CIE AS & A Level Mathematics (9709) ~ 2025 Past Papers

Before you start: show working, keep exact values (π, surds, fractions) until the final line, and label graphs. Method marks are commonly awarded for clear set-up even if a final arithmetic slip appears.

• Q1 ~ Curve vs line (Discriminant & inequalities): Set y expressions equal and form the quadratic in x immediately; label a, b, c. The discriminant tells the graph story: >0 two intersections, =0 tangent, <0 no intersection. Simplify carefully ~ messy coefficients invite sign errors ~ then find the critical values. Express the final k-range in interval notation. Finally, translate the range into the graph context (e.g. “for k < −3 the line does not meet the curve”) so the examiner sees you interpreted the result.

These examiner insights were written specifically for Paper 9709/12 (Feb/Mar 2025) and were cross-checked against the official paper, mark scheme and examiner report. Use them as exam-proof habits, not model answers.

• Q2 ~ Gradient at P and coordinates of M (Differentiation & Rounding): Differentiate first, then substitute x = 1 to find the gradient at P; show dy/dx step by step. For the minimum point, set dy/dx = 0 and solve exactly, then evaluate y ~ keep exact surds and only round the final coordinates to 3 s.f. The examiner report flagged premature decimal rounding and missing steps as common errors, so retain exactness until the final line. Label answers clearly as (x, y).
• Q3 ~ Binomial expansion (minus signs & full expansion): State the general expansion (or expand fully) rather than attempting to multiply four brackets mentally. Pay attention to negative terms ~ examiners frequently penalise dropped minus signs. For part (b) identify which powers of x produce x^2 and list the contributing terms, then add coefficients carefully; write intermediate terms so markers can award method marks where algebra is shown.
• Q4 ~ Perimeter & area of shaded region (trig + sector): Draw and label the triangle and the sector. Use cosine rule or right-triangle trig to find chord/side lengths, and compute arc length using rθ (ensure θ in radians). For area, subtract the sector area from the triangle (use ½ab sinC for the triangle part). The report noted loss of accuracy when candidates mixed radians/degrees ~ stay in radians and keep intermediate exact values where possible.
• Q5 ~ Arithmetic progression sum between bounds: Read carefully: do NOT compute S₁₅₀ and S₄₀₀ then subtract unless those are the intended indices. Instead identify the first and last terms that fall inside (150 ≤ term ≤ 400), find their indices n₁ and n₂, then use S = n/2(2a + (n−1)d) for the partial sum. A shorter method would be using n/2(a + l), less room for error and saves time. Show how you found n₁ and n₂ ~ examiners will credit correct summation set-up even if arithmetic slips occur.
• Q6 ~ Circle centre, radius equation & tangent gradient: Translate the “least distance to axes” conditions: if centre is (h, k) and radius = r, then k − r = 8 and h − r = 5, so (h, k) = (r+5, r+8). Use Pythagoras with distance 15 to form the quadratic in r, solve, and reject negative roots (r is a length). For the tangent at the furthest point, recall radius to point P is radial; tangent ⟂ radius, so find slope of radius and take the negative reciprocal. Show each reasoning step for method marks.
• Q7 ~ Trig identity & equation solving (show that + find solutions): In the ‘show that’ part start from the left-hand side and replace tan²θ by sin²θ/cos²θ, then convert cos²θ to 1−sin²θ to collect terms into a single fraction ~ show every algebraic step. For solving the equation, note the valid range for sin²θ and discard extraneous roots (one quadratic root may be >1). Use symmetry or CAST to list all θ in the specified interval ~ examiners penalise missed solutions and overlooked periodicity.
• Q8 ~ Geometric progression & sum to infinity: Express the second term (ar) and the sum to infinity (a/(1−r)) then eliminate a to get an equation in r. Solve the quadratic and reject r values that make |r| ≥ 1 (sum to infinity only exists for |r|<1 NEVER FORGET THIS). The report shows many mistakenly rejected negative r because of sign confusion ~ keep sign careful and interpret meaning (a negative ratio is allowed if |r|<1), but follow the question’s instruction about which root to accept.
• Q9 ~ Second derivative, integration to y, constants of integration: Use d²y/dx² directly to test the stationary point (positive → minimum). For (b) integrate twice, remembering to introduce C₁ after first integration (dy/dx) and C₂ after second (y). Apply the conditions dy/dx = 0 at x = ½ and y = 9 at that x to find constants. The examiner report highlights lost marks where candidates omitted constants or used calculators with no shown working ~ write the integrals and constants explicitly.
• Q10 ~ Definite integration & area using tangent (substitution + exactness): For the integral part show the substitution and change both limits immediately; missing this step was a common error. When using tangents to find areas, find the tangent equation (dy/dx at the point), then use definite integrals or coordinate geometry (triangle area from intercepts). Keep exact factors (e.g., fractions from integration) and only round at the end to the required s.f. The examiner report flags omitted subtraction (part (b) minus part (a)) as a frequent slip ~ check you combined parts correctly.

Exam-proof habit: after finishing each question, do a 30-second scan ~ confirm you covered all requested interval/branches, retained exact values until final line, and included units where applicable.

Before you start: state domains and restrictions up front, keep exact forms until the final step, and label diagrams. Method marks are often available when the working is transparent ~ show the substitution, the differentiation, and any limits used.

• Q1 ~ Logarithmic equation & domain checks: The question involves manipulating natural logs ~ start by stating the domain explicitly (arguments of ln > 0). After combining log terms, show the algebraic simplification that leads to the exponential form. Examiners mark the algebra and domain checks separately; a correct but domain-invalid root loses credit. Present the final roots and then confirm which satisfy the original log constraints so that the examiner can award both method and accuracy marks.

These examiner insights are written to align with Paper 9709/22 (Feb/Mar 2025) and its mark scheme. Use them as guidance to win method and communication marks. :contentReference[oaicite:1]{index=1}

• Q2 ~ Differential equation from sec x form This question asks you to integrate an expression involving sec x (or similar trig reciprocals). Begin by rewriting sec and other trig functions in sine/cosine if it clarifies the substitution. Integrate carefully, showing each step and the constant of integration. When a particular solution is required (curve passing through a point), apply the point to find the constant only after finding the general solution. Clear labelling of steps protects method marks even if arithmetic slips later.
• Q3 ~ Area between exponentials: For area bounded by exponential curves and an axis, sketch the region and state the intersection x-value you will use as the integration limit. Convert the area to an integral of (upper − lower) with correct limits and show the antiderivative in exact form. The mark scheme rewards correct set-up and exact evaluation; the most common slip is sign errors when determining which curve is on top. Label the region and write the final exact value (simplify fractions or logs where possible).
• Q4 ~ Trig derivative & stationary points: Differentiate using the chain and product rules as appropriate; write dy/dx and show simplification leading to factors you can set to zero. For stationary points, solve dy/dx = 0 and evaluate y ~ include exact radian values and check second derivative (or sign chart) to classify maxima/minima. Examiners penalise missing periodic solutions in the given domain 0 ≤ x < 2π; list every valid x in the interval and annotate which correspond to each stationary value.
• Q5 ~ Sketch & iterative root finding: Part (a): clear sketch axes, label intersections a and b; a neat sketch reduces algebra errors. Part (b): derive the iterative form directly from the rearranged equation and state the convergence condition briefly. When iterating, give each iteration to the requested sig figs and state the stopping rule (3 s.f.). Examiners award method marks for the correct iteration formula and for showing intermediate values even if the final rounded result is slightly off.
• Q6 ~ Trig equation solving & maxima of expression: When solving trig equations across 0°–360°, use identities to reduce the equation, then apply CAST and periodicity to find all solutions. For the part asking for the greatest possible value of an expression involving trig functions, recognise whether it is of the form A sin(θ + ϕ) or can be bounded by Cauchy/AM–GM style manipulations; state the inequality used and show where equality occurs ~ that identifies the angle(s) giving the maximum. Explicitly give the smallest positive angle asked for.
• Q7 ~ Polynomial/log mixture/algebraic simplification: When algebra and logs are mixed, state the rearrangement strategy before plunging into calculation: do you isolate the logarithmic term or convert to exponentials? Examiners reward a one-line plan. If a substitution simplifies the algebra (e.g., set t = ln x), show it and revert to the original variable when interpreting answers. Keep track of domain restrictions and present final results in the required form (exact or decimal to required sf).
• Q8 ~ Derivatives of exponentials & showing no stationary points: For curves defined using sums of exponentials, differentiate term-by-term and simplify. The ‘show that’ part tests algebraic manipulation ~ present that cleanly before concluding about stationary points. If dy/dx simplifies to an expression that cannot be zero for any real x (because it is a sum of positive exponentials, for instance), articulate that inequality clearly and reference why each term’s sign prevents roots. This logical statement secures the mark for “no stationary points.”
• Q9 ~ Integration with substitution & exact evaluation: Often a substitution will convert the integrand to an elementary antiderivative. Make the substitution explicit, compute du (or dx) and change the limits for definite integrals immediately. Show the antiderivative and simplify symbolic factors (avoid premature approximations). When the integral is applied to an area, confirm which function is upper and which is lower over the interval to avoid sign mistakes, and present the exact answer before rounding.
• Q10 ~ Combining methods & answer presentation: Final parts commonly ask you to combine earlier results or to use an approach (algebraic/manipulative or geometric) that is more efficient. Begin with a one-line plan stating the method, then proceed stepwise. Examiners award communication marks for a clear final line with the exact result highlighted. If you use previous parts, reference them explicitly (e.g., “using result from part (b) …”) so markers can trace the logic and allocate method marks accordingly.

Exam-proof habit: after each question, check domain restrictions, list all solutions in the required interval, and keep exact forms until the final step. Write a one-line plan at the top for multi-step parts ~ examiners love readable structure.

Before you start: read each part carefully, state substitutions and domains, and keep exact algebraic forms until the final line. For multi-step parts, write a one-line plan so examiners can follow your method quickly ~ this often secures method marks even when arithmetic slips occur.

• Q1 ~ Parametric / Polar differentiation and tangents: For parametric or polar differentiation, state the derivative formula explicitly (dy/dx = (dy/dt)/(dx/dt) or dy/dθ etc.) before substituting. When finding tangents, evaluate derivatives at the given parameter and present the tangent in clear Cartesian form. Note any parameter values where dx/dt = 0 (vertical tangents) ~ examiners penalise missed singular cases. A precise sketch with parameter markers strengthens the answer.

These insights are written to align with the Pure Mathematics 3 9709/32 syllabus and typical examiner expectations for Feb/Mar 2025. Use them as practical, exam-focused guidance ~ not as model answers.

• Q2 ~ Advanced integration & improper integrals: If the integral has an infinite limit or an integrand with singularities, split the integral and show the limit process for convergence. For substitution-based integrals, perform the substitution and change limits immediately for definite integrals. Examiners award full method marks when the limit-handling is explicit; otherwise candidates risk losing the final accuracy mark even with correct antiderivatives. Keep algebra tidy to avoid arithmetic slips.
• Q3 ~ Arc length, curvature and geometric interpretation: Start by writing the formula for arc length or curvature in the correct parametric form and show the substitution that leads to an integrable expression. If a geometric meaning is required, sketch and label the relevant section so the marker sees the interpretation at a glance. For curvature, show the derivative algebra and simplify before substitution ~ examiners reward clarity of derivation as much as the final numeric expression.
• Q4 ~ Sequences, series & radius/interval of convergence: Identify whether the series is arithmetic, geometric or requires a convergence test (ratio/root/comparison). State the test and why its conditions hold; show limit steps cleanly. For power series, compute the radius and interval of convergence explicitly and check endpoints separately. Examiners often award method marks for correct test selection even if arithmetic evaluation of limits has small errors.
• Q5 ~ Implicit/implicit differentiation & solving for dy/dx: When differentiating implicitly, write the derivative of each term and gather dy/dx terms on one side before solving. Examiners expect the algebra shown in full rather than a single-line result; intermediate simplifications make it easier to award method marks. If asked for normals or tangents, remember normal slope = −1/(dy/dx) at the point ~ state that cleanly and check whether the normal/tangent intersects other given curves as requested.
• Q6 ~ Transformations, inverse functions and mapping: For function transformations or composition, write the transformation chain explicitly (e.g., x→ax+b then f(.)), and show how domain and range change. When finding inverses, swap x and y and solve, then verify by composing f(f^{-1}(x)) = x. Examiners reward a brief verification step ~ it demonstrates correctness and often secures the final accuracy mark.
• Q7 ~ Trig identities & higher-level simplifications: Start with the identity you plan to use (sum-to-product, double-angle etc.) and apply it stepwise. Avoid dividing by expressions that could be zero without stating the restriction. When simplifying to a standard form, annotate each line so markers can credit partial progress. Exactness (sin, cos in radicals, fractions of π) is preferred to decimals unless asked otherwise.
• Q8 ~ Complex numbers ~ roots, arguments and geometry: Convert to polar form where roots or arguments are involved, and use de Moivre’s theorem to list all distinct roots; present them in exact form. For geometry questions, plot Argand diagrams with moduli and principal arguments annotated. Examiners expect both algebraic and geometric reasoning where requested; doing both reduces ambiguity and helps secure method marks even if one representation has a minor slip.
• Q9 ~ Reduction to standard integrals & use of known forms: If an integrand can be matched to a standard table form, state the match (e.g., substitute to reduce to ∫(1/(1+x^2)) dx) and show the substitution clearly. Examiners award method marks for demonstrating why the standard form applies. For definite integrals, remember to change limits; for indefinite integrals, include constants of integration and show how initial conditions determine them if required.
• Q10 ~ Presentation, exactness & answer checking: The finishing touch matters: box your exact final answer, state any domain restrictions, and perform one quick check (substitute back, differentiate, or sanity-check units). Examiners often give the difference between a good and excellent script to candidates who show a brief verification. Keep decimal rounding to the last step and clearly label any approximations to avoid loss of marks.

Exam-proof habit: for multi-part questions write a quick one-line plan (method) at the top, keep exact forms until the final step, and always state domains/restrictions where logs, roots, or denominators appear.

Before you start: draw clear free-body diagrams, define a positive direction and list given constants (use g = 10 m·s⁻² where specified). Examiners reward an explicit plan and labelled diagrams ~ these secure method marks even if algebra becomes heavy.

• Q1 ~ Three coplanar forces (finding i and X): Start with a labelled vector diagram of the three forces and resolve horizontally and vertically. Write ΣFx = 0 and ΣFy = 0 explicitly and solve the resulting simultaneous equations. Examiners award method marks for correct resolution even if trig evaluation has small slips. Check angle conventions (i measured where?) and confirm X is positive. A short concluding sentence explaining physical balance (equilibrium) helps readability.

These examiner insights were written after studying the official paper, mark scheme and examiner report for 9709/42 (Feb/Mar 2025). Use them to guide method and presentation. :contentReference[oaicite:0]{index=0}

• Q2 ~ Cyclist motion with segments of acceleration, steady speed and deceleration: Use SUVAT per segment: acceleration over 42 m, constant-speed over 50 m, deceleration over 16 m. Find V using s = (v^2 − u^2)/(2a) in the deceleration segment or the appropriate SUVAT equation. Then calculate times per segment and sum. Write units for each intermediate result and show substitution ~ examiners frequently give method marks for correct equation choice and consistent use of units.
• Q3 ~ Power, resistance and ascent work (constant speed then climbing): For horizontal cruise, use Power = Resistance × speed to find speed (convert kW/kN consistently). For the climb, include gain in potential energy (mgh) and the work against variable resistance; add these to calculate total work, then divide by time to find average power. Show unit conversions (kW ↔ J s⁻¹) and present the final power to 3 s.f. as required by the rubric.
• Q4 ~ Two masses over a pulley (tension, acceleration & subsequent motion): For the initial release, write separate F = ma equations for each mass and solve simultaneously for tension and acceleration. For the motion until A hits the ground, use energy methods or suvat depending on what is simpler ~ the examiner report shows candidates lost marks when they mixed methods without justification. For the greatest height part, track energy transfer (or use v^2 = u^2 + 2as) and be careful with sign conventions.
• Q5 ~ Series of collisions (1D elastic followed by coalescence): Treat collisions sequentially: apply conservation of momentum and the coefficient of restitution where elastic collision is specified. For the coalescence (perfectly inelastic), combine masses and use momentum conservation to find the final speed. Label velocities before and after clearly and show algebraic steps ~ examiners award partial credit for clear momentum bookkeeping.
• Q6 ~ Block on plane with friction parameter n and threshold forces: Define geometry (angle α) and draw forces: weight resolved, normal, friction. For impending motion up or down, use limiting friction ±μR and state the inequality that produces the condition. Write equations of equilibrium with the friction sign appropriate to impending motion. The mark scheme rewards explicit sign reasoning and substitution checks for multiple cases (up vs down).
• Q7 ~ Velocity as function of time with constants k and p (find constants from acceleration & displacement): Differentiate v(t) to get a(t) and integrate v(t) to find s(t) (or integrate a to get v then s). Apply the conditions (given acceleration at t = 1 and displacement at t = 1) to form two equations for k and p. Show both differentiation and integration steps and solve the linear system ~ examiners give method marks for full working even if algebraic simplification stumbles.
• Q8 ~ Oscillatory / time-dependent force modelling (DEs) or variable forces: When a question leads to differential equations, derive from Newton’s second law and state the DE form before solving. Identify whether the DE is linear with constant coefficients or separable. Show general solution with constants, then apply boundary/initial conditions. The examiner report highlights lost marks when students omitted the derivation from F = ma.
• Q9 ~ Momentum chain and energy checks over multiple interactions: For multi-stage motion, keep a short table of states (velocity, KE, height) at key events. Use momentum conservation where collisions occur, and use energy to relate heights/speeds elsewhere ~ do not mix assumptions. Examiners reward a mix of correct principles applied in the most direct way and penalise inconsistent use of energy and momentum without justification.
• Q10 ~ Presentation & accuracy (final checks): Box final answers, include units, and perform a quick plausibility check (signs, dimensions, magnitudes). For multi-part questions, reference earlier results explicitly when reused. The ER shows many candidates lose marks through careless sign conventions ~ a one-line justification for sign choices is an inexpensive way to capture method credit.

Exam-proof habit: a labelled free-body diagram plus a one-line plan before calculations saves marks.

Before you start: define random variables and state any sampling assumptions. Write distribution names and parameters up front (pmf/pdf), and show intermediate sums; examiners reward explicit structure and units where applicable.

• Q1 ~ Biased coins & discrete distribution (X = number of heads): Begin by constructing the distribution table for X (0–3 heads) and compute each probability using multiplication of independent coin probabilities. The mark scheme rewards a clear pmf first; then compute E(X) and Var(X) by summation showing intermediate steps (E(X^2) if used). Label each column in your table ~ this structured approach secures both method and final marks.

These insights were prepared with reference to the official Paper, Mark Scheme and Examiner Report for 9709/52 (Feb/Mar 2025). :contentReference[oaicite:2]{index=2}

• Q2 ~ Categorical sampling & binomial approximations (multiple parts): For small-sample questions, use exact binomial probabilities with correct p; for larger sample approximations (e.g., n=140), justify a normal approximation and apply continuity correction. Show μ and σ² used for approximation and standardise with z. The ER flags omitted continuity correction as a common error ~ include it and state the approximation assumption explicitly.
• Q3 ~ Expected value & variance from given E(X): If E(X) is provided, compute Var(X) via Var(X) = E(X^2) − [E(X)]^2 when E(X^2) is derivable; show intermediate sums plainly. Examiners award method marks for correct algebraic setup even when arithmetic becomes tedious ~ present intermediate totals to avoid lost marks.
• Q4 ~ Sampling proportions & approximation bounds: When approximating binomial with normal, compute mean np and variance np(1−p), standardise with continuity correction and justify CLT applicability. If computing P(X > k), map to z with (k + 0.5 − μ)/σ. Examiners look for stated assumptions (large n, p not too close to 0 or 1) as part of a correct solution.
• Q5 ~ Hypothesis testing structure (H0/H1, test statistic, p-value): State null and alternative hypotheses in context, compute the test statistic and its sampling distribution, and make a decision at the stated significance level. Conclude in plain language about the claim under test ~ examiners award marks for correct interpretation as well as calculation.
• Q6 ~ Confidence interval setup and interpretation: Present the CI formula using estimator ± critical × SE. Calculate with correct degrees of freedom if using t-distribution or state normal approximation for large samples. Interpret the interval concisely in context (avoid probability phrasing about the parameter).
• Q7 ~ Poisson and exponential relations: When modelling rare events with Poisson, define λ per unit and justify the model assumptions. For waiting-time questions, connect Poisson and exponential distributions and use memoryless property where relevant. Examiners credit unit-consistent parameter handling and interpretation of results.
• Q8 ~ Bivariate summary & correlation interpretation: Compute covariance and Pearson’s r with intermediate sums; comment on strength/direction and possible outliers. If asked for regression, state the regression equation and interpret coefficient units. Examiners expect a short sentence interpreting the slope in context.
• Q9 ~ Generating functions & moment extraction: Define the generating function or mgf, differentiate to obtain moments and evaluate at the specified point. Show the functional differentiation steps rather than only quoting the result to secure method credit.
• Q10 ~ Statistical interpretation & limitations: When asked to interpret results, state the numeric conclusion, the model assumptions, and a short comment on limitations (sample size, model appropriateness). Examiners reward succinct contextual interpretation matched to computed evidence.

Exam-proof habit: always state and justify your model choice (binomial/Poisson/Normal) before computing probabilities.

Before you start: write down the model (distribution family and parameter), label sample size and assumptions, and show the log-likelihood / test-statistic derivation where required. Advanced theory questions need a one-line justification for approximations.

• Q1 ~ Identify distribution & state pmf/pdf and parameters: Begin by writing the pmf/pdf and parameter notation (e.g., λ for Poisson). If asked to derive moments, perform the algebraic summation/integration step-by-step rather than quoting. Examiners reward both correct identification and clear derivations of mean/variance ~ show intermediate manipulation to secure method marks.

These insights align to the official paper and mark scheme for 9709/62 (Feb/Mar 2025). :contentReference[oaicite:4]{index=4}

• Q2 ~ Maximum likelihood estimation: Write the likelihood function for the sample, take logarithms to form the log-likelihood, and differentiate to find the MLE. Show the derivative steps and confirm the second derivative sign or use information to justify maximisation if required. Examiners expect explicit differentiation steps and final estimator expressed in closed form.
• Q3 ~ Sampling distributions & CLT: State whether the sampling distribution is exact or approximate. For CLT approximations, show standardisation (Z = (X − μ)/σ) and justify sample-size adequacy. If exact small-sample distributions apply (t, χ²), state df and formula. Examiners award marks for correct choice and succinct justification of the approximation method.
• Q4 ~ Likelihood-ratio / test construction & χ² approximations: If asked for an LRT, form Λ, compute −2 ln Λ and relate it to χ² with degrees of freedom. Show numeric substitution and interpret p-values in context. Examiners look for both derivation and a clear contextual conclusion about H0/H1.
• Q5 ~ Bayesian update & conjugacy: When Bayesian methods appear, state the prior, multiply by the likelihood, and show the proportional posterior before normalisation. For conjugate priors, identify posterior family and summarise posterior mean/variance. Examiners reward both algebraic clarity and a succinct interpretation of posterior results.
• Q6 ~ Joint distributions, marginalisation & independence tests: Write the joint pdf/pmf and show marginalisation integrals/sums clearly. For independence, test whether f_{X,Y} = f_X f_Y; show steps. Examiners give marks for notation clarity and correct limits in integrals ~ tidy symbolic work convinces markers to award method marks.
• Q7 ~ Regression inference, SEs & predictions: State the linear model and assumptions, present least-squares estimators, and compute standard errors for coefficients where requested. For predictions, provide prediction intervals with SE of prediction, and caution against unjustified extrapolation. Examiners favour both computation and a short contextual interpretation.
• Q8 ~ Asymptotic methods & Delta method: When transforming estimators, state the Delta Method with derivative evaluated at the parameter and compute approximate variance. Justify asymptotic conditions succinctly (n large, differentiability). Examiners want a short justification together with the numeric approximation.
• Q9 ~ Goodness-of-fit & contingency table χ² tests: Compute expected frequencies, combine low counts where required and calculate χ² with df = (rows−1)(cols−1) adjusted for estimated parameters. State p-value and draw a contextual conclusion. Examiners penalise failing to combine small expected cells ~ mention this step when needed.
• Q10 ~ Report-style answer & assumptions: Finish with a concise report: numeric result, model assumptions, and plain-language conclusion. State limitations (e.g., model fit, small n) and propose a short next step if appropriate. Examiners reward a succinct, professional summary that ties numeric inference back to context.

Exam-proof habit: write model + assumptions at the top, show derivations explicitly, and end with a one-line contextual interpretation.

Quick start: write a 1-line plan for each part, keep exact values (surds, π) until the final line, and always label sketches/diagrams. Method marks are everywhere ~ make your steps obvious and tidy.

• Q1 ~ Graph transformations (4 marks): State the transformations in order and use precise language (e.g. “vertical stretch by 2, then translation (0, −14)”). Show the effect on a generic point (x,y) if helpful. Markers award full credit for three distinct, correctly ordered components ~ avoid vague phrasing like “moves up/down” without direction/size.

These insights were written after studying the official May/June 2025 question paper and mark scheme. Use them to focus your practice and presentation. :contentReference[oaicite:0]{index=0}

• Q2 ~ Intersection of curve and line (4 marks): Substitute the line into the curve to form a polynomial in one variable, simplify to a standard quadratic or cubic, and solve. Show algebraic rearrangement (even if you use a calculator) ~ the MS allows small sign errors in rearrangement but method clarity secures M-marks. Present final coordinate pairs clearly.
• Q3 ~ Binomial/general term (4 marks): Identify the general term before hunting for the coefficient. Determine which binomial term contributes x^7, simplify the combinatorial coefficient then equate to 1280 and solve for p. Show each step ~ candidates who list possible contributing terms clearly usually gain method marks even if arithmetic slips occur.
• Q4 ~ Related rates & stationary condition (5 marks total): (a) Differentiate the curve using the chain rule; substitute dx/dt and x = the given value to find dy/dt. Write each derivative step so the marker can award M1 even if simplification is imperfect. (b) For the minimum point, set dy/dx = 0, substitute x = 1/4 and solve for a ~ show the differentiation and algebra explicitly.
• Q5 ~ Trigonometric + algebraic curve (3 parts): Use the known range of cos (−1 to 1) to get quick greatest/least bounds, then refine with differentiation if required. For the sketch, mark turning points and show one or two key sample values; label the x-scale (periods). When asked for number of solutions on an interval, use the sketch plus a short algebraic check ~ the exam allows a diagram-led answer if supported by reasoning.
• Q6 ~ (Series note from MS) The May/June 2025 mark scheme states a series-specific issue: all candidates were awarded full marks for Question 6 and the mark scheme was not used for marking that item. Be aware of this if you compare practice scripts to the real exam. Regardless, practice full algebraic working for similar question types (trig/identities or algebraic manipulation) so reasoning is explicit. :contentReference[oaicite:1]{index=1}
• Q7 ~ Solving trig equations & periodicity: When solving equations that reduce to tan/ sin/ cos, show the principal solution then add the general periodic family (e.g., θ = α + nπ). Check the given domain explicitly to avoid extra solutions; when using calculators, convert to exact radian/degree forms where required. A CAST sketch or short justification reduces chance of missed roots.
• Q8 ~ Geometry (coordinates, angles, sector area): Label points and show coordinate work for distances/gradients; use trig identities or gradient formulas where appropriate. For area/sector parts, show radius identification, compute sector area and triangle area separately, then subtract to get the segment ~ the MS expects this explicit split. Present numerical work to the specified accuracy; show intermediate values (e.g., radius ≈ 5.385…) so markers can award method credit.
• Q9 ~ Calculus mix (derivative → stationary points → integration constant): Differentiate carefully, substitute values to find constants (e.g., set dy/dx = 0 at given x). When integrating to find f(x), show the constant of integration and use any given point to find it. Examiners give follow-through marks (FT) for correctly applied earlier work ~ keep references to earlier parts explicit.
• Q10 ~ Sequences & series (AP / GP parts): For arithmetic sequences form equations from the given terms (use k to get a and d), then apply the sum formula S_n = n/2(2a+(n−1)d). For GP parts, write r and solve using consecutive terms (ar, ar^2). Show the algebra that links k → a,d or k → r; markers reward the correct chain of equations even if final arithmetic has rounding.

Exam-proof habit: box exact final answers, write units, and add a one-line interpretation for geometry/statistics (e.g., “this root means the curve crosses the axis at x = …”). Always include a short check where possible (differentiate to verify, substitute back).

Before you start: state domains/restrictions and give a brief one-line plan for each part. Show substitutions and keep exact forms until the final line ~ method marks follow clear set-up.

• Q1 ~ Log manipulation & exactness: Start by stating domains (arguments > 0). Combine logs using laws stepwise (product/quotient/power), then convert to an exponential to isolate x. Examiners award separate marks for domain-check, algebraic manipulation and final substitution ~ show each so partial credit is visible even if final arithmetic slips occur.

Aligned to the official S25 QP and MS. Use these insights to score method & communication marks. :contentReference[oaicite:1]{index=1}

• Q2 ~ Linear & fractional inequalities + sketch: When solving rational inequalities, bring to one side and factor numerator/denominator. Build a sign table (critical values) or sketch the two curves as required ~ this avoids sign slips. State any excluded values (where denominator = 0) and give final solution in clear interval notation.
• Q3 ~ Stationary points for rational polynomials: Differentiate carefully (quotient rule where needed), simplify numerator and set equal to zero. Solve the resulting polynomial (show factor steps). Use second derivative or sign chart to classify turning points and give coordinates in exact form before rounding ~ examiners reward clear differentiation steps and classification work.
• Q4 ~ Intersection & area between exp/linear curves: Find intersection x by solving the transcendental equation (show rearrangement). For area, integrate (upper − lower) using correct limits; where substitution is used, change limits immediately. Present exact antiderivatives and evaluate symbolically before rounding ~ do not mix degrees/radians.
• Q5 ~ Iterative root-finding (convergence): Rearrange to an iteration x_{n+1} = g(x_n) and state the convergence condition |g'(x)| < 1 near the root briefly. Show each iteration to the required precision, and state your stopping rule (e.g., stability to 4 s.f.). Examiners credit correct iteration formula and intermediate values even if final rounding slightly differs.
• Q6 ~ Trig identities → exact simplification: Start with the identity you will use (double-angle, sum-to-product). Apply transformations line-by-line and avoid dividing by expressions that may vanish. When requested to show equivalence, move from LHS→RHS or RHS→LHS with explicit steps so method marks are clear.
• Q7 ~ Differential equations & initial conditions: Identify the DE class (separable/linear) first. Solve generally, show integration constants, then apply initial condition to find constants. Examiners separate marks for derivation of general solution and correct use of conditions; keep constants labelled C1, C2 to avoid confusion.
• Q8 ~ Parametric arc/gradient problems: Use dy/dx = (dy/dt)/(dx/dt) explicitly, evaluate at the given parameter and present tangents in Cartesian form. If arc length appears, write the integral √( (dx/dt)^2 + (dy/dt)^2 ) and show the substitution; examiners reward neat parametric manipulation and clear labelling of parameter values.
• Q9 ~ Partial fractions then integrate: Check degree and perform polynomial division if needed. Decompose fully (distinct/repeated linear factors accounted for), solve constants either by substitution or equating coefficients, then integrate term-by-term. Show the decomposition step in full ~ MS gives method marks for this even if constants arithmetic is imperfect.
• Q10 ~ Presentation: exact answers and concise checks: Box exact final answers (surds/π). For multi-part questions, reference earlier results explicitly (e.g., “using result from Q3…”). Finish with a short check where practical (substitute back or quick sign/dimensional check) ~ markers reward disciplined presentation.

Exam-proof habit: write a one-line plan for each multi-step part, label intermediate constants, and keep surds/π exact until the last step.

Before you start: choose the correct advanced method (parametric, complex, series test) and write that choice as a one-line plan. Keep algebra tidy ~ examiners follow structured derivations more generously.

• Q1 ~ Parametric/polar differentiation & tangents: Write dy/dx = (dy/dt)/(dx/dt) or the polar derivative as required, then substitute the parameter value. Evaluate carefully where dx/dt = 0 (vertical tangents). Present tangent equations in Cartesian form and box them ~ examiners like neat final forms and checks for singular parameter values.

These insights were aligned with the official S25 QP and MS. Use them for exam-ready practice. :contentReference[oaicite:2]{index=2}

• Q2 ~ Improper integrals & convergence: If limits are infinite or integrand singular, split integral into limits and show convergence steps. Examiners award method marks for explicit limit statements (e.g., lim_{R→∞}). State substitution early for definite integrals and change limits immediately to avoid common slips.
• Q3 ~ Curvature & arc-length set-up: Start by writing curvature/arc-length formula in the correct parametric form, substitute derivatives and simplify integrand before integrating. If algebra simplifies to a standard form, point that out (so markers know you recognised the pattern). A small labelled sketch helps with geometric interpretation marks.
• Q4 ~ Series convergence & radius (power series): Choose ratio/root test for radius of convergence and remember to test endpoints separately. Examiners expect the limit calculations and endpoint checks; don’t skip showing the limit of |a_{n+1}/a_n| or nth-root steps. State the result in interval notation with clear endpoint statements.
• Q5 ~ Implicit differentiation & higher derivatives: Differentiate term-by-term, collect dy/dx terms together and solve symbolically. For second derivatives, differentiate the dy/dx expression carefully and simplify before substituting numeric values. Examiners award marks for the shown chain of algebra even if simplification is messy.
• Q6 ~ Function inversion & verification: When finding inverses, swap x and y and solve; include a short verification f(f^{-1}(x)) = x or comment on domain restriction when necessary. Markers reward this verification step especially when algebra is messy ~ it demonstrates the candidate checked their result.
• Q7 ~ Advanced trig identities & exact simplifications: State upfront which identity you’ll use (Weierstrass, sum-to-product, etc.) and apply stepwise. Avoid dividing by zero issues without stating domain restrictions. Present exact results (in surd or π forms) before any decimalization to secure full marks.
• Q8 ~ Complex roots & de Moivre’s theorem: Convert to polar, apply de Moivre for roots/powers and list all distinct roots. Express answers in exact trigonometric or exponential form. When an Argand representation is helpful, sketch it with labelled moduli and arguments to secure geometric interpretation marks.
• Q9 ~ Reduction to standard integrals & substitution recognition: Match integrals to known forms (arinv, log, arctan etc.) by substitution and state the matching explicitly. For definite integrals, change limits immediately after substitution. Examiners reward correct identification of the matching standard form even if later algebra slips occur.
• Q10 ~ Polished final answers & short checks: Always box exact final answers and include a brief verification ~ differentiate to check, substitute back, or check domain. This small habit often wins the last mark when the main working is right but arithmetic slipped in one line.

Exam-proof habit: write a one-line method choice (substitution / de Moivre / ratio test) at the top ~ it signals correct approach to the marker immediately.

Before you start: draw a labelled free-body diagram (FBD) and declare the positive direction. List constants (e.g., g = 9.8 or 10 as specified) and state any modeling assumptions (massless string, smooth pulley, etc.).

• Q1 ~ Kinematics & SUVAT planning: State which SUVAT equation you will use and why. Use consistent sign convention (positive along motion). For multi-stage motion, label each segment and compute times/displacements then sum; present units and final answer to the required s.f. ~ examiners reward a clear staged approach.

Inspector-aligned insights based on the official S25 QP and MS. Follow the plan to secure method marks. :contentReference[oaicite:3]{index=3}

• Q2 ~ Projectile decomposition & link variable: Resolve initial velocity into horizontal/vertical components; use time as the linking variable. For maximum height use v_y = 0; for range use horizontal speed × time. Write equations used before substituting numbers so method marks are visible.
• Q3 ~ Forces on inclines & friction limits: Draw FBD, resolve weight into components, write ΣF = ma along slope and ΣF⊥ = 0. For impending motion use F = μR appropriately and clearly state whether friction acts up or down the plane. Examiners give marks for correct resolution even if algebra is messy.
• Q4 ~ Energy & work (non-conservative forces): Use ΔKE = work done by net force or conservation with work terms explicit. For variable forces integrate F ds if required; show limits and units. Short justification for chosen method (energy vs Newtonian) is rewarded by examiners.
• Q5 ~ Collisions (coeff. of restitution & momentum): Define the system and apply momentum conservation for the system during collision. Use e to relate relative velocities; label pre/post velocities distinctly. For sequential collisions, keep a tidy table of velocities between stages to avoid confusion ~ markers look for this bookkeeping.
• Q6 ~ Circular motion & centripetal balance: State centripetal acceleration (v^2/r) and resolve forces radially. For non-uniform circular motion include tangential acceleration terms. Examiners credit correct radial/tangential decomposition and sign consistency.
• Q7 ~ Connected particles & pulley constraints: Write the constraint equation linking displacements/velocities/accelerations, label tensions and solve ΣF = ma per mass. State initial conditions where needed and check sign consistency; this explicit constraint step often secures method marks when solving linear systems.
• Q8 ~ Modelling with differential equations: Derive DE from F = ma, identify its type and solve with constants of integration. Apply initial/boundary conditions after general solution to get particular solution. Examiners award marks for the correct derivation even if algebra is long.
• Q9 ~ Moments & equilibrium strategy: Choose a pivot to remove unknown reactions and set ΣM = 0. Show perpendicular distances used for moments and write equilibrium equations clearly. Unit-check and state the assumed sense for clockwise vs anticlockwise.
• Q10 ~ Final checks & answer presentation: Box final numerical answers with units, add a short sanity check (order of magnitude, sign), and reference earlier part values if reused. A one-line justification for sign choices avoids simple penalties in the MS.

Exam-proof habit: always label FBD forces with algebraic expressions (T, R, μR etc.) rather than just arrows ~ this shows your working is ready for marking.

Before you start: define the random variable(s) clearly and state model assumptions (independence, identical trials, rate constancy). Write pmf/pdf form and parameters up-front to avoid model misuse.

• Q1 ~ Discrete distribution setup & pmf table: Build a small probability table showing each outcome and label the pmf. Examiners reward this explicit structure; compute E(X) and Var(X) via summation and show intermediate totals to avoid arithmetic slip penalties.

These notes are aligned with the S25 QP and MS ~ use them to pick the right model and show working that wins method marks. :contentReference[oaicite:4]{index=4}

• Q2 ~ Binomial conditions & p-value logic: State why binomial model applies (fixed n, p constant, independent). For hypothesis tests compute test statistic then p-value or compare critical value. Conclude in context ~ clear interpretation is scored alongside the calculation.
• Q3 ~ Normal approximation & continuity correction: If using normal approx., compute μ = np and σ = √(np(1−p)), apply continuity correction (k±0.5) and standardise. Examiners penalise omitted continuity corrections ~ state it and show z computations stepwise.
• Q4 ~ Expectation from conditional distributions: Use law of total expectation or condition on cases explicitly; write E(X) = Σ x P(X=x) or E(E(X|Y)). Examiners award method marks for clear conditional decomposition even if arithmetics are heavy.
• Q5 ~ Confidence intervals & interpretation: Present CI formula, compute numeric interval and interpret concisely (avoid saying “there is a 95% probability the parameter is in the interval” ~ instead use the standard phrase about confidence level). Examiners score correct calculation and correct interpretation.
• Q6 ~ Poisson modelling & unit consistency: State λ per unit time and convert units early (per hour/day). For sums of independent Poisson variables add λs; for waiting-time questions, link to exponential where needed and invoke the memoryless property if relevant.
• Q7 ~ Correlation & regression basics: Compute covariance and Pearson’s r with intermediate sums, state regression line and interpret slope in context. Examiners expect a short comment on fit or outliers where requested ~ do that briefly to gain communication marks.
• Q8 ~ Generating functions & moment extraction: Define mgf/pgf, differentiate and evaluate at required point to extract moments. Examiners reward explicit functional differentiation rather than shorthand claims ~ show the steps for moment extraction clearly.
• Q9 ~ Sampling distributions & CLT justification: State whether exact or approximate. If using CLT cite sample size reasoning and show standardisation to Z. Examiners value a one-line justification for approximations ~ include it when appropriate.
• Q10 ~ Interpretation & limitations: When asked to interpret results, give a short contextual sentence and state key assumptions/sampling limitations. Examiners award marks for succinct interpretation tied to the numbers, not just restating them.

Exam-proof habit: write the model (Binomial/Poisson/Normal) and its parameters at the top of the solution before any algebra ~ this prevents model misuse and gains method marks.

Before you start: write the model (pmf/pdf) and parameter notation clearly, and indicate whether results will be exact or approximated. Advanced questions require a one-line justification for any asymptotic or approximation method used.

• Q1 ~ Distribution identification & parameter definition: Begin by writing the pmf/pdf with parameter names. If asked to show properties (mean/variance), derive them stepwise ~ show summation/integration and present the final simplified exact form to secure method marks.

These examiner insights are aligned with the S25 QP and MS ~ consult the MS for exact marking points where needed. :contentReference[oaicite:5]{index=5}

• Q2 ~ Maximum likelihood & log-likelihood steps: Write likelihood L(θ), take ln L to form log-likelihood, differentiate and solve for θ̂. Show second derivative or Fisher information briefly if asked to justify a maximum. Examiners award marks for the differentiation chain and final closed-form estimator.
• Q3 ~ Sampling distributions & CLT use: State whether the sampling distribution is exact or approx. For CLT approximations standardise and justify n sufficiently large. If exact small-sample tests apply, state df and exact distribution (t, χ²) and use tables appropriately in conclusion.
• Q4 ~ Likelihood-ratio & hypothesis construction: Form Λ, compute −2 ln Λ and state its χ² approximation with df. Show numeric substitution and interpret p-value in context ~ clear phrasing of H0/H1 conclusions is worth marks in addition to numeric correctness.
• Q5 ~ Bayesian posterior with conjugate priors: Multiply prior × likelihood, identify the posterior family, and present posterior summaries (mean/variance/credible interval). Examiners award marks for algebraic clarity and succinct interpretative commentary (e.g., posterior mean compared to prior).
• Q6 ~ Joint distributions, marginalisation & conditioning: Write joint f_{X,Y}, integrate/sum to get marginals, and present conditional densities f_{Y|X}. Examiners expect correct limits of integration and tidy notation ~ show steps rather than shorthand claims.
• Q7 ~ Regression inference & prediction intervals: Present regression model, compute least-squares estimates, supply SEs and form prediction intervals for new observations. Comment briefly on assumptions (linearity, homoscedasticity) ~ examiners value this short diagnostic note.
• Q8 ~ Asymptotics & Delta method: For transformations of estimators, state the Delta Method, compute the derivative at the parameter and get approximate variance. Include a short comment on the approximation validity (n large) to show exam-level awareness.
• Q9 ~ Goodness-of-fit χ² & combining low cells: Compute expected counts, combine low-frequency cells as necessary and compute χ² with df adjusted for estimated parameters. State p-value and conclude in context; mention limitation if sample size or expected counts are low.
• Q10 ~ Communicative wrap-up & limitations: End with a concise interpretation linking numeric results to context, state assumptions used, and suggest a short next step (e.g., larger sample, check residuals). Examiners reward concise, contextual final sentences as much as a correct numeric answer.

Exam-proof habit: write model + parameter names at the top of each solution and finish with a one-line contextual interpretation to show you know what the numbers mean.