How marks are actually earned

algebraic-proof-attempted-when-exhaustion-required

The most-documented single error in this topic, confirmed independently in two separate series. October 2022: "a significant number of attempts... using logical or algebraic approaches. Such attempts were mixed in quality and often unnecessarily long-winded... Some who obtained a cubic expression for the product just concluded it was even without any justification." October 2023, on a different question: "an attempt at an algebraic solution which proved more demanding than that of a numerical one." When a claim is already restricted to a small, explicitly bounded set of cases, listing and checking them is both the method the spec names and — confirmed twice, independently — the faster and safer route.

numerical-examples-mistaken-for-general-proof

Confirmed as the single worst-performing question on its own paper: an October 2021 report on a "prove true for all positive x, y" question states "though only a third of candidates were able to score the first mark, there were many who attempted an algebraic proof... there was also a large number who considered that several numerical examples of the inequality holding constituted a proof" — 47% of candidates scored 0 out of 4. This is exhaustion's own logic reaching past where it applies: checking a handful of values and finding the claim true every time proves nothing about a domain with no last case to reach, however many values are checked. Only a general argument — one that holds for an arbitrary case, not a list of specific ones — closes an infinite or continuous domain; see the worked algebraic-proof example above for what that argument actually looks like.

multiple-of-k-assertion-without-quotient-shown

Confirmed on a real exhaustion question checking a "multiple of 4" claim — k > 2, so unlike "even" or "multiple of 5," the property is not visible by inspecting a last digit. The real mark scheme is explicit that a bare assertion does not earn the mark: "they can be divided by 4 (on its own) is insufficient without further clarification such as 'to give whole numbers'" (Oct 2023). What it does accept is a minimal but explicit marker that the division genuinely produces a whole number — a calculation such as "$40 \div 4 = 10$," a tick against each value, or the word "prove"/"QED" next to the working — anything that shows the check was actually done, not just declared. For a "multiple of 4" claim, write "$40 \div 4 = 10$, a whole number, so $40$ is a multiple of $4$," not "$40$ is divisible by $4$" on its own; the marker cannot tell a genuine check from a guess unless the quotient — or an equivalent explicit clarification — is shown.

product-misread-as-sum

Confirmed directly: "the definition of 'product' is not widely understood, with many students considering the sum of a, b and c instead" (Oct 2022). This is not an arithmetic mistake inside an otherwise correct method — it is checking the wrong quantity against every single case, so a perfectly executed exhaustive list still proves nothing about the question actually asked. Read what operation the question names before setting up the check, not after.

case-defining-relation-applied-backwards

Confirmed on a real exhaustion question where the cases were generated from a stated relationship between the variables: "sometimes the result of using c = b − 2 instead of c = b + 2" (Oct 2022). A relation like "c is 2 more than b" has a direction; reversing it silently generates a different — and wrong — set of cases from the very first line, so every case checked afterwards is checked against the wrong list, however carefully the rest of the working is done.

column-values-transposed-between-derived-variables

Confirmed on the same well-answered exhaustion question as above, as a separate, less prominent cause of lost marks: "a mix up between columns b and c" (Oct 2023). This is not the relation being applied backwards and not an arithmetic slip — each value can be individually correct, but written under the other variable's column head, so the row no longer records the case it claims to. It shows up specifically when two columns are both derived from the same base variable rather than from each other (here b and c are both computed from a, not from one another) — nothing forces the two labels apart the way a stated relation would, so check each entry against its own column head after filling a row in, not only that the pair of numbers is right somewhere in the row.

case-list-not-exactly-complete

Confirmed as a specific, separately-documented cause of lost marks, distinct from getting a case wrong: "the addition of extra rows" (Oct 2023) costs marks in one direction, and the mechanism above shows why an incomplete list costs them in the other — a bounded set has an exact size, established in the very first step of the proof, and the finished list has to match it exactly, neither padded with cases outside the stated bound nor missing any inside it.

conclusion-omitted

Confirmed on a well-answered exhaustion question, where the cases themselves were rarely the problem: "the solution merely required the sight of three correct rows and a minimal conclusion. Reasons for a loss of marks were:... the omission of a conclusion" (Oct 2023). A completed, correct case list is not yet a finished proof — the sentence stating that every case has been checked and the claim therefore holds is not decoration, it is the step that turns the list into a proof.

counter-example-search-continued-past-the-first-valid-one

Confirmed on a real counter-example question: "most candidates understood the principle of finding a counter-example, although many found far more than necessary" (June 2019) — the same report that states the underlying rule directly: "only one counter-example is required to prove that a statement is not true." Once one failing case has been found and justified, the disproof is complete; continuing to search costs time without earning further credit.

root-sign-taken-straight-from-the-bracket

The dominant error on this spec item, confirmed on a real three-part factor-and-remainder question in both directions at once: candidates were recorded "setting $f(\frac{1}{2}) = 0$ in part (a) and/or setting $f(-2) = -25$ in part (b)" where the correct substitutions were $f(-\frac{1}{2})$ and $f(2)$ (Oct 2023, Q4). The fix is mechanical rather than mnemonic: never read a number off a bracket, solve the bracket for $x$. $(2x+1)=0$ gives $x = -\frac{1}{2}$; $(x+2)=0$ gives $x = -2$; $(3x-2)=0$ gives $x = \frac{2}{3}$. One extra line, and the most expensive error on the topic cannot happen.

remainder-equation-set-to-zero

Confirmed on an accessible opening question: the commonest conceptual errors were "using $f(-2)$ or $f(3)$ rather than $f(2)$ or setting equal to $-3$ or $0$, rather than $3$" (Jan 2024, Q1). Setting the substitution equal to zero is the factor-theorem reflex applied to a remainder question — and it is worth seeing that it is the same identity in both cases, $f(x) \equiv (ax-b)q(x) + r$, with $r$ being whatever the question said it was. Zero is a value of $r$, not the default.

long-division-chosen-when-substitution-was-asked-for

Not wrong, but measurably worse, and the examiners have said so in as many words: "Candidates who chose to use division gave themselves a more difficult task and tended to be less successful, with many giving up part way through the division" (Jan 2024, Q1), on a question where "the remainder theorem was the most efficient approach." Division computes the entire quotient in order to discard it. Reach for it when a later part actually wants the quotient; otherwise substitute.

factorisation-stopped-one-step-short

Confirmed as a specific, separately-recorded cause of lost marks: "leaving the answer to (c) as $(2x+1)(2x^2 + x - 15)$" (Oct 2023, Q4). That quadratic does factorise — $2x^2 + x - 15 = (2x-5)(x+3)$ — which is exactly why the answer was marked short: "factorise completely" is not satisfied while any factor still factorises. After dividing out the known factor, always ask whether the quadratic quotient breaks down further, and check its discriminant if it is not obvious.

hence-factorise-reached-without-using-the-given-factor

On a "hence factorise" part, the word "hence" is a mark-scheme instruction, not decoration: the working must derive the factorisation from the already-established factor, and reaching the identical, completely correct final answer by any other route earns nothing on that part. A real mark scheme is explicit about it, on the same question this lesson's worked chain and marked solution are modelled on: "The question states 'Hence' so there is an expectation that they use the factor of $(2x+1)$ which is given. Solutions that just state $4x^3 + 4x^2 - 29x - 15 = (2x+1)(2x-5)(x+3)$ score M0 and therefore A0 A0" (Oct 2023, Q4(c)(ii)) — a fully correct factorisation, worth zero, purely because the shown working never used the given factor. The matching real error is recorded in the same series' examiner report: candidates "using a calculator to solve $4x^3 + 4x^2 - 29x - 15 = 0$ and writing $x = -3, -\frac{1}{2}, \frac{5}{2}$" — solving the cubic directly, on a paper where calculators are permitted throughout, instead of dividing out the factor the question had already given them. Every worked chain, chain drill and marked solution in this lesson models the division route for exactly this reason: on a "hence" question, showing the division or comparison is not supporting style, it is the entire mark allocation for that part, independent of whether the final product is right.

non-integer-factor-form-given

A genuine "mathematically equivalent but marked wrong" trap, confirmed on a real question whose final step was a factorisation: a candidate "giving the factor as $x - \frac{7}{2}$... This is incorrect as integer values were required" (June 2025 Regional 01R, Q8). $\left(x - \frac{7}{2}\right)$ and $(2x-7)$ have the same root and are the same factor up to a constant, and the scheme still wants the integer-coefficient form. When a root comes out as a fraction $\frac{b}{a}$, write the factor as $(ax - b)$.

answer-left-as-a-list-not-a-product

A real mark scheme states the required form of a factorised cubic explicitly, requiring it "written out as a product of factors (not as a list) and isw" (Jan 2020). Listing the three factors, or listing the three roots, is not the same object as their product — the question asked for $f(x)$ factorised, so the answer is an expression equal to $f(x)$. The isw (ignore subsequent working) note is the encouraging half: once the product is correctly written down, later scribbling does not undo it.

show-that-steps-omitted

On a "show that" part, the printed answer is not the thing being marked — the route to it is. A real report records marks lost for "not showing the $= 0$ or intermediate lines when attempting to prove (a)" (Oct 2023, Q4), and the general marking guidance's cso convention makes the standard explicit: "There must be no errors in this part of the question to obtain this mark." Write the substitution, write the line that clears the fractions, then write the printed result — a solution that leaps from one to the other has demonstrated nothing that could be checked.

centre-sign-not-flipped

A real WMA12 centre/radius question confirms candidates stating the circle's centre with a sign flip — the value read straight off a completed-square bracket, instead of its negative (Jan 2023, Q6). $(x+3)^2$ is zero at $x=-3$, not $x=3$: the bracket has to be interpreted, not transcribed. This is the exact same mechanism the WMA11 pilot documents for a quadratic's turning point, and it costs a mark for the identical reason in both places.

general-form-read-as-centre-radius-form

Completing the square in both $x$ and $y$ is not optional preparation, it is the entire first step whenever a circle's equation is presented multiplied out — reading a centre or radius off general form directly, without converting first, is a genuine error type. It is not, however, the step most candidates actually lose the mark on: a real WMA12 general-form question confirms the vast majority of candidates successfully completed this conversion (Jan 2023, Q6), with the one error that does recur being the sign flip covered in the next trap item, not a failure to attempt the conversion at all.

solved-parameter-not-substituted-into-earlier-general-centre

When a circle's equation is given in terms of an unresolved constant (e.g. centre $(k, 2k)$, found correctly in an earlier part), and a later part requires a SPECIFIC numeric value of that constant — found by solving elsewhere in the same question — the mark for "identifying the centre" in that later part requires substituting the specific solved value back into the earlier general expression, not re-deriving from the original equation and not stopping after substituting the value into the circle's own equation without extracting actual coordinates. A real WMA12 examiner report confirms this directly, on a question where this exact substitution step was the single biggest mark-loss point of the whole question: "after nearly 90% of candidates successfully navigating the first two parts, fewer than 60% managed even the first mark, for identifying the centre of the circle... A few candidates chose the wrong value of k and so lost the mark, while others attempted to use the original equation to find where the circle meets the x axis thus making no progress at all with the question. Others simply put the value of k into the equation of the circle and then stopped. This did not gain any marks" (Jan 2021, Q9). Whenever a constant is left unresolved across multiple parts of the same question, the centre to use from that point on is the one built from the specific value just solved for — not the general symbolic form quoted correctly in an earlier part.

radius-given-as-decimal-not-exact-surd

The general marking guidance's exact-answer rule states plainly 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" — and a real WMA12 circle question specifically required an exact-form radius rather than a decimal approximation (Jan 2023, Q6). A radius of $2\sqrt{5}$ rounded to $4.47$ looks harmless; the mark scheme treats it as a lost mark.

implicit-differentiation-chosen-over-the-property

A tangent's gradient can, in principle, be found by implicitly differentiating the circle's equation instead of using the radius-perpendicularity property — but a real WMA12 report records that "very few attempted to differentiate the equation of the circle, and these often made little progress even though the next method marks were available" (Jan 2021, Q9). Valid on paper is not the same as safe under time pressure: the perpendicularity property reaches the same gradient in two lines, with no unfamiliar technique to execute correctly first.

origin-and-circle-centre-both-labelled-o

A real WMA12 question let a circle's centre and the point O — meaning the origin — sit in the same problem, and a report records "the misunderstanding that O was the centre of the circle rather than the origin... a special case was permitted, and candidates were able to score all but the final A mark" (Jan 2021, Q9). The scheme was generous that one time; reading which O is which before starting is the version of this habit that doesn't rely on generosity.

substitution-direction-not-checked-for-fractions

Before substituting a rearranged line into a circle equation, check which variable's coefficient is smaller — a real WMA12 mark scheme records that "the (−2y−10) substitution for x was more successful than the (−0.5x−5) for y which led to work involving fractions" on a genuine tangent-and-normal intersection question (Jan 2023, Q6). Both directions reach the same answer for the same marks; only one of them does it without an extra fraction-clearing step in the middle.

general-area-formula-chosen-over-axis-aligned-base-height

When a triangle's base lies along an axis, its height is simply the third vertex's coordinate on the other axis — no trig or formula needed. But a real WMA12 examiner report records that "many candidates [were] not using the simple formula for the area of the triangle, electing instead to use ½ab sinC" or even a shoelace formula, "not appreciating the geometry of the situation, with the requisite triangle being right-angled" (Jan 2021, Q9). The mark scheme itself credits either route as a complete method, so no mark is actually lost by reaching for the general formula — but the marked solution above finds T on the x-axis and reads the base OT and height (A's y-coordinate) straight off in two lines precisely because that axis-alignment is there to exploit, while ½ab sinC or a shoelace calculation forces extra side-length and angle work under time pressure for the same credit.

sum-proof-last-term-off-by-one

Confirmed directly on the standard sum-formula proof: "a surprising proportion of candidates struggled to write a correct expression of Sₙ at the start of their proof, often having the last term incorrect" (Jan 2020, Q8(i)). The fix is the same mechanism taught earlier in this lesson: reaching the nth term from the first takes (n − 1) steps of the recurrence, so the last term of an n-term series is a + (n − 1)d, never a + nd.

sum-proof-reversal-skipped

Confirmed on the same question: "Others failed to reverse the terms in the sum correctly, or at all" (Jan 2020, Q8(i)). Without the reversed copy, there is nothing for the forward copy to pair against, and the whole argument — every pair summing to the same constant — has no pairs to point at. This step is not optional decoration on the proof; it is the entire mechanism.

sum-proof-insufficient-terms-shown

Confirmed on the same question: candidates "did not display a sufficient number of terms to make subsequent steps in their proof to complete" (Jan 2020, Q8(i)). Two terms at each end plus an ellipsis — $a + (a+d) + \cdots + (a+(n-2)d) + (a+(n-1)d)$ — is the minimum that actually shows the pattern; a single "..." with no visible terms either side of it gives an examiner nothing to check the pairing against.

induction-attempted-for-standard-proof

Confirmed on the same question: "Attempts at proof by induction were also seen in a few cases, though little progress was made in most of these" (Jan 2020, Q8(i)). Induction can, in principle, prove this result, but it is not the proof the specification names, and the report records it as the less successful choice in practice. Where a question asks to "prove the standard formula," the pairing argument is the direct, expected route.

sum-proof-assumes-the-result

Confirmed on the same question: "Some candidates either started with, or used something that they were trying to prove, e.g. starting with $S_n = \frac{1}{2}n(a+L)$, or attempting to use a summation formula that is a generalisation of the result they needed to show" (Jan 2020, Q8(i)). This scores zero on an "ag" (answer given) proof no matter how cleanly the rest of the working is presented, because it assumes the very fact being proved rather than deriving it — a "prove that" question can never legitimately start from a form of its own target result. The only safe starting point is the definition of the series itself, written out term by term, exactly as the mechanism block above does.

sum-proof-pairing-fails-for-odd-n

Confirmed on the same question: some candidates instead tried pairing terms within a single copy of the sum — first-plus-last, second-plus-second-to-last, and so on — "but they did not give due consideration to what happens when there are an odd number of terms, and were unable to score the accuracy mark as a result" (Jan 2020, Q8(i)). An odd number of terms leaves one unpaired middle term this method has no clean way to handle, which is exactly why the write-forwards/write-backwards/add version taught in this lesson is the safer standard route — it pairs the full sum against a full reversed copy of itself, so it never needs $n$ to be even and never has a leftover term to account for separately.

arithmetic-and-geometric-formulas-confused

Confirmed as a recurring, named error category across at least two series (June 2019 and October 2020): candidates reach for the wrong series formula mid-question — applying the geometric sum formula to an arithmetic series, or the reverse — because the two topics sit close together and use similarly-shaped notation. The one-line check that catches it before it starts: an arithmetic sequence has a constant DIFFERENCE between consecutive terms ($u_{n+1} - u_n = d$, the same value every time); a geometric one has a constant RATIO instead. Confirm which quantity is actually constant before reaching for either sum formula.

arithmetic-and-geometric-formulas-confused

Confirmed as a recurring, named error category across at least two series (June 2019 and October 2020): candidates reach for the wrong series type's formula mid-question, because the two topics sit adjacent in the specification, use similarly-shaped notation, and are printed one above the other on the same formula-booklet page. Seen from the geometric side, the tell is any appearance of $d$, of $(n-1)d$, or of $\frac{1}{2}n$ in working about a series whose terms are multiplying. The two-line check that catches it before a single mark is at stake: compute $u_{n+1} - u_n$ and $u_{n+1}/u_n$ for two consecutive pairs, and see which of the two comes out the same both times.

overlap-term-double-counted

Confirmed on a January 2022 mixed arithmetic-then-geometric question, verified as that paper's biggest discriminator. The examiner records that candidates "were not able to correctly process the overlap of the 20th term of the arithmetic series with the first term of the geometric series, and double counted this term" (Jan 2022). When one series ends on the same term another begins with, that term belongs to exactly one of the two sums. The report itself names both legal fixes: the geometric series starts at (shared term) × r, "or alternatively" the shared term is subtracted out of the arithmetic total — either one, never neither, never both.

geometric-series-restarted-from-the-original-first-term

Confirmed on the same January 2022 question, and distinct from the double-count above: "many candidates failed to understand the context with many assuming they had to start the geometric series from the first term of $a = 100$ again" (Jan 2022). Where a sequence changes character partway through, the geometric part inherits its first term from wherever the arithmetic part left off — not from the beginning of the whole sequence. Reading the question's "from the $k$th term onwards" literally, and writing down what that term actually equals before doing anything else, is the fix.

sum-to-infinity-first-term-misidentified

Confirmed on a January 2024 sum-to-infinity question: candidates "often used a = 6 instead of a = 1.5 in the sum to infinity formula" (Jan 2024, Q5(i)) — taking the sequence's original first term rather than the first term of the series actually being summed. This bites whenever a question asks for the sum of everything after some term, or re-expresses a series starting from a later index. The $a$ in $\frac{a}{1-r}$ is not "the first term of the sequence"; it is the first term of the sum you are being asked for. The self-check: the answer to "sum everything after the first term" must be smaller than the answer to "sum everything" by exactly that first term.

calculator-summation-shown-without-formula

Confirmed on the same January 2024 question: "a number resorted to using the summation button on their calculator which, even if [the value] was found, no marks were scored" (Jan 2024, Q5(i)). Calculators are permitted throughout this paper, which makes this trap easier to fall into rather than harder — a correct number with no visible formula and no visible substitution earns nothing, because the marks on offer are method marks and there is no method on the page to award them to. The mark scheme's standing principle says the same thing in general terms: where a learnt formula is used, "the formula should be quoted first."

standard-proofs-not-known

Confirmed on the January 2020 arithmetic sum-formula proof, with the examiner's advice naming both series explicitly: students "would be well advised to make sure they know the standard proofs for the arithmetic and geometric series to ensure they are able to give should it be required" (Jan 2020, Q8(i)). The specification's own guidance for item 4.4 says the same — "the proof of the sum formula for a finite series should be known." These are the only marks in the topic that a formula booklet cannot rescue: the formulae are printed, the proofs are not.

geometric-proof-subtraction-direction-reversed

Confirmed on the October 2020 geometric sum-formula proof: "A significant number started correctly and knew they had to subtract but subtracted the wrong way round to obtain e.g. $S_n - rS_n = ar^n - a$ and a significant number of candidates obtained $S_n - rS_n = a + ar^n$" (Oct 2020, Q8(a)). The correct line is $S_n - rS_n = a - ar^n$: the survivors of the subtraction are whichever terms belong to the line that was NOT shifted (the $a$ from $S_n$) minus whichever belong to the line that WAS (the $ar^n$ from $rS_n$), in that order — reverse which line each survivor is taken from, or turn the subtraction into an addition under exam pressure, and the result no longer factorises to the printed answer no matter how the rest of the algebra is handled. The examiner report records this immediately alongside the same question's other confirmed proof error (importing the arithmetic proof's reverse-and-add method), making the two the two most-recorded ways this specific proof goes wrong.

convergence-condition-read-as-r-less-than-one

Derived from the mechanism rather than quoted — the facts bank records no examiner finding on this specific error, and this entry carries no citation. Reading $|r| < 1$ as "$r < 1$" admits every negative ratio, including $r = -2$ and $r = -1.5$, whose terms grow without limit while alternating in sign. The modulus is a statement about size: $S_\infty$ exists precisely when each term is smaller in size than the one before it, regardless of sign. The related boundary case is $r = -1$, where $r^n$ alternates between $-1$ and $1$ forever and the partial sums alternate between $a$ and $0$ — never settling, which is why the specification's inequality is strict.

term-index-mismatched-to-power

Confirmed directly on a real WMA12 question: common mistakes included "using the wrong power (usually 4 or 6 instead of 5 for the fractional part) or using the wrong binomial coefficient (¹²C₆, ¹²C₈ and ⁷C₅ instead of ¹²C₇)" (Jan 2024, Q2). The power of $x$ in the general term $\binom{n}{r}a^{n-r}(bx)^r$ is $r$ itself — no "one more" or "one less." State $r$ explicitly, on its own line, before touching any other number, and this error becomes almost impossible to make by accident.

general-series-form-chosen-over-combinatorial-form

The P2 formula booklet page prints two binomial formulas side by side: the combinatorial $(a+b)^n$ expansion for positive integer $n$, and the general series $(1+x)^n = 1+nx+\ldots$ for real $n$ and $|x|<1$ — the second is P4 content, not required here, and genuinely harder to apply. An examiner report confirms most candidates default correctly: "Part (a) was generally done very well with candidates mostly using the (a+b)ⁿ form of the binomial expansion where n is a positive integer rather than (1+x)ⁿ where n is a rational number" (June 2025, Q1) — but a regional variant paper shows exactly what the harder route costs once a question adds a second unknown: "Many candidates struggled to make a start with this question due to the fact that there were two unknowns, n and k, within the binomial expression (1+kx)ⁿ" (June 2025 Regional 01R, Q7). On P2, $n$ is always a known positive integer — default to $(a+b)^n$ every time.

bx-term-not-raised-as-a-whole

Confirmed directly on a real WMA12 question, not just derived from the arithmetic: the examiner report for the same anchor question cited elsewhere in this lesson names this as the most common way of losing the accuracy mark for the unsimplified term — "using invisible brackets that were not recovered in further work (4x⁷ instead of (4x)⁷) resulting in multiplication by 4 instead of 4⁷" (Jan 2024, Q2). $(bx)^r$ means both $b$ and $x$ are raised to the power $r$ — $(4x)^3 = 4^3x^3 = 64x^3$, not $4x^3$. Losing the power on the numeral while keeping it on the variable is easy to miss on the page, because the $x^3$ still looks correct; only the coefficient in front of it is wrong. Write $(bx)^r$ as $b^r x^r$ before evaluating $b^r$, so the two pieces are never handled in the same breath.

sign-lost-on-odd-powers-of-a-negative-b

When $b$ is negative, $(bx)^r = b^r x^r$ carries the sign of $b^r$ — negative when $r$ is odd, positive when $r$ is even, for the same reason $(-1)^3 = -1$ while $(-1)^4 = 1$. The terms of an expansion like $(3-2x)^7$ therefore alternate strictly by the parity of $r$, not by any other pattern — dropping the sign on just one term, usually by writing $(-2)^r$ as $2^r$ out of habit, produces an answer that looks structurally right and is numerically wrong by exactly a factor of $-1$.

pascal-triangle-row-miscounted

Row $n$ of Pascal's triangle — counting the single "1" at the top as row 0 — has exactly $n+1$ entries, $\binom{n}{0}$ through $\binom{n}{n}$. Starting the count at row 1 instead of row 0 reads off the coefficients for $n-1$, one whole row too early, and every coefficient taken from it is wrong even though every addition inside the triangle was done correctly. Before reading off a row, count its entries: $n+1$ numbers, not $n$.

required-term-not-isolated-from-a-list

Confirmed directly on the anchor WMA12 question: the examiner report states the final accuracy mark "was commonly not awarded to candidates who failed to isolate the required term from a list" (Jan 2024, Q2), and the mark scheme itself warns "the term or coefficient must be identified if they have more than one term." Writing several terms, a full expansion, or a whole row of Pascal's triangle is not itself an error — but once the value asked for has been found, it has to be pointed to on its own, not left sitting unlabelled inside the longer list. State it on its own final line (or circle/box it) the moment it's reached.

log-subtraction-rewritten-as-division

Confirmed directly on a real WMA12 log-manipulation question: "A fairly common mistake was to rewrite log₃(5x+7) − log₃4x as log₃(5x+7)/log₃4x. Some then crossed out log₃ in the numerator and denominator as though simplifying a fraction" (Jan 2023, Q4(i)). The subtraction law says $\log_a x - \log_a y = \log_a(x/y)$ — the two ARGUMENTS are divided, the two log SYMBOLS never are. Writing $\log_3(5x+7)/\log_34x$ treats "log₃" as a common factor that can be cancelled, which it is not: $\log_a$ is a function name, not a multiplier.

log-of-a-sum-expanded-as-sum-of-logs

Confirmed on the same question: "Another mistake, common for this type of question, was to expand the brackets within the log function, writing log₃(5x+7) as log₃5x + log₃7" (Jan 2023, Q4(i)). No law distributes a log across a sum or difference inside its own argument — the addition law only COMBINES two separate logs into one, it never SPLITS a single log open. $\log_a(x+y)$ has no simpler form; treating it like $\log_a(xy)$ applies the multiplication law to an expression that is added, not multiplied.

log-squared-confused-with-log-of-the-square

Confirmed on a harder question, described as "one of the most challenging parts on the paper": "understanding the difference between logₐ(y²) and (logₐy)²... rewriting (logₐy)² as 2logₐy was a very common mistake" (Jan 2023, Q4(ii)). The power law moves an exponent sitting INSIDE the log — it says nothing about an exponent sitting OUTSIDE the whole expression. $(\log_a y)^2$ means "square the number $\log_a y$ produces," and no law turns that squaring into a multiplication by 2 — the two expressions are built from the same symbols in a different order and are not generally equal.

exponent-rearrangement-left-unfinished

Confirmed on a solving-aˣ=b question with a measured mark-by-mark drop-off: "A fall off from nearly 90% to just over 70% between the first two marks is revealing" (Jan 2021, Q3(i)) — the gap between the M1 for setting up $(x+2)\log 7 = \log 3$ and the first A1 for reaching ANY correct exact expression for $x$. Taking logs correctly is not the same as finishing the algebra — once the exponent has been isolated as an expression like $2x-1$ or $x+1$, every remaining operation (adding, subtracting, dividing) still has to be carried out on both sides before an exact expression for $x$ exists at all. (A separate, later-stage error — reaching a correct exact expression for $x$ but leaving it uncombined into a single required log form — is a distinct trap: see "single-log-form-left-uncombined" below.)

single-log-form-left-uncombined

Confirmed on the same question, describing the steeper of its two mark-by-mark drops — "with only 50% scoring all three. For example, [many] achieved x = log₇3 − 2 but then did not reach the required form, failing to deal with the −2" (Jan 2021, Q3(i)). This is NOT an unfinished-rearrangement error: $x = \log_7 3 - 2$ is already a fully isolated, fully-credited exact expression for $x$ — it earns the A1 the trap above describes. The mark actually lost here is a further, distinct accuracy mark for combining that expression into the single logarithm the question specifically demanded ("giving your answer in the form $x = \log_7 a$, $a$ rational"): $\log_7 3 - 2 = \log_7 3 - \log_7 49 = \log_7\dfrac{3}{49}$, using $\log_7 49 = 2$ since $7^2 = 49$ — i.e. rewriting the constant $2$ via $\log_a a = 1$, here $2 = 2\log_7 7 = \log_7(7^2) = \log_7 49$. Whenever a question specifies the answer form this precisely, a correct but unsimplified expression like $\log_a b - c$ earns only the intermediate accuracy mark, never the final one.

logs-taken-before-the-equation-is-rearranged

A direct piece of technique advice from the same report, given as general guidance rather than a specific error: "Although the question asked for an answer in log form, it is not always advisable to take logs first when the equation is presented in index form" (Jan 2021, Q3(i)). Where a constant multiplies or divides the exponential term, isolating it algebraically first — before any logarithm is taken — removes an entire log law (usually the addition or subtraction law) from the working. Taking logs immediately is not wrong, but it adds a step, and every added step is an added chance for one of the other traps on this list to strike.

divides-out-a-common-trig-factor

Confirmed on a real WMA12 identity-substitution question: after multiplying through by cosθ and factorising, the correct path reaches sinθ(2 + 3cosθ) = 0 — but "many however cancelled the sinθ and hence only produced solutions from cosθ = −⅔" (Oct 2023, Q3), silently discarding the whole sinθ = 0 branch. This is structurally the identical error to dividing a polynomial equation by x instead of factoring it out — cancelling a factor from both sides of an equation throws away every solution where that factor is zero. Factorise; never divide by an expression that could itself be zero.

wrong-identity-substituted-for-cos-squared

Confirmed on a real WMA12 question, in an examiner report quoted as: "cos2θ was occasionally replaced with 1 − sinθ" (Oct 2022, Q5) — read in context, immediately alongside a separate note about the sin²θ + cos²θ = 1 identity, this is cos²θ being substituted with 1 − sinθ instead of the real identity, 1 − sin²θ: the SQUARED sine dropped to a plain sine. It is the same shape of error the prequestion in this lesson previews with a clean fraction (cosθ = 1 − sinθ) — a real, recorded exam mistake, not a hypothetical one. The identity relates the squares of sin and cos, never the functions themselves directly.

coefficient-dropped-when-distributing-after-substitution

Confirmed on the same question: "3(1 − sin²θ) expanded to 3 − sin²θ" (Oct 2022, Q5) — the leading coefficient multiplied only one of the two terms inside the bracket instead of both. This is exactly the error modelled on the wrong path of the marked-solution above, just with a 3 in place of a 2. The check that catches it costs five seconds: expand the resulting expression back out and confirm it returns what you started with.

incomplete-multiply-through-after-tan-substitution

The same distribute-across-every-term hazard as the coefficient-dropped trap above, but at the fraction-clearing step rather than a bracket-expansion step: once tan θ = sin θ/cos θ has been substituted into a mixed equation, EVERY term has to be multiplied through by cos θ to clear the fraction, not just the term that already had a cos θ sitting in it. The real WMA12 mark scheme for a tan-substitution question of this shape names this as its own credited action, worded to allow for exactly this slip: "Multiplies through by cosθ (on at least 2 of the 3 terms in the equation) and attempts to use ±sin²θ±cos²θ=±1 to set up a quadratic equation in cosθ" (June 2025, Q8(ii) mark scheme) — "on at least 2 of the 3 terms" is a direct acknowledgement that clearing the fraction on only some of the terms is the common failure mode being marked around. The worked-chain above models the full clear-then-substitute sequence and flags exactly where that slip would happen.

compound-angle-not-isolated-before-inverse-function

Confirmed on a compound-angle equation, tan(2x + π/5) = k, solved in radians: "some did not recognise that they should isolate the tan(2x + π/5) term before attempting to take arctan" (June 2025, Q8(i)) — a small structural failure with a large consequence, since every later step depends on the bracket being isolated first. There were also candidates who worked in degrees on this exact question despite the radian-form interval — the units of an interval have to be matched throughout, not switched to whichever is more familiar partway through.

coefficient-distributed-across-the-whole-compound-angle

Confirmed on the same question: "distributing the coefficient of the tangent term throughout the angle resulting in tan(6x + 3π/5)" (June 2025, Q8(i)) — multiplying a coefficient into the angle INSIDE the trig function, when it should only ever multiply the function's output. tan(2x + π/5) has no coefficient to distribute into its argument at all; the 3 in "3tan(2x + π/5) = k" multiplies the tan value once it is found, never the angle itself.

compound-angle-bracket-range-scaled-by-coefficient

Confirmed in the mark scheme for the same compound-angle question, not the examiner report: "Correct order of operations 2x + π/5 = π/6, 7π/6, 13π/6 ... to find a value for x which does not need to be in the given range (e.g. condone −π/60)" (June 2025, Q8(i) mark scheme) — three candidate bracket-values, not the usual one or two, and the dM1 mark is for the METHOD of generating them, before anything is checked against the range. Every "compound angle" example elsewhere in this lesson has a coefficient of 1 on x, where the bracket's range matches x's own interval exactly — here the coefficient is 2, so solving (2x + π/5) = k over 0 ⩽ x < π means the BRACKET sweeps an interval TWICE as wide, two full periods of tan (period π) rather than one, and the calculator's value needs tan's period added repeatedly across that whole doubled range before anything is converted back. From arctan(1/√3) = π/6: adding π once gives 7π/6, adding it again gives 13π/6 — both additional candidates a coefficient of 1 would never produce. Converting each via x = (bracket − π/5)/2 gives x = −π/60 (from π/6), x = 29π/60 (from 7π/6) and x = 59π/60 (from 13π/6); only the accuracy mark, A1, actually enforces 0 ⩽ x < π — which is exactly why −π/60 is explicitly condoned at the method stage and only rejected once the final answer is stated. The method mark rewards generating enough candidates across the scaled range; the accuracy mark is what filters them.

decimal-given-instead-of-exact-kπ-radian-form

Confirmed on the same compound-angle question: "A few did not take note of the required form of the answer kπ (with k rational) and used decimals. There were also candidates who worked in degrees" (June 2025, Q8(i) examiner report) — a distinct failure from the traps above about this equation, none of which turn on whether the final answer is left decimal or kept exact. Even after correctly isolating and solving tan(2x + π/5) = √3/3, the calculator's decimal output (2x + π/5 ≈ 0.628…) has to be recognised as a standard angle (arctan(1/√3) = π/6) and carried through in that exact form — "in the form kπ" is an explicit instruction that a decimal, however accurately rounded, does not satisfy; the mark scheme's own accuracy line states plainly that the final answer "must be exact and in radians" (June 2025, Q8(i) mark scheme). This is not the same rule as "don't round early": that rule protects a genuinely decimal final answer (as in Q8(ii)) from accumulating rounding error; this one requires recognising when the final answer isn't decimal at all.

premature-rounding-before-final-angle

Confirmed on a three-term quadratic in cosθ where the roots did not factorise: "some candidates rounded the exact solution for cosθ to 3 significant figures which caused rounding errors when finding θ and a loss of the final accuracy mark" (June 2025, Q8(ii)) — directly confirming the WMA12 general marking guidance's exact-answer rule for this exact topic. Keep the un-rounded value (a surd, a fraction, or extra decimal places) all the way through and round only the final angle, to whatever precision the question actually asks for.

second-derivative-found-but-never-substituted

Confirmed on a real WMA12 question testing exactly this mark: having correctly found the second derivative, examiners recorded that "there were many who did not then substitute the values for x found in part (a) into their second derivative" (Jan 2021 Q2) — the expression itself was right, but the one step that turns it into an actual answer, plugging in the specific x-value already found, was skipped. An expression in x has no sign at all until a number goes into it.

y-coordinates-found-when-only-x-was-asked

Confirmed on the same real WMA12 question (Jan 2021 Q2), and about part (a) rather than part (b): "candidates often went on to do unnecessary work by finding the y coordinates of the stationary points or assumed this was what was required of part (b)" — but part (a) on that exact question asked only for the x-coordinates, and part (b) (justify the nature of each stationary point) needs no y-value at all, only the x-value substituted into the second derivative. Read what a part is actually asking before doing the extra work: finding a coordinate you weren't asked for costs real time on a timed paper and earns nothing extra, and assuming a later part needs something it doesn't is a comprehension error, not a calculus one.

second-derivative-set-to-zero-and-solved

Confirmed on the same real WMA12 question: "a common error was to try to solve for the second derivative" (Jan 2021 Q2) — treating d²y/dx²=0 as a fresh equation to solve for x, the way dy/dx=0 was solved to find the stationary points in the first place. This finds a genuinely different feature of the curve (where its concavity itself changes, its point of inflection) and answers nothing about the nature of the stationary points the question actually asked about — this lesson's own marked-solution above shows exactly what that produces when carried all the way through.

y-values-eyeballed-instead-of-gradient-tested

Confirmed on the same real WMA12 question, contrasting two methods for classifying a stationary point when the second-derivative test is unavailable or unused: testing the sign of the gradient either side earned "full credit as it used further calculus", but "a rarely used method looking at the shape of the curve by looking at the y values either side of the stationary point was not given credit as it did not use further calculus as asked for in the question" (Jan 2021 Q2). The two methods can look similar on the page — both compare 'either side' of the point — but only one of them is actually calculus, and the mark scheme cares about the method, not just a correct-looking conclusion.

critical-value-inequality-wrong-region-chosen

Confirmed on a real cross-topic WMA12 question combining differentiation, factorisation and integration to test a decreasing-function region (spec 7.1): a documented wrong final answer read "x < −5 or x > ⅔" (Jun 2025, WMA12/01A, Q8) — the region OUTSIDE the two critical values, the opposite side from the one actually being asked for. dy/dx, as a quadratic in x, is positive outside its own two roots and negative between them (for a positive leading coefficient), or the reverse for a negative one — and the only way to know which side is correct is to test the sign of dy/dx at one value actually inside the interval in question, not to guess from which side 'looks like' the answer.

second-derivative-test-inconclusive-mishandled

Not attached to a real examiner-report quote in the facts bank behind this course — the WMA12 differentiation questions reviewed for this lesson document the failure to substitute into an already-nonzero second derivative (see above), not a case where the test itself returns exactly zero. It is included anyway because it is genuine, checkable spec-7.1 content, and follows directly from this lesson's own mechanism: d²y/dx²=0 says only that the gradient's own rate of change happens to be zero at that one instant — not that 'there is no information available' and not that the point must automatically be a point of inflection, both conclusions the test never supported. The only fix is to go back to testing the sign of dy/dx directly, either side, which this lesson's method-comparison above shows earning exactly as much credit as the second-derivative route in the first place.

stationary-point-x-coordinate-rounded-instead-of-exact

Not every dy/dx = 0 quadratic reduces to the tidy, monic, integer-rooted case most of this lesson's own worked examples use: $3x^2-2x-8=(3x+4)(x-2)=0$ factorises just as cleanly — no formula needed — and still gives $x=-\dfrac{4}{3}$, a genuine fraction. The method mark for solving the quadratic is earned by the factorisation itself, regardless of how the root is then written down; it is the accuracy mark for the coordinate that the fraction protects and a decimal forfeits. The general marking guidance's exact-answer rule states plainly 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" — so $x=-\dfrac{4}{3}$ (or an equivalent single fraction) is required, and $x=-1.33$ loses the mark even though the decimal is correctly rounded.

answer-produced-with-no-method-shown

Confirmed on a real area-between-a-line-and-a-curve question: "it is noteworthy that there were a number of students who relied on a calculator to evaluate the definite integral for them, which is not advisable when an exact answer is required as permitted calculators are supposed to only be able to carry out numerical integration, which will commonly give a decimal answer only... full method must be shown for full credit" (Jan 2020, Q4). Permitted calculators on this paper can do numerical integration — approximate, decimal — but not the symbolic integration a written method has to show; the two are not interchangeable when the question asks for an exact value.

curve-and-line-subtracted-in-the-wrong-order

Confirmed on the same question: "it was not uncommon to see the difference the wrong way round, though students would generally realise the final answer must be positive at the end and recover the marks" (Jan 2020, Q4). The mechanism is exactly this lesson's diagram: subtracting (line − curve) instead of (upper − lower) flips the sign of the whole answer. It is self-correcting only for a student who actually checks the sign of the final number against "area must be positive" — silently accepting a negative result loses the mark this trap describes.

line-boundary-ignored-entirely

Confirmed independently on two different real series, both testing this exact spec point — a genuinely recurring trap, not a one-off. On a two-region area question with a curve and a line: "Many attempted solutions simply integrated the equation of the curve and evaluated between the limits 0 and 1, thus completely ignoring the requirement to subtract the area under the line l. This gained no credit as it was not a correct strategy for R1" (Jan 2023, Q9(c)). And on a separate curve-and-line area question: "One common error, aside from bracketing mistakes, made in this question was that only the area under the curve being found, 31√5⁄3 being given as the area of the region R" (Jan 2020, Q4) — echoed again on a third series, "Some candidates, however, did not know how to deal with the additional line and just found the area under the curve" (June 2025, Q3). Reading the question fully before starting — is the region bounded by the curve ALONE, or by the curve and something else — is the entire fix, and it has to happen before any integral is written down, not after.

sign-change-inside-the-interval-not-split

Not attached to a real examiner-report quote in the facts bank behind this course — the archived area questions reviewed for WMA12 all use curve-and-line or curve-and-curve regions that stay one sign throughout, not a region where the curve itself crosses the axis. It is included anyway because it is genuine, checkable spec-8.2 content, and it follows directly from the mechanism derived earlier in this lesson: integrating straight across a root mixes a positive and a negative contribution into one running total before either can be made positive, producing a number that is neither the honest signed integral nor the true area. See the marked-solution's common wrong path for exactly what number that mistake produces.

slowest-valid-method-chosen-under-time-pressure

Confirmed as a real, quoted efficiency comparison between mathematically equivalent methods on an area-between-curve-and-line question: one valid strategy "was a more time-consuming approach requiring either five terms to be integrated or terms to be collected resulting in occasional sign errors" (June 2025, Q3) than the other creditable strategies for the same question. All three methods in this lesson's method-comparison earn full marks — but under a 90-minute paper with 75 marks across the whole thing, picking the slower of two equally valid routes is a real cost even when it produces the right answer.

y-values-used-as-x-limits

Confirmed on the same real anchor question as this lesson's y-axis-boundary worked example above: "Another mistake seen on numerous occasions was to use the y values of 7 and 17 as their limits to an integration with respect to x, which was incorrect and usually meant only the first two marks were scored" (Jan 2020, Q4). The two numbers named there are the curve's value at x = 0 and the height of the bounding line — real numbers on the diagram, but y-values, never x-values, and ∫(…)dx only ever takes x-values as its limits. It is exactly the shape the worked-chain above is built to guard against: one limit found by solving curve = line, the other supplied directly by the y-axis as x = 0 — never by reading off the curve's or the line's own y-coordinate at that point.

strip-width-from-point-count-not-gap-count

Confirmed directly, and described as something every examiner marking the series saw: "All examiners saw candidates who thought that the strip width was 0.2, obtained from dividing 1 by the 5 points rather than the 4 spaces" (October 2020). Five ordinates bound four strips, not five — the number of x-values in a table is always one more than the number of strips it describes, because each interior ordinate is shared between two neighbouring strips rather than belonging to a strip of its own.

endpoints-and-interior-ordinates-swapped

Confirmed as the commonest single arithmetic error on a real trapezium-rule question: "The commonest error was to double the three given values" (October 2020) — losing track of which ordinates sit at the two ends of the region (used once each in the sum) and which sit strictly between them (used twice each). The mechanism block above derives exactly why the split runs this way: an interior ordinate belongs to two adjacent strips at once when the strips' individual areas are added, an endpoint ordinate belongs to only one.

bracket-closed-before-scaling

Confirmed, and structurally distinct from the doubling error above even though both look similar on the page: "A few candidates closed the brackets after the initial addition of two terms, then adding the rest afterwards, thus gaining no credit" (January 2023). The $\frac{1}{2}h$ has to multiply the ENTIRE bracket — endpoints and doubled interior ordinates together — because it was factored out of every individual trapezium's area at once. Splitting the expression into two separately-scaled pieces breaks the algebra that produced the compressed formula in the first place, and the mark scheme gives no partial credit for the resulting number.

conclusion-restated-instead-of-reasoned

Confirmed as the hardest single mark on the whole paper to earn in the series it was checked in: "Part (c) was perhaps found to be the most difficult mark for candidates to achieve. Stating that the answer... is an underestimate because it is less than the true area is not giving a reason as to why it is less. It was important for candidates to allude to the fact that the sum of the areas of trapezia found in part (a) was less than the shaded area, hence it is an underestimate" (October 2023). "It's smaller, so it's an underestimate" repeats the definition of the word; the credited reason names the mechanism — the chords sit below the curve because of how the curve bends — that makes the sum come out smaller in the first place. This mark is also, on every verified instance, ONE mark rather than two: the scheme reads "B1: States underestimate AND gives a valid reason" (October 2023 Q6c), a single combined requirement, not a label mark plus a separate reason mark. The same pattern holds outside this topic too — June 2021's analogous "state direction, give reason" mark for a binomial-expansion truncation error is likewise a single combined B1 with no partial credit for the label alone. Do not expect a mark for writing "underestimate" (or "overestimate") on its own; the mechanism has to be there in the same sentence.

trapezium-rule-assumed-tested-in-isolation

The trapezium rule is regularly combined with other spec content rather than tested as a standalone calculation — confirmed across at least two series. One examiner report on a genuinely unusual question format states directly: "Part (i) was a novel way of testing the trapezium rule via logarithms" (January 2025); a separate report on a different series records the same rule embedded in a "real life" water-flow context (October 2023, Q6). A student who has only ever practised the trapezium rule attached to a bare polynomial or root function is unprepared for a question that requires setting up the ordinate table from a log expression, or from a context described in words, before the trapezium-rule arithmetic itself even begins.