AS Maths (Edexcel) exam technique
How the exams work
Pearson Edexcel AS level Mathematics (8MA0): two written papers at the end of the course, on the first year of A level Mathematics. Paper 1 is Pure Mathematics (2 hours, 100 marks). Paper 2 is Statistics and Mechanics (1 hour 15 minutes, 60 marks), with a Statistics section and a Mechanics section of 30 marks each. Both allow a calculator and the formulae booklet (with statistical tables). AS grades run from A to E; there is no A*.
| Paper | Time | Marks | Calculator | What’s on it |
|---|---|---|---|---|
| Paper 1: Pure Mathematics | 2 h | 100 marks | Allowed | Any Pure topic (1 to 8 and 10). |
| Paper 2: Statistics and Mechanics | 1 h 15 min | 60 marks | Allowed | Section A: Statistics (30 marks). Section B: Mechanics (30 marks). |
Command words
| Word | What it means | How to answer | Example | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Find | Work out a value, an expression, a set of values or a point. Any correct method is allowed unless the question says otherwise. | Show the key steps so method marks can be given, then give the answer in the form asked for (coordinates, exact value, equation). | Find the coordinates of the turning point of \(y = 2x^2 - 12x + 7\). \(\frac{dy}{dx} = 4x - 12 = 0\), so \(x = 3\) and \(y = -11\): the turning point is \((3, -11)\). | ||||||||||||
| Calculate | Work out a numerical answer, usually with a calculator. | Write the formula or expression with the numbers substituted, then the answer to a suitable accuracy with units. | Calculate the area of a sector of radius 6 cm and angle 1.2 radians. \(\frac{1}{2} \times 6^2 \times 1.2 = 21.6\) cm2. | ||||||||||||
| Determine | Find something and justify it. Often used when a decision is needed: 'determine whether' or 'determine the nature of'. | Do the working, then state your conclusion clearly with the reason that supports it. | Determine the nature of the stationary point of \(y = x^3 - 6x^2 + 9x\) at \(x = 3\). \(\frac{d^2y}{dx^2} = 6x - 12 = 6 > 0\) when \(x = 3\), so it is a minimum. | ||||||||||||
| Solve | Find every value that satisfies the equation or inequality, within any interval given. | Show the algebra, adjust the interval if the variable is transformed, and list all solutions in the right units. | Solve \(\tan 2x = 1\) for \(0 \le x < \pi\). For \(0 \le 2x < 2\pi\), \(2x = \frac{\pi}{4}\) or \(\frac{5\pi}{4}\), so \(x = \frac{\pi}{8}\) or \(\frac{5\pi}{8}\). | ||||||||||||
| Show that | The result is given; you must show clearly and logically how it follows. The marks are for the working. | Start from what you are given, show every step, and end on exactly the printed result. You may use the result in later parts even if you couldn't show it. | Show that \(2\cos^2 x + 3\sin x = 3\) can be written as \(2\sin^2 x - 3\sin x + 1 = 0\). Replace \(\cos^2 x\) with \(1 - \sin^2 x\): \(2 - 2\sin^2 x + 3\sin x = 3\), so \(2\sin^2 x - 3\sin x + 1 = 0\), as required. | ||||||||||||
| Prove | Give a complete, rigorous argument that the statement is always true (or, with 'disprove', that it is not). | Use general algebra or cover every case, make each step follow from the last, and finish with a sentence stating what has been proved. | Prove that the sum of any three consecutive integers is a multiple of 3. \(n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)\), which is a multiple of 3 for every integer \(n\). | ||||||||||||
| Hence | You must use the result you have just found. Another method will usually score no marks. | Quote or use the previous answer directly and build on it. | (a) Given that \((x + 1)\) is a factor, factorise \(x^3 - 7x - 6\) fully: \((x + 1)(x - 3)(x + 2)\). (b) Hence solve \(x^3 - 7x - 6 = 0\): \(x = -1\), 3 or \(-2\). | ||||||||||||
| Hence or otherwise | Using the previous result is expected to be the quickest route, but any correct method earns full marks. | Use the earlier part if you can; if you choose another method, show it fully. | Given that \(\cos 2x = 1 - 2\sin^2 x\), hence or otherwise find \(\int \sin^2 x \, dx\). \(\int \frac{1}{2}(1 - \cos 2x) \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + c\). | ||||||||||||
| Write down / State | Give the answer or fact without working. Usually worth 1 mark, and the answer should follow directly from what you know or from earlier work. | Give a short, precise answer. If you find yourself doing a long calculation, you have probably missed something simpler. | Write down the centre and radius of the circle \((x - 2)^2 + (y + 5)^2 = 49\). Centre \((2, -5)\), radius 7. | ||||||||||||
| Sketch | Draw the general shape of a graph with its key features shown. It doesn't need to be to scale or plotted. | Get the shape right and mark the intercepts with coordinates, asymptotes with equations, and turning points when asked. | Sketch \(y = \frac{1}{x - 2}\), showing the asymptotes \(x = 2\) and \(y = 0\) and the point where the curve meets the \(y\)-axis, \(\left(0, -\frac{1}{2}\right)\). | ||||||||||||
| Draw / Complete | Fill in a table, tree diagram or Venn diagram, or draw accurately on a given diagram (a box plot, a staircase diagram). | Fill every gap, use the accuracy asked for, and use a ruler and dark pencil for diagrams. | Complete the table of values of \(y = \sqrt{1 + x^2}\) for the trapezium rule, giving values to 4 decimal places.
| ||||||||||||
| Explain / Give a reason | Say why something is true, in words, linked to the maths or the context. | Make one clear point for each mark and connect it to the question, using 'because' or 'so'. | Explain why the trapezium rule gives an overestimate of \(\int_0^1 e^x \, dx\). The curve is convex (it bends upwards), so the top of each trapezium lies above the curve. | ||||||||||||
| Interpret | Say what a value, parameter or result means in the real-life context. | Refer to the context and the units; don't just repeat the number. | The number of bacteria after \(t\) hours is modelled by \(N = 200e^{0.03t}\). Interpret the value 200. It is the number of bacteria at the start, when \(t = 0\). | ||||||||||||
| Evaluate | Either find the value of an expression (such as an integral or a sum), or, for a model, judge how well it fits the situation. | For a value, show the method and give the number. For a model, compare its predictions with the facts and give specific strengths or limitations. | Evaluate \(\sum_{r=1}^{20} (3r + 2)\). An arithmetic series with first term 5 and last term 62: \(\frac{20}{2}(5 + 62) = 670\). | ||||||||||||
| Estimate | Give an approximate value using a model, a graph, a regression line or an approximation method. | Show which method you used and comment on reliability if asked (interpolation or extrapolation, overestimate or underestimate). | Use the regression line \(y = 4.2 + 0.85x\) to estimate \(y\) when \(x = 30\). \(4.2 + 0.85 \times 30 = 29.7\). If 30 lies within the range of the data, this is interpolation, so it is fairly reliable. | ||||||||||||
| Express / Write in the form | Rearrange into exactly the form given and find the constants. | Match your working to the given form and state each constant, meeting any conditions (such as \(R > 0\)). | Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), \(R > 0\), \(0 < \alpha < 90^\circ\). \(R\cos\alpha = 3\), \(R\sin\alpha = 4\), so \(R = 5\) and \(\tan\alpha = \frac{4}{3}\), giving \(\alpha = 53.1^\circ\). | ||||||||||||
| Deduce | Use a result you have already found to reach a new one with little extra work. | Say which earlier result you are using and show the short step that follows from it. | The curve \(y = f(x)\) has a minimum point at \((3, 2)\). Deduce the minimum point of \(y = f(x + 1) - 4\). Move left 1 and down 4: \((2, -2)\). | ||||||||||||
| Comment on / Describe | Make a relevant observation about data, a result or a model, backed by the numbers. | Say what the evidence shows in context. In statistics, use the correct words (positive or negative correlation, skew, location, spread). | The product moment correlation coefficient between daily hours of sunshine and daily maximum temperature is 0.62. Comment on this. There is positive correlation: days with more sunshine tended to be warmer. | ||||||||||||
| Test | Carry out a full hypothesis test at the stated significance level. | Write the hypotheses in terms of the parameter, state the distribution under \(H_0\), find the probability or critical region, compare with the significance level, and conclude in context. | A spinner should land on red with probability 0.5. In 20 spins it lands on red 15 times. Test, at the 5% level of significance, whether the spinner is biased towards red. \(H_0: p = 0.5\), \(H_1: p > 0.5\), \(X \sim B(20, 0.5)\), \(P(X \ge 15) = 0.0207 < 0.05\), so reject \(H_0\): there is evidence that the spinner is biased towards red. | ||||||||||||
| Use / Using (algebra, calculus, the substitution ...) | The question names the method you must use. Answers found another way, such as by calculator, trial and improvement or reading a graph, can score nothing. | Show the named method clearly, step by step, even if you check the answer another way. | Use algebra to solve \(\frac{3}{x} = x - 2\). \(x^2 - 2x - 3 = 0\), so \((x - 3)(x + 1) = 0\), giving \(x = 3\) or \(x = -1\). |
Find
Work out a value, an expression, a set of values or a point. Any correct method is allowed unless the question says otherwise.
How to answer. Show the key steps so method marks can be given, then give the answer in the form asked for (coordinates, exact value, equation).
Example. Find the coordinates of the turning point of \(y = 2x^2 - 12x + 7\). \(\frac{dy}{dx} = 4x - 12 = 0\), so \(x = 3\) and \(y = -11\): the turning point is \((3, -11)\).
Calculate
Work out a numerical answer, usually with a calculator.
How to answer. Write the formula or expression with the numbers substituted, then the answer to a suitable accuracy with units.
Example. Calculate the area of a sector of radius 6 cm and angle 1.2 radians. \(\frac{1}{2} \times 6^2 \times 1.2 = 21.6\) cm2.
Determine
Find something and justify it. Often used when a decision is needed: 'determine whether' or 'determine the nature of'.
How to answer. Do the working, then state your conclusion clearly with the reason that supports it.
Example. Determine the nature of the stationary point of \(y = x^3 - 6x^2 + 9x\) at \(x = 3\). \(\frac{d^2y}{dx^2} = 6x - 12 = 6 > 0\) when \(x = 3\), so it is a minimum.
Solve
Find every value that satisfies the equation or inequality, within any interval given.
How to answer. Show the algebra, adjust the interval if the variable is transformed, and list all solutions in the right units.
Example. Solve \(\tan 2x = 1\) for \(0 \le x < \pi\). For \(0 \le 2x < 2\pi\), \(2x = \frac{\pi}{4}\) or \(\frac{5\pi}{4}\), so \(x = \frac{\pi}{8}\) or \(\frac{5\pi}{8}\).
Show that
The result is given; you must show clearly and logically how it follows. The marks are for the working.
How to answer. Start from what you are given, show every step, and end on exactly the printed result. You may use the result in later parts even if you couldn't show it.
Example. Show that \(2\cos^2 x + 3\sin x = 3\) can be written as \(2\sin^2 x - 3\sin x + 1 = 0\). Replace \(\cos^2 x\) with \(1 - \sin^2 x\): \(2 - 2\sin^2 x + 3\sin x = 3\), so \(2\sin^2 x - 3\sin x + 1 = 0\), as required.
Prove
Give a complete, rigorous argument that the statement is always true (or, with 'disprove', that it is not).
How to answer. Use general algebra or cover every case, make each step follow from the last, and finish with a sentence stating what has been proved.
Example. Prove that the sum of any three consecutive integers is a multiple of 3. \(n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)\), which is a multiple of 3 for every integer \(n\).
Hence
You must use the result you have just found. Another method will usually score no marks.
How to answer. Quote or use the previous answer directly and build on it.
Example. (a) Given that \((x + 1)\) is a factor, factorise \(x^3 - 7x - 6\) fully: \((x + 1)(x - 3)(x + 2)\). (b) Hence solve \(x^3 - 7x - 6 = 0\): \(x = -1\), 3 or \(-2\).
Hence or otherwise
Using the previous result is expected to be the quickest route, but any correct method earns full marks.
How to answer. Use the earlier part if you can; if you choose another method, show it fully.
Example. Given that \(\cos 2x = 1 - 2\sin^2 x\), hence or otherwise find \(\int \sin^2 x \, dx\). \(\int \frac{1}{2}(1 - \cos 2x) \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + c\).
Write down / State
Give the answer or fact without working. Usually worth 1 mark, and the answer should follow directly from what you know or from earlier work.
How to answer. Give a short, precise answer. If you find yourself doing a long calculation, you have probably missed something simpler.
Example. Write down the centre and radius of the circle \((x - 2)^2 + (y + 5)^2 = 49\). Centre \((2, -5)\), radius 7.
Sketch
Draw the general shape of a graph with its key features shown. It doesn't need to be to scale or plotted.
How to answer. Get the shape right and mark the intercepts with coordinates, asymptotes with equations, and turning points when asked.
Example. Sketch \(y = \frac{1}{x - 2}\), showing the asymptotes \(x = 2\) and \(y = 0\) and the point where the curve meets the \(y\)-axis, \(\left(0, -\frac{1}{2}\right)\).
Draw / Complete
Fill in a table, tree diagram or Venn diagram, or draw accurately on a given diagram (a box plot, a staircase diagram).
How to answer. Fill every gap, use the accuracy asked for, and use a ruler and dark pencil for diagrams.
Example. Complete the table of values of \(y = \sqrt{1 + x^2}\) for the trapezium rule, giving values to 4 decimal places.
At \(x = 1\), \(y = \sqrt{2} = 1.4142\); at \(x = 2\), \(y = \sqrt{5} = 2.2361\).
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| \(y\) | 1 | 1.1180 | 1.8028 |
Explain / Give a reason
Say why something is true, in words, linked to the maths or the context.
How to answer. Make one clear point for each mark and connect it to the question, using 'because' or 'so'.
Example. Explain why the trapezium rule gives an overestimate of \(\int_0^1 e^x \, dx\). The curve is convex (it bends upwards), so the top of each trapezium lies above the curve.
Interpret
Say what a value, parameter or result means in the real-life context.
How to answer. Refer to the context and the units; don't just repeat the number.
Example. The number of bacteria after \(t\) hours is modelled by \(N = 200e^{0.03t}\). Interpret the value 200. It is the number of bacteria at the start, when \(t = 0\).
Evaluate
Either find the value of an expression (such as an integral or a sum), or, for a model, judge how well it fits the situation.
How to answer. For a value, show the method and give the number. For a model, compare its predictions with the facts and give specific strengths or limitations.
Example. Evaluate \(\sum_{r=1}^{20} (3r + 2)\). An arithmetic series with first term 5 and last term 62: \(\frac{20}{2}(5 + 62) = 670\).
Estimate
Give an approximate value using a model, a graph, a regression line or an approximation method.
How to answer. Show which method you used and comment on reliability if asked (interpolation or extrapolation, overestimate or underestimate).
Example. Use the regression line \(y = 4.2 + 0.85x\) to estimate \(y\) when \(x = 30\). \(4.2 + 0.85 \times 30 = 29.7\). If 30 lies within the range of the data, this is interpolation, so it is fairly reliable.
Express / Write in the form
Rearrange into exactly the form given and find the constants.
How to answer. Match your working to the given form and state each constant, meeting any conditions (such as \(R > 0\)).
Example. Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), \(R > 0\), \(0 < \alpha < 90^\circ\). \(R\cos\alpha = 3\), \(R\sin\alpha = 4\), so \(R = 5\) and \(\tan\alpha = \frac{4}{3}\), giving \(\alpha = 53.1^\circ\).
Deduce
Use a result you have already found to reach a new one with little extra work.
How to answer. Say which earlier result you are using and show the short step that follows from it.
Example. The curve \(y = f(x)\) has a minimum point at \((3, 2)\). Deduce the minimum point of \(y = f(x + 1) - 4\). Move left 1 and down 4: \((2, -2)\).
Comment on / Describe
Make a relevant observation about data, a result or a model, backed by the numbers.
How to answer. Say what the evidence shows in context. In statistics, use the correct words (positive or negative correlation, skew, location, spread).
Example. The product moment correlation coefficient between daily hours of sunshine and daily maximum temperature is 0.62. Comment on this. There is positive correlation: days with more sunshine tended to be warmer.
Test
Carry out a full hypothesis test at the stated significance level.
How to answer. Write the hypotheses in terms of the parameter, state the distribution under \(H_0\), find the probability or critical region, compare with the significance level, and conclude in context.
Example. A spinner should land on red with probability 0.5. In 20 spins it lands on red 15 times. Test, at the 5% level of significance, whether the spinner is biased towards red. \(H_0: p = 0.5\), \(H_1: p > 0.5\), \(X \sim B(20, 0.5)\), \(P(X \ge 15) = 0.0207 < 0.05\), so reject \(H_0\): there is evidence that the spinner is biased towards red.
Use / Using (algebra, calculus, the substitution ...)
The question names the method you must use. Answers found another way, such as by calculator, trial and improvement or reading a graph, can score nothing.
How to answer. Show the named method clearly, step by step, even if you check the answer another way.
Example. Use algebra to solve \(\frac{3}{x} = x - 2\). \(x^2 - 2x - 3 = 0\), so \((x - 3)(x + 1) = 0\), giving \(x = 3\) or \(x = -1\).
Exam tips
The weeks before the exams
- Revise the whole AS course. Both papers test all of it, and many long questions start from skills such as quadratics, indices, straight lines and the binomial expansion.
- Move from topic-by-topic practice to timed, mixed practice. In the exam nothing tells you which topic a question is from, and Pure questions often combine two (a tangent with a circle, logs with a model, differentiation with an area).
- Know what the formulae booklet (Mathematical Formulae and Statistical Tables) gives you and where to find it, but don't rely on it: learn by heart the results you use all the time, such as the laws of logarithms, the sine and cosine rules, the area of a triangle \(\dfrac{1}{2}ab\sin C\), the discriminant, the equation of a circle and the suvat equations.
- Get used to the statistical tables in the booklet and the binomial cumulative distribution function on your calculator. Know how to read a cumulative probability and how to find a critical region.
- Study the large data set Pearson publishes for the course. Questions can assume you know its variables, their units, where and when the data was collected and how missing values are recorded, so a student who has used it can answer questions that others can't.
- Keep an error log: for every mark you drop, write the topic, what went wrong and the correct method. Re-read it every week; the same few mistakes usually cost most of the marks.
- Make flashcards for things you must recall instantly (laws of indices and logs, the sine and cosine rules, differentiation and integration of powers of \(x\), the hypothesis-test layout, modelling assumptions) and review them little and often.
- In the last month, sit at least one full paper each week under exam conditions, alternating Paper 1 (2 hours) and Paper 2 (1 hour 15 minutes), and mark it strictly.
The night before and the morning of the exam
- Pack black pens (more than one), an HB or B pencil for sketches and diagrams, a ruler, a rubber, a protractor, and your calculator with fresh or charged batteries.
- Check your calculator is allowed: Pearson does not allow calculators that can do symbolic algebra, symbolic differentiation or integration, or that have formulae or notes stored in them. Clear any stored programs or text.
- Don't use coloured pens or highlighters on the paper. Underline key words in pen or pencil instead.
- Do light revision only: your flashcards, your error log and the layout of the formulae booklet. Don't start a topic you haven't learnt.
- Before Paper 2, run through the binomial hypothesis-test layout, the conditions for a binomial model and the standard modelling assumptions in mechanics.
- Get a proper night's sleep. Tiredness shows up as sign errors and misread questions, and one slip early in a 10-mark question can cost most of it.
- In the morning, eat, arrive early and do two or three quick warm-up questions (a differentiation, a quadratic, a trig equation) to get your brain into gear.
The first five minutes
- Fill in your details on the front, check you have the formulae booklet, and check how many questions there are so you know the paper is complete.
- Look at the angle mode on your calculator: AS trigonometry is in degrees, so make sure it is set to degrees.
- Spend about a minute flicking through the whole paper: see where the long questions are and which topics come up, so nothing surprises you with ten minutes left.
- The marks for each part are shown in brackets. Use them to judge how much working each part needs.
- Start with question 1. The papers usually open with shorter, more routine questions, so bank those marks and build confidence. If a later question looks hard, note it and come back; don't let it unsettle you.
- On Paper 2, write down the time you will switch sections (about 37 minutes in). You can answer Section A or Section B first, as long as each gets its fair share of time.
Timing: about 1.2 minutes a mark
- Paper 1 is 100 marks in 120 minutes, 1.2 minutes per mark: a 5-mark part is worth about 6 minutes and a 10-mark question about 12 minutes. Paper 2 is 60 marks in 75 minutes, about 1.25 minutes per mark.
- Work a little faster than this on routine questions so you save time for the long questions near the end and for checking.
- Check your pace against the clock: roughly a quarter of the marks done every 30 minutes on Paper 1. If you are well behind at the one-hour mark, speed up on routine parts rather than rushing the hard ones.
- On Paper 2, give each section about 37 minutes and stick to it. A statistics section that overruns takes marks away from mechanics, and the other way round.
- If a part has taken its time and you're not close, write down the method you would use, leave a gap and move on. A fresh look at the end often works.
- The later questions are longer and harder, but their first parts are often routine (differentiate, find a coordinate, write down a value). Never leave a whole long question blank.
- Don't over-write: a 1-mark 'write down' needs no working, and a 2-mark 'explain' needs one or two clear points, not a paragraph.
Reading the question
- Read the whole question, all parts, before you start. Later parts often show why an earlier part is there and which method is intended.
- Underline the command word and the form of the answer: 'exact', 'in the form \(a + b\sqrt{3}\)', 'to 3 significant figures', 'where \(a\), \(b\) and \(c\) are integers'. An answer in the wrong form can lose the final mark.
- In trig questions, check the interval: \(0 \le \theta < 360^\circ\) is the usual AS interval, and if the angle is \(2\theta\) or \(\theta + 30^\circ\), change the interval to match before you solve.
- Look for a method instruction: 'use algebra', 'using calculus', 'hence', or 'In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.' Another method may score nothing.
- In context questions, pin down what each variable means and its units (\(t\) in hours or years, \(P\) in thousands) before you use the model.
- Read the figure and its labels: which curve is which, which region is shaded, which coordinates are marked. Don't measure from diagrams; they are not always drawn to scale.
- Watch the small words: 'positive', 'integer', 'non-zero', 'for all', 'at least', 'more than', 'given that', 'the exact value'.
- When you finish a part, re-read the question and check you answered what was asked: coordinates rather than just \(x\), the equation of the tangent rather than its gradient, the probability rather than a table value.
Showing working and how the marks are given
- There is no multiple choice on these papers. Every question is answered in the space in the question paper, and most marks are for the method as well as the answer.
- Edexcel mark schemes give M marks for a correct method, A marks for accurate answers (which depend on the method mark), and B marks for correct results that don't need a method. A dM mark needs the earlier M mark too.
- A correct answer with no working is risky: where a method is asked for it can score nothing, and a wrong answer with no working always scores nothing. A wrong answer after a correct method still earns the M marks.
- Write your method so an examiner can follow it: state the rule, substitute, then simplify. For example, write \(\dfrac{dy}{dx} = 0\) before solving for a stationary point, and \(X \sim B(20, 0.3)\) before a probability.
- Follow-through (ft) marks mean a wrong answer in part (a), used correctly in part (b), can still earn marks in (b). Always carry on.
- If you can't do a 'show that' part, use the printed answer in the next part. You can score every mark in the later part.
- Marks labelled cso (correct solution only) need a fully correct argument with no wrong lines. They often sit at the end of 'show that' and proof questions.
- Answers that round to the correct value are usually accepted, but only if you kept enough accuracy: write at least 4 significant figures in intermediate working and keep the full value in your calculator.
'Show that' and proof questions
- In a 'show that' question the answer is printed, so every mark is for the working. Show each step; don't jump from the start to the given result.
- Start from the expression or the information in the question, not from the answer. End with a line that exactly matches the printed result and a short statement such as 'as required'.
- Proof by deduction: represent general cases with algebra, e.g. \(2n\) for an even number, \(2n + 1\) for an odd number, \(n\) and \(n + 1\) for consecutive integers, then simplify to show the property.
- Proof by exhaustion: split into cases that cover every possibility (e.g. \(n\) even and \(n\) odd, or checking each of a small number of values) and prove each one.
- Disproof by counter-example: give one specific example, show the working that proves it fails, and state that the claim is therefore false.
- Finish every proof with a sentence stating what you have proved. Without it the last mark is often lost.
Modelling and problem-solving questions
- Many questions are set in a context: population growth, cooling, a drug in the bloodstream, a car's journey, the volume of a container. The maths is standard; the skill is turning the words into equations.
- Write the model down and define any variable you introduce. Check the units: hours or minutes, metres or kilometres, thousands or single items.
- In an exponential model such as \(P = Ae^{kt}\) or \(y = ab^x\), \(A\) or \(a\) is the value when the variable is 0, and the sign and size of \(k\) (or whether \(b\) is above or below 1) describe growth or decay. Use logarithms to find a constant or a time.
- Give answers in context and with units, e.g. 'the population first exceeds 5000 in the 13th year', not just \(t = 12.4\)
- Expect to criticise or refine the model. Give a specific limitation linked to the context (e.g. 'the model predicts the fish population grows without limit, but the lake can only support a limited number'), not 'it may not be accurate'.
- Unstructured problems give a goal but no steps. Write down what you know, what you need and what links them (an area needs an integral, a maximum needs a derivative, a length might need a vector's magnitude), then work step by step.
- Sense-check the answer: a negative length, a probability above 1 or a time before the start means something has gone wrong.
Sketches, graphs and diagrams
- A sketch shows the shape and the key features, not plotted points: where the graph meets the axes (with coordinates), turning points if asked for, and asymptotes with their equations.
- Transformations: follow one key point and each asymptote through every step. \(y = f(x + a)\) moves the graph left by \(a\), \(y = af(x)\) stretches it vertically by factor \(a\), \(y = f(ax)\) stretches it horizontally by factor \(\dfrac{1}{a}\) and \(y = -f(x)\) reflects it in the \(x\)-axis.
- Statistics: draw box plots on the scale given and mark outliers with a cross. In a histogram, frequency density = frequency ÷ class width, so the area of a bar, not its height, gives the frequency.
- Mechanics: draw a force diagram in every forces question, even when it isn't asked for, and a velocity–time graph for a journey in stages. The gradient of a velocity–time graph is the acceleration and the area under it is the displacement.
- Use the diagram in the question: mark lengths, angles and coordinates on it as you find them.
- Draw graphs and diagrams in dark pencil (HB or B) and use a ruler for straight lines.
Paper 2 Section A: statistics
- Lay out every binomial hypothesis test the same way: define the parameter, write \(H_0\) and \(H_1\) in terms of it (e.g. \(H_0: p = 0.3\), \(H_1: p < 0.3\)), state the distribution under \(H_0\), find the probability or the critical region, compare with the significance level, and conclude in context.
- A test gives evidence, not proof. Write 'there is sufficient evidence at the 5% level to suggest that the proportion of ... has decreased', never 'this proves'. A bare 'reject \(H_0\)' without context loses the final mark.
- AS tests are one-tailed: compare the probability with the significance level, or use the critical region. For a critical region from the binomial tables, choose the region whose probability is as close as possible to the significance level without going over, unless the question tells you otherwise, and be ready to give the actual significance level.
- Know that the significance level is the probability of rejecting \(H_0\) when it is actually true.
- Binomial: when asked why the model suits, give the conditions in context (a fixed number of trials, two outcomes, independent trials, a constant probability of success). Use cumulative probabilities, e.g. \(P(X \ge 5) = 1 - P(X \le 4)\)
- Data questions are about interpretation: compare distributions using a measure of location and a measure of spread, in context; explain the effect of outliers; say whether a prediction is interpolation (more reliable) or extrapolation (unreliable).
- When a question says 'using your knowledge of the large data set', your answer must use something specific about it, such as a variable, a unit, a location or how the data was recorded.
- Probability: use Venn and tree diagrams and test independence with \(P(A \cap B) = P(A) \times P(B)\)
Paper 2 Section B: mechanics
- Draw a large, clear force diagram with every force labelled: weight \(mg\), normal reaction \(R\), tension \(T\) and any driving or resisting force. Show the direction of the acceleration.
- Say which direction you are taking as positive and apply \(F = ma\) in the direction of motion. In equilibrium, the resultant force in each direction is zero.
- Take \(g = 9.8\) m s−2 unless told otherwise and give answers that use \(g\) to 2 or 3 significant figures, as the front cover says. Using 9.81 or 10 loses accuracy marks.
- Use the suvat equations only when the acceleration is constant. When acceleration varies with time, use calculus: \(v = \dfrac{ds}{dt}\), \(a = \dfrac{dv}{dt}\), \(s = \int v \, dt\), with constants found from the starting conditions.
- Connected particles: write \(F = ma\) for each particle separately. A light inextensible string over a smooth pulley has the same tension on both sides, and both particles have the same magnitude of acceleration. When the string breaks or goes slack, start a new stage of the motion.
- Vectors: work in \(\mathbf{i}\) and \(\mathbf{j}\) components. Speed is the magnitude of the velocity vector; give a direction as an angle measured from a stated reference, such as the vector \(\mathbf{i}\) or north.
- Know what each modelling assumption means for the maths: particle (size and rotation ignored), light (mass ignored, so the tension is the same throughout), inextensible (connected objects have the same acceleration), smooth (no friction), uniform (weight acts at the centre), air resistance ignored.
- Give units with every answer: m, m s−1, m s−2, N, kg, s.
Exact answers, accuracy and units
- 'Exact' means no rounded decimals: leave the answer as a fraction, a surd, or in terms of \(\ln\) and \(e\) (e.g. \(x = \dfrac{1}{2}\ln 5\)). A calculator decimal loses the mark.
- Unless the question or the front cover says otherwise, give inexact answers to 3 significant figures. Answers that use \(g = 9.8\) are given to 2 or 3 significant figures.
- Don't round in the middle of a calculation. Store values in your calculator's memory and write at least 4 significant figures in your working; an angle rounded early to \(53^\circ\) can push the final answer out of the accepted range.
- Give angles in degrees, as the AS course does, and give a solution of a trig equation to the accuracy asked (usually 1 decimal place).
- Round sensibly in context: you can't have 12.7 fish or 3.4 buses, so say whether the model gives 12 or 13, and why.
- Include units in the final answer of every context question, and keep them consistent (don't mix minutes with hours, or grams with kilograms).
Using your calculator
- Many calculators allowed in the exam have functions that save time: table mode, equation solvers and the binomial distribution. Learn them on the model you will use in the exam, not on the day.
- Use the calculator to check your algebra, not to replace it. When a question says solutions relying entirely on calculator technology are not acceptable, write out every algebraic step; a calculator answer alone scores nothing there.
- Table mode: evaluate a function at many points for checking the shape of a sketch or a stationary point.
- Statistics: use the distribution functions for binomial probabilities and statistics mode for the mean, standard deviation and, if needed, a regression line. Always write the probability you are finding, e.g. \(P(X \le 3) = 0.3823\) for \(X \sim B(10, 0.4)\), not just the number.
- Check the angle mode at every trig question: degrees at AS.
- Check solutions by substituting them back into the original equation.
Checking, extra space and crossing out
- Check as you go: substitute solutions back into the equation, differentiate an integral to see if you get back to the start, test a sketch with a couple of values, and ask whether each answer is sensible.
- At the end, check every answer against the question: the form asked for, the accuracy, the units, all solutions in the interval, both coordinates.
- Where you can, check by a different route: a stationary point with table mode, a probability using the complement, a vector answer by drawing it.
- The answer space is not a guide to how much to write: there may be more space than you need, so judge by the marks.
- If you run out of space, continue on a spare page or additional paper, write the question number beside the continued work, and write 'continued on page ...' in the original space so the examiner finds it.
- To change an answer, cross it out with a single neat line and write the new answer and working below it. Don't leave two different answers standing: make it clear which one you want marked.
- Don't rub out or scribble over working that might be right. Even an unfinished method can earn M marks.
When you are stuck
- Write down what you know in maths form: the given information, a diagram, the formula that links the quantities. This often earns the first method mark and shows you the way forward.
- Look at the marks: a 2-mark part needs one or two steps. If you are filling half a page, you have probably missed a quicker route.
- Look back at the earlier parts. A 'show that' or 'hence' is a signpost to the method, and you can use a printed answer even if you couldn't prove it.
- Try the standard moves for the topic: in trig, use \(\sin^2\theta + \cos^2\theta = 1\) or \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\) to get a single trig function; in algebra, factorise or complete the square; in mechanics, write \(F = ma\) for each particle in turn.
- If you are still stuck after a couple of minutes, leave a gap and move on. Come back at the end with fresh eyes.
- Don't leave any part blank. A correct first step, such as differentiating or setting up an equation, can earn marks even if you can't finish.
- Stay calm: every paper has some parts designed to stretch the strongest students, and dropping one part won't decide your grade.
Learning from mocks
- Sit mocks as if they were real: the full 2 hours for Paper 1 or 1 hour 15 minutes for Paper 2, only the formulae booklet, your exam calculator and no notes.
- Mark strictly against a mark scheme. Don't give yourself an A mark for an answer that is 'nearly right' or a method mark for a method you didn't write down.
- Sort every lost mark by cause: didn't know the content, chose the wrong method, careless slip, misread the question, or ran out of time. Each needs a different fix.
- Redo every question you dropped marks on within a few days, without looking at the solution, and again a week later.
- Track topics across several mocks. The topics that keep costing marks become next week's revision.
- Look at your timing: where you were at 30, 60 and 90 minutes on Paper 1, and which questions took too long.
- Compare your written answers with model solutions for layout and wording, especially hypothesis-test conclusions, proofs, 'show that' steps and comments on models.
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