HITL Monitor

Recent warnings & errors (60)

Problems the run loop would otherwise only print to the terminal — a generation slot dropped on a JSON parse failure, a rubric-mining reply that would not parse, a generation/ingest that errored. They pile up here (newest first) so you can spot and debug them async.

2026-08-19 21:16:18 [warning] generator.paper_style — empty generation (question and mark scheme missing) — slot skipped, no audit written. Usually max_tokens truncation: the model spent its budget before reaching the deliverable fields.

2026-08-19 21:12:21 [info] generator.paper_style — structured tool-use generation failed (emit_question: hit max_tokens (16000) before the tool input was complete. Raise max_tokens, or shorten what the prompt asks the model to produce before the deliverable fields.); falling back to free-text JSON parse.

2026-08-19 21:12:02 [info] visual.autofigure — mark-scheme autofigure: solution was graphical but carried no figure — drew one (GDC).

2026-08-19 21:11:29 [warning] visual.matplotlib — matplotlib sandbox — matplotlib code errored: Q_2\!\left(-\tfrac{\sqrt{21}}{3},\,\tfrac{2\sqrt{21}}{3}\right) ^ ParseSyntaxException: Expected '\\right', found '\' (at char 12), (line:1, col:13)

2026-08-19 18:35:56 [warning] generator.paper_style — Generator JSON parse failed — slot skipped (Expecting ',' delimiter: line 2 column 15588 (char 15589))

2026-08-19 18:31:41 [info] generator.paper_style — structured tool-use generation failed (emit_question: hit max_tokens (16000) before the tool input was complete; empty required field(s): model_answer. Raise max_tokens, or shorten what the prompt asks the model to produce before the deliverable fields.); falling back to free-text JSON parse.

2026-08-19 18:27:54 [info] visual.autofigure — mark-scheme autofigure: solution was graphical but carried no figure — drew one (GDC).

2026-08-19 18:27:24 [warning] visual.matplotlib — matplotlib sandbox — matplotlib code errored: S_n^B = 8000\left(1-(\tfrac{3}{4})^n\right) ^ ParseSyntaxException: Expected '\\right', found '\' (at char 21), (line:1, col:22)

2026-08-15 12:11:44 [info] visual.autofigure — mark-scheme autofigure: solution was graphical but carried no figure — drew one (GDC).

2026-08-15 11:57:46 [info] visual.autofigure — mark-scheme autofigure: solution was graphical but carried no figure — drew one (GDC).

2026-08-09 07:57:28 [info] visual.autofigure — mark-scheme autofigure: solution was graphical but carried no figure — drew one.

2026-08-09 07:51:26 [warning] visual.matplotlib — matplotlib sandbox — matplotlib code errored: \displaystyle\sum_{k=1}^{n} k\cdot u_k ^ ParseFatalException: Unknown symbol: \displaystyle, found '\' (at char 0), (line:1, col:1)

2026-08-04 04:50:42 [info] generator.paper_style — structured tool-use generation failed (Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.); falling back to free-text JSON parse.

2026-08-04 03:01:14 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-04 02:15:30 [info] generator.paper_style — structured tool-use generation failed (Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.); falling back to free-text JSON parse.

2026-08-03 18:16:15 [error] generate.paper_style — generation failed: Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.

2026-08-03 17:26:51 [info] generator.paper_style — structured tool-use generation failed (Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.); falling back to free-text JSON parse.

2026-08-03 15:44:43 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-03 15:26:32 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-03 15:11:42 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-03 14:54:07 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-03 13:39:04 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-03 13:18:30 [warning] visual.matplotlib — matplotlib sandbox — matplotlib code errored: /home/ubuntu/.config/matplotlib is not a writable directory Matplotlib created a temporary cache directory at /tmp/matplotlib-9tuey6tw because there was an issue with the default path (/home/ubuntu/.c

2026-08-03 13:12:37 [info] generator.paper_style — structured tool-use generation failed (Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.); falling back to free-text JSON parse.

2026-08-03 07:56:42 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 18:21:50 [warning] visual.matplotlib — matplotlib sandbox — matplotlib code errored: /home/ubuntu/.config/matplotlib is not a writable directory Matplotlib created a temporary cache directory at /tmp/matplotlib-0v3ltb9t because there was an issue with the default path (/home/ubuntu/.c

2026-08-02 18:12:45 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 17:42:10 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 17:36:34 [info] generator.paper_style — structured tool-use generation failed (Request timed out or interrupted. This could be due to a network timeout, dropped connection, or request cancellation. See https://docs.anthropic.com/en/api/errors#long-requests for more details.); falling back to free-text JSON parse.

2026-08-02 16:46:57 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 13:22:25 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 12:48:27 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 12:27:00 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 10:45:08 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 10:36:05 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 10:30:10 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 08:35:02 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-02 08:33:25 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-01 05:21:29 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-01 05:19:54 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-01 05:02:00 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-08-01 04:44:08 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)

2026-07-31 04:36:41 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-31 04:36:37 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:55:01 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:54:57 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:54:53 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:14:42 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:02:52 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:02:47 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:02:43 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:01:56 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:01:52 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:01:45 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:01:42 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 16:01:40 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 15:53:42 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 15:53:37 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 15:53:34 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

2026-07-30 15:44:57 [error] generate.paper_style — generation failed: name 'slot_num_in_batch' is not defined

Constitution — v003

Core principles are hand-authored or curriculum-seeded rules, always applied. Mined principles are learned from reviewer notes; each carries a Beta(α, β) belief, where α = 1 + supporting reviews and β = 1 + contradicting reviews (so α=3, β=1 means 2 reviews backed it and 0 went against it). Its estimated support rate is α/(α+β). A mined principle is promoted to active — and only then fed into generation prompts — once it has at least 5 supporting reviews and the Wilson 95% lower bound on its support rate clears the promotion threshold (about 5 clean supports). An active principle is retired only if it later collects 2 or more contradictions. "Inactive" below means simply not-yet-promoted.

Core principles (87)

core_001 — Every variable, function, and constant used in a question must be defined within that same question.
core_002 — Every question must start with a clear instruction verb (Solve, Prove, Find, Evaluate, Determine, Show that, Calculate, Sketch, Given that ... find).
core_003 — Never write a bare equation or expression as a question. A question must instruct the student to do something with it.
core_004 — A question must be self-contained — solvable without seeing any other question or external context.
core_005 — Critical information must not be missing: required units when units matter, diagrams when referenced, table values when used.
core_006 — The provided model answer must be mathematically correct and consistent with the question statement.
core_007 — A question must not reference another question's answer (no "using your answer from part (a)" in a standalone context).
core_008 — Standard IB/AP notation (g = 9.8 m/s², e, π, sin, cos, ln) may be used without explicit definition.
core_009 — Standard math/physics conventions (m for mass, v for velocity, x for an independent variable) may be used without re-definition.
core_010 — LaTeX is required for all mathematics: `$...$` inline, `$$...$$` display.
core_011 — Units must be consistent throughout the question.
core_012 — The model answer must show full working, not just a final numeric value.
core_013 — Each mathematical step must follow logically from the previous one.
core_014 — The final answer must be clearly stated and (where applicable) include units.
core_015 — Verifying anything: just substitute into the equation/expression. You don't need to re-derive anything you can check by substitution.
core_016 — Solving any equation of type f(x) = g(x): draw f(x) and g(x) and find intersections; or draw f(x) − g(x) and find roots.
visual_core_question_demands_visual_mark_scheme — If the question asks the student to 'sketch', 'draw', 'plot', 'graph', or otherwise produce a visual artefact, the MARK SCHEME MUST contain a corresponding visual diagram (graph, sketch, or plot) showing the canonical answer. A worded mark scheme without a visual for a visual question is incomplete and will be rejected.
visual_core_complex_number_argand_in_mark_scheme — For questions involving complex numbers (Argand diagrams, polar form, roots of unity, modulus/argument, complex exponentials, De Moivre's theorem, geometric operations on complex numbers), the MARK SCHEME MUST include an Argand-diagram-style visual showing: (a) the unit circle when the modulus is 1 or relevant, (b) the position vector(s) of every complex number named in the question (z, z_1, z_2, w, etc.), (c) any angle(s) referenced (argument, principal value), and (d) the geometric operation if applicable (rotation, reflection, scaling). Use the visual schema: {type: 'argand_diagram', spec: {points: [{label, re, im}], angles: [...], unit_circle: true|false}}.
concept_t02_limit_001 — MARK SCHEME — Limit evaluation (first step): the model answer must show substitution of the given value into the limit expression as the very first step, before any other technique is applied.
concept_t02_limit_002 — MARK SCHEME — Indeterminate form $\frac{\infty}{\infty}$: structure the working as (1) identify that both numerator and denominator grow without bound; (2) divide numerator and denominator by the highest-speed expression in the denominator to compare speeds; (3) if both are polynomials, compare degrees directly since the highest degree dominates; (4) apply the speed ordering $\log < p(x) < \exp$ where needed to resolve the limit.
concept_t02_limit_008 — MARK SCHEME — Indeterminate form $\frac{0}{0}$: the model answer must resolve a $\frac{0}{0}$ form by factorizing the numerator and/or denominator to cancel the offending factor; if the factorization is non-obvious, series expansion should be invoked to make it possible; note that factorization is always possible.
concept_t02_limit_009 — MARK SCHEME — L'Hôpital's rule: the model answer must (1) verify the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$; (2) confirm $f$ and $g$ are differentiable near $a$ and $g'(x) \neq 0$ near $a$; (3) apply $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$ only if the right-hand limit exists; (4) note that L'Hôpital must NOT be used when $\sin x$ and $\cos x$ appear in both numerator and denominator together, as this causes cycling — the mark scheme must penalise use of L'Hôpital in such cases.
concept_t02_limit_010 — MARK SCHEME — Limit as $x \to -\infty$: the model answer must apply the substitution $x = -t$ (so that $t \to +\infty$) and rewrite the entire limit in terms of $t$ before evaluating.
concept_t02_limit_011 — MARK SCHEME — Converting $x \to \infty$ to $t \to 0$: the model answer must apply the substitution $x = \frac{1}{t}$ (so that $t \to 0$) and rewrite the limit entirely in terms of $t$ before evaluating.
concept_t02_limit_012 — MARK SCHEME — Indeterminate form $\infty - \infty$: the model answer must handle the $\infty - \infty$ form by (1) rationalizing if the expression appears as a fraction, or (2) factorizing otherwise.
concept_t03_general-functions_001 — MARK SCHEME — Classifying a relation as a function: when the model answer determines whether a relation is a function, it must (1) apply the vertical line test graphically; (2) state rigorously that every input $x \in X$ must have an output, and that output must be unique, i.e. $\forall x \in X, \exists! y \in Y$ s.t. $(x,y) \in f$.
concept_t03_general-functions_009 — MARK SCHEME — Verifying a function has an inverse: the model answer must (1) state that the function must be one-to-one; (2) identify the geometric property that the function must be monotonically increasing or decreasing throughout its domain; (3) link this to differentiation as the standard verification method.
concept_t03_general-functions_003 — MARK SCHEME — Drawing the reciprocal function $\frac{1}{f(x)}$: when the model answer draws or describes $\frac{1}{f(x)}$, it must apply these properties in order: (1) every root of $f(x)$ becomes a vertical asymptote of $\frac{1}{f(x)}$; (2) every maximum of $f(x)$ becomes a minimum of $\frac{1}{f(x)}$ and vice versa; (3) evaluate one-sided limits at each asymptote to determine the behaviour around them.
concept_t03_general-functions_004 — MARK SCHEME — Calculating a horizontal asymptote: when the model answer finds a horizontal asymptote, it must evaluate the limit of the function at the endpoints of the domain, typically $\lim_{x \to -\infty} f(x)$ and $\lim_{x \to +\infty} f(x)$.
concept_t03_general-functions_005 — MARK SCHEME — Solving a function equation $f(x) = g(x)$: when the model answer solves such an equation, it must follow one of two equivalent approaches: (1) draw $f(x)$ and $g(x)$ separately on the same axes and find their intersections graphically/algebraically, OR (2) form $f(x) - g(x)$ and find its zeroes.
concept_t03_general-functions_010 — MARK SCHEME — Proving an absolute inequality $f(x) < k$: when the model answer proves such an inequality, it must (1) check whether the formulation $f(x) < k$ is achievable; (2) if $k$ is separable, verify that the maximum of $f(x)$ lies below $k$ (using the imaginary-line visual trick or calculus); and (3) if $k$ is not separable from $f(x)$, use function properties such as the number of solutions to reason about the inequality.
concept_t03_general-functions_011 — MARK SCHEME — Drawing a general function: when the model answer sketches any general function, it must follow this flowchart in order: (1) identify all asymptotes and intercepts; (2) determine increasing and decreasing intervals via differentiation and locate all critical points; (3) identify all inflection points; (4) draw the function using the information gathered.
concept_t03_general-functions_007 — MARK SCHEME — Proving $f(x) > g(x)$ always holds: the model answer must use calculus to show that $f(x) - g(x) > 0$ for all $x$ in the domain, rather than relying solely on a graphical argument.
concept_t04_quadratic-functions_004 — MARK SCHEME — Solving a quadratic equation: structure the working by (1) attempting factorization first; (2) if factorization does not work, applying the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for the general form $ax^2 + bx + c$.
concept_t04_quadratic-functions_005 — MARK SCHEME — Drawing a quadratic function: the model answer must show (1) converting $f(x)$ into vertex form $a(x-b)^2 + c$; (2) identifying the vertex $(b, c)$; (3) identifying the $y$-intercept; (4) sketching the parabola from those key features.
concept_t04_quadratic-functions_003 — MARK SCHEME — Similar triangles: when the model answer uses similar triangles, it must (1) explicitly state which triangles are similar in formal notation (e.g. $\triangle ABC \sim \triangle DEF$) before any ratio work; (2) then read off the ratio equations directly from the named triangles: $\dfrac{AB}{DE} = \dfrac{AC}{DF} = \dfrac{BC}{EF}$.
concept_t05_polynomial-functions_001 — MARK SCHEME — Number of zeroes of a cubic function: when the model answer determines the number of real zeroes of a cubic, structure the working by (1) finding the two critical values $c_1, c_2$ (local max and min); (2) computing the product $c_1 \cdot c_2$ of their function values; (3) concluding: if $c_1 c_2 > 0$ then 1 real solution, if $c_1 c_2 = 0$ then 2 real solutions (one touching), if $c_1 c_2 < 0$ then 3 real solutions.
concept_t06_rational-functions_001 — MARK SCHEME — Sketching linear-over-linear rational functions: structure the working as (1) calculate the horizontal and vertical asymptotes; (2) find the intercepts to determine which partition the curve lies in; (3) draw the graph.
concept_t06_rational-functions_002 — MARK SCHEME — Sketching linear-over-quadratic rational functions: structure the working as (1) factorize the quadratic denominator; (2) calculate the horizontal and vertical asymptotes; (3) calculate limits around the asymptotes; (4) draw the graph.
concept_t06_rational-functions_003 — MARK SCHEME — Sketching quadratic-over-linear rational functions: structure the working as (1) divide the quadratic by the linear expression to obtain the oblique asymptote; (2) find the intercepts to determine which partition the curve lies in; (3) draw the graph.
concept_t07_modulus-functions_001 — MARK SCHEME — Solving modulus equations: when the model answer solves a modulus equation, present the fastest method as squaring both sides to eliminate the modulus sign, using $|x|^2 = x^2$; if an alternative method is shown, use the piecewise definition of $|x|$ to reason through the different cases separately.
concept_t08_exponential-and-logarithmic-functions_001 — MARK SCHEME — Sketching an exponential function: when the model answer sketches an exponential function, structure the working as: (1) apply limits as $x \to \pm\infty$ to identify horizontal asymptotes and end behaviours; (2) find intercepts; (3) draw the curve.
concept_t08_exponential-and-logarithmic-functions_002 — MARK SCHEME — Sketching a logarithmic function: when the model answer sketches a logarithmic function, structure the working as: (1) find the domain; (2) apply the applicable end limit to find end behaviours; (3) find intercepts; (4) draw the curve.
concept_t09_trigonometric-functions_001 — MARK SCHEME — Simplifying trig expressions with $n\pi \pm \theta$: when the model answer simplifies $\sin(n\pi \pm \theta)$, $\cos(n\pi \pm \theta)$, or $\tan(n\pi \pm \theta)$ for $0 < \theta < \frac{\pi}{2}$, structure the working by (1) determining the quadrant implied by $n\pi \pm \theta$ to fix the sign of the result, (2) identifying whether $n$ is odd or even to decide whether the function name changes (sin↔cos for odd multiples, same function for even multiples), then (3) writing the simplified expression with the correct sign and function — e.g. $\sin(3\pi + \theta) = -\sin(\theta)$.
concept_t09_trigonometric-functions_002 — MARK SCHEME — Deriving product-to-sum formulas: the model answer must show (1) start from the compound angle (addition/subtraction) formulae that contain the required product; (2) add or subtract the two chosen equations to eliminate the unwanted product term; (3) state the resulting product-to-sum identity — e.g. adding $\cos(A-B)$ and $\cos(A+B)$ yields $2\cos A\cos B$.
concept_t09_trigonometric-functions_003 — MARK SCHEME — Deriving sum-to-product formulas: the model answer must show (1) begin from the product-to-sum formulae; (2) introduce the substitutions $u = \frac{A+B}{2}$, $v = \frac{A-B}{2}$ so that $A = u+v$, $B = u-v$; (3) rewrite the sum or difference in terms of $u$ and $v$ and apply compound angle identities; (4) substitute back to express the result in terms of $A$ and $B$ — e.g. $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$.
concept_t09_trigonometric-functions_004 — MARK SCHEME — Applying the sine and cosine rules: the model answer must (1) state the appropriate rule — sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ when two angles and one side or two sides and a non-included angle are known, cosine rule $c^2 = a^2+b^2-2ab\cos C$ when three sides or two sides and the included angle are known; (2) justify the choice of rule based on the given information before substituting values.
concept_t10_differentiation_006 — MARK SCHEME — Related rates: when the model answer solves a related-rates problem, structure the working as this flowchart: (1) write down all given conditions (e.g. $\frac{dh}{dt}$) and draw the situation if possible; (2) state what is to be found (WTS, e.g. $\frac{dV}{dt}$); (3) identify the formula/relationship that incorporates all variables involved; (4) differentiate that formula, isolate the target quantity, then substitute the particular values.
concept_t10_differentiation_002 — MARK SCHEME — Optimization problems: when the model answer solves an optimization problem, structure the working as: (1) draw the diagram; (2) label the variables; (3) write the objective function; (4) find the minimum or maximum accordingly.
concept_t10_differentiation_003 — MARK SCHEME — Identifying stationary points: when the model answer classifies stationary points, structure the working as: (1) find where $f'(x) = 0$; (2) use the first-derivative sign test (most reliable) by drawing a sign diagram of $f'(x)$ around the point; (3) if the second-derivative test is used, note that $f''(x) < 0$ indicates a local maximum, $f''(x) > 0$ indicates a local minimum, and $f''(x) = 0$ requires further analysis.
concept_t10_differentiation_004 — MARK SCHEME — Identifying inflection points: when the model answer locates inflection points, structure the working as: (1) find all points where $f''(x) = 0$; (2) draw a sign diagram of $f''(x)$ around each such point; (3) confirm that the sign of $f''(x)$ actually changes — if it does NOT change, that point is NOT an inflection point.
concept_t10_differentiation_005 — MARK SCHEME — Tangent lines: when the model answer finds a tangent line, always begin by establishing the point of contact, then structure the working as: (1) find the point of contact $(a, f(a))$; (2) compute the slope $f'(a)$; (3) write the tangent line as $f(x) - f(a) = f'(a)(x - a)$.
concept_t11_integration_009 — MARK SCHEME — Integration tricks (algebraic simplification): when the model answer integrates a complicated or fractional expression, show the algebraic rewrite first — (1) convert roots/surds to exponent form where possible (e.g. $\frac{1}{\sqrt{2x-1}} = (2x-1)^{-1/2}$); (2) split fractions into separate terms where possible; (3) expand low-power bracket expressions if simpler — before applying any integration rule.
concept_t11_integration_010 — MARK SCHEME — Integration of composite linear functions $f(ax+b)$: when the model answer integrates a function of the form $f(ax+b)$, structure the working as (1) identify the outer function $f$ and inner linear function $ax+b$; (2) confirm the inner function differentiates to a constant $a$; (3) apply $\int f(ax+b)\,dx = \frac{1}{a}F(ax+b) + C$, showing the $\frac{1}{a}$ scaling factor explicitly, where $F$ is any antiderivative of $f$.
concept_t11_integration_011 — MARK SCHEME — Integration by substitution (choosing substitution): when the model answer performs integration by substitution, show the substitution choice justified by the case following this priority table: (1) for $\sqrt{f(x)}$, use $f(x)=t$; (2) for a product $f(x)g(x)$, use $g(x)=t$; (3) for a product $f(x)^n g(x)^m$, substitute inside the higher-order term; (4) for fractions, handle the denominator first using the same list; (5) for $x^2+a^2$, use $x=a\tan\theta$; (6) for $\sqrt{x^2-a^2}$, use $x=a\sec\theta$; (7) for $\sqrt{a^2-x^2}$, use $x=a\sin\theta$.
concept_t11_integration_004 — MARK SCHEME — Integration by parts: when the model answer applies integration by parts, show the formula $\int f(x)g'(x)\,dx = f(x)g(x) - \int f'(x)g(x)\,dx$ explicitly, identify $f$ and $g'$ (or equivalently $u$ and $dv$, or using $d(g(x))$ notation to organise the working), then evaluate the remaining integral.
concept_t11_integration_012 — MARK SCHEME — Special-form integration tricks: when the model answer integrates a special-form expression, identify the applicable case and apply the corresponding trick: (1) $\sin^2 x$ or $\cos^2 x$ → use the $\cos(2x)$ double-angle identity to reduce power; (2) $\tan x$ → rewrite as $\frac{\sin x}{\cos x}$ then substitute; (3) $\tan^2 x$ → replace with $\sec^2 x - 1$ using the identity $1+\tan^2 x = \sec^2 x$; (4) $\frac{P(x)}{(x-a)(x-b)}$ → use partial fractions $\frac{A}{x-a}+\frac{B}{x-b}$; (5) $\frac{P(x)}{(x-a)(x-b)(x-c)}$ → decompose as $\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}$; (6) $\frac{P(x)}{(x-a)(x^2+bx+c)}$ → decompose as $\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}$.
concept_t11_integration_013 — MARK SCHEME — Volume of revolution: when the model answer computes a volume of revolution, structure the working as: (1) draw or describe the function and the axis of rotation; (2) identify the radius of the representative disk/shell as a function of the integration variable; (3) write the area of the circle formed by that radius; (4) set up and evaluate the integral over the appropriate bounds summing those circular areas.
concept_t11_integration_014 — MARK SCHEME — Definite integral by substitution (bound conversion): when the model answer evaluates a definite integral using substitution $g(x)=t$, show explicitly (1) the substitution and that $g'(x)\,dx = dt$; (2) the conversion of bounds $a \to g(a)$ and $b \to g(b)$; (3) the new definite integral $\int_{g(a)}^{g(b)} f(t)\,dt$ evaluated directly without converting back to $x$.
concept_t11_integration_015 — MARK SCHEME — Inverse trig integration ($\sec x$, $\csc x$): when the model answer integrates $\sec x$ or $\csc x$, show the multiplication trick explicitly: (1) for $\sec x$, multiply numerator and denominator by $\sec x + \tan x$ (i.e. write $\sec x \cdot \frac{\sec x + \tan x}{\sec x + \tan x}$) then apply substitution on the resulting fraction; (2) for $\csc x$, apply the analogous multiplication by $\frac{\csc x + \cot x}{\csc x + \cot x}$ and substitute.
concept_t12_kinematics_001 — MARK SCHEME — Displacement vs distance calculation: when the model answer computes displacement or distance over $[t_1, t_2]$, show both integrals explicitly: displacement $= \int_{t_1}^{t_2} v(t)\,dt$ and distance $= \int_{t_1}^{t_2} |v(t)|\,dt$, and note that distance $\geq |\text{displacement}|$.
concept_t13_differential-equation_001 — MARK SCHEME — Solving first-order ODEs (three cases): when the model answer solves a first-order ODE, identify which of the three types applies and structure the working accordingly. (1) Separable: rearrange to $\frac{1}{g(y)}\,dy = f(x)\,dx$, then integrate both sides and add constant $C$. (2) Dimensionally homogeneous: substitute $v = y/x$ so $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$, rewrite as $\frac{dv}{F(v)-v} = \frac{dx}{x}$, then solve the resulting separable equation in $v$ and $x$, and back-substitute $v = y/x$. (3) Homogeneous linear: find the integrating factor $\mu(x) = e^{\int P(x)\,dx}$, multiply both sides by $\mu$, recognise the left side as $d(\mu y)/dx$, then integrate both sides; equivalently, when $q(x)=0$ rearrange to $\frac{1}{y}\,dy = -p(x)\,dx$, integrate to get $\ln|y| = -\int p(x)\,dx + C$, and write $y = C\exp\!\left(-\int p(x)\,dx\right)$.
concept_t13_differential-equation_002 — MARK SCHEME — Euler's method: when the model answer applies Euler's method to $\frac{dy}{dx}=f(x,y)$, $y(x_0)=y_0$, structure the working as: (1) state the initial condition $(x_0, y_0)$ and the step size $h$; (2) compute the slope $f(x_n, y_n)$ from the ODE; (3) step forward via $x_{n+1} = x_n + h$ and $y_{n+1} = y_n + h\,f(x_n, y_n)$, using the slope at the current point to estimate the next value; (4) repeat until the desired $x$ value is reached, clearly tabulating each $(x_n, y_n)$ pair.
concept_t13_differential-equation_005 — MARK SCHEME — Characteristic equation method for linear ODEs with constant coefficients: when the model answer solves a homogeneous linear ODE with constant coefficients $a_n y^{(n)} + \cdots + a_1 y' + a_0 y = 0$, structure the working as: (1) substitute the trial solution $y = e^{\lambda x}$ to derive the characteristic (auxiliary) polynomial $a_n \lambda^n + a_{n-1}\lambda^{n-1} + \cdots + a_1 \lambda + a_0 = 0$; (2) find all roots $\lambda_i$ (real distinct, repeated, or complex conjugate); (3) write the general solution as the appropriate linear combination — real distinct roots give $\sum C_i e^{\lambda_i x}$, a repeated root $\lambda$ of multiplicity $m$ contributes $(C_1 + C_2 x + \cdots + C_m x^{m-1})e^{\lambda x}$, and complex roots $\alpha \pm \beta i$ give $e^{\alpha x}(A\cos\beta x + B\sin\beta x)$; (4) apply any initial/boundary conditions to determine the constants.
concept_t13_differential-equation_004 — MARK SCHEME — Power series method: when the model answer solves an ODE using the power series method, structure the working as: (1) assume a solution of the form $y(x) = \sum_{k=0}^{\infty} a_k(x-x_0)^k$ (expanding about $x_0$; use $x_0=0$ unless otherwise indicated, giving $y = \sum_{n=0}^{\infty} a_n x^n$); (2) differentiate term by term to obtain $y'$, $y''$, etc.; (3) substitute into the ODE and collect terms by equal powers of $(x-x_0)$; (4) equate each coefficient of $(x-x_0)^n$ to zero to generate a recurrence relation for the $a_k$; (5) express the general solution in terms of the free constants $a_0$ and $a_1$, writing out enough terms of the series to make the pattern clear.
concept_t14_maclaurin-expansion-hl_001 — MARK SCHEME — Deriving the Maclaurin expansion: when the model answer derives the Maclaurin series, structure the working as: (1) assume $f$ can be represented near $x=0$ by $f(x)=\sum_{n=0}^{\infty}a_n x^n$; (2) differentiate $n$ times and set $x=0$ to obtain $f^{(n)}(0)=n!\,a_n$, hence $a_n=\frac{f^{(n)}(0)}{n!}$; (3) substitute back to state $f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$; (4) state the existence condition: $f$ must be analytic at $0$ (infinitely differentiable near $0$ and $\lim_{n\to\infty}R_n(x)=0$), noting that infinite differentiability alone is not sufficient.
concept_t15_complex-number_001 — MARK SCHEME — Cartesian to Euler form conversion: when the model answer converts a complex number from cartesian to Euler form, structure the working as: (1) draw the complex number $a + bi$ on the Argand diagram; (2) find the modulus using geometry ($r = \sqrt{a^2 + b^2}$); (3) find the argument angle using geometry; (4) write in Euler form $re^{i\theta}$.
concept_t15_complex-number_002 — MARK SCHEME — nth root method: when the model answer solves an equation of the form $z^n = \text{SMTH}$, structure the working as: (1) convert SMTH to Euler form; (2) generalise the angle by adding $2k\pi$, writing $\text{SMTH} = re^{i(\theta + 2k\pi)}$; (3) solve for $z$ by applying the power of $\frac{1}{n}$; (4) substitute $k = 0, 1, \ldots, n-1$ to obtain all $n$ distinct roots.
concept_t15_complex-number_003 — MARK SCHEME — Structure of nth root solutions: when the model answer interprets the solution set of $z^n = \alpha$, explicitly identify (1) the modulus $r^{1/n}$ as the common radius of all roots; (2) the offset angle $\frac{\theta}{n}$ as the argument of the first root ($k=0$); (3) the rotation increment $\frac{2\pi}{n}$ between successive roots, noting the roots form a regular $n$-gon in the Argand diagram.
concept_t15_complex-number_004 — MARK SCHEME — Cartesian vs Euler form choice: when the model answer selects a form for a complex number calculation, state that cartesian form is used only when computing real/imaginary parts or rationalising; all other situations (multiplication, powers, roots, etc.) should use Euler form.
concept_t15_complex-number_006 — MARK SCHEME — Using complex numbers in series/integration with $\cos$ and $\sin$: when the model answer tackles a series or integral involving $\cos$ or $\sin$, express the trigonometric functions via $e^{i\theta}$ (using $e^{i\theta} = \cos\theta + i\sin\theta$), perform the calculation in exponential form, then extract the real or imaginary part as required.
concept_t16_vectors_001 — MARK SCHEME — Shortest distance from a point to a line: structure the working as (1) parameterize the line, calling the general point on it $N$; (2) calculate the vector $\overrightarrow{PN}$; (3) use the perpendicularity condition $\overrightarrow{PN} \cdot \mathbf{d} = 0$ (where $\mathbf{d}$ is the direction vector) to find the parameter $\lambda$; (4) substitute back and compute $\|\overrightarrow{PN}\|$.
concept_t16_vectors_005 — MARK SCHEME — Shortest distance from a point to a plane: structure the working as (1) construct a line through the point $P$ using the plane's normal vector as direction; (2) parameterize that line, with the starting point at $P$ walking toward the plane; (3) find the point $N$ where the line meets the plane by substituting the parametric coordinates into the plane equation; (4) substitute back and compute $\|\overrightarrow{PN}\|$.
concept_t16_vectors_003 — MARK SCHEME — Intersection of two lines: parameterize both lines separately, introducing a distinct parameter for each, then equate the two expressions component-wise and solve for the parameters, verifying the solution is consistent across all components.
concept_t16_vectors_004 — MARK SCHEME — Vector equation of a line through two points: express the line as $\mathbf{r} = \overrightarrow{OA} + \lambda\,\overrightarrow{AB}$, identifying $\overrightarrow{OA}$ as the position vector (starting point) and $\overrightarrow{AB} = B - A$ as the direction vector, then convert to Cartesian or parametric form as required.
concept_t17_probability-and-statistics_001 — MARK SCHEME — Probability distribution pipeline: when the model answer solves any probability distribution question, structure the working as: (1) define the random variable; (2) identify the distribution; (3) write out the probability statement such as $P(X > r) = p$; (4) substitute the distribution and evaluate.
concept_t17_probability-and-statistics_005 — MARK SCHEME — Unknown parameter in normal distribution: when the model answer solves for an unknown parameter (e.g. $\mu$ or $\sigma$) from a given probability statement involving a normal distribution, structure the working as: (1) standardise by writing $Z = \frac{X - \mu}{\sigma} \sim N(0,1)$; (2) rewrite the probability statement entirely in terms of $Z$; (3) express as a CDF equation using $\Phi$ then apply $\Phi^{-1}$ to isolate the unknown parameter; (4) evaluate using the GDC.
concept_t17_probability-and-statistics_003 — MARK SCHEME — Fair game problems: when the model answer determines whether a game is fair or finds a missing value for fairness, the working must (1) explicitly define $X$ as the net gain of the player, (2) state the condition $E(X) = 0$, and (3) use $\text{gain} = \text{profit} - \text{cost}$ when constructing the distribution of $X$.
concept_t17_probability-and-statistics_006 — MARK SCHEME — Combination $\binom{n}{r}$ calculation: when the model answer evaluates $\binom{n}{r}$, rewrite it as $\frac{n(n-1)(n-2)\cdots(n-r+1)}{r!}$ (permutation over factorial) so terms cancel easily, and apply the identity $\binom{n}{r} = \binom{n}{n-r}$ first whenever $r > n/2$ to minimise the number of factors before expanding.
concept_t18_proofs_005 — MARK SCHEME — Proof by contradiction: when the model answer proves a statement by contradiction to establish $p \Rightarrow q$, structure the working as follows: (1) state the goal $p \Rightarrow q$ and assume $\neg q$ (alongside $p$), framing the assumed scenario as $p \land \neg q$; (2) derive a contradiction $C$ (a known false proposition) from $p \land \neg q$; (3) conclude that since a true statement cannot imply a false one, $p \land \neg q$ must be false, and since $p$ is true, $\neg q$ must be false, hence $q$ is true.
concept_t18_proofs_006 — MARK SCHEME — Mathematical induction: when the model answer proves a statement by induction, structure the working as: (1) write down the conjecture $P_n$ and state the domain/condition for $n$; (2) prove the base case $n = n_0$; (3) state the induction hypothesis using 'suppose' or 'assume' — e.g. 'Suppose $P_k$ is true'; (4) show $P_{k+1}$ is true using the induction hypothesis; (5) write the conclusion: 'Since the base case is true, and $P_{k+1}$ is true whenever $P_k$ is true, we conclude that $P_n$ must be true for all $n$.'
concept_t18_proofs_003 — MARK SCHEME — Direction of proof for implications: when the model answer addresses why one must start from the LHS when proving $x \Rightarrow y$, it must note that equality ($=$) is bidirectional so either side may be the starting point, but a unidirectional implication $\Rightarrow$ does not guarantee the truth of its converse, so the proof must always proceed from $x$ (LHS) toward $y$ (RHS).
concept_t19_sequences-and-series_001 — MARK SCHEME — Sequences and series (general): when the model answer solves any sequences and series problem, structure the working as: (1) write down the general term first; (2) write down everything the question specifies; (3) solve using the relevant formula.
concept_t19_sequences-and-series_002 — MARK SCHEME — Proving $a_n = S_n - S_{n-1}$: when the model answer proves this identity, structure the working by (1) expanding $S_n = a_1 + a_2 + \cdots + a_n$ and $S_{n-1} = a_1 + a_2 + \cdots + a_{n-1}$, then (2) subtracting to show all terms except $a_n$ cancel, yielding $S_n - S_{n-1} = a_n$.
concept_t20_binomial-expansion_001 — MARK SCHEME — Binomial expansion (general): when the model answer solves any binomial expansion question, structure the working as: (1) write the general term form; (2) separate the coefficient and variable terms; (3) adjust the expression to meet the given conditions stated in the problem; (4) substitute and simplify.
concept_t21_factor-theorem-remainder-theorem-vietas-formula_001 — MARK SCHEME — Synthetic division: when the model answer applies synthetic division, structure the working as: (1) write down the coefficients of the polynomial in order (including any zero coefficients for missing terms); (2) apply the synthetic division algorithm using the root value; (3) read off the coefficients of the quotient polynomial and identify the remainder from the final entry.
concept_t21_factor-theorem-remainder-theorem-vietas-formula_002 — MARK SCHEME — Polynomial long division and partial fractions: when the model answer performs polynomial long division to split a rational expression, structure the working as: (1) set up the long division of the numerator by the denominator; (2) carry out the division step by step to obtain a quotient and remainder; (3) write the result in the form quotient + remainder/divisor; (4) simplify or split further as required.

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Pending review queue (15)

q_ps_004 in t02limit-limit • score 0.61
q_ps_005 in t02limit-limit • score 0.42
q_ps_006 in t02limit-limit • score 0.47
q_ps_007 in t02limit-limit • score 0.17
q_ps_008 in t02limit-limit • score 0.59
q_ps_009 in t02limit-limit • score 0.64
q_ps_003 in t02_limit • score 0.48
q_ps_003 in • score 0.00
q_ps_t19_007 in t19_sequences-and-series • score 0.63
q_ps_t19_008 in t19_sequences-and-series • score 0.45
q_ps_t19_009 in t19_sequences-and-series • score 0.55
q_ps_t19_010 in t19_sequences-and-series • score 0.24
q_ps_t19_011 in t19_sequences-and-series • score 0.27
q_ps_t02_017 in t02_limit • score 0.21
q_ps_t19_012 in t19_sequences-and-series • score 0.30

High-score items are highest information-gain. Walk top-down.

Per-question log (172)

Click any row to expand the human feedback that was submitted for it.

H — the model's uncertainty about the question's difficulty: the Shannon entropy (in bits, 0 to about 1.58) of its Easy/Medium/Hard posterior. H near 0 means the model is confident the question sits in one bucket; a high H means it is torn between buckets.
Score — the active-learning priority: how much the system expects to learn from a human review of this question. It blends H, Δ (the gap between the model's estimate and the difficulty the slot asked for) and novelty (how unlike the existing canonical examples the question is). Higher = review sooner — the pending queue is sorted by it.

ID	Pattern / Section	Declared	Estimated	Reviewed	H	Score	Outcome	Feedback	Preview
q_ps_001	Paper 1 — Multiple Choice	Medium	Medium	—	0.0	0.475	discarded	—	The function $f$ is defined by $$f(x) = \begin{cases} \dfrac{\sin(5x)}{2x} & x \neq 0 \\[6pt] k & x = 0 \end{cases}$$ Find the value of $k
No human feedback submitted yet.
q_ps_002	Paper 1 — Multiple Choice	Medium	Hard	—	0.0	0.55	discarded	—	Evaluate $\displaystyle\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$. $\textbf{(A)} \quad 0$ $\textbf{(B)} \quad \dfrac{1}{4}$ $\textbf{(C)
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.477	needs_human	reviewed	Consider the function $$f(x) = \begin{cases} \dfrac{\sin(3x)}{x} & x \neq 0 \\[6pt] k & x = 0 \end{cases}$$ where $k$ is a real co
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Easy	Medium	—	0.0	0.642	accepted	—	Use the Squeeze Theorem to evaluate $$\lim_{x \to 0} \, x^4 \sin\!\left(\frac{1}{x^2}\right).$$ [3 marks]
No human feedback submitted yet.
q_ps_005		Easy	Medium	—	0.0	0.636	accepted	—	Evaluate $$\lim_{x \to 0} \frac{e^{3x} - 1}{5x},$$ given that $\displaystyle\lim_{t \to 0} \frac{e^{t} - 1}{t} = 1$. [3 marks]
No human feedback submitted yet.
q_ps_006		Medium	Hard	—	0.0	0.486	accepted	—	Consider the function $$f(x) = \begin{cases} \dfrac{\ln(1+2x)}{x} & x \neq 0 \\[8pt] k & x = 0 \end{cases}$$ where $k$ is a real constant.
No human feedback submitted yet.
q_ps_007		Medium	Hard	—	0.0	0.477	accepted	—	Consider the function $f(x) = e^{2x} - 2x - 2$. (a) Show, using the Intermediate Value Theorem, that the equation $f(x) = 0$ has at lea
No human feedback submitted yet.
q_ps_008		Medium	Hard	—	0.0	0.442	accepted	—	Let $f(x) = \sqrt{x^2 + 6x} - x$ for $x > 0$. (a) Evaluate $\displaystyle\lim_{x \to \infty} f(x)$, showing full working. [3 marks]
No human feedback submitted yet.
q_ps_009		Medium	Hard	—	0.0	0.449	accepted	—	Consider the expression $\dfrac{2x^2 + x}{2x^2 - x + 3}$ for $x > 0$. (a) Show that $$\frac{2x^2 + x}{2x^2 - x + 3} = 1 + \frac{2x - 3
No human feedback submitted yet.
q_ps_010		Hard	Hard	—	0.0	0.206	accepted	—	For $x > 0$, define $h(x) = \sqrt{x^2 + 4x} - x$. (a) Show that $$h(x) = \frac{4x}{\sqrt{x^2 + 4x} + x},$$ and hence evaluate $\displ
No human feedback submitted yet.
q_ps_011		Hard	Hard	—	0.0	0.25	accepted	—	Evaluate each of the following limits, showing full working. (a) Using L'Hôpital's rule, evaluate $$\lim_{x \to \infty} \frac{x^3 + 2x
No human feedback submitted yet.
q_ps_012		Hard	Hard	—	0.0	0.204	accepted	—	Consider the function $f(x) = \sqrt{x^2 + 6x + 2}$. (a) Evaluate $$\lim_{x \to \infty} \bigl(f(x) - x - 1\bigr),$$ showing that the e
No human feedback submitted yet.
q_ps_013		Hard	Hard	—	0.0	0.255	accepted	—	Consider the following limits. (a) Let $a$ be a real constant. Given that $$\lim_{x \to 2} \frac{x^2 + ax - 6}{x^2 - x - 2}$$ exists
No human feedback submitted yet.
q_ps_014		Easy	Medium	—	0.0	0.62	accepted	—	Given that $\displaystyle\lim_{u \to 0} \frac{\sin u}{u} = 1$ and $\displaystyle\lim_{u \to 0} \frac{\tan u}{u} = 1$, (a) evaluate $$\
No human feedback submitted yet.
q_ps_015		Medium	Hard	—	0.0	0.497	accepted	—	Consider the following three limits, each involving the exponential function. Use the technique indicated in each part. (a) Given that
No human feedback submitted yet.
q_ps_016		Hard	Medium	—	0.0	0.596	accepted	—
No human feedback submitted yet.
q_ps_t02_017		Hard	Hard	—	0.0	0.209	needs_human	—	Consider the following four limits. (a) Using the identity $\sin(3x) = 3\sin x - 4\sin^3 x$, evaluate $$\lim_{x \to 0} \frac{\sin(3x)
No human feedback submitted yet.
q_ps_001		Easy	Easy	—	0.0	0.43	accepted	—	The rational function $f$ is defined by $$f(x) = \frac{2x + 4}{x - 3}.$$ Find (a) the equation of the vertical asymptote of $f$, (b) the
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.435	accepted	—	Consider the function $f$ defined by $$f(x) = \frac{3x + 4}{x - 3}, \quad x \neq 3.$$ (a) Write down the equations of the asymptotes o
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.497	accepted	reviewed	Consider the function $f$ defined by $$f(x) = \frac{x^3 - x}{x^2 + x - 2}, \quad x \neq 1, \; x \neq -2.$$ (a) Show that $f(x) = \dfra
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.466	accepted	—	Let $f$ and $g$ be functions defined for all $x \in \mathbb{R}$ by $$f(x) = e^x - e^{-x}, \qquad g(x) = x^2 + 1.$$ (a) Find and simpli
No human feedback submitted yet.
q_ps_005		Medium	Hard	—	0.0	0.466	accepted	—	Let $f$ and $g$ be functions defined for all $x \in \mathbb{R}$ by $$f(x) = \ln(x^2 + 1), \qquad g(x) = e^x - 3.$$ (a) Show that $f$ i
No human feedback submitted yet.
q_ps_006		Easy	Medium	—	0.0	0.62	accepted	—	The graph of $y = f(x)$ is transformed to give the graph of $y = 2f(x + 3)$. (a) Describe this transformation as a sequence of two simp
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.192	accepted	—	Consider the functions $f$ and $g$ defined by $$f(x) = x^2 - 4, \quad x \in \mathbb{R}, \qquad g(x) = \ln x, \quad x > 0.$$ (a) **(i)
No human feedback submitted yet.
q_ps_008		Hard	Hard	—	0.0	0.219	accepted	—	Consider the functions $f$ and $g$ defined by $$f(x) = \frac{x}{x^2 + 1}, \quad x \in \mathbb{R}, \qquad g(x) = \sqrt{x + 1}, \quad x \geq
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Let $P(x) = x^3 - 2x^2 - 5x + 6$. (a) Show that $(x - 1)$ is a factor of $P(x)$. (b) Hence fully factorise $P(x)$.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.486	accepted	—	The polynomial $P(x) = x^3 + ax^2 + bx - 6$, where $a$ and $b$ are constants, satisfies the following two conditions: - $(x - 2)$ is a fact
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.396	accepted	reviewed	The rational function $f(x) = \dfrac{P(x)}{(x-2)(x+1)}$, where $P(x) = x^3 + ax^2 + bx - 4$ and $a, b \in \mathbb{R}$, satisfies the followi
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.459	accepted	—	Consider the quadratic function $f(x) = x^2 - 2kx + (k + 6)$, where $k \in \mathbb{R}$ is a constant. (a) Find the values of $k$ for wh
No human feedback submitted yet.
q_ps_005		Medium	Hard	—	0.0	0.477	accepted	—	Consider the quadratic function $f(x) = 2x^2 - (2m+1)x + m$, where $m \in \mathbb{R}$ is a constant. (a) Find the values of $m$ for whi
No human feedback submitted yet.
q_ps_006		Easy	Medium	—	0.0	0.637	accepted	—	A farmer has $24$ m of fencing to enclose a rectangular pen against a straight wall. The wall forms one side of the pen, so only three sides
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.273	accepted	—	Consider the quadratic function $f(x) = x^2 - 2(k+1)x + k^2$, where $k \in \mathbb{R}$. (a) Show that the discriminant of $f$ is $\Delt
No human feedback submitted yet.
q_ps_008		Hard	Hard	—	0.0	0.227	accepted	—	Consider the quadratic function $f(x) = x^2 - 2mx + (m + 6)$, where $m \in \mathbb{R}$. (a) Show that the discriminant of $f$ is $\Delt
No human feedback submitted yet.
q_ps_009		Hard	Hard	—	0.0	0.251	accepted	—	Consider the function $f(x) = x^2 - 4tx + (3t^2 + 2t - 1)$, where $t \in \mathbb{R}$. (a) Show that the discriminant of $f$ is $\Delta
No human feedback submitted yet.
q_ps_010		Hard	Hard	—	0.0	0.245	accepted	—	Consider the function $f(x) = 2x^2 - 4kx + (k^2 + 3k - 4)$, where $k \in \mathbb{R}$. (a) Express $f(x)$ in the form $2(x - h)^2 + c$,
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Let $P(x) = x^3 + 2x^2 - 5x - 6$. (a) Show that $(x + 1)$ is a factor of $P(x)$. (b) Hence factorise $P(x)$ completely.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.466	accepted	—	Let $P(x) = x^3 - 3x^2 + ax + b$, where $a, b \in \mathbb{R}$. It is given that $(x - 2)$ is a factor of $P(x)$, and that when $P(x)$ is di
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.477	accepted	reviewed	Let $P(x) = 2x^3 + ax^2 + bx - 6$, where $a, b \in \mathbb{R}$. When $P(x)$ is divided by $(x - 1)$, the remainder is $5$. When $P(x)$ is d
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.449	accepted	—	Let $f(x) = x^4 - 8x^2 + 12$. (a) Find the coordinates of all turning points of $f$. [3 marks] (b) Sketch the graph of $y = f(x)$,
No human feedback submitted yet.
q_ps_005		Medium	Hard	—	0.0	0.435	accepted	—	Let $f(x) = x^3 - 3x^2 - 9x + k$, where $k \in \mathbb{R}$. (a) Find the $x$-coordinates of the turning points of $f$, and hence find t
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.206	accepted	—	Let $f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12$. (a) Show that $f(-1) = 0$, and hence factorise $f(x)$ completely over $\mathbb{R}$. [4 marks]
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.227	accepted	—	Let $f(x) = x^4 - 8x^2 + k$, where $k \in \mathbb{R}$. (a) Find the $x$-coordinates of all critical points of $f$, and hence write down
No human feedback submitted yet.
q_ps_008		Hard	Hard	—	0.0	0.231	accepted	—	Let $f(x) = 2x^3 - 9x^2 + 12x + k$, where $k \in \mathbb{R}$. (a) Find the coordinates of the turning points of $f$ in terms of $k$, an
No human feedback submitted yet.
q_ps_009		Hard	Hard	—	0.0	0.227	accepted	—	Let $f(x) = x^4 - 10x^2 + 9$. (a) Find the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points of $f$. [5 mark
No human feedback submitted yet.
q_ps_010		Hard	Hard	—	0.0	0.227	accepted	—	Let $f(x) = x^3 - 6x^2 + 9x + k$, where $k \in \mathbb{R}$. (a) Find the coordinates of the turning points of $f$ in terms of $k$. Usin
No human feedback submitted yet.
q_ps_011		Hard	Hard	—	0.0	0.245	accepted	—	Let $f(x) = x^4 - 10x^2 + k$, where $k \in \mathbb{R}$. (a) Find the coordinates of all critical points of $f$. Using the second deriva
No human feedback submitted yet.
q_ps_012		Hard	Hard	—	0.0	0.227	accepted	—	Let $f(x) = x^3 - 3px + 2p$, where $p > 0$ is a real constant. (a) Find the coordinates of the critical points of $f$ in terms of $p$.
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.64	accepted	—	Consider the rational function $f(x) = \dfrac{2x + 4}{x - 1}$. (a) Write down the equations of all asymptotes of $f$. [2 marks] *(b)
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.459	accepted	—	Consider the rational function $$f(x) = \frac{x^3 - x^2 - 4x + 4}{x^2 - x - 2}.$$ (a) Show that $f$ has a removable discontinuity at $
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.435	accepted	reviewed	Consider the rational function $$f(x) = \frac{3x^2 - x - 2}{x^2 - 4}, \quad x \neq \pm 2.$$ (a) Express $f(x)$ in the form $$p + \fra
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.282	accepted	—	Consider the rational function $$f(x) = \frac{5x+7}{x^2+2x-3}, \quad x \neq 1,\ x \neq -3.$$ (a) Factorise $x^2+2x-3$ and hence expres
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.214	accepted	—	Consider the rational function $$f(x) = \frac{x^2 - 3x + 6}{x - 1}, \quad x \neq 1.$$ (a) Perform polynomial long division to express
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.214	accepted	—	Consider the rational function $$f(x) = \frac{x^3 - 2x^2 - x + 2}{x^2 + x - 6}.$$ (a) Show that $f$ has a removable discontinuity at $
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.214	accepted	—	Consider the rational function $$f(x) = \frac{x^2 - 5x + 10}{x - 2}, \quad x \neq 2.$$ (a) Perform polynomial long division to express
No human feedback submitted yet.
q_ps_008		Hard	Hard	—	0.0	0.242	accepted	—	Consider the rational function $$f(x) = \frac{3x - 1}{x^2 - 4x + 3}.$$ (a) Factorise $x^2 - 4x + 3$. Hence write down the equations of
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.636	accepted	—	Solve $\|3x + 1\| = 7$.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.555	accepted	—	Let $f(x) = \|2x - 3\| - \|x + 1\|$. (a) Express $f(x)$ as a piecewise linear function, and hence sketch the graph of $y = f(x)$. Your sket
No human feedback submitted yet.
q_ps_003		Hard	Hard	Medium	0.0	0.228	accepted	reviewed	Let $f(x) = \big\|\|x^2 - 4\| - 3\big\|$. (a) Show that $f$ is an even function, and hence express $f(x)$ as a piecewise function, clearly
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.435	accepted	—	Let $f(x) = \|x^2 - 2x - 3\|$. (a) Sketch the graph of $y = f(x)$ for $-3 \leq x \leq 5$. Your sketch must clearly show the coordinates o
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.236	accepted	—	Let $f(x) = \|x + 2\| - 2\|x - 1\| + \|x - 4\|$. (a) Express $f(x)$ as a piecewise linear function, showing clear working for each interval.
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.219	accepted	—	Let $g(x) = \big\|\,\|x^2 - 4\| - 3\,\big\|$. (a) Express $g(x)$ as a piecewise function, showing clear working for each step of the simpli
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.624	accepted	—	Let $f(x) = e^x$ and $g(x) = \ln(x + 3)$. (a) Find $(f \circ g)(x)$, simplifying your answer fully. [2 marks] (b) State the do
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.459	accepted	—	Let $f(x) = e^x + 1$ and $g(x) = \ln(x - 1)$, where $x > 1$. (a) Find and simplify $(g \circ f)(x)$. State the domain of $g \circ f$. *
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.427	accepted	reviewed	Let $f(x) = \ln\!\left(\dfrac{1+x}{1-x}\right)$ for $-1 < x < 1$, and let $g(x) = \dfrac{e^x - 1}{e^x + 1}$ for $x \in \mathbb{R}$. (a)
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.255	accepted	—	(a) Solve the equation $4^x - 6 \cdot 2^x + 8 = 0$. [4 marks] (b) Solve the equation $\log_8 x + \log_4 x + \log_2 x = \dfrac{1
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.246	accepted	—	(a) Solve the equation $3 \cdot 4^{x} - 10 \cdot 2^{x} + 3 = 0$, giving your answers in the form $x = \pm \log_2 k$ where $k$ is an inte
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.636	accepted	—	(a) State the Pythagorean identity that relates $\sec\theta$ and $\tan\theta$. [1 mark] (b) Given that $\tan\theta = \dfrac{5}{12}$
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.538	accepted	—	Let $\alpha = \arctan\!\left(\dfrac{3}{4}\right)$ and $\beta = \arctan\!\left(\dfrac{5}{12}\right)$, where $0 < \alpha < \dfrac{\pi}{2}$ and
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.466	accepted	reviewed	Let $f(\theta) = 5\sin\theta - 12\cos\theta$. (a) Express $f(\theta)$ in the form $R\sin(\theta - \phi)$, where $R > 0$ and $0 < \phi <
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.191	accepted	—	Consider the finite sum $\displaystyle S_n(x) = \sum_{k=1}^{n} \cos(kx)$, where $n \in \mathbb{Z}^+$ and $x \in \mathbb{R}$. (a) Starti
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.266	accepted	—	Let $f(\theta) = 2\sin\theta\cos\theta - \sqrt{3}(1-2\sin^2\theta)$. (a) Show that $f(\theta) = 2\sin\!\left(2\theta - \dfrac{\pi}{3}\r
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.266	accepted	—	Let $f(x) = \cos\!\left(x + \dfrac{\pi}{6}\right)\cos\!\left(x - \dfrac{\pi}{6}\right)$. (a) Starting from the compound angle formulae
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.206	accepted	—	Let $f(x) = \sin 5x + \sin 3x$. (a) Starting from the compound angle formulas for $\sin(P+Q)$ and $\sin(P-Q)$, derive the sum-to-produc
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.642	accepted	—	The function $f$ is defined by $$f(x) = \begin{cases} kx + 1 & x < 2 \\ x^2 - 1 & x \geq 2 \end{cases}$$ where $k$ is a constant. Find th
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.486	accepted	—	The function $f$ is defined by $$f(x) = \begin{cases} ax^2 + b & x \leq 1 \\ 3x - 7 & x > 1 \end{cases}$$ where $a$ and $b$ are real const
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.502	accepted	reviewed	The curve $C$ has equation $x^2 - xy + y^2 = 12$. (a) Show that the point $P(2, -2)$ lies on $C$. [1 mark] (b) Show that $$\frac{
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Easy	Easy	—	0.0	0.169	accepted	—	Consider the function $f(x) = x^2 e^x$. (a) Find $f'(x)$. [2 marks] (b) Find the equation of the tangent to the curve $y = f(x)$ a
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.227	accepted	—	Consider the function $f(x) = \dfrac{x^2}{e^x}$, defined for all $x \in \mathbb{R}$. (a) Show that $f'(x) = \dfrac{x(2-x)}{e^x}$. Henc
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.206	accepted	—	Consider the function $f(x) = \dfrac{\ln x}{x}$, defined for all $x > 0$. (a) Show that $f'(x) = \dfrac{1 - \ln x}{x^2}$. [2 marks] **
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.25	accepted	—	Consider the function $f(x) = e^{-x}(\sin x + \cos x)$, where $x \in \mathbb{R}$. (a) Show that $f'(x) = -2e^{-x}\sin x$. [2 marks] **
No human feedback submitted yet.
q_ps_t10_009		Hard	Hard	—	0.0	0.167	accepted	—	The curve $C$ is defined by the equation $$x^2 + xy + y^2 = 7.$$ (a) Show that the point $P(1, 2)$ lies on $C$. [1 mark] (b) Show
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Evaluate $\displaystyle\int_0^1 x e^x \, dx$ using integration by parts.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.497	accepted	—	Consider the function $f(x) = \dfrac{x}{\sqrt{x^2+1}}$. (a) Find $\displaystyle\int \frac{x}{\sqrt{x^2+1}}\, dx$ using the substitution
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.381	accepted	reviewed	Consider the function $f(x) = e^{2x}\cos(3x)$. (a) Find $\displaystyle\int e^{2x}\cos(3x)\,dx$. [4 marks] (b) Hence evaluate $\dis
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.239	accepted	—	Let $I_n = \displaystyle\int_0^{\pi/2}\cos^n x\,dx$, where $n$ is a non-negative integer. (a) Show that, for integers $n \geq 2$, $$I_
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.183	accepted	—	Let $g(x) = \dfrac{4x^2 - 3x + 1}{(x-1)(x^2+1)}$. (a) Express $g(x)$ in partial fractions. [3 marks] (b) Hence evaluate $\displays
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.642	accepted	—	Find the gradient of the curve $x^2 + 2y^2 = 9$ at the point $(1, 2)$.
No human feedback submitted yet.
q_ps_002		Easy	Medium	—	0.0	0.62	accepted	—	A particle moves along a straight line. Its displacement from a fixed origin at time $t$ seconds ($t \geq 0$) is given by $$s(t) = t^3 - 6t
No human feedback submitted yet.
q_ps_003		Easy	Medium	Medium	0.0	0.642	accepted	reviewed	A particle moves along a straight line. Its velocity at time $t$ seconds, for $0 \leq t \leq 4$, is given by $$v(t) = 6 - 2t \quad \text{m
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.459	accepted	—	A particle moves along a straight line. Its acceleration, $a(t)$ m s$^{-2}$, at time $t$ seconds is given by $$a(t) = \begin{cases} 3 & 0 \
No human feedback submitted yet.
q_ps_005		Medium	Hard	—	0.0	0.486	accepted	—	A particle moves along a straight line. Its displacement from a fixed point $O$, in metres, at time $t$ seconds is given by $$s(t) = \begin
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.209	accepted	—	A particle $P$ moves along a straight line. Its velocity, $v(t)$ m s$^{-1}$, at time $t$ seconds is given by $$v(t) = \begin{cases} t^2 - 4
No human feedback submitted yet.
q_ps_007		Hard	Hard	—	0.0	0.239	accepted	—	A particle $P$ moves along a straight line. Its displacement from a fixed origin $O$, measured in metres, at time $t$ seconds ($0 \le t \le
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.642	accepted	—	Solve the differential equation $$\frac{dy}{dx} = 2xy,$$ given that $y = 3$ when $x = 0$.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.47	accepted	—	A cup of coffee is placed in a room where the air temperature is a constant $20°C$. At time $t = 0$ minutes, the temperature of the coffee i
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.459	accepted	reviewed	A function $y(x)$ satisfies the differential equation $$\frac{dy}{dx} + \frac{2}{x}\,y = x^2, \quad x > 0,$$ with initial condition $y = 2
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.219	accepted	—	Answer all four parts. Each part is independent. (a) Consider the differential equation $$\frac{dy}{dx} = \frac{x^2 + y^2}{xy}, \q
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.218	accepted	—	A mathematician conducts a systematic investigation of four differential equations, each requiring a different solution technique. In each p
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.642	accepted	—	Using the Maclaurin series for $\sin x$, find $$\lim_{x \to 0} \frac{\sin x - x}{x^3}.$$
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.566	accepted	—	Consider the function $f(x) = \ln(1 + x^2)$. (a) Write down the Maclaurin series for $\ln(1 + x)$, giving the first four non-zero terms
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.485	accepted	reviewed	Consider the function $f(x) = e^x \sin x$. (a) Using the Maclaurin series definition $f(x) = \displaystyle\sum_{n=0}^{\infty} \frac{f^{
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.282	accepted	—	Consider the function $f(x) = \arctan(2x)$. (a) Using the geometric series $\dfrac{1}{1-t} = \displaystyle\sum_{n=0}^{\infty} t^n$, sub
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.251	accepted	—	Consider the function $f(x) = \ln\!\left(\dfrac{1+x}{1-x}\right)$. (a) Write down the Maclaurin series for $\ln(1+x)$, giving the first
No human feedback submitted yet.
q_ps_001		Easy	Easy	—	0.0	0.43	accepted	—	Find the modulus and argument of the complex number $z = 1 + \sqrt{3}\,i$, giving the argument in radians.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.485	accepted	—	Consider the equation $z^3 = -8i$, where $z \in \mathbb{C}$. (a) Express $-8i$ in Euler form $re^{i\theta}$, where $r > 0$ and $\theta
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.459	accepted	reviewed	Let $\theta \in \mathbb{R}$. (a) Using De Moivre's theorem, show that $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta \quad \text{and} \q
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.198	accepted	—	Consider the sum $\displaystyle\sum_{k=0}^{n-1} e^{ik\theta}$, where $n \in \mathbb{Z}^{+}$, $n \geq 2$, and $\theta \in \mathbb{R}$ with $\
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.198	accepted	—	Let $\omega = e^{2\pi i/5}$. (a) Find all solutions to $z^5 = 1$ in Euler form $z = e^{i\theta}$, and sketch the solutions on an Argand
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.624	accepted	—	Let $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$. (a) Find $\
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.47	accepted	—	Two lines $l_1$ and $l_2$ have vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.486	accepted	reviewed	A plane $\Pi$ has vector equation $$\mathbf{r} = \begin{pmatrix}1\\0\\2\end{pmatrix} + s\begin{pmatrix}1\\2\\1\end{pmatrix} + t\begin{pmatr
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.261	accepted	—	The points $A$, $B$ and $C$ have coordinates $(1,\, 0,\, -1)$, $(2,\, 2,\, 5)$ and $(0,\, 1,\, -1)$ respectively. The plane $\Pi$ passes thr
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.206	accepted	—	Two particles $P$ and $Q$ move through three-dimensional space. At time $t$ seconds ($t \geq 0$), their position vectors $\mathbf{r}_P$ and
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Two bags contain coloured balls. Bag 1 contains $4$ red balls and $1$ blue ball. Bag 2 contains $2$ red balls and $3$ blue balls. A bag is c
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.508	accepted	—	A carnival game involves spinning a wheel that awards a player $X$ prize tokens, where $X$ is a discrete random variable with the following
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.475	accepted	reviewed	A continuous random variable $X$ has the probability density function $$f(x) = \begin{cases} k(4x - x^2) & 0 \leq x \leq 4 \\ 0 & \text{oth
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.227	accepted	—	A factory produces electronic components. Each component is sourced from one of two suppliers: $40\%$ of components come from Supplier $A$ a
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.264	accepted	—	A city has two taxi companies: Alpha Cabs and Beta Taxis. Of all taxis operating in the city, $60\%$ belong to Alpha Cabs and the remaining
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.642	accepted	—	Prove by mathematical induction that $7^n - 1$ is divisible by $6$ for all $n \in \mathbb{Z}^+$.
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.47	accepted	—	This question asks you to apply three different proof techniques. (a) Consider the statement: "For $n \in \mathbb{Z}^+$, if $n^2$ is no
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.448	accepted	reviewed	This question asks you to use several proof techniques related to divisibility by $3$. (a) Consider the statement $P$: "For all $n \in
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.196	accepted	—	This question explores divisibility by $5$ through a range of proof techniques. (a) Consider the statement $S$: "For all $n \in \mathb
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.255	accepted	—	This question asks you to apply a range of proof techniques. (a) Consider the statement $S$: "For all $x \in \mathbb{R}$, if $x^3 + x
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.63	accepted	—	Prove by mathematical induction that, for all $n \in \mathbb{Z}^+$, $$\sum_{r=1}^{n} r = \frac{n(n+1)}{2}.$$
No human feedback submitted yet.
q_ps_002		Easy	Medium	—	0.0	0.63	accepted	—	The expressions $k + 1$, $3k - 2$, and $2k + 7$ are three consecutive terms of an arithmetic sequence. (a) Find the value of $k$. **(b
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.52	accepted	reviewed	A geometric sequence has second term $u_2 = 6$ and fifth term $u_5 = \dfrac{2}{9}$. (a) Find the common ratio $r$ and the first term $a
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Medium	Hard	—	0.0	0.538	accepted	—	The sum of the first $n$ terms of a sequence $\{u_n\}$ is given by $$S_n = 4\!\left(1 - \left(\frac{1}{3}\right)^{\!n}\right), \quad n \in
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.204	accepted	—	An arithmetic sequence $\{u_n\}$ has first term $p$ and common difference $d$, where $d > 0$. A geometric sequence $\{v_n\}$ has first term
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.266	accepted	—	Parts (a) and (b) concern the sequence $\{u_n\}$ whose partial sum is given by $$S_n = 3n^2 - n, \quad n \in \mathbb{Z}^+.$$ (a) F
No human feedback submitted yet.
q_ps_t19_007		Easy	Medium	—	0.0	0.635	needs_human	—	Three consecutive terms of a geometric sequence are $k$, $k + 6$, and $4k$, where $k \neq 0$. Find the possible values of $k$.
No human feedback submitted yet.
q_ps_t19_008		Medium	Hard	—	0.0	0.449	needs_human	—	The sum of the first $n$ terms of a sequence $\{u_n\}$ is given by $$S_n = 10 - 10\left(\frac{2}{5}\right)^n, \quad n \in \mathbb{Z}^+.$$
No human feedback submitted yet.
q_ps_t19_009		Medium	Hard	—	0.0	0.555	needs_human	—	Consider the sequence with general term $u_k = k(k+2)$, where $k \in \mathbb{Z}^+$. (a) Show that $$\sum_{k=1}^{n} k(k+2) = \frac{n(n+
No human feedback submitted yet.
q_ps_t19_010		Hard	Hard	—	0.0	0.237	needs_human	—	The partial sum of the first $n$ terms of a sequence $\{u_n\}$ is given by $$S_n = 2n^2 + 3n, \quad n \in \mathbb{Z}^+.$$ (a) Find an
No human feedback submitted yet.
q_ps_t19_011		Hard	Hard	—	0.0	0.273	needs_human	—	A geometric series has first term $a > 0$ and common ratio $r$, where $\|r\| < 1$. The sum to infinity of the series is $40$. The sum of the
No human feedback submitted yet.
q_ps_t19_012		Hard	Hard	—	0.0	0.297	needs_human	—	A sequence $\{u_n\}$ is defined by the recurrence relation $u_{n+1} = \frac{u_n}{2u_n - 1}$, with $u_1 = 2$. (a) Find the values of $u_2$,
No human feedback submitted yet.
q_ps_t19_013		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{u_n\}$ is given by $$S_n = 4n^2 + 3n - 12\cdot\left(\frac{2}{3}\right)^n + 12, \qquad n \in
No human feedback submitted yet.
q_ps_t19_014		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_015		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{u_n\}$, where $n \in \mathbb{Z}^+$, is given by $$S_n = 3n^2 - n + 5 \cdot 2^n - 5.$$ **(a
No human feedback submitted yet.
q_ps_t19_016		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_017		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{u_n\}$, where $n \in \mathbb{Z}^+$, is given by $$S_n = \frac{n(n+1)(2n+1)}{3} - \frac{12}{
No human feedback submitted yet.
q_ps_t19_018		Hard	Hard	—	0.0	0.167	accepted	—	A sequence $\{u_n\}$ is defined by the recurrence relation $$u_{n+1} = \frac{3}{4}\,u_n + 5, \quad n \in \mathbb{Z}^+,$$ with first term $
No human feedback submitted yet.
q_ps_t19_019		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_020		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_021		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_022		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_023		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_024		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_025		Hard	Hard	—	0.0	0.167	accepted	—	A sequence $\{u_k\}$ is defined by $u_k = k(3k+1)$ for $k \in \mathbb{Z}^+$. Let $T_n = \displaystyle\sum_{k=1}^{n} u_k$. (a) Show that
No human feedback submitted yet.
q_ps_t19_026		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_027		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_028		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_029		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_030		Hard	Hard	—	0.0	0.167	accepted	—
No human feedback submitted yet.
q_ps_t19_031		Hard	Hard	—	0.0	0.167	accepted	—	The partial sum of the first $n$ terms of a sequence $\{a_n\}$ is given by $$S_n = 5(3^n - 1), \quad n \in \mathbb{Z}^+.$$ (a) Find $a
No human feedback submitted yet.
q_ps_t19_032		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{u_n\}$ is given by $$S_n = (2n+1) \cdot 2^n - 1, \quad n \in \mathbb{Z}^+.$$ (a) Using
No human feedback submitted yet.
q_ps_t19_033		Hard	Hard	—	0.0	0.167	accepted	—	A sequence $\{u_n\}$ is defined by the recurrence relation $$u_{n+1} = 2u_n + 3^n, \quad n \in \mathbb{Z}^+,$$ with $u_1 = 5$. (a) Fi
No human feedback submitted yet.
q_ps_t19_034		Hard	Hard	—	0.0	0.167	accepted	—	The total number of pages read by a student in the first $n$ weeks of a reading programme is modelled by the partial sum $$S_n = 3n^2 + n +
No human feedback submitted yet.
q_ps_t19_035		Hard	Hard	—	0.0	0.167	accepted	—	A company offers employees two monthly savings plans, Plan A and Plan B. Plan A is an arithmetic sequence of monthly deposits. The depo
No human feedback submitted yet.
q_ps_t19_036		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{a_k\}$ is given by $$S_n = 2n^3 + 3n^2 - 5n, \quad n \in \mathbb{Z}^+.$$ (a) (i)
No human feedback submitted yet.
q_ps_t19_037		Hard	Hard	—	0.0	0.167	accepted	—	A sequence $\{u_k\}$ is defined by $$u_k = k \cdot \left(\frac{2}{3}\right)^{k-1}, \quad k \in \mathbb{Z}^+.$$ (a) Write down $u_1$, $
No human feedback submitted yet.
q_ps_t19_039		Hard	Hard	—	0.0	0.167	accepted	—	The sum of the first $n$ terms of a sequence $\{u_k\}$ is given by $$S_n = 2n^2 - n + 12\!\left(1 - \left(\frac{1}{2}\right)^{\!n}\right),
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Consider the expansion of $(2x + 3)^5$. (a) Write down the general term, $T_{r+1}$, of this expansion. [1 mark] (b) Find the coeff
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.583	accepted	—	Consider the expansion of $\left(x^2 + \dfrac{k}{x}\right)^9$, where $k$ is a positive constant. (a) Write down the general term $T_{r+
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.435	accepted	reviewed	Consider the expansion of $(1 + 2x)^n$, where $n$ is a positive integer. (a) Given that the coefficient of $x^2$ in the expansion is $1
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.231	accepted	—	Consider the binomial expression $\left(x^3 - \dfrac{k}{x}\right)^8$, where $k$ is a positive constant. (a) Write down the general term
No human feedback submitted yet.
q_ps_005		Hard	Hard	—	0.0	0.245	accepted	—	Consider the expansion of $\left(x^2 + \dfrac{k}{x}\right)^n$, where $n$ is a positive integer and $k$ is a positive real constant. (a)
No human feedback submitted yet.
q_ps_001		Easy	Medium	—	0.0	0.637	accepted	—	Let $P(x) = x^3 + ax^2 + 3x - 4$, where $a$ is a constant. (a) Given that the remainder when $P(x)$ is divided by $(x - 2)$ is $14$, fi
No human feedback submitted yet.
q_ps_002		Medium	Hard	—	0.0	0.426	accepted	—	Let $f(x) = 2x^3 + px^2 + qx - 18$, where $p, q \in \mathbb{R}$. The polynomial $f(x)$ has three real roots $\alpha$, $\beta$, $\gamma$. It
No human feedback submitted yet.
q_ps_003		Medium	Hard	Medium	0.0	0.47	accepted	reviewed	Let $P(x) = x^4 - 3x^3 + bx^2 + cx - 10$, where $b, c \in \mathbb{R}$. It is given that $2 + i$ is a root of $P(x)$, and that the remaining
Reviewer: rev_4twhx7s07ii818x • Difficulty: Medium • Pattern verdict: Positives (none) Negatives (none) raw FeedbackRecord JSON { "feedback_id": "f_web_q_ps_003_ms8v8cj4", "question_id": "q_ps_003", "reviewer": "rev_4twhx7s07ii818x", "timestamp": "2026-07-31T11:35:02.512Z", "difficulty_human": "Medium", "description": { "positives": [], "negatives": [] } }
q_ps_004		Hard	Hard	—	0.0	0.211	accepted	—	Let $P(x) = 2x^4 - 6x^3 + bx^2 + cx - 20$, where $b, c \in \mathbb{R}$. It is given that $1 - 2i$ is a root of $P(x)$, and that the remainin
No human feedback submitted yet.
q_ps_005		Hard	Medium	—	0.0	0.596	accepted	—
No human feedback submitted yet.
q_ps_006		Hard	Hard	—	0.0	0.206	accepted	—	Let $P(x) = x^4 + ax^3 + bx^2 + 2x - 15$, where $a, b \in \mathbb{R}$. (a) Given that $(x + 1)$ is a factor of $P(x)$, and that the rem
No human feedback submitted yet.

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