How marks are actually earned

cancel-and-simplify-step-abandoned

Confirmed directly, verbatim, against both the examiner report AND the real mark scheme for the same question. Oct 2020 Q9(a) — divide $\frac{x^4-x^3-10x^2+3x-9}{x^2-x-12}$ — awards the last two of its four marks as "M1: Writes the given expression in the required form using $x^2-x-12=(x-4)(x+3)$... A1: Correct answer... Note that Q = 5 is given so it must be shown from correct work, not just stated." Candidates who divided correctly still lost both marks: "candidates did not continue to factorise the denominator and cancel (x+3) and hence not prov[e]... that Q is 5, they just stated it instead" — over 70% scored full marks, but the report names this specific abandonment as the reason the rest did not.

given-value-not-verified-by-substitution

A genuinely different failure from the one above — worth telling apart precisely, because the two are marked differently and fixed differently. Not every division question ends in a fraction to simplify: some GIVE you one of the constants outright and mark it as an independent, "shown" accuracy mark for proving that stated value, not for reaching it — the mark-scheme convention from §4 of the facts bank for "the answer is printed on the paper." Verified verbatim, Jan 2025 Q4(a)(ii) — divide $\frac{4x^3+2x^2+3x+8}{x^2+4} \equiv Ax+B+\dfrac{Cx+D}{x^2+4}$ and "show that $D=0$" — the real mark scheme awards this as "B1*: Fully shows that D = 0 from clear and correct work... they would need to set up (at least) two correct equations and solve, with appropriate substitutions seen, to show that D = 0." The examiner report confirms candidates who found $A$, $B$ and $C$ correctly still lost this mark: "did not subsequently establish that D = 0 as they did not show the substitution." There is no factor to cancel here — $x^2+4$ has no real linear factor, so "factorise and cancel" is not the fix. The fix is the same "show that" discipline applied to a different target: write down the actual equation the given value must satisfy, substitute into it, and show it holds — not assert the printed value because it is, in fact, printed.

combined-fraction-not-fully-justified

A separate, independently confirmed pattern across at least three series (Oct 2020, Jan 2024, Jan 2025): candidates combine rational expressions over a common denominator and reach the correct final simplified form, but without showing the intermediate working that justifies it — and lose the mark attached to the justification even though the answer on the page is right. This is the general "show that" rule from the paper's own general marking guidance applied to this specific topic: an answer that happens to be correct isn't the same thing as an answer that's been shown to be correct, and only the second earns a 'show that' mark.

method-not-set-up-before-the-arithmetic

The general marking guidance, verified verbatim from the January 2023 mark scheme and cross-checked against October 2023 and June 2022: "Where a method involves using a formula that has been learnt, the advice given in recent examiners' reports is that the formula should be quoted first... Where the formula is not quoted, the method mark can be gained by implication from correct working with values but may be lost if there is any mistake in the working." Algebraic division has no single formula to quote, but the identity it's built on does: writing $f(x) \equiv D(x)Q(x)+R(x)$ before diving into the subtraction protects the method mark the same way quoting the quadratic formula does — implied method from unlabelled working is real credit, but it is credit that a single early slip can destroy entirely.

exact-form-abandoned-for-a-decimal

Verified verbatim, same general marking guidance: "Examiners' reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals." A simplified rational expression or a constant found by cancelling factors is an exact algebraic answer by nature — decimalising it (turning $\frac{5}{x+3}$ into something like "$\approx 1.67$ when $x=0$", or rounding a found constant) answers a question that wasn't asked and drops marks a correct exact form would have kept.

calculator-technology-cited-as-the-method

The paper-wide rubric, verified verbatim from a real WMA13 question paper: "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable." Simplification and division questions are exactly where this lands hardest, because a calculator can often produce the simplified fraction directly. Examiner reports across the paper document candidates losing marks for a correct final answer with no algebraic method shown on a question carrying this instruction — the number being right is not the thing being marked.

domain-of-inverse-omitted

The single most consistently reported error anywhere in the WMA13 archive for this sub-topic, confirmed across at least three separate series. Verified verbatim, Jun 2022 Q2: "Despite being a standard question, the majority of candidates still failed to state the domain for their inverse function and did not achieve the B mark." Verified verbatim, Jan 2022 Q6: "a majority of candidates were not aware that the domain was required, and therefore by far the most common score seen was 2/3" (out of 3) — independently confirmed again in Oct 2021 Q1(b). The mechanism is structural, not carelessness alone: the domain of f⁻¹ is an independent B mark (see the next trap), so getting every line of algebra correct still leaves it unclaimed unless it is written down as its own separate statement.

domain-of-inverse-derived-from-scratch-not-recognised-as-range-of-f

A distinct pattern from simply omitting the domain: some candidates DO attempt to state a domain for f⁻¹, but re-derive it from first principles on the new expression rather than recognising it is already sitting in the answer to an earlier part of the question, as the range of f. Verified, Oct 2022 Q2(b) — the report notes candidates deriving the inverse's domain "from scratch" instead of using the range of f they had, or could have had, already. This costs time even where it does not cost the mark outright, and it is the exact opposite of the efficient route the worked chain above demonstrates: find the range of f early, and the domain of f⁻¹ is already answered before the algebra for f⁻¹(x) itself has even begun.

inverse-algebra-and-its-domain-are-independently-marked

Not an error a candidate makes so much as a fact about the mark scheme that, misunderstood, produces one: the domain of an inverse function is not a bonus tacked onto the M1 A1 for finding its formula, and it is not lost automatically if the algebra goes wrong, or gained automatically if it goes right. This course's own research verifies this directly from a real mark scheme, Jan 2023 Q1(c) (there, finding g⁻¹(x) for a fraction g rather than f⁻¹(x) for a quadratic, but the structure is identical to every example in this lesson): the scheme awards M1 A1 for finding the inverse and a SEPARATE, independent B1 for the domain. Treating the domain as "the last part of finding the inverse" rather than its own distinct, separately-earned statement is the root cause behind both traps above.

inverse-rearrangement-shown-as-calculator-output-with-no-algebra

WMA13 carries an explicit, paper-wide, verbatim rubric on exactly this kind of algebra: certain questions state "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable" (verified verbatim from the January 2023 question paper, and confirmed recurring across multiple series). Finding f⁻¹(x) is a rearrangement — precisely the kind of step a calculator or an equation solver can shortcut silently. Where a question carries this instruction, arriving at a correct f⁻¹(x) with no visible swap-and-rearrange working risks the method mark outright, whatever the final expression says.

inverse-answer-attached-to-the-wrong-function-letter

A distinct, purely notational failure mode, confirmed directly in a real mark scheme rather than an examiner report. When a question defines more than one function — exactly the situation in every worked example, chain drill and marked solution in this lesson, since composition and inverses are normally taught and tested together — a correctly-derived inverse can still lose its accuracy mark if it is labelled with the wrong function's letter. Verified verbatim, Jan 2023 Q1(c), which defines both f(x) and g(x) and asks specifically for g⁻¹(x): the scheme's note on the A1 mark reads "Condone y = (3−x)/(2x) o.e and even g⁻¹ = (3−x)/(2x) but NOT f⁻¹ = (3−x)/(2x) o.e." Right rearrangement, right final expression, zero marks — because f is the letter more often inverted in practice, and writing "f⁻¹(x) = ..." out of habit when the question asked for g⁻¹(x) is an easy, silent slip that the algebra itself never reveals.

positive-x-branch-not-mirrored-for-f-of-modulus-x

Confirmed directly on a real f(|x|) reflective-symmetry question: "very few students understood that... the negative x part of the graph is a reflection of the positive x part... so [the second solution] is also a solution" (Jun 2023, Q6(d)). The mechanism is the one this lesson derives rather than states: for x < 0, f(|x|) equals f evaluated at the corresponding positive value, so the entire left-hand branch is a rebuild, not a survival of the curve's original left half.

only-one-branch-of-modulus-equation-solved

A confirmed, recurring pattern across at least three series (Jan 2022 Q7, Oct 2020 Q4(c), Jun 2023 Q6(c)): candidates solve only one of the two cases a modulus equation or inequality actually splits into, or find both critical values correctly but then fail to select the correct combined region at the end. Jan 2022's own examiner report calls the specific question this pattern showed up on "very demanding by many." The fix is structural, not a matter of care: always write out both cases before solving either.

critical-values-correct-but-wrong-region-selected

Confirmed by the mark scheme's own design, not just examiner commentary: Pearson attaches a dedicated, separately-earned mark to the single step of choosing which region satisfies a modulus inequality — distinct from, and dependent on, the earlier mark(s) that found the critical values in the first place. Verbatim from a plain linear modulus-inequality question of exactly the shape this lesson teaches: "dM1: Selects outside region for their critical values... It is dependent upon having attempted to solve one correct equation" (Jan 2022, Q7(b)). On a harder WMA13 question that combines the modulus with a second technique — so that TWO separate method marks, not one, precede this same step — the dependency escalates and the mark scheme uses a doubly-dependent ddM1 instead, verified verbatim elsewhere in this paper's mark schemes: "ddM1: Chooses the outside region for their values... It is dependent on both previous method marks" (Oct 2023, Q9(c)). Either way, the region-selection step is never folded into the mark that found the values — it is always its own separate checkpoint, exactly because getting the two boundary values right and then picking the wrong side of them is common enough to need one.

combined-transformation-only-partially-applied

Confirmed on a real combined-transformation coordinate question: common errors included giving "an error of (−8, −3) or (−8, −6)" and, separately, that "a common error was to subtract 1 from the x coordinate to give (−5, −9)" (Jan 2024, Q1). Both are the same underlying failure: applying only one of the two combined moves correctly, or applying the right move to the wrong coordinate. The check from the mechanism above is the direct fix — track one point through both moves independently and confirm each one only touched the coordinate it was supposed to.

calculator-used-with-no-algebraic-method-shown

WMA13 carries an explicit no-calculator-methods rubric on specific questions, verified verbatim from the January 2023 question paper: "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable." The facts bank confirms this same requirement recurred specifically on the January 2024 modulus simultaneous-equation question (Q8) — the exact topic this lesson covers. Treat the rubric as a cue that the algebra itself, not just the final numbers, is what earns the marks on that question.

cosec-written-as-reciprocal-of-cosine

Confirmed directly on a real WMA13 identity question: candidates "struggled writing cosec θ = 1/cos θ" (Jan 2022, Q2) — reaching for cosine because of the shared first syllable, when the correct pairing is cosec θ = 1/sin θ. The same report records the flip side too: "even some of the weakest candidates were able to earn a mark for stating the identity cosec θ = 1/sin θ" (Jan 2022, Q2) — this is a genuinely accessible mark, and losing it to a mispairing rather than a harder step is the most avoidable loss in this whole topic. Check any pairing by confirming it multiplies to 1: cosec θ · sin θ = 1, not cosec θ · cos θ.

coefficient-folded-into-the-reciprocal

A second, distinct error on the exact same real WMA13 question named above, re-verified directly against the primary mark scheme (not just the examiner-report prose): the same series' report records candidates "sometimes" writing "3 cosecθ = 1/ 3sinθ" (Jan 2022, Q2) — folding the coefficient into the denominator alongside sin θ, rather than leaving it multiplying the finished reciprocal. The real mark scheme is explicit that this scores nothing for the identity mark: "Note that 3cosecθ = 1/(3sinθ) is B0 unless there is an aside that does state cosecθ = 1/sinθ." The fix is mechanical, not a new idea: reciprocate the trig function alone first — cosec θ = 1/sin θ — then multiply by whatever coefficient sits in front, e.g. 3 cosec θ = 3 × (1/sin θ) = 3/sin θ. The coefficient never moves inside the denominator next to sin θ.

quadratic-in-tan-or-cot-missing-a-second-solution

Documented on a different WMA13 trig equation, not one reducing via sec²θ or cosec²θ specifically, but the exact same shape of failure: "it was disappointing that many candidates failed to identify the second solution here to gain the full marks" (Jun 2022, Q7). Every equation in this lesson that reduces to a quadratic in tan θ or cot θ produces TWO values of the trig ratio, and each of those typically produces TWO angles in a full 0°–360° range — up to four solutions from one equation. Stopping after the first branch, or after one solution within a branch, is this trap in miniature.

solved-ratio-mistaken-for-the-angle-itself

Confirmed on a WMA13 question that solved for sin x rather than x directly: "a few interpreted the solution to their equation as being the value for x rather than for sin x, thereby losing both marks" (Jan 2024, Q6(c)). The identical failure is available here with tan θ or cot θ: solving cot²θ − cotθ − 6 = 0 gives values of cot θ (3 and −2), not values of θ. The step from "cot θ = 3" to "θ = 18.4° or 198.4°" is not optional bookkeeping — it is a required, markable step that a genuine number of candidates skip on the analogous sin-based question.

identity-not-quoted-before-use

General marking guidance for this paper states the advice explicitly: "where a method involves using a formula that has been learnt, the advice given in recent examiners' reports is that the formula should be quoted first... where the formula is not quoted, the method mark can be gained by implication from correct working with values but may be lost if there is any mistake in the working" (Jan 2023 general marking guidance, cross-checked against Oct 2023 and Jun 2022). Applied here: writing "sec²θ ≡ 1 + tan²θ" as its own line before substituting is what makes the method mark secure even if the next line contains a slip — skipping straight to the substituted equation makes the method mark depend on everything downstream staying correct.

restricted-domain-of-the-inverse-left-unstated

Confirmed as the single most consistently dropped mark on real inverse-function questions on this paper: "despite being a standard question, the majority of candidates still failed to state the domain for their inverse function and did not achieve the B mark" (Jun 2022, Q2), and independently, "a majority of candidates were not aware that the domain was required, and therefore by far the most common score seen was 2/3" (Jan 2022, Q6). Those quotes are about inverse functions generally (spec 1.2), not about arcsin/arccos/arctan specifically — but the failure is identical in shape: a question asking for the domain OR range of arcsin, arccos or arctan is asking for exactly the kind of stated restriction these reports record candidates omitting, even when the rest of the working is correct.

missing-intermediate-line-in-a-prove-that

A general pattern confirmed across a WMA13 series' full set of reports: "it was noticeable in this series that many candidates omitted important lines when proceeding to the given solution resulting in the loss of some vital marks" (Jan 2022, general summary). Applied to this topic specifically: a "show that sec²θ ≡ 1 + tan²θ" question is asking for the division-by-cos²θ derivation shown above written out, not just asserted — jumping from cos²θ + sin²θ ≡ 1 straight to the answer, with the dividing-through step invisible, is exactly the omission these reports repeatedly penalise.

wrong-double-angle-cosine-form

Confirmed verbatim on a real WMA13 exam question of exactly this lesson's worked-chain shape — "Solve, for 0° ⩽ x < 360°, the equation 2cos2x = 7cosx" (Oct 2020, Q1, 5 marks; NOT a "prove"/"show that" question — an equation to solve by the same substitute-then-3TQ route as this lesson's own worked chain). The examiner report records it as "generally well done with about two-thirds of the candidates scoring full marks" — so this is a minority-but-real error, not the modal one — "where errors occurred, these were mainly writing cos 2x as cos²x − 1 or 1 − cos²x, or incorrectly stating 4 cos²x − 2 = 2 cos x as the first step, implying an incorrect identity due to a lack of brackets": half-remembering one of the three real forms (cos²x−sin²x, 2cos²x−1, 1−2sin²x) and dropping the factor of 2, or — the second, distinct error — writing 2×2cos²x−1 without the bracket around (2cos²x−1) it needs, which the real mark scheme flags by name: "2 × 2cos²x − 1 = 7cosx is M0 unless the correct identity has been previously stated or recovery occurs." Both errors cost the same mark for the same underlying reason: the coefficient of 2 outside the bracket has to survive the substitution, and it is exactly that 2 — as a factor, or as a bracket protecting it — that goes missing.

incomplete-working-on-a-prove-that

The single most repeated failure mode on this topic across multiple series, and not a maths error at all: "It was noticeable in this series that many candidates omitted important lines when proceeding to the given solution resulting in the loss of some vital marks" (Jan 2022, general summary). The identical discipline is documented independently outside this topic too — a Jan 2022 report on a differentiation "show that" question notes "many good candidates lost marks here for merely writing down the given answer from a correct dx/dy without any intermediate lines," cited here only as evidence the same general marking principle (every line on the way to a GIVEN answer is part of what is marked, not optional scaffolding) recurs across different question types, not as a trig-specific incident in its own right. On a "prove" or "show that" instruction, the double-angle substitution line itself is one of the lines being marked.

dividing-away-a-common-trig-factor

A genuinely common algebra trap once a double-angle form has been substituted into an equation like sin2θ = sinθ: 2sinθcosθ = sinθ looks, on sight, like something to simplify by dividing both sides by sinθ, leaving 2cosθ = 1. That step is illegal exactly when sinθ could be zero, and it silently discards every solution where sinθ = 0 actually satisfies the ORIGINAL equation. Rearrange to zero and factorise instead — sinθ(2cosθ − 1) = 0 — so both branches, sinθ = 0 and cosθ = ½, survive. The rule isn't specific to trig: never divide an equation by an expression that could itself be zero, only ever factor it out.

unhelpful-cos2a-form-chosen

All three forms of cos2A are equally TRUE, so picking the 'wrong' one never produces a wrong answer — it produces an equation that still has two different trig functions mixed together, when a better choice would have collapsed it to one. Substituting cos²A − sin²A into an equation that is otherwise entirely in sinA doesn't simplify anything; it introduces a fresh cos²A term that then has to be eliminated with sin²A + cos²A ≡ 1 anyway — the same identity a better initial choice would have used once, not twice. Before substituting, check what function the rest of the equation is already written in, and match the cos2A form to it.

half-angle-substitution-misread

"Half the angle" and "half the value" are not the same operation, and the notation makes them easy to blur: sin(θ/2) means the sine of half the angle, an entirely different number from (sinθ)/2, half the sine's value. The half-angle identities in this lesson — sin²(θ/2) ≡ (1−cosθ)/2 and its cosine equivalent — only come out right if the /2 is read as dividing the ANGLE going into the substitution A=θ/2, not as an operation performed on sinθ or cosθ afterwards. Write the substitution out explicitly ('let A = θ/2') before touching the double-angle formula, rather than trying to halve an angle and a trig function in the same mental step.

missing-second-solution-in-range

Confirmed verbatim on a real "solve in a given range" trig question: "It was disappointing that many candidates failed to identify the second solution here to gain the full marks" (Jun 2022, Q7). Cosine (and sine) are two-to-one over a full 360° turn for any target value strictly between −1 and 1, so an equation of the form cos(θ−α) = k always has a second solution in a 360° interval — finding the first one algebraically correctly is not the same as finding the answer.

degrees-radians-mismatch

A documented pattern across five separate series (Oct 2020, Jan 2022, Jun 2022, Jun 2023, Jan 2024): candidates lose the final accuracy mark on a 'solve in a given range' trig equation by giving the answer in degrees when the question specified radians, or vice versa. The interval itself is the tell — 0° ≤ θ < 360° wants degrees, 0 ≤ θ < 2π wants radians — and it is worth checking that match before writing a single answer down, not after.

exact-R-decimal-rounded-or-alpha-given-in-wrong-angle-unit

This trap sits earlier than the one above — on the R and α accuracy marks THEMSELVES, while building the harmonic form, not on the final solved θ. A real WMA13 harmonic-form question can specify both a required FORM for R (exact, not decimal) and a required UNIT for α (radians, not degrees), and a numerically correct value in the wrong form or unit earns nothing on that mark. Confirmed verbatim on a real question testing exactly this spec point, Oct 2020 Q7(a): "Express cos x + 4 sin x in the form R cos(x − α) where R > 0 and 0 < α < π/2. Give the exact value of R and give the value of α, in radians, to 3 decimal places." The real mark scheme is explicit on both counts. For R = √17: "Condone R = ±√17 (Do not allow decimals for this mark...)" — a correct decimal such as 4.123 scores zero on that mark, because the question asked for the EXACT value, and √17 is not the same answer as its rounded decimal for marking purposes. For α = awrt 1.326: "Note that the degree equivalent α = awrt 75.96° is A0" — the identical angle, correctly computed, in the wrong unit, still earns nothing. Read the question’s own instructions on form and unit before writing R and α down, not after computing them the way that feels automatic.

extra-out-of-range-solutions-kept

The same five-series pattern's third face: including a solution generated by the ±360n general form that actually falls outside the stated interval once it is written out. Generating θ = α ± cos⁻¹(k) + 360n is the right method; the last step — checking every value produced against the stated bounds before it goes in the answer — is not optional bookkeeping, it is the mark.

stops-after-the-shifted-angle-never-recovers-theta

The specific error this lesson's marked solution builds around — solving for (θ−α) or (θ+α) and reporting that as if it answered the question — is one instance of a broader, independently documented failure: stopping at an intermediate variable instead of converting back to the one actually asked about. A different WMA13 equation type shows the identical pattern one layer earlier, at the point of substitution rather than shifting: "A few interpreted the solution to their equation as being the value for x rather than for sin x, thereby losing both marks" (Jan 2024, Q6(c)). Different equation, same discipline missing: know which variable you have just solved for, and know which one the question asked for.

minimising-the-original-expression-confused-with-minimising-the-wave

When a harmonic-form term sits inside a denominator (or is being subtracted from something), minimising the WHOLE expression can require MAXIMISING the wave, not minimising it — the two optimisations point in opposite directions. Confirmed directly on a real paper, where a fraction of the shape (constant)/(constant + wave) had to be minimised: "this did cause some confusion for many in that they did not realise that for the fraction to be a minimum the denominator had to be a maximum" (Oct 2020, Q7). Before optimising anything built out of R cos(θ−α), ask explicitly which direction — max or min of the WAVE — actually produces the extreme of the thing the question asked about.

incomplete-working-on-a-prove-that-identity

Spec 2.3's own guidance names an example identity students should be able to prove — "cos x cos 2x + sin x sin 2x ≡ cos x" — and identity proofs of this kind are marked on the presence of every intermediate line, not just a correct method plus a correct final line. Confirmed, a general summary from a real series: "many candidates omitted important lines when proceeding to the given solution resulting in the loss of some vital marks" (Jan 2022). On a 'show that' or 'prove' instruction, the compound-angle expansion line itself is not optional working to skip past — it is one of the lines being marked.

modelling-answer-missing-units-or-context-word

The single most consistently reported error anywhere in the WMA13 exponentials/logs archive, confirmed near-identically across at least four separate series: "the units (tonnes) were often omitted" (Oct 2021, Q3(b)); "only about half of them remembered that... they needed to also state the units" (Jan 2022, Q4(c)); "many lost the accuracy mark by either omitting the units '£' or not referring to a 'loss'" (Oct 2022, Q5(a)); "many did not gain the mark as they did not include the units (m2)" (Jan 2025, Q2(a)). A correct number with no unit or no context word attached is treated as an incomplete answer, not a minor presentation slip — write the unit or the contextual word every time a modelling question's answer is a quantity, not just an $x$.

calculator-equation-solver-used-with-no-algebra-shown

WMA13 papers carry an explicit, verbatim, paper-wide rubric on specific parts — "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable" (verified verbatim from the question paper itself) — and equations of the form $e^{ax+b}=p$ or $\ln(ax+b)=q$ are exactly where it lands. An examiner report confirms the consequence directly: "candidates should make sure that they do not resort to using a general equation solver... as this may not necessarily score full marks... it may be that full marks will not be awarded in the future, despite a correct answer" (Jun 2022, §5.3). Show the $\ln$ or $e^{(\ldots)}$ line explicitly, every time, on a question carrying this instruction.

log-base-confused-on-a-log-log-graph

Verified real context, Jun 2023 Q2: a log-log graph set explicitly in base $6$ — the examiner report records "being confused by log base 6, with some attempts to use log base 10 or 'e' and 'ln'" as the main documented error. The mechanism block above proves the gradient of a log-log graph is the same whatever base is chosen (true because the exponent $n$ in $y=ax^n$ doesn't depend on which base the logs are taken in — this is NOT true for a log-linear graph's gradient, $\log_c b$ for $y=kb^x$, which does change with the base $c$) — but the INTERCEPT is not, because it equals $\log_c(\text{constant})$ for whichever base $c$ the graph was actually plotted in. Un-logging an intercept with the wrong base — reading a base-6 intercept as if it were base 10 — produces a wrong constant even when every other step was correct.

exact-value-required-but-answer-rounded-early

Verified verbatim, from the general marking guidance itself, applying across every WMA13 topic but landing especially hard here because solving $e^{ax+b}=p$ so often produces an answer that is only exact as a logarithm: "Examiners' reports have emphasised that where, for example, an exact answer is asked for... marks will normally be lost if the candidate resorts to using rounded decimals". Where a question asks for an exact value of $k$ or $x$, stop at the $\ln(\ldots)$ or $\dfrac{\ln(\ldots)}{\ldots}$ form — a rounded decimal answers a different question than the one asked, even if a later part of the same question does then want 3 significant figures.

domain-restriction-from-a-non-positive-argument-not-explained

Confirmed real context: a temperature-decay model (Jan 2024, Q5) carries a hard lower-bound domain restriction that has to be explained using log-of-a-non-positive-number reasoning — the facts bank describes this context without quoting the examiner report verbatim on this specific point, so it is presented here as a described real question type, not a direct quote. The underlying mechanism is exact, though: $\ln(ax+b)$ is only defined where $ax+b>0$, so a model built around a $\ln(\ldots)$ term is only valid on the part of its domain where that argument stays positive — and a question that asks WHY a model breaks down, or for how long it remains valid, is asking for exactly this restriction to be identified and explained, not for more algebra on the equation itself.

log-notation-dropped-or-misplaced-when-reading-a-log-log-graph

Verified verbatim, Jun 2023 Q2(a)(i) examiner report — a distinct notation failure from the base-confusion trap above, on the very same real question: writing down the linear equation from a given log-log graph "proved to be straight forward and was generally done correctly for one mark. If this was not achieved, for example, a few students simply wrote $y = 4 - 2x$" — dropping every $\log_6$ entirely and treating the two labelled axes as if they were plain, unlogged $x$ and $y$. A second, separate slip the same report names: "the argument was often written as a superscript within the log" — writing something that reads as $\log 6^x$ rather than $\log_6 x$, putting the base in the wrong position. Neither is a calculation error — the axis labels on the graph already state the base and which variable is logged; the failure is not reading them onto the equation correctly, not anything that happens afterward.

rule-not-quoted-before-use

The single most consistently repeated piece of exam-technique advice anywhere in the WMA13 archive, confirmed independently across at least five series on this exact topic. Verbatim, Oct 2020 Q3: "A small number of candidates failed to quote the quotient rule formula and then gave an incorrect differentiation so were unable to gain credit for their method... Candidates should be advised to quote the formulae they use in their method." Verbatim, Jun 2023 Q10(a): "it is always advisable for candidates to quote the rule they are using before applying it to a particular function in case of slips in substitution." The mechanism is structural, not stylistic: the general marking guidance states that where a formula is not quoted, the method mark can only be gained BY IMPLICATION from correct working — so a single slip anywhere in the working can cost the method mark too, not just the accuracy mark, precisely because there was no quoted formula on the page to prove the method was ever correct in the first place.

answer-left-in-the-wrong-variable

Specific to spec 4.3. Verified verbatim, Oct 2020 Q8(ii), on an equation requiring $\frac{dy}{dx} = \frac{1}{dx/dy}$: "it was not uncommon to see the cosy term missing... many were unable to write cos y in terms of e^x" and, on the alternative method, "it was surprising to see the large proportion of candidates... who reached an answer eˣ/cos y and proceeded no further to replace cos y in terms of x." The failure is not algebraic — the candidates named here had already done the hard part correctly. It is stopping one line early: $\frac{dx}{dy}$ or $\frac{dy}{dx}$ found and inverted correctly, but never converted back into the variable the question actually asked for.

show-that-missing-intermediate-lines

Confirmed on a "show that" question in this exact topic area, Jan 2022: "many good candidates lost marks here for merely writing down the given answer from a correct dx/dy without any intermediate lines" — and independently, the same series' general summary: "many candidates omitted important lines when proceeding to the given solution resulting in the loss of some vital marks." When the answer is printed on the paper (a "show that" question), the mark scheme is explicitly marking the WORKING that reaches it, not the fact that the candidate's final line happens to match — reaching the printed answer proves nothing on its own.

calculator-relied-on-with-no-method-shown

Verified from the assessment structure itself, not one specific question: individual WMA13 parts carry an explicit rubric line, verbatim from the January 2023 question paper: "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable." Jun 2022's examiner report confirms candidates losing marks under this exact rubric: "there were a number of parts which stated that either relying on or entirely relying on the use of calculator technology was not allowed." Differentiation questions are exactly where this rubric lands, because the algebra itself — which rule, applied correctly — is the thing being examined, not just the final numerical or symbolic value a calculator could produce without it.

exact-form-given-as-a-decimal

A general marking convention, verified verbatim from the Jan 2023 general guidance and independently corroborated by examiner-report commentary in nearly every series reviewed: "Examiners' reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals." Directly relevant here: a spec 4.3 answer such as $\frac{1}{4\sqrt{1-x^2}}$ is an exact form, not an instruction to reach for a decimal approximation the moment a square root appears — the square root is the answer, not an obstacle to clear before writing one down.

rate-of-change-answered-by-substitution-not-differentiation

The single most directly on-topic, most consistently repeated error for spec 4.4 specifically, confirmed across three series with a genuine rate-of-change part (Jan 2022 Q8(c), Jan 2024 Q5(c), Jun 2023 Q7(b)) — candidates asked for a rate of change instead calculate an average or finite difference, or simply substitute a value into the model with no differentiation attempted at all, scoring zero either way. Verified verbatim, Jun 2023 Q7(b): "Some, perhaps missing, or misunderstanding, the reference to 'rate of decrease' substituted t = 5 directly into N... Both approaches earned no marks." The phrase to watch for is exactly this: "rate of change," "rate of increase," "rate of decrease" — each is a direct instruction to differentiate first and only then substitute. A value of the function itself, however accurately computed, and an average change over a time interval, however close the two numbers look, are both a different quantity from the one being asked for, and both are credited nothing.

modelling-rate-missing-its-unit-or-direction-word

A confirmed, cross-cutting pattern for exactly this style of modelling answer — This course's own research documents it directly across at least four series (Oct 2021 Q3(b): "the units (tonnes) were often omitted"; Jan 2022 Q4(c): "only about half of them remembered that... they needed to also state the units"; Oct 2022 Q5(a): "many lost the accuracy mark by either omitting the units '£' or not referring to a 'loss'"; Jan 2025 Q2(a): "many did not gain the mark as they did not include the units (m2)"), and independently confirmed in a differentiation-specific context too (Jan 2024 Q4(b): "Not all candidates appeared to be confident about what was expected of them to fully justify that f(x) was decreasing... Others did not proceed to a conclusion such as 'hence the function is decreasing'."). For a rate-of-change answer specifically, this trap has two parts at once: the unit itself (£ per year, m² per day, whatever the model's own variables are measured in) and the direction word ("increasing" or "decreasing"). A numerically correct rate with neither is an incomplete answer, not a presentation nicety — write both, every time a question's answer is a rate of something, not just a bare number.

domain-restriction-not-recognised-as-a-genuine-answer

Verified directly, verbatim, from the real Jan 2024 Q5(d) mark scheme and examiner report — a temperature model, $T=10+8e^{-Bt}$, with exactly the hard lower-bound restriction this lesson's chain drill models, asking candidates to explain why $T=5$ has no solution. The mark scheme's own allowed and disallowed wording is precise and worth teaching directly: credit was given for "which is not possible", "cannot be done", or "you cannot find the log of a negative number" — but explicitly NOT for "logs cannot be negative" or "you cannot have a negative time", both flagged in the scheme itself as "ambiguous/incorrect statements". "Logs cannot be negative" is genuinely wrong, not just imprecise — $\ln(0.5)\approx-0.693$ is a perfectly ordinary negative log; what a positive-based exponential term can never produce is a zero or negative ARGUMENT for the log to act on, not a negative log value. Verified verbatim, the examiner report: "Some clear explanations of why the temperature could not reach 5 degrees were seen by either explaining that the lower limit was 10 or by trying to solve the equation and explaining that the log of a negative number could not be found. Common incorrect answers were to give the minimum value as 8 or 18." It follows directly from the mechanism derived earlier in this lesson: an exponential term is always strictly positive, so a model built around one can never actually cross the floor or ceiling that term is added to. Recognising "this equation has no real solution" AS the answer — not as a sign to go back and re-check the arithmetic — is exactly what spec 4.4's own guidance means by "consider whether the predicted range of values is appropriate," and stating WHY with the precise reasoning the mark scheme actually credits, not the imprecise version it explicitly rejects, is what separates full marks from none here.

negative-exponent-sign-dropped-during-differentiation

A second, independently confirmed differentiation slip specific to a DECAYING model written with a negative exponent, $N=N_0a^{-t}$ or $N=N_0e^{-Bt}$, rather than every model met so far in this lesson, $N=N_0a^t$ with $0<a<1$ — confirmed across two series (Jan 2022 Q8(c), Jan 2024 Q5(c)). The chain rule (already a prerequisite from differentiation-rules.ts, applied here to the exponent $-t$ itself rather than restated) adds an extra factor of $-1$ that a bare application of $a^t\to a^t\ln a$ does not produce on its own: $\frac{d}{dt}\left[a^{-t}\right]=-a^{-t}\ln a$, not $+a^{-t}\ln a$. Verified verbatim, Jan 2022 Q8(c) examiner report: "Common errors were to miss off the negative sign and treated the t as a constant power and subtracting 1 from it (this incorrect method achieved no marks for this part)." Verified verbatim, the real Jan 2024 Q5(c) mark scheme itself: "If they lose the minus sign in ...e−Bt they obtain ±0.0518… and this scores M0" — in that scheme the sign is folded directly into the METHOD mark, so losing it can cost the entire part, not just a final accuracy line. The fix is mechanical: after differentiating a model with a negative exponent, check that the sign of the exponent's own coefficient survived into the derivative, before substituting anything.

modulus-dropped-in-log-integration

Confirmed directly on a real recognition-integration question involving negative values: "many did not have the modulus and left the values as 5ln(-2) and 5ln(-4) not realising these are undefined" (Oct 2020, Q9(c)); independently confirmed in Jun 2023, Q9(c). The mechanism this lesson's own derivation makes explicit: $\ln|f(x)|$ is defined and differentiates correctly for every $f(x) \neq 0$, including negative values, while $\ln(f(x))$ alone breaks the moment $f(x)$ turns negative. Whether the modulus is essential or just safe depends entirely on whether $f(x)$ can actually be negative on the domain in question — check that before deciding it can be dropped, never assume.

constant-of-integration-omitted

Confirmed across at least two series on this exact topic: "A fair number of candidates could not be awarded the final mark, despite having obtained the correct algebraic expression, as they had forgotten to include the constant of integration" (Oct 2021, Q5(ii)); independently confirmed, "a substantial number of candidates neglected to include the constant of integration and were penalised" (Jan 2022, Q3(i)). It costs one mark on an otherwise perfect answer, on an indefinite integral specifically — a definite integral (like part (d) of the marked solution above) has no constant to lose, which is worth noticing precisely so the two cases are not confused.

double-angle-form-confused

A real double-angle "show that" question records candidates who "commonly wrote cos 2x as cos²x − 1 or 1 − cos²x" (Oct 2020, Q1) — misapplying or confusing the three equivalent double-angle-for-cosine forms with each other, or with the plain Pythagorean identity. This lesson's identity-rewrite technique for $\sin^2 x$, $\cos^2 x$ and $\tan^2 x$ depends entirely on quoting the RIGHT rearranged form with the RIGHT coefficient — get the identity wrong (as in the marked solution's own common wrong path above) and the integration method that follows can be flawless and still reach the wrong answer.

intermediate-lines-omitted-on-hence-or-show-that

Confirmed as a general pattern across multiple series, not tied to one question: "many candidates omitted important lines when proceeding to the given solution resulting in the loss of some vital marks" (Jan 2022, general summary); the same report separately records "many good candidates lost marks here for merely writing down the given answer from a correct dx/dy without any intermediate lines." A "hence, show that" part with a printed target answer — exactly the shape of part (d) in the marked solution above — is where this costs the most: every line from the antiderivative to the printed target has to actually appear.

booklet-result-not-recognised

Confirmed on a real recognition-type integral: "some students did not realise the result could be found from their Formula Booklet, and instead used a substitution" (Jun 2023, Q9(c), on ∫cosec x dx). Substitution is a fully valid, mark-scheme-credited alternative route — the method-comparison block above shows exactly that — but re-deriving from scratch under time pressure, when a result is already printed and the question does not demand a derivation, spends time and adds a line where a sign or arithmetic slip can happen that a direct quote could not have introduced.

continuity-statement-omitted

The single most repeated specific error across the entire numerical-methods topic, confirmed independently in general summary AND question-specific commentary across at least five series. "the omission of reference to 'continuity' or 'continuous' being the most common error" (Oct 2022, general summary); "there was a failure to also comment on the fact that the function was continuous" (Jan 2022, Q5(a)); "students often lost a mark in question 2(a) for not making a reference to the continuity of the function" (Jan 2024, general summary, referring to Q2(a)). Encouragingly, the most recent series checked shows this improving: "The mention of continuity in some form... now exceeds the lack of mention as candidates have adapted to this requirement being needed" (Jan 2025, Q1(a)) — but the fix costs one clause ("f is a polynomial, so continuous") and there is no reason to be part of the group still losing it.

hence-root-conclusion-missing

A real mark scheme's stated requirements for the sign-change accuracy mark name three separate components, verbatim: both function values correct, "reason, which must mention continuity and state or indicate sign change in some way," and "conclusion, 'hence root'" (Oct 2023, Q1(a)). Values and reasoning with no final sentence tying them together is, by this wording, still an incomplete answer — the write-up needs to explicitly land on "hence there is a root in this interval," not leave the reader to infer it.

iterate-given-to-fewer-decimal-places-than-asked

Confirmed across at least two series: "there were candidates who did not give the value of their iterate to the required number of decimal places" (Oct 2020, Q6(c)); "A common error in part (c)(ii) was an error in rounding or premature rounding of intermediate steps" (Jan 2024, Q2(c)(ii)). This is two failure modes wearing one name: stating fewer decimal places than requested (loses the mark even if the digits shown are a fair rounding), and rounding an INTERMEDIATE value before using it in the next substitution (which changes the final answer, not just its presentation).

ans-key-substitution-not-shown

Confirmed verbatim: "it is even more important that candidates show the method of embedding the values in the iterative formula to demonstrate they understand how to generate the values" (Jun 2022, Q8(c)) — the report explicitly frames a correct final iterate with no substitution shown as "an answer not implying a correct method to solving had been shown." A calculator's stored-answer key can chain a whole sequence of iterates silently; the mark scheme is checking for the written line that proves the chain, not just its last link.

calculator-solver-used-despite-the-no-calculator-instruction

Verified verbatim from an actual WMA13 question paper, printed directly above a question of this type: "In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable." (Jan 2023, Q5). An examiner report independently corroborates the consequence: "there were a number of parts which stated that either relying on or entirely relying on the use of calculator technology was not allowed" (Jun 2022) — and this warning recurs across multiple series specifically on numerical-methods-style questions, where a calculator's built-in equation solver can produce the final answer directly, bypassing exactly the method the question is testing.