MATH40007 Seven Year Effective Coverage Matrix REVISED

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: 64e9cae7b1c3700efc8e466138e99272a6914340593aa5d77502c6ed234de951
Source date: 2026-08-06

Summary

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

MATH40007 Revised Historical-Coverage Audit

Metric	Value	Core correction
Official question-level ability units	28	2026 determines difficulty, not content slots.
Old effective A/B units	16	2020-2026 determines knowledge, entry cues and method chains.
Old label-only/missing C/D units	12	C = label-only false coverage; excluded from effective coverage.
New effective A/B units	28	Only Paper 1 remains a 2026-structure reinforcement paper.
Paper-level difficulty	A32 / B20 / C12 / D16	Papers 2-6 rotate historical ability families.

OfficialAbilityUnits

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Year	Question	Ability unit / subparts	Recognition cue	First key step	Course method chain	Old grade	New paper coverage	New grade
2020	Q1	Circuit Laplacian; Dirichlet voltages; unit-current response	large symmetric network / terminal data	write KCL or Lx=f	Laplacian -> symmetry -> voltages -> currents	C	Paper 4 Q1	B
2020	Q2	Random walk with competing targets and changed graph	journey stops at target/return	set harmonic boundary values	harmonic equations -> first-step conditioning	C	Paper 4 Q1 + Paper 5 Q2	B
2020	Q3	Four-mass spring network; constrained modes; modal equilibrium	fixed nodes and external forces	form reduced K/M	generalised eigenproblem -> modal expansion	B	Paper 5 Q3 + Paper 6 Q2	B
2020	Q4	First-quadrant complex potential; two grounded walls; boundary flux	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	B	Paper 6 Q1	A
2021	Q1	Cycle graph; circulant eigenvectors; effective conductance; voltage expansion	cycle/circulant symmetry	write Fourier modes	circulant spectrum -> modal voltage -> conductance	C	Paper 2 Q1 + Paper 3 Q3	B
2021	Q2	Subway random walk; escape and hitting a set	uniform choices on graph	assign hitting probabilities	harmonic system -> starting-edge average	C	Paper 2 Q2 + Paper 5 Q2	A
2021	Q3	Incidence and Laplacian null spaces; symmetry-built eigenvectors	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	C	Paper 5 Q1	A
2021	Q4	Upper-half-plane source/sink; insulated boundary; cross-line current	source-sink image form	differentiate h(z)	boundary component -> line integral -> conservation	B	Paper 3 Q4	A
2022	Q1	Grounded path Green matrix; reciprocity; inverse reduced Laplacian	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	B	Paper 4 Q2	A
2022	Q2	Incidence nullity; random-walk escape; internal hitting probability	graph plus multiple absorbing events	set different boundary values for each part	rank-nullity -> harmonic equations -> first-step average	C	Paper 2 Q2	A
2022	Q3	Six-node block Laplacian; symmetric/antisymmetric eigenvectors	two repeated blocks	try (e,e) and (e,-e)	base spectrum -> block lifting	B	Paper 4 Q3	A
2022	Q4	Circular electrode near grounded wall; limiting point source	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	B	Paper 4 Q4	A
2023	Q1	Tree conductance recurrence; generation voltages; infinite profiles	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	B	Paper 3 Q2	A
2023	Q2	Large ladder/chain spectral expansion and effective conductance	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	C	Paper 4 Q2 + Paper 6 Q4	B
2023	Q3	Non-uniform mass-spring generalised eigenproblem; symmetry reduction	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	D	Paper 5 Q3	A
2023	Q4	Grounded slit/electrode complex map; edge singularity	square-root branch at electrode endpoint	choose branch and check boundary	log derivative -> source check -> endpoint asymptotics	C	Paper 5 Q4	A
2024	Q1	Parameter-weighted circuit; limiting regimes; network substitution	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	B	Paper 3 Q1	A
2024	Q2	Incidence/Laplacian null spaces; edge deletion; random walk	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	C	Paper 5 Q1 + Paper 2 Q2	B
2024	Q3	Complete graph spectrum; effective conductance; voltage expansion	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	C	Paper 3 Q3	A
2024	Q4	Multiple sources in half-plane; wall-current graph and total flux	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	B	Paper 3 Q4	A
2025	Q1	Tridiagonal/circulant spectra; Thomson minimisation and check	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	B	Paper 2 Q1	A
2025	Q2	Incidence dimensions; escape probability; internal hitting Q2(c)	same graph, different absorbing sets	reset boundary values for each part	incidence rank -> distinct harmonic systems	C	Paper 2 Q2	A
2025	Q3	Two-mass equilibrium, reaction, frequencies and full perturbation	equilibrium then perturbation	separate x=x_e+hat{x}	force balance -> eigenmodes -> initial-value projection	B	Paper 2 Q3	A
2025	Q4	Piecewise 1D conductor with source jump; interfaces; no-current condition	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	B	Paper 2 Q4	A
2026	Q1	Four fundamental spaces plus weighted circuit	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	A	Paper 1 Q1	A
2026	Q2	Two conductance reductions plus escape-before-return	two networks and random walk	reduce networks then use C/degree	series-parallel -> effective C -> escape identity	A	Paper 1 Q2	A
2026	Q3	Forced path diffusion; steady/transient modal solution	linear diffusion with boundaries	subtract steady state	K eigensystem -> initial deviation -> exponentials	A	Paper 1 Q3	A
2026	Q4	Uniform electrothermal rod and insulated-end variant	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	A	Paper 1 Q4	A

OldMockDisposition

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Old file role	Question	Old content	Disposition	Reason
Old Paper 1	Q1	2026 subspaces/circuit	RETAIN as 2026 reinforcement	Directly serves the single permitted 2026-structure role.
Old Paper 1	Q2	2026 conductance/random walk	RETAIN as 2026 reinforcement	Directly serves the single permitted 2026-structure role.
Old Paper 1	Q3	2026 diffusion	RETAIN as 2026 reinforcement	Directly serves the single permitted 2026-structure role.
Old Paper 1	Q4	2026 electrothermal rod	RETAIN as 2026 reinforcement	Directly serves the single permitted 2026-structure role.
Old Paper 2	Q1	incidence/circuit	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 2	Q2	bridge/random walk	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 2	Q3	spring modes	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 2	Q4	half-plane source	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 3	Q1	spectra/Thomson	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 3	Q2	tree/random walk	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 3	Q3	block spectrum	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 3	Q4	piecewise rod	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 4	Q1	inverse Laplacian/circuit	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 4	Q2	bridge/random walk	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 4	Q3	forced diffusion	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 4	Q4	complex half-plane	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 5	Q1	subspaces/circuit	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 5	Q2	ladder/random walk	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 5	Q3	coupled oscillation	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 5	Q4	variable rod	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 6	Q1	incidence/circuit	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 6	Q2	cube/random walk	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 6	Q3	forced diffusion	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.
Old Paper 6	Q4	source-sink potential	REPLACE/REMAP	Usable mathematics, but the system fixed the same four 2026 slots and overcounted label coverage.

NewSixPaperMap

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Paper	Role	Historical basis	Coverage purpose
1	2026 structure reinforcement	2026 Q1-Q4	Controlled familiarity only; explicitly not seven-year coverage
2	2025 historical bridge	2025 Q1-Q4	Direct coverage of spectra/Thomson, internal hitting, springs, piecewise conduction
3	2023-2024 historical abilities	2023 Q1; 2024 Q1,Q3,Q4	Tree recurrence, parameter circuits, complete-graph spectrum, multi-source potential
4	2020-2022 historical abilities	2020 Q1; 2022 Q1,Q3,Q4	Large network response, Green matrix, block lifting, electrode limit
5	Seven-year gap closing	2021 Q3; 2023 Q3,Q4; 2024 Q2	Wheel nullspaces, generalised modes, slit singularity, changed-graph hitting
6	Mixed transfer pressure test	2020 Q4; 2021/22/23 mixed	Topic order deliberately breaks the 2026 slot-recognition habit

DifficultyCalibration

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Paper	A marks	B marks	C marks	D marks	Total	Role
1	32	20	12	16	80	2026 structure reinforcement
2	32	20	12	16	80	2025 historical bridge
3	32	20	12	16	80	2023-2024 historical abilities
4	32	20	12	16	80	2020-2022 historical abilities
5	32	20	12	16	80	gap closing
6	32	20	12	16	80	mixed transfer pressure test
Six-paper total	192	120	72	96

VersionMap

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Old artifact	Status	Replacement	Website instruction
MATH40007_All_6_Questions_FINAL_V6.pdf	OBSOLETE - over-anchored	MATH40007_6_Papers_REVISED_HISTORICAL_COVERAGE_Questions.pdf	Remove from active whitelist; preserve in quarantine
MATH40007_All_6_Solutions_FINAL_V6.pdf	OBSOLETE - over-anchored	MATH40007_6_Papers_COURSE_METHOD_AUDITED_Solutions.pdf	Remove from active whitelist; preserve in quarantine
MATH40007_Six_Paper_Coverage_FINAL.pdf	OBSOLETE - label coverage	MATH40007_Seven_Year_Effective_Coverage_Matrix.xlsx	Do not display old coverage claim
MATH40007_Coverage_and_Use_Plan.pdf	HISTORICAL RECORD	MATH40007_Over_Anchoring_Diagnosis_and_Correction.pdf	Keep only as audit trail
MATH40007 Q2(c) Specialist practice package	MERGED	Revised Paper 2 Q2	Retain only if clearly labelled optional bridge
Tony Paper 1 marking report	RETAIN	unchanged	Keep in feedback section

OfficialSubpartMap

Source cell order and numeric values are retained; prose labels are translated. Scroll within the table to view all rows and columns.

Year	Question	Part	Marks	Subpart ability	Recognition cue	First key step	Course method chain	New paper coverage	New grade
2020	Q1	a	4	graph Laplacian	large symmetric network / terminal data	write KCL or Lx=f	Laplacian -> symmetry -> voltages -> currents	Paper 4 Q1	B
2020	Q1	b	8	Dirichlet voltages and effective conductance	large symmetric network / terminal data	write KCL or Lx=f	Laplacian -> symmetry -> voltages -> currents	Paper 4 Q1	B
2020	Q1	c	8	unit-current response and edge current	large symmetric network / terminal data	write KCL or Lx=f	Laplacian -> symmetry -> voltages -> currents	Paper 4 Q1	B
2020	Q2	a	6	best starting station for target before stop	journey stops at target/return	set harmonic boundary values	harmonic equations -> first-step conditioning	Paper 4 Q1 + Paper 5 Q2	B
2020	Q2	b	6	escape probability after leaving start	journey stops at target/return	set harmonic boundary values	harmonic equations -> first-step conditioning	Paper 4 Q1 + Paper 5 Q2	B
2020	Q2	c	8	changed graph after station closure	journey stops at target/return	set harmonic boundary values	harmonic equations -> first-step conditioning	Paper 4 Q1 + Paper 5 Q2	B
2020	Q3	a	4	mass-spring graph Laplacian	fixed nodes and external forces	form reduced K/M	generalised eigenproblem -> modal expansion	Paper 5 Q3 + Paper 6 Q2	B
2020	Q3	b	6	fixed-mass natural modes	fixed nodes and external forces	form reduced K/M	generalised eigenproblem -> modal expansion	Paper 5 Q3 + Paper 6 Q2	B
2020	Q3	c	6	modal equilibrium setup	fixed nodes and external forces	form reduced K/M	generalised eigenproblem -> modal expansion	Paper 5 Q3 + Paper 6 Q2	B
2020	Q3	d	4	equilibrium displacements	fixed nodes and external forces	form reduced K/M	generalised eigenproblem -> modal expansion	Paper 5 Q3 + Paper 6 Q2	B
2020	Q4	a	5	harmonicity and source	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	Paper 6 Q1	A
2020	Q4	b	2	grounded x-axis boundary	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	Paper 6 Q1	A
2020	Q4	c	2	grounded y-axis boundary	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	Paper 6 Q1	A
2020	Q4	d	5	boundary current component	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	Paper 6 Q1	A
2020	Q4	e	6	integrated boundary flux	log potential in quadrant	check boundary modulus	differentiate log -> boundary current -> integrate flux	Paper 6 Q1	A
2021	Q1	a	3	cycle Laplacian	cycle/circulant symmetry	write Fourier modes	circulant spectrum -> modal voltage -> conductance	Paper 2 Q1 + Paper 3 Q3	B
2021	Q1	b	5	circulant eigenvectors/eigenvalues	cycle/circulant symmetry	write Fourier modes	circulant spectrum -> modal voltage -> conductance	Paper 2 Q1 + Paper 3 Q3	B
2021	Q1	c	5	effective conductance	cycle/circulant symmetry	write Fourier modes	circulant spectrum -> modal voltage -> conductance	Paper 2 Q1 + Paper 3 Q3	B
2021	Q1	d	7	voltage eigen-expansion	cycle/circulant symmetry	write Fourier modes	circulant spectrum -> modal voltage -> conductance	Paper 2 Q1 + Paper 3 Q3	B
2021	Q2	a	5	hit A before B from X	uniform choices on graph	assign hitting probabilities	harmonic system -> starting-edge average	Paper 2 Q2 + Paper 5 Q2	A
2021	Q2	b	8	escape A to B before return	uniform choices on graph	assign hitting probabilities	harmonic system -> starting-edge average	Paper 2 Q2 + Paper 5 Q2	A
2021	Q2	c	4	hit any of B,Y,Z before return	uniform choices on graph	assign hitting probabilities	harmonic system -> starting-edge average	Paper 2 Q2 + Paper 5 Q2	A
2021	Q2	d	3	qualitative probability comparison	uniform choices on graph	assign hitting probabilities	harmonic system -> starting-edge average	Paper 2 Q2 + Paper 5 Q2	A
2021	Q3	a	2	incidence left-null dimension	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	Paper 5 Q1	A
2021	Q3	b	2	Laplacian null dimension	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	Paper 5 Q1	A
2021	Q3	c	3	explicit Laplacian	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	Paper 5 Q1	A
2021	Q3	d	10	four symmetry-form eigenvectors	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	Paper 5 Q1	A
2021	Q3	e	3	remaining two eigenvectors	incidence/nullity request	use rank-nullity and graph components	null spaces -> explicit L -> symmetry eigenspaces	Paper 5 Q1	A
2021	Q4	a	4	identify point source	source-sink image form	differentiate h(z)	boundary component -> line integral -> conservation	Paper 3 Q4	A
2021	Q4	b	5	no-flux boundary	source-sink image form	differentiate h(z)	boundary component -> line integral -> conservation	Paper 3 Q4	A
2021	Q4	c	5	current across finite y-axis interval	source-sink image form	differentiate h(z)	boundary component -> line integral -> conservation	Paper 3 Q4	A
2021	Q4	d	6	total cross-line current	source-sink image form	differentiate h(z)	boundary component -> line integral -> conservation	Paper 3 Q4	A
2022	Q1	a.i	2	node-1 unit-voltage response	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	a.ii	2	node-1 effective conductance	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	b.i	2	node-2 unit-voltage response	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	b.ii	2	node-2 effective conductance	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	c.i	2	node-3 unit-voltage response	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	c.ii	2	node-3 effective conductance	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	d	4	reciprocity identity	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q1	e	4	Green matrix/inverse reduced Laplacian	multiple unit-voltage experiments	write reduced K	response columns -> symmetry reciprocity -> K inverse	Paper 4 Q2	A
2022	Q2	a	4	incidence dimensions/nullity	graph plus multiple absorbing events	set different boundary values for each part	rank-nullity -> harmonic equations -> first-step average	Paper 2 Q2	A
2022	Q2	b	8	escape N to S before return	graph plus multiple absorbing events	set different boundary values for each part	rank-nullity -> harmonic equations -> first-step average	Paper 2 Q2	A
2022	Q2	c	8	internal hitting N before S	graph plus multiple absorbing events	set different boundary values for each part	rank-nullity -> harmonic equations -> first-step average	Paper 2 Q2	A
2022	Q3	a	2	explicit block-graph Laplacian	two repeated blocks	try (e,e) and (e,-e)	base spectrum -> block lifting	Paper 4 Q3	A
2022	Q3	b	3	block form	two repeated blocks	try (e,e) and (e,-e)	base spectrum -> block lifting	Paper 4 Q3	A
2022	Q3	c	5	base 3x3 eigensystem	two repeated blocks	try (e,e) and (e,-e)	base spectrum -> block lifting	Paper 4 Q3	A
2022	Q3	d	10	symmetric/antisymmetric eigenvectors	two repeated blocks	try (e,e) and (e,-e)	base spectrum -> block lifting	Paper 4 Q3	A
2022	Q4	a	3	grounded wall	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2022	Q4	b	6	constant-voltage circular electrode	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2022	Q4	c	3	wall current density	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2022	Q4	d	2	maximum current point	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2022	Q4	e	3	total wall current	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2022	Q4	f	3	point-source limiting process	Möbius log and circle boundary	identify constant-modulus boundary	geometry -> derivative -> wall current -> flux -> limit	Paper 4 Q4	A
2023	Q1	a	1	tree incidence dimensions	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q1	b.i	4	finite tree effective conductance	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q1	b.ii	6	generation voltages	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q1	b.iii	1	infinite unit-conductance limit	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q1	c	4	profile c_j=1/j	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q1	d	4	profile c_j=j	regular branching by generation	replace each layer by parallel resistance	layer resistances -> series sum -> limits	Paper 3 Q2	A
2023	Q2	a	2	block matrix construction for ladder	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	Paper 4 Q2 + Paper 6 Q4	B
2023	Q2	b	4	spectral coefficients	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	Paper 4 Q2 + Paper 6 Q4	B
2023	Q2	c	6	two-parameter recurrence representation	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	Paper 4 Q2 + Paper 6 Q4	B
2023	Q2	d	4	closed-form response	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	Paper 4 Q2 + Paper 6 Q4	B
2023	Q2	e	4	effective conductance	long structured graph	use supplied sine/circulant modes	spectral coefficients -> voltage/current observable	Paper 4 Q2 + Paper 6 Q4	B
2023	Q3	a	1	symmetry matrix square	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q3	b	4	symmetry eigensystem	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q3	c	3	mass-scaled eigenproblem	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q3	d	2	commutation calculation	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q3	e	2	commuting-eigenvector argument	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q3	f	8	natural frequencies	unequal masses and commuting symmetry	mass-scale K and split parity spaces	M^-1/2 K M^-1/2 -> symmetry -> modes	Paper 5 Q3	A
2023	Q4	a	6	grounded slit electrode	square-root branch at electrode endpoint	choose branch and check boundary	log derivative -> source check -> endpoint asymptotics	Paper 5 Q4	A
2023	Q4	b	6	verify finite point source	square-root branch at electrode endpoint	choose branch and check boundary	log derivative -> source check -> endpoint asymptotics	Paper 5 Q4	A
2023	Q4	c	4	complex current density	square-root branch at electrode endpoint	choose branch and check boundary	log derivative -> source check -> endpoint asymptotics	Paper 5 Q4	A
2023	Q4	d	4	endpoint singularity	square-root branch at electrode endpoint	choose branch and check boundary	log derivative -> source check -> endpoint asymptotics	Paper 5 Q4	A
2024	Q1	a	2	weighted Laplacian	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q1	b	2	c to zero limit	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q1	c	2	c to infinity limit	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q1	d	5	internal voltages as functions of c	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q1	e	4	general effective conductance	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q1	f	5	substitution into six-node circuit	conductance parameter c	solve KCL symbolically	weighted L -> limits -> exact C(c) -> substitute module	Paper 3 Q1	A
2024	Q2	a	2	incidence left-null dimension	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	Paper 5 Q1 + Paper 2 Q2	B
2024	Q2	b	2	Laplacian null dimension	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	Paper 5 Q1 + Paper 2 Q2	B
2024	Q2	c	2	edge deletion raising components	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	Paper 5 Q1 + Paper 2 Q2	B
2024	Q2	d	7	escape from W	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	Paper 5 Q1 + Paper 2 Q2	B
2024	Q2	e	7	hit E before W from X	topology change after deleting edge	count components/cycles first	rank-nullity -> modified graph -> harmonic probabilities	Paper 5 Q1 + Paper 2 Q2	B
2024	Q3	a	2	complete-graph Laplacian	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	Paper 3 Q3	A
2024	Q3	b	4	effective conductance	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	Paper 3 Q3	A
2024	Q3	c	6	real orthonormal eigensystem	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	Paper 3 Q3	A
2024	Q3	d	6	voltage expansion under unit current	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	Paper 3 Q3	A
2024	Q3	e	2	conductance check	complete graph symmetry	use L=nI-J	orthogonal complement spectrum -> current expansion	Paper 3 Q3	A
2024	Q4	a	2	grounded wall	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	b	2	source locations and strengths	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	c	3	complex current density	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	d	1	zero tangential/normal component	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	e	2	wall-current formula	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	f	4	total wall current	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2024	Q4	g	6	wall-current sketch	product/ratio of image factors	locate every log singularity	differentiate -> boundary restriction -> integrate and sketch	Paper 3 Q4	A
2025	Q1	a.i	1	tridiagonal rank	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	a.ii	3	tridiagonal eigenvalues	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	b.i	1	circulant rank	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	b.ii	4	circulant eigenvalues	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	c.i	4	Thomson minimisation	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	c.ii	3	equivalent conductance	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q1	c.iii	4	energy/conductance check	two matrix structures plus current energy	use standard spectra then parameterise flows	spectrum -> energy minimisation -> equivalent circuit check	Paper 2 Q1	A
2025	Q2	a.i	1	incidence dimensions	same graph, different absorbing sets	reset boundary values for each part	incidence rank -> distinct harmonic systems	Paper 2 Q2	A
2025	Q2	a.ii	2	right/left null dimensions	same graph, different absorbing sets	reset boundary values for each part	incidence rank -> distinct harmonic systems	Paper 2 Q2	A
2025	Q2	b	8	escape W to S	same graph, different absorbing sets	reset boundary values for each part	incidence rank -> distinct harmonic systems	Paper 2 Q2	A
2025	Q2	c	9	internal hitting W before S from N	same graph, different absorbing sets	reset boundary values for each part	incidence rank -> distinct harmonic systems	Paper 2 Q2	A
2025	Q3	a	4	static equilibrium	equilibrium then perturbation	separate x=x_e+hat{x}	force balance -> eigenmodes -> initial-value projection	Paper 2 Q3	A
2025	Q3	b	2	ceiling reaction	equilibrium then perturbation	separate x=x_e+hat{x}	force balance -> eigenmodes -> initial-value projection	Paper 2 Q3	A
2025	Q3	c	6	natural frequencies	equilibrium then perturbation	separate x=x_e+hat{x}	force balance -> eigenmodes -> initial-value projection	Paper 2 Q3	A
2025	Q3	d	8	full perturbation solution	equilibrium then perturbation	separate x=x_e+hat{x}	force balance -> eigenmodes -> initial-value projection	Paper 2 Q3	A
2025	Q4	a	3	piecewise differential equations/interfaces	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	Paper 2 Q4	A
2025	Q4	b	8	piecewise voltage solution	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	Paper 2 Q4	A
2025	Q4	c	3	zero-current condition	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	Paper 2 Q4	A
2025	Q4	d	2	specialised solution	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	Paper 2 Q4	A
2025	Q4	e	4	voltage sketch	piecewise F(x) and interface	integrate J'=F and enforce continuity	piecewise integration -> interface -> parameter condition -> sketch	Paper 2 Q4	A
2026	Q1	a.i	1	column space	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	a.ii	3	right null space	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	a.iii	1	row space	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	a.iv	3	left null space	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	b.i	4	internal voltage	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	b.ii	4	edge current and direction	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q1	b.iii	4	effective conductance	subspaces and small circuit	rank-nullity then KCL	linear spaces -> harmonic voltages -> current	Paper 1 Q1	A
2026	Q2	a	8	effective conductance network 1	two networks and random walk	reduce networks then use C/degree	series-parallel -> effective C -> escape identity	Paper 1 Q2	A
2026	Q2	b	6	effective conductance network 2	two networks and random walk	reduce networks then use C/degree	series-parallel -> effective C -> escape identity	Paper 1 Q2	A
2026	Q2	c	6	escape-before-return probability	two networks and random walk	reduce networks then use C/degree	series-parallel -> effective C -> escape identity	Paper 1 Q2	A
2026	Q3	a	4	steady/transient decomposition	linear diffusion with boundaries	subtract steady state	K eigensystem -> initial deviation -> exponentials	Paper 1 Q3	A
2026	Q3	b	4	steady-state modal coefficients	linear diffusion with boundaries	subtract steady state	K eigensystem -> initial deviation -> exponentials	Paper 1 Q3	A
2026	Q3	c	8	explicit transient solution	linear diffusion with boundaries	subtract steady state	K eigensystem -> initial deviation -> exponentials	Paper 1 Q3	A
2026	Q3	d	4	sketch and limiting values	linear diffusion with boundaries	subtract steady state	K eigensystem -> initial deviation -> exponentials	Paper 1 Q3	A
2026	Q4	a	2	electrical BVP	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A
2026	Q4	b	4	thermal BVP with dissipation	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A
2026	Q4	c	6	solve voltage and temperature	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A
2026	Q4	d	4	heat fluxes	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A
2026	Q4	e	3	insulated-end temperature	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A
2026	Q4	f	1	insulated-end heat flux	electrical heating drives thermal BVP	solve current first	phi/J -> heat source -> BVP -> energy balance	Paper 1 Q4	A