MATH40006 Revision techniques 137

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: 6742f63d354154b66ede5be183d3c33b29921f2a97cfb5e4d9c1e86bd417a516
Source date: 2026-08-06

Source page 1

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_1_Worked_Solutions Page 1

English passage 2 (62% down the source page) - Accessible course-material basis: the official 2025-26 marking scheme.

English passage 3 (64% down the source page) - Historical competency gap addressed: retain the sole paper reinforcing the current structure, practising the complete sequence of grids, colour maps, contours, vector fields and wireframes.

English passage 4 (66% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark visualisation unit).

English passage 5 (69% down the source page) - Reason for equivalence: 8 marking steps, two grid resolutions and 5 types of figures; computational workload, abstraction and the estimated 18-20 minutes align directly.

Source page 2

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

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Source page 3

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

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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 (95% down the source page) MATH40006_Practice_Paper_1_Worked_Solutions Page 4

English passage 2 (71% down the source page) - Accessible course-material basis: the official 2025-26 marking scheme.

English passage 3 (74% down the source page) - Historical competency gap addressed: retain the complete implementation of the current Pritchard sieve and its boundary tests.

English passage 4 (76% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark long-algorithm unit).

English passage 5 (78% down the source page) - Reason for equivalence: initialisation, single-step updates, loop wrapping and five categories of tests comprise 5 stages; code length and the estimated 25 minutes align directly.

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 (95% down the source page) MATH40006_Practice_Paper_1_Worked_Solutions Page 5

English passage 2 (54% down the source page) - Accessible course-material basis: the official 2025-26 marking scheme.

English passage 3 (56% down the source page) - Historical competency gap addressed: retain practice of the current short-sequence and exact rational-number structures.

English passage 4 (59% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short integrated unit).

English passage 5 (61% down the source page) - Reason for equivalence: four steps involving adjacent entries, interleaving, iteration and Rational; approximately 10-12 minutes, with the same marks and workload as the main examination.

Source page 6

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_2_Worked_Solutions Page 1

English passage 2 (64% down the source page) - Historical examination-question basis: 2026 Alternative Q1(i-a to ii-h).

English passage 3 (66% down the source page) - Accessible course-material basis: the 2026 Alternative paper (no marking scheme was supplied).

English passage 4 (69% down the source page) - Historical competency gap addressed: row operations, elimination and the general REF sequence; although these appeared in the six earlier papers, they need to be independent of the 2026 question slots.

English passage 5 (71% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark multi-step implementation unit).

English passage 6 (73% down the source page) - Reason for equivalence: the original 25-mark Alternative sequence is reduced to 20 marks, retaining in-place row operations, column-by-column elimination, rectangular matrices and tests that the input is unchanged; estimated time: 20 minutes.

Source page 7

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_2_Worked_Solutions Page 2

English passage 2 (41% down the source page) - Historical examination-question basis: 2026 Alternative Q2(i-a to ii-h).

English passage 3 (44% down the source page) - Accessible course-material basis: the 2026 Alternative paper (no marking scheme was supplied).

English passage 4 (46% down the source page) - Historical competency gap addressed: determinants by elimination, partial pivoting and row-swap signs.

English passage 5 (48% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark long-algorithm unit).

English passage 6 (51% down the source page) - Reason for equivalence: comparable long functions, boundary matrices, numerical checks and complexity explanations; a 4-mark symbolic proof is removed to keep the total time within 60 minutes.

Source page 8

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_2_Worked_Solutions Page 3

English passage 2 (14% down the source page) - Accessible course-material basis: the 2026 Alternative paper (no marking scheme was supplied).

English passage 3 (16% down the source page) - Historical competency gap addressed: the complete data sequence of nested timing dictionaries, pickle, DataFrame and CSV.

English passage 4 (18% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short integrated unit).

English passage 5 (21% down the source page) - Reason for equivalence: 4 direct steps, a small number of file operations and structural checks; estimated time: 8-10 minutes.

Source page 9

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_3_Worked_Solutions Page 1

English passage 2 (71% down the source page) - Accessible course-material basis: the official 2023-24 marking scheme.

English passage 3 (74% down the source page) - Historical competency gap addressed: Eratosthenes, conditional appending, construction by trial division and timing comparisons.

English passage 4 (76% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark integrated-algorithm unit).

English passage 5 (79% down the source page) - Reason for equivalence: two algorithms, testing and timing comprise 5 stages; reducing the explanation at the million-element scale keeps code length and the estimated 22 minutes comparable.

Source page 10

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_3_Worked_Solutions Page 2

English passage 2 (47% down the source page) - Accessible course-material basis: the official 2023-24 marking scheme.

English passage 3 (49% down the source page) - Historical competency gap addressed: recursive branches, base representations, run-length compression and transfer of methods.

English passage 4 (51% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark multi-step integrated unit).

English passage 5 (54% down the source page) - Reason for equivalence: greater abstraction but less plotting work; base cases, recursion, inverse-function tests, list compression and complexity take approximately 20 minutes in total.

Source page 11

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_3_Worked_Solutions Page 3

English passage 2 (68% down the source page) - Historical examination-question basis: 2023-24 Assessment 3 Q3(i-a to ii-e).

English passage 3 (71% down the source page) - Accessible course-material basis: the official 2023-24 marking scheme.

English passage 4 (73% down the source page) - Historical competency gap addressed: file reading, sequential search, dictionaries and Series; the complete 35-mark examination question is reduced to a short bridging exercise.

English passage 5 (75% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short data unit).

English passage 6 (78% down the source page) - Reason for equivalence: only the core reading step, two container types and successful/unsuccessful search tests are selected; estimated time: 9-10 minutes.

Source page 12

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_4_Worked_Solutions Page 1

English passage 2 (79% down the source page) - Accessible course-material basis: the official 2022-23 marking scheme.

English passage 3 (82% down the source page) - Historical competency gap addressed: the Legendre recurrence, symbolic-to-numerical conversion, roots and weights, and Gaussian quadrature.

English passage 4 (84% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark highly abstract integrated unit).

English passage 5 (86% down the source page) - Reason for equivalence: the original 35-mark question is reduced to 20 marks, retaining the recurrence, lambdify, roots/weights and one integration; approximately 20 minutes.

Source page 13

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

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Source page 14

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_4_Worked_Solutions Page 3

English passage 2 (6% down the source page) - Accessible course-material basis: the official 2022-23 marking scheme.

English passage 3 (9% down the source page) - Historical competency gap addressed: the two historical method sequences for integer square roots using bit operations and discrete Newton iteration.

English passage 4 (11% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark long-algorithm unit).

English passage 5 (14% down the source page) - Reason for equivalence: each algorithm includes tests and counting; abstraction is slightly greater, but the specified n=120 hand calculation and complete closed-form proof are removed; estimated time: 25 minutes.

English passage 6 (66% down the source page) - Accessible course-material basis: the official 2021-22 marking scheme.

English passage 7 (68% down the source page) - Historical competency gap addressed: exact LCM counts, worst-case input experiments and asymptotic complexity.

English passage 8 (71% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short-algorithm unit).

English passage 9 (73% down the source page) - Reason for equivalence: select the core counters, scanning and complexity from the original 25-mark sequence; 4 marking stages, with an estimated time of 10 minutes.

Source page 15

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

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Source page 16

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_5_Worked_Solutions Page 1

English passage 2 (65% down the source page) - Accessible course-material basis: the official 2021-22 marking scheme.

English passage 3 (67% down the source page) - Historical competency gap addressed: complex iteration, Boolean masks and escape-time plots, which are high-mark historical competencies absent from the 2026 paper.

English passage 4 (69% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark array/graphics unit).

English passage 5 (72% down the source page) - Reason for equivalence: the same use of two-dimensional grids, vectorisation and graphical output; the iteration is more abstract but there are fewer figure types; estimated time: 20 minutes.

Source page 17

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Source page 18

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_5_Worked_Solutions Page 3

English passage 2 (20% down the source page) - Accessible course-material basis: the official 2021-22 marking scheme.

English passage 3 (23% down the source page) - Historical competency gap addressed: fixed points, period-2 points, derivative-based stability and boundary contours.

English passage 4 (25% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark highly abstract method-sequence unit).

English passage 5 (27% down the source page) - Reason for equivalence: symbolic solving, lambdify, composite mapping and contours comprise 6 steps; code length is controlled; estimated time: 23-25 minutes.

Source page 19

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_5_Worked_Solutions Page 4

English passage 2 (69% down the source page) - Historical examination-question basis: 2023-24 Assessment 3 Q3(i-d to ii-e).

English passage 3 (72% down the source page) - Accessible course-material basis: the official 2023-24 marking scheme.

English passage 4 (74% down the source page) - Historical competency gap addressed: four search approaches (sequential, binary, dict and Series) and transfer of complexity analysis.

English passage 5 (76% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short comparison unit).

English passage 6 (79% down the source page) - Reason for equivalence: only core search and small-scale timing are required, with 4-5 steps and an estimated time of 10 minutes.

Source page 20

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

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Source page 21

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_6_Worked_Solutions Page 1

English passage 2 (69% down the source page) - Accessible course-material basis: the official 2021-22 marking scheme.

English passage 3 (71% down the source page) - Historical competency gap addressed: the complete historical sequence of copying, shuffling, pairing, bidirectional mapping, and recursive binary-search encoding/decoding.

English passage 4 (73% down the source page) - Equivalent 2026 difficulty unit: 2026 Q1 (20-mark integrated multi-function unit).

English passage 5 (76% down the source page) - Reason for equivalence: several short functions and a round-trip self-check, with the information presented in a different order; approximately 20 minutes for all code and tests.

Source page 22

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_6_Worked_Solutions Page 2

English passage 2 (40% down the source page) - Accessible course-material basis: the official 2022-23 marking scheme and the Alternative paper.

English passage 3 (42% down the source page) - Historical competency gap addressed: a combination across years of Newton iteration counts, performance data, pickle, DataFrame and CSV.

English passage 4 (44% down the source page) - Equivalent 2026 difficulty unit: 2026 Q2 (20-mark long-algorithm and data unit).

English passage 5 (47% down the source page) - Reason for equivalence: approximately 12 marks for the main Newton algorithm and 8 marks for the data pipeline; greater integration but short individual functions; estimated time: 24 minutes.

Source page 23

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

English passage 1 (95% down the source page) MATH40006_Practice_Paper_6_Worked_Solutions Page 3

English passage 2 (24% down the source page) - Accessible course-material basis: the official 2023-24 marking scheme.

English passage 3 (27% down the source page) - Historical competency gap addressed: practise recognition of binary runs and transfer of run lengths and run heads through an unfamiliar formulation.

English passage 4 (29% down the source page) - Equivalent 2026 difficulty unit: 2026 Q3 (10-mark short recursion/list unit).

English passage 5 (31% down the source page) - Reason for equivalence: base cases, traversal, two outputs and example self-checks comprise 4 steps; approximately 10 minutes.