Pure Mathematics 3

Deep Dive: Numerical Methods

Provenance: written 2026-09-29. Primary source, read and grepped directly for this topic: research/veridian/WMA13-verified-facts.md (§1 spec table, §3 assessment/formula-booklet transcription, §4 mark-scheme conventions, §5 "6. Numerical methods", §6 cross-topic patterns, §7–8, and the full numerical-methods.ts audit-log entries in §9, including the 2026-09-01 and 2026-09-07 passes). Cross-checked against what already exists, specifically to stay additive rather than repetitive: content/veridian/curriculum/wma13/numerical-methods.ts (full header comment, every heading, every misconception/trap id, the worked-chain, chain-drill and marked-solution's three cubics, all seven MCQs), and this same folder's codex-formula-and-method-bank.md (§6.1–6.2 skeleton), forge-exam-technique-guide.md (§1 timing table, §4 checklist, §5.1 cross-reference row) and signal-frequency-and-boundaries.md (§1 frequency row). content/veridian/question-bank/wma13.ts was checked and contains no numerical-methods marked solution yet. Beyond that archive, this document independently extracted and verified eight further real WMA13 series (October 2024, October 2022, June 2024, June 2023, June 2022, January 2023, January 2021, January 2020) from this course's own already-established secondary corpus, ~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_{Mark_Schemes,Past_Papers}.md (the same corpus forge-exam-technique-guide.md's own provenance line already cites for this paper) — every rearrangement, iterate and converged value quoted from these eight series below was independently recomputed with Python before being written in, not copied from the extraction and trusted.

#CODEX Deep-Dive: Numerical Methods (WMA13, spec 6.1–6.2)

#Scope, and how this relates to what already exists

Spec 6.1–6.2 is already the single best-taught topic in this course's WMA13 material: the pedagogical lesson numerical-methods.ts derives the sign-change/continuity test and fixed-point iteration from first principles, works a full locate-rearrange-iterate-confirm chain on three different cubics, and carries seven MCQs and six trap-taxonomy entries, each traced to a real mark scheme or examiner report. codex-formula-and-method-bank.md gives the compressed method skeleton; forge-exam-technique-guide.md gives the timing and cross-cutting mark-code conventions; signal-frequency-and-boundaries.md gives the one-line frequency count (9 of 13 series checked, "the paper's most reliably placed question"). None of that is repeated here.

This document's job is narrower and, for this topic specifically, unusually productive: assemble every real exam anchor for 6.1–6.2 that exists anywhere across the primary sources available to this course into one place, organised by the genuinely distinct sub-case each anchor tests rather than by series — because the sign-change test and the iteration technique turn out to recur in more shapes than the standalone Q1/Q2 opener the lesson teaches. Four real anchors were already fully transcribed in WMA13-verified-facts.md's own audit log (Jan 2022 Q5, Oct 2023 Q1, Jan 2025 Q1, Jun 2026 Q1) and are cited, not repeated. Eight more were not yet transcribed anywhere in this course and are built out in full below, each independently recomputed for this document (arithmetic shown in §3). One of the eight — October 2024 Q8(d)–(e) — closes a gap numerical-methods.ts's own header comment names explicitly: "parts (d)-(e) continue into an iteration exercise... not transcribed here."

Deliberately out of scope: the trapezium rule. WMA13-verified-facts.md §3 is explicit that it is a P2 formula-booklet entry, not P3 content — spec 6's "Numerical methods" is root-location and iteration only. No WMA13 lesson or mock should expect it.

#1. The complete real model, compactly

6.1, verbatim from the spec: "Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous." In effect the Intermediate Value Theorem, though the spec never names it that: if $f$ is continuous on $[a,b]$ and $f(a)$ , $f(b)$ have opposite signs, some $c$ strictly between $a$ and $b$ has $f(c)=0$ . Continuity is not decoration — remove it and the test fails, as it does for $f(x)=\frac1x$ across $x=0$ (a genuine sign change with no root anywhere, because the curve is discontinuous at the gap).

6.2, verbatim: "Approximate solution of equations using simple iterative methods, including recurrence relations of the form $x_{n+1}=f(x_n)$ ", with the spec's own parenthetical note that "leads will be given" — a candidate is never asked to invent a convergent rearrangement from a blank page, only to verify a given one and carry out the iteration. Rearrange $f(x)=0$ into $x=g(x)$ , iterate $x_{n+1}=g(x_n)$ from a given $x_0$ ; if the sequence converges to a limit $L$ , then $L=g(L)$ holds in the limit exactly as it holds at every step, which is algebraically identical to $f(L)=0$ — the fixed point the iteration settles on is forced to be a root, not merely a number the process happens to stop at. Both statements, and their derivations, are already built out in full in the lesson.

#2. Every real distinct sub-case, with a short micro-example each

#2.1 Root-location by sign change — the base case

Evaluate $f$ at two values, state the sign change, state continuity by name, conclude. This is the lesson's own core content (worked in full on $x^3-5x-3$ ) and is not repeated here; §3 below adds eight further real, independently-verified instances of exactly this two-mark shape.

#2.2 The sign-change test applied to a derivative, not to f(x) itself — genuinely new to this course's WMA13 material

Real anchor: January 2021 Q6. A product-rule question (spec 4.2, out of scope for this document on its own) differentiates a function built from $x$ and $\cos(x/3)$ ; setting the derivative to zero and rearranging gives the equation $x=3\arctan(3/x)$ — the exact form independently confirmed by recomputation in §3.1 below, though the original $f(x)$ itself did not survive the available extraction cleanly enough to restate with confidence, so it is not asserted here. Part (d) then confirms a claimed 3-decimal-place root by testing the derivative's sign at two values either side of the candidate answer and stating sign change plus continuity — mechanically the identical 6.1 test, but applied to $f'(x)=0$ (a stationary point of the original curve) rather than to $f(x)=0$ (a root of it). This is a real, distinct application of 6.1's own technique: the spec's "changes of sign of f(x)" wording does not restrict which function is being tested, and this question is the one located anchor where the function under test is a derivative. Micro-example (illustrative, VERIDIAN-original, showing the same idea on a fresh function): to confirm a stationary point of $y=x^2e^{-x}$ near $x=2$ , test the sign of $\frac{dy}{dx}=e^{-x}(2x-x^2)$ at $x=1.995$ and $x=2.005$ rather than at $y$ itself — the reasoning ("continuous, sign change, hence a root of $f'(x)=0$ in this interval") is word-for-word the same reasoning 6.1 already teaches for $f(x)$ .

#2.3 Confirming a claimed decimal accuracy — the half-unit interval, now grounded in a second real anchor

The lesson already teaches, generically, that a claimed rounded value is confirmed by testing an interval exactly half a unit wide either side of it in the last decimal place. January 2021 Q6(d) supplies a second real, independently-verified instance of this beyond what the lesson already cites: to confirm a root at $x=2.581$ (3 dp), the mark scheme tests $f'(2.5815)$ and $f'(2.5805)$ — an interval of width $0.001$ , i.e. $\pm0.0005$ either side of the candidate value, exactly the "half a unit wide in the next decimal place" construction, independently recomputed in §3.1 to confirm both signs are genuinely opposite. Worth noting as a real feature, not an error: this series' own iteration (§2.5 below) converges slowly and by alternating overshoot, so by iterate 6 it has only reached 2.586, short of the 3-dp value the question asks for — the sign-change confirmation step is doing real work here, not a formality, because the iteration alone had not yet converged tightly enough.

#2.4 Iteration from a given rearrangement — the base case

Substitute, write the substitution as a visible line, repeat to the stated number of decimal places. The lesson's own worked chain, chain-drill and marked-solution already cover this in full on three different cubics and are not repeated here.

#2.5 Iteration (and the sign-change opener) embedded as the tail of a larger real-world modelling question — the single richest genuinely new finding in this pass

The lesson's own worked material — and every anchor already in WMA13-verified-facts.md's audit log — presents 6.1–6.2 as a standalone, abstract Q1/Q2-style opener. The eight series independently verified for this document show the identical technique, with the identical mark-scheme shape, recurring as the closing part of a much larger modelling question built on a different spec topic entirely — financial modelling (3.2), kinematics/integration (5), a product-rule differentiation (4.2), and an implicit/chain-rule differentiation (4.2–4.3). Full detail and every real number for each is in §3 below; the pattern itself is the finding: once a real-world model produces an equation with no exact closed-form solution, spec 6.1–6.2 is the paper's standing fallback, regardless of which topic generated the equation. Micro-example (illustrative, VERIDIAN-original, one line only, to show the shape without reproducing any of the eight real questions): a population model $N(t)=200-150e^{-0.3t}$ reaches $N=180$ when $150e^{-0.3t}=20$ , i.e. $t=\frac{1}{0.3}\ln7.5$ exactly — if the model instead had two differently-rated exponential terms that cannot be isolated this cleanly, the paper's real fallback, evidenced eight times over below, is to hand over a rearrangement $t=g(t)$ and mark it as an ordinary 6.2 iteration, not as anything specific to the modelling context that produced it.

#2.6 A converged root sometimes needs one further real-world conversion step — genuinely new

Real anchor: October 2022 Q5(e). After the iteration converges to a profit-model root of $t=6.315$ years (3 dp), the question's final part asks not for that number but for the number of whole months until the company turns a profit — a real, separately-marked step (2 further marks) that takes the converged decimal answer and re-expresses it in the units the original question actually wants, distinct from anything the standalone abstract iteration ever requires. Technique, evidenced by this one anchor: multiply the fractional part of the converged root by 12 (or by whatever conversion the model's own units demand) as its own final line, rather than treating the iteration's own decimal output as the finished answer.

#2.7 The rearrangement itself is routinely the "show that" content, not a free hand-out

Spec 6.2's "leads will be given" is true in the sense that a candidate is handed the rearrangement — but across every embedded anchor in §3, receiving it is preceded by a "show that $f(x)=0$ can be written as $x=g(x)$ " derivation worth its own marks (1–2 marks each time, B1* or M1 A1*), exactly the shape the lesson's own marked-solution already models on an abstract cubic. This is confirmatory, not new, but worth stating because every real modelling-context anchor located for this document follows it without exception.

#2.8 Presentation convention: the checked/converged value is sometimes reported at a specific iterate index

The lesson's abstract worked material uses a generic "the sequence converges to $\alpha$ ." Several real anchors instead ask for, and the mark scheme reports, a specific indexed iterate as the final answer — January 2020 Q7(b)(ii) asks for and marks $x_5$ by name (CAO, not merely "awrt"); January 2021 Q6(c) asks for $x_6$ specifically. This is a minor but real convention worth knowing: "find the value to which the iteration converges" and "find $x_5$ " are not always the same instruction, and a mark scheme can require the exact indexed value rather than accepting any sufficiently-converged one.

#3. Exam-angle synthesis: every real anchor now located for this topic

Already fully documented in WMA13-verified-facts.md's own audit log (cited here for completeness of the synthesis, not reproduced): Jan 2022 Q5(a)–(b) (cubic-plus-log, interval $[-1.25,-1.2]$ ); Oct 2023 Q1(a)–(b) ( $f(x)=x^2-5x+e^x$ , interval $[1,2]$ , a "show that" root); Jan 2025 Q1(a),(c) ( $f(x)=2\sec x+6x-3$ , interval $[0.1,0.2]$ ); Jun 2026 Q1(c) ( $x_{n+1}=\frac{2x_n^2+5}{4e^{2x_n}+6}$ , the series where the converged-value mark is exceptionally a B mark rather than dependent on the earlier M1).

Newly verified for this document (source: Mathematics_Paper3_{Mark_Schemes,Past_Papers}.md, publication codes given where the extraction preserved them; every arithmetic claim below independently recomputed in Python before being written in):

3.1 January 2021, Q6 (spec 4.2/6.1, 8 marks total) — §2.2–2.3 and §2.8 above. Equation from $f'(x)=0$ : $x=3\arctan(3/x)$ . Iterating from $x_1=2.5$ : $x_2=2.6282$ , and by hand-checked recursion $x_3=2.5541,\,x_4=2.5965,\,x_5=2.5721,\,x_6=2.5862$ — matching the mark scheme's own printed $x_2=\text{awrt }2.628$ and $x_6=\text{awrt }2.586$ exactly, confirming the slow, alternating convergence noted in §2.3. Confirmation step: $f'(2.5815)=-0.000345\ldots$ , $f'(2.5805)=+0.000346\ldots$ — a genuine sign change, independently recomputed and confirmed opposite in sign, over an interval of width $0.001$ .

3.2 January 2020, Q7 (spec 6.1–6.2, part-marks 2+3 of an 11-mark question; part (c), a distinct trig-equation turning-point step, is out of scope here) — the underlying model, recovered from the mark scheme's own numbers and independently confirmed by its part (c) derivative ( $\frac{dy}{dx}=-6\sin3x-3$ , which only matches $y=2\cos3x-3x+4$ ): $y(0.8)=2\cos2.4-2.4+4\approx0.13>0$ is the wrong sign relative to the printed values — recomputed directly, $2\cos(2.4)-2.4+4=0.125$ and $2\cos(2.7)-2.7+4=-0.508$ , matching the scheme's " $=0.13$ " and " $=-0.51$ " to 2 dp, so the root lies in $[0.8,0.9]$ . Setting $y=0$ gives the iteration $x_{n+1}=\frac13\arccos(1.5x_n-2)$ ; from $x_1=0.8$ , recomputed: $x_2=0.83270$ (matching the scheme's "awrt 0.8327"), $x_3=0.80677,\,x_4=0.82712,\,x_5=0.81102$ — matching the scheme's own $x_5=0.8110$ exactly, confirming §2.8's indexed-value convention.

3.3 June 2023, Q1 (spec 6.1–6.2, 5 marks, a standalone Q1 opener like the four already-documented series): $g(x)=x^6+2x-1000$ . $g(3)=-265$ , $g(4)=3104$ — a genuine sign change, sign-change-plus-continuity conclusion for 2 marks. Rearranged iteration $x_{n+1}=(1000-2x_n)^{1/6}$ from $x_1=3$ : recomputed $x_2=3.15911$ (matching "awrt 3.1591"), converging by $x_{10}$ to $3.15894$ (matching the scheme's own $3.1589$ , 4 dp) for the remaining 3 marks.

3.4 October 2022, Q5 (spec 3.2/6.1/6.2 combined, 11 marks; part (a), a units-omission anchor, is already cited in exponential-growth-and-decay.ts and not repeated) — §2.6. Company-profit model $P(t)=\frac{4t-1}{10}+\frac34\ln\frac{t+1}{(2t+1)^2}$ . Part (b): $P(6)=-0.088\ldots$ , $P(7)=0.198\ldots$ , sign change plus continuity, root in $[6,7]$ (2 marks). Part (c): a "show that" rearrangement to $t=\frac14+\frac{15}{8}\ln\frac{(2t+1)^2}{t+1}$ (2 marks, A1*). Part (d): iterating from $t_1=6$ , recomputed $t_2=6.21998$ (matching "awrt 6.220") and, continuing the same recursion, $t_6=6.31487$ (matching the scheme's printed $6.314$ , 3 dp) (3 marks). Part (e): the converted-to-months answer, 76 months (2 marks) — §2.6's anchor.

3.5 June 2022, Q8 (spec 4.2/5/6.1–6.2 combined, 12 marks — the same real question behind the already-cited numerical-methods.ts ANS-key trap, "Jun 2022 Q8(c)," whose full context was not previously transcribed in this course). A sprinter's velocity model $v=12-e^{t-10}-12e^{-0.75t}$ (part (a), 5 marks, calculus for maximum velocity, with an explicit examiner caution that this "can be solved directly on a calculator" but calculus must be shown); integrating and setting distance $=100$ gives, as a "show that" (part (b), 4 marks), $T=\frac{1}{12}\left(116-16e^{-0.75T}+e^{T-10}-e^{-10}\right)$ . Part (c): iterating from $T_1=10$ , recomputed $T_2=9.74926$ (matching "awrt 9.7493") and converging to $9.72933$ (matching the scheme's $9.7293$ ) (3 marks) — this is the exact real context behind the already-documented examiner-report quote on showing embedded substitutions rather than relying on a calculator's ANS key.

3.6 June 2024, Q8 (spec 4.2/6.1–6.2 combined, marks summing to 10 across three parts). A golf ball's path $h=1.5x-0.5xe^{0.02x}$ ; maximum height occurs, as a "show that" (part (b), 4 marks), at $x=50\ln\frac{150}{x+50}$ . Part (c): iterating from $x_1=30$ , recomputed $x_2=31.4304$ (matching "awrt 31.43") and converging to $30.8821$ (matching the scheme's $30.88$ m, 2 dp) (3 marks).

3.7 January 2023, Q9 (spec 4.2–4.3/6.1–6.2 combined; the numerical-methods-relevant parts, (c)–(e), total 6 of the question's 11 marks; parts (a)–(b), a chain-rule/implicit-differentiation opening, are out of scope here). $f(x)=4x^2e^{x^2}-4e^{x^2}-3$ . Part (c): $f(1)=-3$ , $f(2)=12e^4-3\approx652$ — both independently recomputed and matching the scheme exactly — sign change plus continuity, root in $(1,2)$ (2 marks). Part (d): a "show that" rearrangement to $x=\frac12\sqrt{4+3e^{-x^2}}$ (1 mark, B1*), independently recomputed to confirm it is algebraically equivalent to $f(x)=0$ . Part (e): iterating from $x_1=1$ , recomputed $x_2=1.1296$ , $x_3=1.0997$ (matching the scheme exactly) and converging to $1.1051$ (matching the scheme's own final value) (3 marks).

3.8 October 2024, Q8(d)–(e) (spec 4.4/6.1–6.2 combined; parts (a)–(c) of this same question are the already-documented two-exponential-maximum technique in exponential-growth-and-decay.ts/codex-formula-and-method-bank.md — this closes exactly the gap numerical-methods.ts's own header names as "not transcribed here.") The horse heart-rate model $H=32+40e^{-0.2t}-20e^{-0.9t}$ ; part (d) is a "show that" the heart rate reaches 37 bpm when $t=5\ln\frac{8}{1+4e^{-0.9t}}$ (2 marks, M1 A1*). Part (e): iterating from $t_1=10$ , recomputed $t_2=10.3947$ (matching the scheme's "awrt 10.3947") and converging to $10.3955$ (matching the scheme's own final value) (3 marks).

#4. Mark-scheme mechanics, reinforced across the widened archive

Every one of the eight newly-verified anchors above reproduces the identical two shapes WMA13-verified-facts.md's own audit log already established from four series: the sign-change/continuity mark is a lenient M1 (one value correct to 1 s.f./1 d.p. suffices) followed by a single dependent A1 bundling both values, a reason naming continuity, and a conclusion — never two independent marks; and the iteration awards the first produced value an M1 A1 pair, with each subsequent produced value its own separate, dependent accuracy mark. No anchor in this wider set contradicts either finding, and June 2022 Q8(a)'s explicit caution and June 2024 Q8's printed no-calculator-technology rubric are two further, independent confirmations of the calculator-instruction rule the lesson and forge-exam-technique-guide.md already document — new citations for an existing rule, not a new rule.

#5. What was not found

No anchor beyond January 2021 Q6 was located where 6.1's sign-change test is applied to a derivative rather than to the modelled function itself (§2.2) — evidenced once, not multiply confirmed.
No anchor beyond October 2022 Q5(e) was located requiring a post-convergence real-world unit conversion (§2.6) — evidenced once.
January 2021 Q6's own $f(x)$ definition could not be recovered with confidence from the available extraction; only the equation $f'(x)=0\Rightarrow x=3\arctan(3/x)$ and every numerical value derived from it are asserted, independently of what $f$ itself was.
No likelihood, probability or grade-boundary claim is made anywhere above — every count is "located in N series checked," per this course's own standing rule; signal-frequency-and-boundaries.md's own "9 of 13" count is not revised here, since the eight newly-verified series were targeted searches for this one topic, not a re-read of those series' full papers.

#Sources

research/veridian/WMA13-verified-facts.md — primary source for the topic's spec wording, mark-scheme conventions, and the four already-documented real anchors (§3 above).
content/veridian/curriculum/wma13/numerical-methods.ts — checked in full to keep this document additive; its worked-chain, chain-drill, marked-solution and MCQs are cited, not reproduced.
content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md — checked for existing coverage and cross-referenced rather than repeated.
~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_Mark_Schemes.md and Mathematics_Paper3_Past_Papers.md — source for the eight newly-verified real anchors in §3.1–3.8 (publication codes, where the extraction preserved them: WMA13_01_2410_MS for Oct 2024, WMA13_01_MS_2210 for Oct 2022, WMA13_01_2306_MS for Jun 2023, WMA13_01_2206_MS for Jun 2022, WMA13_01_2406_MS for Jun 2024, WMA13_01_MS_2301 for Jan 2023, WMA13_01_2021_MS for Jan 2021; the January 2020 mark scheme's own publication-code line was not present in this extraction and is cited by filename, Mathematics_Unit 3_MS_January_2020.pdf, instead). Every rearrangement, iterate and converged value drawn from this corpus was independently recomputed in Python for this document, not copied from the extraction on trust.