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This lesson teaches you a systematic framework for tackling the long, multi-part questions that appear at the end of each Edexcel A-Level Mathematics paper. These questions typically carry 10-16 marks and combine multiple topics. Students who approach them without a strategy often lose marks unnecessarily. A structured approach maximises your score even when the question is difficult.
Use this four-step process for every multi-step question.
Read the entire question — all parts — before writing anything.
Why this matters:
What to look for:
Before writing any mathematics, plan your approach.
For each part, identify:
Example planning: Question: A curve has equation y = x²e^(-x). (a) Find dy/dx. (b) Find the coordinates of the stationary points. (c) Determine the nature of each stationary point. (d) Sketch the curve.
Plan:
Work through each part, showing full working. Apply these principles:
| Principle | How to Apply |
|---|---|
| Use the result from the previous part | If part (b) says "hence", you must use your answer from part (a) |
| Show every step | Especially in "show that" parts — do not skip anything |
| Label your work | Write "Part (a)", "Part (b)" clearly |
| Keep it organised | Start each part on a new line; do not mix parts together |
| State your intermediate results | "So the gradient is 3" or "Therefore x = 2 or x = -1" |
After completing the question, quickly verify:
Multi-step questions combine topics. Recognising common combinations helps you plan your approach.
| Combination | What It Looks Like |
|---|---|
| Differentiation + coordinate geometry | Find the gradient of a curve, then the equation of the tangent/normal, then where it meets a line |
| Integration + area | Find an integral, use it to calculate the area between curves or between a curve and the x-axis |
| Trigonometry + calculus | Use a trig identity to rewrite an expression, then integrate or differentiate it |
| Sequences + proof | Use the formula for a geometric series, then prove a result about the sum |
| Parametric equations + calculus | Convert between parametric and Cartesian, differentiate, find tangent/normal, find area |
| Exponential models + logarithms | Set up an exponential model, use logarithms to solve for unknowns, interpret in context |
| Differential equations + integration | Set up a differential equation from a word problem, separate variables, integrate both sides |
| Mechanics + calculus | Use variable acceleration with v = ds/dt and a = dv/dt, integrate to find displacement |
Many multi-step questions use a chain of "hence" instructions where each part builds on the previous one.
Example structure:
Strategy:
This is a critical situation. Part (a) of a multi-step question often provides a result needed for the rest of the question.
What to do:
| Situation | Action |
|---|---|
| Part (a) is "show that" | Attempt it. If you cannot complete it, use the given result for subsequent parts — you lose the marks for part (a) but can still earn all marks for parts (b) onwards. |
| Part (a) asks you to find something | Attempt it. If you get stuck, look at part (b) for clues about what the answer should be. |
| You have no idea how to start | Skip the entire question, do other questions, and come back with fresh eyes. |
Key Point: Never leave parts (b), (c), and (d) blank just because you could not do part (a). If part (a) says "show that f(x) = 2x³ - x + 1", then use f(x) = 2x³ - x + 1 in the subsequent parts even if you could not prove it.
Mechanics multi-step problems often involve connected particles, multiple stages of motion, or forces at angles.
Some questions describe motion that changes (e.g., constant acceleration for 5 seconds, then constant velocity, then deceleration).
Statistics multi-step problems often follow a pattern: model setup, calculation, hypothesis test, interpretation.
Each step typically earns marks, so even partial completion is worthwhile.
A curve C has equation y = 2x³ - 9x² + 12x - 4.
(a) Find dy/dx. [2 marks] (b) Find the coordinates of the stationary points of C. [4 marks] (c) Determine the nature of each stationary point. [3 marks] (d) Show that C crosses the x-axis at x = 2. [1 mark] (e) Sketch C, showing the coordinates of the stationary points and any axis intercepts. [4 marks]
Planning this question:
Total time budget: approximately 17 minutes (14 marks × 1.2 minutes/mark).
| Step | Time | Action |
|---|---|---|
| READ | 1-2 min | Read all parts. Identify topics, key words, links between parts. |
| PLAN | 1 min | Identify the technique for each part. Note which results carry forward. |
| EXECUTE | Main time | Work through each part with full working. Use previous results when told to. |
| CHECK | 1-2 min | Verify every part is answered. Check forms, units, context. |
| Situation | What to Do |
|---|---|
| Stuck on part (a) "show that" | Use the given result for later parts |
| Stuck on part (a) "find" | Look at later parts for clues, then move on |
| Running out of time | Prioritise first parts of remaining questions (easiest marks) |
| "Hence" instruction | Must use previous result |
| "Hence or otherwise" | Encouraged to use previous result, but alternatives accepted |
Multi-step questions are where the most marks are won and lost. A systematic approach ensures you extract maximum marks even from questions you find challenging.
Long synoptic questions on Edexcel 9MA0 papers — the 8, 10, and 12-mark items that dominate the back end of Papers 1, 2, and 3 — are not designed to test whether you can recall a technique. They are designed to test whether you can plan a route through unfamiliar terrain when several techniques have to interlock. This deeper-strategy section gives you a planning framework, a worked specimen, and a vocabulary for the failure modes that cost candidates marks. It is intentionally process-focused rather than topic-focused, because the topics themselves are covered elsewhere in the course; what is rarely taught explicitly is the meta-skill of orchestrating those topics under pressure.
George Pólya's How to Solve It (1945) remains the most influential framework for mathematical problem-solving. His four phases — understand the problem, devise a plan, carry out the plan, look back — map cleanly onto the demands of an A-Level synoptic question. The adaptation matters because A-Level differs from competition or research mathematics in three respects: the time budget is tight (roughly 1.5 minutes per mark), the toolkit is bounded by the specification, and the mark scheme rewards method as well as final answers. Pólya's framework therefore needs to be tightened rather than relaxed.
Phase 1 — Understand. Read the entire question, including all sub-parts, before writing anything. Identify what is given, what is asked, and which sub-parts feed forward into later sub-parts (the words hence and use the result of part (a) are explicit signals). Underline or circle quantities on the question paper. Note any constraints — "for x>0", "where k is a positive integer", "in the interval 0≤θ<2π". Constraints almost always govern which branch of a square root or which solutions of a trig equation are valid.
Phase 2 — Plan. Before committing to algebra, sketch the route in the margin. For a 12-mark question this might be three or four bullet points: "differentiate implicitly; set dxdy=0; solve for x; substitute back". Planning costs 30-60 seconds and saves multiples of that downstream. The plan also gives you a recovery anchor if you get stuck: you can return to it and ask which step is failing.
Phase 3 — Execute. Work the plan one step at a time, writing each line clearly enough that an examiner can follow your reasoning without your supervision. The 9MA0 mark scheme awards method marks (M), accuracy marks (A), and occasionally B marks for stand-alone results. Skipping algebra to save time is false economy — the M marks are the densest part of the question's value-per-line.
Phase 4 — Review. With perhaps 30 seconds spare per long question, look back. Does the answer have the right units? The right sign? Does it satisfy the original constraint? If the question asked for an exact value and you have produced a decimal, you have lost the A mark. The review phase catches more marks than candidates expect.
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