑥 ↓ 1. (4 marks) (Total: 20 marks) 20

Worked solution and marking guidance

Question 4 For |𝑥| < 1 approached from above, √︁ 𝑥2 − 1 = 𝑖 √︁ 1 − 𝑥2, and |𝑥 + 𝑖 √ 1 − 𝑥2| = 1, so 𝜙 = 0. At infinity ℎ(𝑧) ∼ − 𝑚 2𝜋 log(2𝑧), which supplies the net far-field current associated with strength 𝑚. Since 𝑑 𝑑𝑧 log(𝑧 + √︁ 𝑧2 − 1) = 1√ 𝑧2 − 1 , 𝐽𝑥 − 𝑖𝐽 𝑦 = 𝑚 2𝜋 √ 𝑧2 − 1 . On the electrode, 𝐽𝑥 = 0, 𝐽 𝑦 = 𝑚 2𝜋 √ 1 − 𝑥2 , so the inward flux into the conductor is ∫ 1 −1 𝐽𝑦 𝑑𝑥 = 𝑚 2 . For 𝑥 > 1, 𝐽𝑦 = 0 and 𝐽𝑥 = 𝑚 2𝜋 √ 𝑥2 − 1 ∼ 𝑚 2𝜋 √︁ 2(𝑥 − 1) . Question difficulty allocation: 𝐴 = 12, 𝐵 = 4, 𝐶 = 0, 𝐷 = 4. 10