MATH40006 Practice Paper 4 Solutions

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: ff57e2619f8d6ea197721a3744633110f1f60c75df36a9139b0654d47895cf1b
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_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 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 (95% down the source page) MATH40006_Practice_Paper_4_Worked_Solutions Page 2

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 (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 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_4_Worked_Solutions Page 4