A-Level Maths

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OCR B MEI · A-Level Maths

Comprehension

Practise reading a mathematical passage and answering questions about it. Each passage uses a hypothetical model; use the stated information and assumptions.

Choose a passage

Original comprehension · 15 marks

Designing a bounded scanning path

Connect two control settings, a parametric curve and its reachable positions. A calculator may be used.

The passage

A designer is testing an arch-shaped scanning path for a small plotting head. Its position is measured in centimetres. A non-negative control setting determines the position using

The setting is a dimensionless control value, not elapsed time. Each finite setting gives one point on the path. The model describes geometry only; it does not specify how quickly the head moves.

As the control setting increases, the head moves from left to right. The horizontal coordinate can also be written as

This form helps distinguish a coordinate that is approached from one that is actually reached. To describe the path without the control setting, the designer rearranges the equation for and substitutes into the equation for .

A second controller uses an angle instead, with

All angles are in radians. The designers want the two controllers to trace the same path. They also need the exact greatest height of the plotting head.

Assume the equations describe the intended path exactly. Motor limits, vibration and the size of the plotting head are ignored.

1. Check a control setting [2 marks]

Find the exact coordinates of when .

Show worked solution

2. Eliminate the control setting [4 marks]

Show that the path can be written as

Show worked solution

Multiply the numerator and denominator by .

3. Reachable positions [3 marks]

State the range of for finite settings . Find the limiting position as , and explain whether the head reaches it at any finite setting.

Show worked solution

Since at every finite setting, remains below .

The limiting position is . It is approached but never reached at a finite setting.

4. Use the angular controller [3 marks]

Use and a double-angle identity to show that .

Show worked solution

5. Locate the highest point [3 marks]

Find the exact greatest height of the path and the corresponding values of , and .

Show worked solution

On , the largest value of is , attained when .

The highest point is .

Original comprehension · 15 marks

Tracking the intersection of two rotating beams

Connect a geometric model, a curved path and motion through a camera window. A calculator may be used.

The passage

Two beam projectors are mounted on a horizontal baseline in a fictional motion-tracking experiment. Their positions are and . Both coordinates are measured in metres. The beams lie in one vertical plane. A fine mist makes their paths visible to a camera facing that plane.

The projectors rotate together. The beam from makes an angle with the positive horizontal direction. The beam from makes an angle with that direction. Thus the beams meet at a right angle. Restrict attention to , so their intersection lies above the baseline. All angles are in radians.

The equations of the two beam lines can be written as

A controller locates the intersection by solving these equations. It uses the formulae

The sketch shows one illustrative position and is not to scale. The projectors rotate at a constant angular rate . Constant angular rate does not by itself imply constant horizontal or vertical velocity. The speed of the intersection is calculated from

The tracking camera only records the intersection while . Assume the beam lines are exact, the mounting points stay fixed, and both beams have sufficient range. The model ignores the time taken by light to travel.

1. Check a position [2 marks]

Find the exact coordinates of when .

Show worked solution

2. Derive the controller formulae [4 marks]

Starting from the two beam-line equations, derive the expressions for and used by the controller.

Show worked solution

Multiplying by is valid throughout the stated interval.

3. Identify the path [3 marks]

Eliminate to show that lies on a circle. Give its centre and radius, and identify which part is traced.

Show worked solution

The centre is and the radius is . The upper semicircle is traced from right to left. The two endpoints are excluded because the angle interval is open.

4. Find the speed [3 marks]

Show that the speed of is constant, and find its value.

Show worked solution

The horizontal and vertical components vary, but their combined magnitude is constant.

5. Time in the camera window [3 marks]

Find the time for which the camera records during one sweep. Give your answer to 3 significant figures.

Show worked solution

Since decreases, the intersection enters at and leaves at . Use .

Original comprehension · 15 marks

Estimating a recorder’s energy use

Compare a discrete total with an integral estimate, then check a practical budget. A calculator may be used.

The passage

A fictional wildlife recorder sends a short burst of data whenever it wakes. During a test, each successful transmission allows the next burst to use less energy. The model assigns the energy

to burst , where the first burst is numbered . This reduction is a modelling assumption for the test; it is not a general law for batteries. For a positive integer , the energy used by the first bursts is

An engineer wants a quick estimate without adding the terms individually. Extend the formula to real values of the burst index:

The curve is decreasing. A rectangle of width 1 with its height taken at the left endpoint represents one term of the sum. The diagram illustrates the first five rectangles.

The proposed estimate is

This integral is a mathematical approximation to a sum over burst numbers. It does not represent a continuously operating transmitter. The engineer accepts an estimate when its percentage error, measured relative to the exact sum, is no more than 5%. A separate test budget allows at most 100 millijoules for the bursts; other energy use is excluded.

1. Reading the burst model [2 marks]

Find the energy of the first burst and the ratio of the energy of each burst to that of the previous burst.

Show worked solution

The ratio is approximately 0.9231.

2. An exact total [3 marks]

Use a geometric series to show that

Show worked solution

There are terms, beginning with 12 and with common ratio .

3. Comparing rectangles and the curve [3 marks]

Find . Using the rectangles described in the passage, explain whether this estimate is too large or too small.

Show worked solution

On each unit interval the left-endpoint rectangle lies above the strictly decreasing curve. Thus for every positive integer : the estimate is too small.

4. Is the accuracy affected by the number of bursts? [4 marks]

Show that the percentage error of does not depend on . Calculate this error and decide whether the estimate meets the 5% tolerance.

Show worked solution

The common factor is non-zero for . It cancels, so the ratio is independent of .

This is below 5%, so the estimate meets the stated tolerance for every positive integer number of bursts in this model.

5. Staying within the energy budget [3 marks]

Use the exact total to find the greatest number of bursts allowed by the 100-millijoule budget. Justify the integer choice.

Show worked solution

Each extra burst uses positive energy, so the total increases. The greatest permitted number is .

Original comprehension · 15 marks

Approximating a zoom-control scale

Compare two formulae, assess their errors and calibrate a setting.

The passage

A designer is testing a zoom control for a fictional drawing app. The control multiplies both the width and height of the displayed image by . Thus each linear dimension increases by per cent; this is not the percentage increase in area. The control is restricted to .

The design assigns the exact slider score

The score is dimensionless. For example, doubling both the width and height corresponds to and score . The designer is comparing two simpler formulae:

These are candidate approximations to the score, not alternative definitions of magnification. An approximation meets the design tolerance at a particular setting when its absolute error is at most 0.01:

The designer also needs to work backwards from a requested score to a zoom setting. A formula that underestimates the score at a fixed setting may produce too much zoom when used in reverse. The questions explore both kinds of use.

1. Interpreting the scale [2 marks]

A setting has score . Find and state the percentage increase in each linear dimension of the displayed image.

Show worked solution

The width and height are each 60% greater than their original values.

2. Comparing the approximations [3 marks]

Show that . Hence compare the two approximations when .

Show worked solution

Both numerator and denominator are positive, so .

3. Does the rational formula overestimate? [4 marks]

Let . Differentiate and show that . Use to determine whether overestimates or underestimates for .

Show worked solution

For , the derivative is positive. Since , we have . Thus : the rational formula underestimates the exact score.

4. Checking the tolerance [3 marks]

At , find the absolute error of each approximation to 4 decimal places. Which formula meets the stated tolerance at this setting?

Show worked solution

Only has absolute error at most 0.01 at this setting. This single check does not establish accuracy over the whole interval.

5. Calibrating a requested score [3 marks]

The requested score is 0.5. Solve exactly, then solve . Does calibration using produce too much or too little magnification?

Show worked solution

The exact value is approximately 0.6487, less than . Using therefore produces too much magnification.

Original comprehension · 15 marks

Choosing a viewing position in a gallery

Interpret an angle model, use inverse tangents and recommend a practical position. A calculator may be used.

The passage

A gallery is deciding where to put a viewing mark for a tall wall panel. Making the panel subtend a large angle at the viewer’s eye may make fine details easier to see, but standing extremely close also makes the viewing direction steep.

In the model, the viewer stands directly in front of the panel. The lower and upper edges are respectively 1.2 m and 4.8 m above the viewer’s eye level. Let be the horizontal distance, in metres, from the eye to the wall. The angle between the two sight lines is . All angles are in radians.

For a sight line at angle above the horizontal, . The inverse tangent returns an angle in . You may use

The team compares possible positions using . The panel’s viewing angle is acute for every , so increasing increases . For practical reasons, the floor mark can be placed only at 2.0 m, 2.5 m or 3.0 m from the wall.

1. Model the viewing angle [2 marks]

Write as a difference of two inverse tangents. Calculate when , to 3 significant figures.

Show worked solution

2. Simplify the comparison function [3 marks]

Show that .

Show worked solution

3. Check the angle convention [2 marks]

Explain why is valid in this model without adding a multiple of . What happens to the viewing angle as becomes very large?

Show worked solution

Both sight lines make acute angles with the horizontal, and the upper one has the larger angle. Their difference satisfies , within the range of the inverse tangent, so no adjustment is required.

As , and hence .

4. Find the best unrestricted position [3 marks]

Differentiate and find the distance which maximises the viewing angle. Justify that it gives a maximum.

Show worked solution

For the derivative is positive; for it is negative. Thus m gives the global maximum. Maximising also maximises .

5. Choose a permitted floor mark [3 marks]

Compare the three permitted positions using and recommend one. Find its viewing angle to 3 significant figures.

Show worked solution

The greatest value occurs at 2.5 m, so choose this permitted position.

6. Assess the recommendation [2 marks]

A colleague claims that the chosen mark is best for every visitor. Give two reasons why the model does not establish that claim.

Show worked solution

Visitors have different eye heights, which changes the two heights used in the formula and can change the best distance.

A large viewing angle alone does not guarantee comfort or readability. For example, eyesight, glare, panel detail or the steepness of the upward viewing direction may also affect the best position. Other relevant, distinct limitations are acceptable.

Original comprehension · 15 marks

Balancing detail and effort in a site survey

Interpret a model, optimise an efficiency index and apply practical constraints. A calculator may be used.

The passage

An archaeological team uses a scanner to inspect marked patches of ground before excavation. A longer scan reveals more detail, but staff time and data processing also increase. The team wants to compare scanning settings using an efficiency index.

Let be the number of additional hours spent scanning one patch after an initial setup. The team proposes a dimensionless detail score and a dimensionless resource score . A resource score of 1 represents the setup alone. These are model scores, not measurements of the number of objects found or the cost in pounds.

The efficiency index is the detail score divided by the resource score: . The scores are intended to model diminishing gains in detail and an increasing processing burden. Maximising the detail score alone would favour longer scans; maximising the efficiency index balances both scores.

For a pilot survey, the equipment permits only three additional scan durations: 0.4, 0.6 and 0.8 hours. A later version may allow any non-negative duration. A separate resource limit may also restrict which durations are available.

The model treats all patches as equally difficult to scan. In practice, soil composition, depth and interference may affect the result. The team will compare the predictions with observations before using the index to plan a complete survey.

1. Read the two scores [2 marks]

Calculate and . Explain why these two scores cannot be interpreted directly as a count of discoveries and a cost in pounds.

Show worked solution

Both are dimensionless model scores. Their definitions provide no conversion to discoveries or pounds.

2. Differentiate the efficiency index [3 marks]

Show that .

Show worked solution

3. Find the best continuous setting [3 marks]

Find the exact duration at which is greatest for . Justify that it is a maximum.

Show worked solution

The denominator of is positive. Its numerator is positive before and negative afterwards. Therefore the efficiency increases and then decreases, giving the global maximum.

4. Choose an available pilot setting [3 marks]

Calculate the efficiency index for each of the three pilot durations, to four decimal places. State which setting should be used if this index is the only criterion.

Show worked solution

Use 0.6 hours. It gives the largest index among the three available choices.

5. Apply a resource limit [2 marks]

The later version permits any duration, but the resource score must satisfy . Find the permitted interval for and hence choose the duration giving the greatest efficiency under this limit.

Show worked solution

This interval lies below , so the efficiency increases throughout it. Choose hours.

6. Assess the model [2 marks]

Give two distinct limitations of using the efficiency index as the only basis for planning the survey.

Show worked solution

For example: different patches may require different models because their ground conditions differ; maximising a ratio may select a short scan with insufficient total detail to meet the survey objective. Other sensible, distinct limitations are acceptable.

Original comprehension · 15 marks

Planning a drainage trench

Build a geometric volume model and use it to plan soil removal. A calculator may be used.

The passage

A site team is estimating how much soil must be removed to dig a straight drainage trench across level ground. The trench will be 12 metres long. To make the side walls more stable, the cross-section is designed as a trapezium rather than a rectangle.

Let be the distance in metres from the shallow end, with . The planned depth, in metres, is . This steady change in depth allows water to flow towards the deeper end.

Every cross-section has a bottom width of 0.60 metres. From the bottom to ground level, each side extends outwards horizontally by twice the vertical depth. Thus a cross-section of depth widens by on each side. The sketch illustrates the shape; it is not to scale.

The area of a trapezium is half the sum of its parallel sides multiplied by the perpendicular distance between them. If the cross-sectional area at position is square metres, a short slice of length has approximate volume . Adding many thin slices leads to an integral for the volume of the complete trench.

The model assumes the ground is level, the bottom width is constant and the side slopes follow the design exactly. It ignores roots, stones and local changes in ground conditions.

After excavation, the soil occupies more space because it is no longer packed as tightly. The team estimates that this increases its volume by 18%. A truck can carry at most 2.5 cubic metres of the loosened soil on each load. The final load need not fill the truck.

1. Interpret the dimensions [2 marks]

Find the depth and the top width of the trench at .

Show worked solution

2. Derive the cross-sectional area [3 marks]

Show that the cross-sectional area at depth is .

Show worked solution

3. Express the area along the trench [2 marks]

Show that .

Show worked solution

4. Calculate the excavation volume [4 marks]

Use integration to find the volume of soil to be excavated, in cubic metres.

Show worked solution

5. Plan the removal [4 marks]

Calculate the minimum number of truckloads required under the model. Give one reason why the actual number of loads might differ from your calculation.

Show worked solution

The minimum is 4 loads: three full loads would be insufficient. The actual soil may expand by more or less than 18%, or the excavated sides may differ from the designed slopes, changing the volume to be carried.

Original comprehension · 15 marks

Average speed on a two-leg shuttle route

Develop a journey model, optimise it and account for a practical delay. A calculator may be used.

The passage

A test shuttle travels from a loading bay to a checkpoint and returns along the same route. Each leg has length kilometres. Its outward speed is kilometres per hour and its return speed is kilometres per hour. Both speeds are positive. The model treats each leg as being travelled at its stated constant speed, ignoring the time taken to accelerate, turn and slow down.

For a complete journey, the average speed is the total distance divided by the total elapsed time. When the two legs have different speeds, equal distances do not take equal times. The average of the two speed settings therefore need not be the average speed of the journey.

First consider journeys with no stop at the checkpoint. Let be the average speed of the complete journey, in kilometres per hour. The outward and return travel times, in hours, are given below.

The controller is programmed to allow only pairs of speed settings satisfying . This is a rule for the test, rather than a claim about fuel use or energy. The engineer wants to choose the settings that give the greatest average speed.

A later safety review limits the outward speed to at most 12 kilometres per hour and the return speed to at most 20 kilometres per hour. The controller rule still applies. The engineer must choose the best settings within these restrictions.

Finally, the shuttle must stop for loading at the checkpoint. In this final test each leg is 0.9 kilometres long. The stop adds time but no distance, and it must be included when calculating the average speed of the complete journey.

1. Calculate a journey average [2 marks]

Without a stop, the shuttle travels outward at and returns at . Calculate its average speed, giving an exact value.

Show worked solution

2. Derive a model [3 marks]

Show that, without a stop, . Explain why the leg length does not appear in this expression.

Show worked solution

The common factor cancels. Changing both equal leg lengths by the same factor changes the distance and travel time by that same factor.

3. Find the unrestricted optimum [4 marks]

Write and , where . Show that . Hence find the greatest possible value of and the speed settings that achieve it.

Show worked solution

Since , , with equality when . Thus the maximum average speed is 15 kilometres per hour, achieved by .

4. Apply the safety restrictions [3 marks]

Now impose the two speed limits from the passage. Find the best permitted values of and , and the resulting average speed without a stop.

Show worked solution

The smallest possible value of is 9, achieved when . Therefore and .

5. Allow for a loading stop [3 marks]

For the final test, use the speed settings from question 4. Find the longest loading stop for which the average speed of the complete journey is at least 14 kilometres per hour. Give your answer in seconds to 3 significant figures.

Show worked solution

If the loading stop lasts hours, the average-speed requirement gives

This is the exact upper limit rounded to 3 significant figures. An actual stop must not exceed seconds; a stop lasting exactly 12.9 seconds would be slightly too long.

Original comprehension · 15 marks

The hidden time in a staged file check

Use a repeated process to compare an ideal workload with a practical time limit. A calculator may be used.

The passage

An archive uses a program that checks a file in progressively smaller sections. Stage 0 consists of one check. Each check leads to two checks at the next stage, so stage contains checks. All checks are carried out one after another on the same computer.

The first check takes 12 minutes of processing time. The program is designed so that each check at the next stage needs only of the processing time of its parent check. This is an ideal model for the processing time, before allowing for the work needed to start a check.

For example, stage 1 has two checks taking minutes each, so its total processing time is 9 minutes. Let be the total processing time of stage , and let be the combined processing time of stages 0 through , inclusive.

The archive would like to stop when the processing time omitted from all later stages is less than 2 minutes, according to the ideal model. This is a rule for choosing the number of stages; it does not mean that the remaining sections have actually been checked.

A trial reveals a second cost: every check takes a further 3 seconds to initialise. This time is additional to its processing time and does not become smaller at later stages. Let be the total running time, in minutes, including initialisation, for stages 0 through .

The archive can allow at most 50 minutes for the program to run. It needs to choose the greatest complete stage that fits this time allowance, and determine whether the original rule about omitted processing time is still achievable.

1. Read the stage rule [2 marks]

For stage 2, state the number of checks and calculate the total processing time.

Show worked solution

2. Find the ideal workload [3 marks]

Show that . Hence find a formula for and its limiting value as increases.

Show worked solution

Since , the limiting processing time is 48 minutes.

3. Apply the stopping rule [3 marks]

In the ideal model, find the smallest integer for which the combined processing time of all stages after stage is less than 2 minutes. Show how your answer meets the strict inequality.

Show worked solution

The inequality reverses on division because . Hence the smallest integer is .

4. Include initialisation [3 marks]

Show that . Explain why has no finite limit even though does.

Show worked solution

The number of checks increases without bound, and each has a fixed positive initialisation time. The initialisation term therefore increases without bound, despite the finite limit of the processing term.

5. Choose a feasible stage [4 marks]

Find the greatest complete stage for which . Justify that no greater value can work. Determine whether this choice also meets the omitted-processing-time rule from question 3.

Show worked solution

Every further stage adds positive processing and initialisation times, so is strictly increasing. The greatest permitted stage is therefore .

This does not meet the original stopping rule. Stage 11 would be needed for that rule, but it cannot fit the 50-minute allowance.

Original comprehension · 15 marks

Choosing an exposure setting for a sensor

Use a response model to balance timing sensitivity and exposure duration. A calculator may be used.

The passage

A laboratory sensor records a signal while its shutter is open. A longer exposure gives a stronger reading, but the response gradually levels off. For a fixed sample, a proposed model is

where is the exposure duration in seconds and is the signal reading in the device’s units. The model describes the response to a constant source; it does not account for changes in the source itself.

The shutter timing has a small proportional error. The engineer wants to know how much this affects the reading. The relative sensitivity is defined by

For a small change in exposure duration, the percentage change in the signal is approximately times the percentage change in the duration. For example, a relative sensitivity of 0.4 means that a small 1% increase in duration produces about a 0.4% increase in the reading. This is a local approximation; a larger change should be checked using the signal model directly.

The engineer requires a relative sensitivity of at most 0.2. The device allows exposure settings in steps of 0.01 seconds, and each exposure must take no more than 8 seconds. The aim is to find the shortest available exposure setting that satisfies both requirements.

1. Interpret the model [2 marks]

Calculate the reading when . You are told that . Estimate the percentage increase in the reading caused by a small 1% increase in exposure duration.

Show worked solution

The estimated percentage increase is .

2. Find a sensitivity formula [3 marks]

Differentiate and hence show that

Show worked solution

3. Check a larger timing change [3 marks]

The exposure duration increases from 5 seconds to 5.5 seconds. Calculate the percentage increase in the signal from the model. Compare it with the estimate obtained using , and explain why the two answers differ.

Show worked solution

The local estimate overstates the increase by about 0.29 percentage points. It uses the rate of change at the start, whereas the response becomes less steep as the exposure lengthens. A 10% timing change is large enough for this difference to matter.

4. Establish that sensitivity decreases [4 marks]

Put and let . Show that and for . By differentiating your formula for , deduce that the relative sensitivity decreases for all .

Show worked solution

Therefore for .

5. Choose an exposure setting [3 marks]

Using a numerical method or a calculator, solve for . Find the shortest available exposure setting that meets the sensitivity requirement, and check the time limit. Explain why ordinary rounding to two decimal places is unsuitable here.

Show worked solution

Choose 6.66 seconds. Sensitivity decreases as time increases, so this is the shortest permitted setting that meets the requirement. It is below the 8-second limit. Ordinary rounding would give 6.65 seconds, which fails the sensitivity requirement.

Original comprehension · 15 marks

Reciprocal settings in a signal filter

Read the mathematical model, then use a new variable to select a setting. A calculator may be used.

The passage

An engineer is exploring a simplified model for a signal filter. Its dimensionless tuning parameter is positive. The model assigns the filter a dimensionless response score

Two settings are called reciprocal if their product is 1. In this model, replacing by leaves the score unchanged. The engineer calls this reciprocal symmetry.

Directly solving for at a specified score would usually produce a quartic equation. Reciprocal symmetry suggests introducing a new variable

The identity can then be used to rewrite the score as a quadratic expression in . A proposed value of is useful only if it corresponds to an allowed positive value of .

To recover a setting from , multiply by . This gives . When there are two positive solutions, their product is 1, so they describe a reciprocal pair.

For a new device, the engineer wants a score of 26 and must use a setting . The formula is a proposed model; actual filter performance must still be checked experimentally.

1. Check the symmetry [2 marks]

Calculate and . Explain why equality of these scores is consistent with the passage.

Show worked solution

The settings are reciprocal. Exchanging and exchanges terms in the score without changing their sum.

2. Find the allowed range [3 marks]

Use and to show that . State when equality holds. Hence show that .

Show worked solution

Equality holds only when .

For , and , so .

3. Choose the required setting [4 marks]

Use the new variable to solve . Give all positive values of exactly, then choose the one permitted for the new device.

Show worked solution

Reject because . Thus .

Both are positive, but only satisfies .

4. Interpret a nearby score [3 marks]

A different device needs a score of 20. Without finding either setting, show that there are exactly two positive settings and that exactly one exceeds 1.

Show worked solution

The other root is negative and must be rejected. Since , the equation has two distinct real roots. Their sum is positive and their product is 1, so both are positive. They are unequal reciprocals, so one is below 1 and one above 1.

5. Check uniqueness on the permitted branch [3 marks]

Show that is strictly increasing for . Explain what this says about the chosen setting for score 26, and give one reason to test the physical device rather than rely only on the model.

Show worked solution

For , both and are positive. Therefore the score is strictly increasing.

There is only one setting on with score 26. Measurements are still needed because the proposed formula may omit physical effects or may not match the real device accurately.

Original comprehension · 15 marks

Drawing a smooth curve with straight segments

Read the model, then balance drawing accuracy against a memory limit. A calculator may be used.

The passage

A designer wants a small display to draw a smooth trace. In a simplified model, the trace has equation for . Both coordinates are measured in millimetres.

The display stores selected points on the curve and joins consecutive points with straight segments. More segments give a smoother image, but storing more points uses more memory.

Consider one segment whose endpoints have horizontal coordinates and , where and . The height of the straight segment at horizontal coordinate is

The vertical drawing error is the height of the segment minus the height of the curve at the same horizontal coordinate. It is a vertical difference, not the shortest distance to the curve.

A detail of the trace. The orange line marks the vertical error at the midpoint of this segment.

The designer initially uses four equal-width segments across the full interval. A refinement routine doubles the number of segments each time. Alternatively, a custom setting can use any positive integer number of equal-width segments.

The pixel pitch is 0.08 mm. The designer accepts a trace only if its maximum vertical drawing error is no more than half this pitch. The device has room to store at most 25 points, including the two endpoints. Assume that the only drawing error is caused by replacing the curve with straight segments.

1. Check one segment [3 marks]

Take and . Find the equation of the straight segment. Calculate its vertical drawing error at the midpoint.

Show worked solution

2. Find the greatest error [4 marks]

Show that . By completing the square, or otherwise, show that the maximum error on a segment is , attained at its midpoint.

Show worked solution

The squared term is non-negative. The maximum is at , the midpoint.

3. Run the refinement routine [3 marks]

The display uses equal-width segments. Explain why the maximum error is mm. Starting at and doubling each time, find the first number of segments that meets the accuracy requirement.

Show worked solution

The allowed error is mm. Each doubling divides the error by 4.

Segments	Maximum error (mm)
4	1.25
8	0.3125
16	0.078125
32	0.01953125

The first acceptable doubling is .

4. Choose a custom setting [3 marks]

Find the smallest positive integer number of equal-width segments that meets the accuracy requirement. Check that one fewer segment would not be sufficient.

Show worked solution

5. Make a recommendation [2 marks]

Decide whether the doubling routine or the custom setting can meet both the accuracy requirement and the memory limit. State one limitation of this drawing model.

Show worked solution

A trace with segments needs stored points. The custom setting uses 23 segments and 24 points, so it meets both requirements. The doubling routine needs 32 segments and 33 points, exceeding the limit of 25.

For example, rounding stored coordinates or lighting individual pixels can introduce further error. The calculation assumes exact coordinates and straight segments, so it does not measure every source of visible distortion.

Original comprehension · 13 marks

A faster image-processing queue

Read the passage, then use it to make and justify a decision. A calculator may be used.

The passage

A laboratory processes a queue of microscope images. Earlier images help the software prepare reusable calculations, so later images take less time. One image must finish before the next begins.

The modelled time for image number is seconds, for . The total time for the first images is

To estimate a long queue without adding every term, define the continuous function , for . It is positive and decreasing.

A unit-width rectangle ending at an integer has area greater than if its height is taken from the left end. Rectangles whose heights are taken from the right end lie below the curve. These comparisons give, for ,

where . The comparisons are strict because is strictly decreasing.

A sharper upper bound can be obtained by calculating the first image times directly and estimating only the rest. For ,

The laboratory wants to know whether a queue of nine images is guaranteed to finish within 150 seconds according to this model. No interruption or additional set-up time is included.

1. Read the model [2 marks]

Find and . Interpret the limit in this setting.

Show worked solution

The time per image approaches 4 seconds as more images are processed. It does not approach zero; the model retains a fixed amount of work for every image.

2. Obtain the integral estimate [3 marks]

Show that . Hence find .

Show worked solution

3. Decide what the first bounds tell you [3 marks]

Use the bounds in the passage to find a lower and an upper bound for . Do these bounds alone guarantee that nine images finish within 150 seconds? Explain your answer.

Show worked solution

No. These bounds allow totals on either side of 150 seconds. An upper bound above 150 does not show that the queue will exceed 150 seconds.

4. Improve the decision [5 marks]

(a) Calculate , keeping unrounded values for later working.

(b) Use the sharper upper bound with to decide whether the model guarantees completion within 150 seconds. Give your upper bound to 3 decimal places.

Show worked solution

(a) Add the six modelled image times.

(b) Estimate the remaining three images using the integral.

The unrounded upper bound is below 150 seconds, so the model guarantees completion in time.

(c) For example, an interruption, extra set-up work, or unusually complex images could make actual processing slower than the model.

Pure

Proof

Reason with statements and implications, use counterexamples, and prove results by exhaustion, deduction and contradiction.

Proof: Start with an explicit assumption, explain each implication, name the contradiction and conclude the original claim. The irrationality proofs use fractions in lowest terms; the prime proof requires an unlisted prime divisor, whether the constructed number is prime or composite.

Pure → Algebra & functions

Graph Transformations

Translation, reflection and stretch recognition first; then standard functions, graphical and coordinate mappings, and finally ordered pairs of transformations.

Horizontal and vertical cases are deliberately mixed within each skill: distinguishing them is part of the mathematics. The card header also stays neutral so it never gives away whether the transformation is a translation, reflection or stretch.

House wording for transformation descriptions

Horizontal stretch

Stretch, parallel to the x-axis, scale factor 2

Vertical stretch

Stretch, parallel to the y-axis, scale factor 3

Reflection

Reflection in the x-axis

Translation

Translation by \(\begin{pmatrix}3\\4\end{pmatrix}\)

Generator safeguard: some standard functions have symmetries or identities that can make a reverse transformation question non-unique. The generators will use unambiguous cases unless recognising equivalent transformations is itself the intended skill.

Pure → Coordinate geometry

Straight Lines

Coordinate methods and linear graphs, beginning with midpoints, distances and gradients. Circles are kept as a separate topic.

Pure → Coordinate geometry

Circles

Build and sketch circles, use circle geometry and distances, and investigate intersections, tangents and normals.

Pure → Algebra & functions

Quadratics

Factorising and completing the square first, then symmetry, turning points, exact solutions, the discriminant, graph sketching and quadratic modelling.

Pure → Algebra & functions

Hidden quadratics

Recognise a quadratic structure hidden inside powers of x, trigonometric functions, arbitrary-base exponentials and natural exponentials, then make an appropriate substitution and solve.

Hidden quadratics involving powers of x

Solve equations such as x^6-9x^3+8=0 by substituting for the repeated power of x.

Hidden quadratics involving trigonometric functions

Solve equations such as 2sin²x−3sin x+1=0 on a stated interval, including rejection of impossible trigonometric values.

Hidden quadratics involving arbitrary-base exponentials

Use the same exponential hidden-quadratic generator as ELZQ, with substitutions such as y=2^x.

Hidden quadratics involving e^x

Use the same natural-exponential hidden-quadratic generator as EXHQ, substituting y=e^x or a shifted form such as y=e^(x+2).

Mixed hidden quadratics

Randomly mix powers of x, trigonometric functions, arbitrary-base exponentials and e^x so the appropriate substitution must first be recognised.

Hidden quadratics: Worked answers explicitly introduce the substitution, solve the resulting quadratic, substitute back, and reject impossible values where appropriate. The arbitrary-base and e^x skills share their generators with ELZQ and EXHQ respectively.

Pure → Algebra & functions

Indices

Move from recognising negative and fractional index notation to common-base fluency, calculus-ready power forms and fluent simplification, then finish by diagnosing common misconceptions about the index laws.

Write expressions using index notation

Progress from 1/x = x^-1 through roots and reciprocal roots to expressions such as 1/(fourth root of x^3) = x^-3/4.

Write expressions in the form ax^n

Prepare expressions for calculus by separating the numerical coefficient, for example 1/(2x) = (1/2)x^-1, before progressing to roots and fractional indices.

Evaluate numerical powers without a calculator

Positive, negative and fractional powers, including fractional bases.

Simplify using index laws

Start with x^4x^7, (x^5)^3 and quotients, then combine several laws and introduce negative and fractional indices.

Simplify expressions involving several variables

Build from products and powers in two variables to expressions such as a^3b^5(c^6)^3/(abc)^4.

Simplify expressions involving numerical coefficients and variables

Simplify products, powers and quotients with numerical coefficients, several variables and negative or fractional indices.

Write an expression in the form 2^k

High-volume procedural practice rewriting powers, roots, fractional powers and quotients using base 2 before combining the indices.

Write an expression in the form a^k, where a>2

The target base changes between cards, for example 3^k, 8^k, 9^k, 16^k and 25^k.

Index laws: Always true, Sometimes true or Never true

Target misconceptions such as adding powers when terms are added, mishandling zero or negative indices, and treating roots as though they distribute over addition.

Indices: Practise numerical and algebraic powers, including negative and fractional indices.

Pure → Algebra & functions

Surds

Simplify and calculate exactly with surds, rationalise numerical denominators, then extend the same methods to algebraic expressions involving a parameter.

Simplify surds

Progress from sqrt(12) to larger multiples such as 3sqrt(108), always giving the surd in simplest form.

Write an expression as a single square root

For example, 5√3 = √75.

Calculate with surds

Add, subtract, multiply and expand exact surd expressions, including conjugate-style products.

Rationalise a denominator containing a single surd

Examples include 1/sqrt(5) and 5sqrt(2)/(2sqrt(3)).

Rationalise a denominator using a conjugate

Progress from 1/(2+sqrt(3)) to expressions with surds in both numerator and denominator.

Mixed exact surd manipulation

Combine several techniques in one expression, with exact simplified answers throughout.

Rationalise an algebraic denominator containing a single surd

For example, 1/sqrt(k) = sqrt(k)/k, before moving to more involved powers and coefficients.

Rationalise an algebraic binomial denominator using a conjugate

Examples include 1/(sqrt(k)-2) and later expressions with k in more than one place.

Surds: The six numerical skills are non-calculator practice; two further skills cover algebraic surds.

Pure → Algebra & functions

Inequalities

Convert between inequality, set and interval notation; solve linear, quadratic and cubic inequalities; find parameter values; then sketch and shade regions bounded by lines and curves.

Inequalities: All 16 skills are available, including three optional Challenges. Solving cards randomly require answers in set or interval notation. Two-dimensional sketches use solid boundaries for inclusive inequalities and dashed boundaries for strict inequalities; the final interpretation skill uses labelled regions on a supplied diagram.

Pure → Trigonometry

Basic trigonometry

Exact values and ratios, triangle methods, bearings and problem solving, with a separate radians pathway.

Basic trigonometry: Degree and radian triangle skills remain separate. Exact radian values involving π are retained where appropriate; otherwise numerical answers are given to 3 significant figures.

Pure → Trigonometry

Arcs and sectors

Arc length, sector area and perimeter, reverse problems, annular sectors and segments.

Arc length and sector area

Use s = θr and A = ½θr² directly. Exact π values and exact decimal radian angles are both included.

Sector perimeter

Find sector perimeters, including major sectors and questions combining area and perimeter.

Find a radius or angle

Work backwards from arc length, sector area or sector perimeter to find a missing radius or angle.

Use arc length and sector area together

Use two pieces of sector information together, for example arc length and area, to determine both the radius and the angle.

Sector area and perimeter equations

Form and solve a quadratic for the radius, then check which sectors are possible.

Annular sectors

Find the area and perimeter of a sector of a ring.

Segment areas

Find minor and major segment areas, exactly or to 3 significant figures.

Arcs and sectors: Angles are in radians throughout. Exact values involving π and simple terminating decimals are kept exact.

Pure → Trigonometry

Trigonometric graphs & equations

Sketch sine, cosine and tangent, identify periods, and solve equations using graph symmetry and periodicity.

Trigonometric graphs & equations: Exact special-angle values are retained exactly. Non-exact degree solutions are given to 1 decimal place and non-exact radian solutions to 3 significant figures. Worked equations use sketches, symmetry and periodicity rather than a CAST diagram.

Pure → Trigonometry

Trigonometric identities

Simplify expressions, solve equations using identities, and prove identities through substitution, factorisation, expansion and algebraic fractions.

Simplify trigonometric expressions

Simplify expressions involving sin x, cos x and tan x, including use of sin²x + cos²x = 1.

Solve equations using trigonometric identities

Use sin²x + cos²x = 1, then factorise and solve every valid branch in the stated interval.

Prove straightforward trigonometric identities

Start with the left-hand side and use direct substitutions involving tan x = sin x / cos x and sin²x + cos²x = 1.

Prove identities by expanding and factorising

Use expansions, common factors, difference of two squares and sums or differences of cubes before applying sin²x + cos²x = 1.

Prove identities involving algebraic fractions

Combine fractions and simplify complex fractions, working from the left-hand side to reach the right-hand side.

Trigonometric identities: This first section is deliberately restricted to sine, cosine and tangent.

Pure → Trigonometry

Compound angle formulae

Use the geometric construction to express key lengths, then recall and apply the compound-angle formulae.

Prove the sine addition formula using triangle areas

Use a perpendicular height and the areas of two smaller triangles to derive the sine addition formula.

Find lengths in the compound-angle construction

Use one highlighted right-angled triangle at a time to express a labelled length in terms of x and y.

Write and recognise compound-angle formulae

Expand sin(x + y), cos(A - B) or tan(θ + φ), and recognise the same formulae written backwards.

Find exact values using special angles

Use combinations of 30°, 45°, 60° and their radian equivalents to find exact sine, cosine and tangent values.

Find an exact compound value from known ratios

Use acute-angle information such as sin x = 3/5 and cos y = 5/13 to calculate a compound value exactly.

Simplify right-angle shifts

Use compound-angle formulae to simplify sine, cosine and tangent expressions involving 90° or π/2.

Rewrite using complementary angles

Rewrite a numerical sine value as a cosine value, or a cosine value as a sine value, in degrees and radians.

Solve equations using compound angle formulae

Either recognise an expanded compound angle or expand one, reduce the equation to a familiar trigonometric equation, then solve on a stated interval.

Compound angle formulae: Formula-booklet identities are used throughout. Exact values stay exact, and equation solutions are given on the stated degree or radian interval.

Pure · Differentiation

Choosing Differentiation Methods

Read the structure, choose an efficient route and explain your reasoning.

Choose differentiation rules

Recognise Chain, Product and Quotient Rules, including all seven combinations.

Rewrite to choose a simpler method

Rearrange negative powers in algebraic, exponential, logarithmic and trigonometric functions to choose a rule.

Compare Product and Quotient Rule methods

Differentiate the same function in two forms and compare the working.

Choose a method and differentiate

A full mix of single-rule and combined-rule questions using standard functions.

Identify errors in differentiation methods

Distinguish products from composites and correct missing or misused rules.

✦ Challenge

Count stationary points using mixed rules

Use a simplified derivative and its discriminant to justify the number of stationary points.

Pure · Differentiation

Quotient Rule

Differentiate quotients, simplify carefully and explore the shape of a curve.

Domain matters: A denominator zero is not a stationary point or a point of inflection. Retain original restrictions when cancelling factors. Trigonometric inputs are in radians.

Pure · Differentiation

Product Rule

Differentiate products, factorise derivatives and apply them to the shape of a curve.

Curve shape: Positive second derivative means convex; negative means concave. A point of inflection requires a change in concavity. Trigonometric inputs are in radians.

Pure · Functions

Inverse functions

Reverse functions, connect domains and ranges, and reflect graphs in the line y = x.

Inverse functions: Check the original domain, reverse the mapping and state the domain of the inverse.

Pure · Functions

Composite functions

Compose functions in the correct order, preserve input restrictions and find the attainable outputs.

Composition: Apply the rightmost function first. Keep every original domain restriction, even after simplifying.

Pure · Functions

Domain and Range

Classify functions, describe input and output sets, then explore how restricting a domain changes the range.

Notation: Domains use x. Ranges use y or the function named in the question. Check which endpoints are attained and keep excluded values out of the final set.

Pure · Functions and graphs

Modulus Functions

Read and sketch modulus graphs, find their equations, then use linear branches to solve equations and inequalities.

Pure · Exponential models

Exponential Growth & Decay

Build and interpret models, calculate times and rates, and explain their long-term predictions.

For the prerequisite algebra and graph work, use Exponentials & logarithms. Basic exponential differentiation is in Differentiating exponentials.

Pure · Algebra & functions

Partial fractions

Combine algebraic fractions, handle distinct and repeated factors and improper fractions, then apply the results to integration and expansions.

Pure · Trigonometry

Small-angle approximations

Use radians to approximate trigonometric expressions, explore limits and derive the sine and cosine derivatives.

Understand the approximations and their conditions

Recall the three formulae, explain the radian condition, interpret the graphs near zero, derive the cosine approximation and correct common errors.

Approximate trigonometric functions of small angles

Begin with sine, cosine and tangent of multiples of an angle, then include fractional angles and coefficients.

Approximate powers and products

Square the whole approximation and multiply expressions. Explicitly discard powers higher than the square of the angle.

Simplify expressions using small-angle approximations

Combine terms and simplify fractions. Later include cosine quotients and use a short binomial approximation where needed.

Solve equations using small-angle approximations

Form a linear or quadratic approximation and find a small positive angle in radians.

Evaluate trigonometric limits

Use the small-angle formulae to simplify ratios near zero, including sine, tangent and differences involving cosine.

Differentiate sine and cosine from first principles

Use compound-angle identities and the standard limits. Begin with sine and cosine, then include coefficients and multiples of the angle.

Assess the accuracy of an approximation

Compare a small-angle estimate with the calculator value, calculate percentage error and identify overestimates or underestimates.

✦ Optional challenge

Improve an approximation using supplied series

Use supplied higher-power terms for sine and cosine, then compare percentage errors. This optional extension follows the final slides.

Pure → Trigonometry

Double angle formulae

Derive and apply double-angle formulae, prove identities, solve equations and explore triple-angle Challenge skills.

Double angle formulae: Formula questions use the five forms taught in the source lesson. Exact values remain exact, quadrant information is used explicitly, and direct double-angle equations are solved over the stated interval.

Pure → Trigonometry

Trigonometric modelling

Interpret, use and construct realistic models involving sine and cosine.

Trigonometric modelling: Contexts are chosen so a sine or cosine model is plausible. The sequence moves from straightforward use and interpretation through solving models and threshold durations, before students construct increasingly less-signposted models. The final multistep modelling skill is marked as an optional challenge.

Statistics

Probability

Use Venn diagrams, probability notation, conditional probability, two-way tables, probability trees and probability modelling.

Probability: Frequencies are converted to exact fractions in simplest form. Venn-diagram fluency and the addition rule are established before conditional probability is introduced. Conditional probability remains a core exam skill, including complements and more involved event expressions. Independent and mutually exclusive events then build directly on these relationships, followed by two-way tables and probability trees as alternative representations of the same probability ideas. Tree work includes route calculations, complements, constructing trees, conditional probability and unknown probabilities. Probability modelling finishes the topic by critiquing assumptions and considering the likely effect of more realistic assumptions; none of these skills is labelled as Challenge.

Statistics → Statistical distributions

Discrete random variables

Construct and represent discrete probability distributions, find missing values and parameters, calculate event probabilities, and solve harder algebraic distribution problems.

Choose a probability model

Select binomial, normal or neither and justify the assumptions in context

Discrete random variables: Every generated probability distribution has non-negative probabilities summing exactly to 1. Event-probability cards often use irregular support sets. Piecewise probability functions are treated as an ordinary exam skill and are included in normal revision packs.

Statistics → Statistical distributions

Binomial distribution

Model binomial situations, interpret probability statements, use calculator probabilities, and work with the mean and variance.

Binomial distribution: Probability cards use calculator methods; the hand calculation of the binomial probability formula is not treated as a separate skill. Binomial hypothesis testing remains in the Statistical hypothesis testing section.

Statistics → Data presentation & interpretation

Linear regression and correlation

Calculate, estimate, interpret and use the product moment correlation coefficient and linear regression models.

Linear regression and correlation: Scatter diagrams cover visual estimates, comparisons and the limitations of PMCC. Context questions use “predicted” language for regression models and do not treat correlation as evidence of causation.

Statistics

The Normal Distribution

Use calculator methods for probabilities and inverse normal; standardise when finding unknown parameters.

From a normal probability to a binomial count

Find the probability for one observation, then model a count of independent observations

Normal curves: points of inflection

Use the inflection points to find the mean and standard deviation, or vice versa

Show where a normal curve changes concavity

Differentiate a supplied curve equation and justify its points of inflection

Normal models from histograms

Assess a histogram and estimate normal parameters from its shape and central spread

Statistics

The Normal Distribution: Approximations

Normal approximations to binomial distributions and continuity corrections.

Statistics

Hypothesis Testing: Sample Means Hypothesis Testing

Sample means, standard errors and tests for a population mean.

Statistics

Hypothesis Testing: Binomial Hypothesis Testing

Form hypotheses, assess evidence and find critical regions.

Statistics → Statistical hypothesis testing

PMCC hypothesis testing

Form and carry out hypothesis tests for a population product moment correlation coefficient.

Write hypotheses for a PMCC test

From a contextual scenario, define ρ as the parent population correlation coefficient and write H₀ and H₁ for a positive, negative or two-tailed test.

Critical values and critical regions

Use a supplied table extract to select the critical value and state the critical region.

Decisions from critical-region number lines

Read the shaded critical region and sample PMCC, decide whether to reject H₀, and conclude in context.

Conduct a PMCC hypothesis test

Use a given sample PMCC and the correct AQA critical value to test for positive, negative or any correlation, with a conclusion in context.

PMCC hypothesis tests from raw data

Calculate the PMCC from paired data and complete a hypothesis test with a conclusion in context.

Spot and correct PMCC test mistakes

Correct errors involving hypotheses, critical values, significance levels and conclusions.

PMCC hypothesis testing: Hypothesis-only cards require students to define ρ in context and write the hypotheses. Critical values are supplied or shown in a table extract. Raw-data tests require students to calculate the sample PMCC; number-line cards ask for a decision without drawing a diagram.

Pure → Algebra & functions

Polynomials

Expand and factorise, divide with the grid method, investigate real roots, and sketch cubic and quartic graphs.

Polynomials: Factor-theorem cards use only the factor theorem. Polynomial division is demonstrated with the grid method; Challenge cards are excluded from ordinary random packs.

Pure → Algebra & functions

Sketching rational functions

Sketch reciprocal and reciprocal-square families using intercepts and dashed vertical and horizontal asymptotes.

Sketch y=a/x

Sketch positive and negative reciprocal graphs, with x=0 and y=0 shown as dashed asymptotes.

Sketch y=a/x+c

Use the horizontal asymptote y=c, the vertical asymptote x=0 and any exact x-intercept.

Sketch y=a/(x-b)

Translate the reciprocal graph horizontally, identify x=b as the dashed vertical asymptote, and label the exact y-intercept.

Sketch y=a/(x-b)+c

Combine horizontal and vertical translations, with both dashed asymptotes and exact axis intercepts marked when they exist.

Rewrite and sketch a linear-over-linear rational function

Rewrite an expression such as (x+5)/(x+3)=1+2/(x+3), then sketch from its asymptotes and intercepts.

Sketch y=a/x²

Use symmetry about the y-axis and the dashed asymptotes x=0 and y=0, for either sign of a.

Sketch y=a/(x-b)²+c

Identify both dashed asymptotes and mark exact intercepts, including paired surd roots where appropriate.

✦ Optional challenge

Sketch a less standard rational function

Sketch forms based on x/(x-b) and x/(x-b)² by locating intercepts and dashed asymptotes and considering end behaviour.

Available

Sketching rational functions: Vertical and horizontal asymptotes are always dashed. Graphs use only significant intercept/asymptote labels, no grid, and Challenge cards are excluded from ordinary random packs.

Mechanics

2D Vectors

Vector notation, arithmetic, position vectors, magnitude, unit vectors, parallelism, collinearity and magnitude–direction conversion.

2D Vectors: Typeset vector variables use bold upright notation; directed line vectors use arrows. Ordinary skills deliberately mix column-vector and i, j component form wherever either representation is mathematically appropriate. Challenge cards are excluded from ordinary random packs.

Mechanics

3D Vectors

Three-dimensional vector notation, directed vectors, magnitude and unit vectors, parallelism, collinearity and 3D problem solving.

3D Vectors: Ordinary skills mix column-vector and i, j, k component notation wherever appropriate. Bold k is the third basis vector; italic k is reserved for a scalar. Challenge cards are excluded from ordinary random packs.

Mechanics → Kinematics

Graphs of motion

Interpret position–time and velocity–time graphs, connect gradients with rates of change, and connect signed areas with displacement.

Sketch a motion graph from words

Translate a described journey into a displacement–time or velocity–time graph

Interpret curved velocity–time graphs

Compare gradients and areas, identify greatest acceleration and assess models

Graphs of motion: Position and displacement use metres; time uses seconds. Areas above and below the time axis are distinguished clearly, and worked graphs highlight only the gradient or area needed.

Mechanics → Kinematics

Variable acceleration

Connect displacement, velocity and acceleration using calculus, then explore standard-function models, direction changes and motion extrema.

Variable acceleration: The cards use straight-line motion with displacement in metres and time in seconds. Trigonometric arguments are in radians. Worked answers distinguish displacement from distance and signed velocity from speed.

Mechanics

Forces and equilibrium

Identify forces, draw force diagrams, use equilibrium and apply Newton’s second law to accelerating bodies.

Form differential equations from forces

Draw the forces and use F = m dv/dt with velocity-dependent resistance

Solve motion with velocity-dependent resistance

Form and solve the equation of motion, use the initial velocity and interpret the limit

Mechanics quantities and units

Recognise SI units, equivalent notation and quantities without units

Convert units before calculating

Convert speed, time, distance or mass before applying a mechanics formula

Mechanics → Further kinematics

SUVAT

Choose and apply the constant-acceleration formulae, then practise AQA vertical-motion conventions and other applications.

Derive the SUVAT formulae from a velocity–time graph

Use gradient for acceleration and trapezium area for displacement, then eliminate variables algebraically.

Choose the appropriate SUVAT formula

List s, u, v, a, t, identify the known values and choose the formula with only one unknown.

Choose and use the appropriate SUVAT formula

Questions are mixed across all five formulae and include the distinction between velocity and speed.

Vertical projection and maximum height

Use v=0 at the greatest height and apply the accuracy implied by the stated value of g.

Vertical motion from a height

Find either an impact time or the speed at the moment of impact, using one consistent sign convention.

Powered motion followed by motion under gravity

Treat the powered and free-flight stages separately, carrying exact or unrounded values into the second stage.

Simultaneous vertical particles

Find either when or where the particles meet, using the same time variable for both particles.

Model reaction and braking distances

Add constant-speed reaction distance to constant-deceleration braking distance and interpret whether a collision occurs.

AQA gravity convention: Treat every stated value as exact except g. Round final answers to 1, 2 or 3 significant figures when g is given as 10, 9.8 or 9.81 respectively. Carry unrounded values between parts. If no numerical value of g is given, leave the answer in terms of g unless it cancels.

Pure · Trigonometry

Reciprocal Trigonometric Functions

Sec, cosec and cot: exact values, graphs, identities, equations and proofs.

Pure · Trigonometry

Inverse Trigonometric Functions

Principal branches, exact values and equations, with sketches of arcsine, arccosine and arctangent.

Pure

Differential Equations

Separate variables, apply initial conditions and solve differential-equation models in context.

Form a differential equation from words

Translate proportionality and rates of change into an equation with a positive constant

Pure

Trigonometry: Harmonic Forms

Combine sine and cosine, solve equations and find extreme values.

Mechanics

Projectiles

Resolve motion, apply SUVAT and find trajectory equations.

Mechanics

Kinematics: 2D SUVAT

Apply constant-acceleration formulae to vectors, positions and bearings.

Mechanics

Kinematics: 2D Variable Acceleration

Differentiate and integrate vector motion models, using initial conditions and directions.

Pure

Reduction to Linear Form

Use transformed variables to fit exponential and power models, recover constants and make predictions.

Pure

Binomial Series

Expand, combine and apply series with negative and fractional powers.

Pure

Parametric Equations: Introduction

Explore parametric relationships through calculation and reasoning.

Pure

Parametric Equations: Differentiation

Explore parametric relationships through calculation and reasoning.

Pure

Parametric Equations: Integration

Explore parametric relationships through calculation and reasoning.

Form a parametric integral

Include the change from x to the parameter

Parameter limits and branches

Select the limits for the specified region

Polynomial areas

Substitute, expand and integrate

Exponential areas

Combine exponentials before integrating

Trigonometric areas

Use identities or reverse the chain rule

Rational areas

Simplify or divide before integrating

Signed integrals

Connect signs, direction and geometric area

Enclosed and loop areas

Use signed boundary contributions

✦ Challenge

Areas using integration by parts

Choose with LATE and retain the limits

✦ Challenge

Pure

Differentiation: Connected Rates of Change

Use the chain rule to connect changing quantities.

Pure

Differentiation: Optimisation

Build a model, apply a constraint and find the optimum.

Pure

Differentiation: Implicit Differentiation

Differentiate both sides, find gradients and explore inverse functions.

Mechanics

Forces: Blocks on Slopes

Resolve forces, describe motion and connect particles on inclined planes.

Mechanics

Connected Particles

Strings, rods, pulleys, lifts and contact forces between stacked bodies.

Mechanics

Friction

Limiting friction, rough horizontal surfaces, angled pushes and pulls, and stopping motion.

Mechanics

Moments

Calculate turning effects and use force and moment equations to solve rigid-body equilibrium problems.

Mechanics → Mathematical modelling

Modelling assumptions

Recall what standard mechanics assumptions mean, then apply them in traditional A-Level Maths exam contexts and identify invalid conclusions.

Interpret and apply mechanics modelling assumptions

Move between the stated assumption, its mathematical consequence and the correct use of that consequence in a traditional A-Level Maths question.

Modelling assumptions: A conclusion is used only when the necessary assumption has been stated. Cards distinguish assumptions that are often confused, such as particle versus negligible resistance, smooth versus light, and limiting equilibrium versus ordinary equilibrium.

Pure → Sequences & series

Inductive definitions & recurrence relations

Generate and interpret recursively defined sequences, find limits, classify sequence behaviour, investigate periodicity, and establish increasing or decreasing behaviour.

Inductive definitions: Limit questions state when convergence may be assumed. Fixed-point solutions are not automatically treated as actual limits, and increasing/decreasing claims use the strict definitions uₙ₊₁>uₙ and uₙ₊₁<uₙ for all positive integers n.

Pure → Sequences & series

Arithmetic sequences & series

Find and use the nth term of an arithmetic sequence, work with sigma notation, calculate arithmetic-series sums, and model realistic situations.

Arithmetic sequences & series: The nth-term formula u_n=a+(n−1)d is used explicitly. Sum questions choose the most natural of the two standard arithmetic-series formulae. Modelling cards use realistic fixed-change contexts and require students to decide whether a term or a cumulative sum is needed.

Pure → Sequences & series

Geometric sequences & series

Build fluency with geometric sequences first: identify the common ratio, use u_n=ar^(n−1), solve consecutive-term problems, and recover a sequence from two stated terms.

Geometric sequences & series: Questions use exact values. Negative ratios are bracketed clearly, fractional ratios remain exact, non-consecutive-term questions use an odd index gap where needed, and series cards use S_n=\frac{a(1-r^n)}{1-r} and S_∞=\frac{a}{1-r} only when valid.

Pure

Exponentials & logarithms

Move between exponential and logarithmic notation, use the laws of logarithms, solve logarithmic and exponential equations, and sketch exponential and logarithmic functions.

Exponentials & logarithms: Logarithm-law cards preserve exact forms. Equation cards show the natural logarithm being applied to both sides before rearranging. Inequality cards state when division by a negative logarithm reverses the inequality, domain restrictions are checked when roots may be invalid, and hidden-quadratic cards show the substitution before factorising.

Pure → Sequences & series

Pascal's triangle and nCr

Calculate combinations, simplify expressions involving combinations, and solve equations in n.

Pascal's triangle and nCr: Non-calculator calculations are chosen to cancel cleanly. Equation cards use only values of n for which every combination shown is defined.

Pure → Sequences & series

Binomial expansion

Calculate combinations using nCr, expand binomials, find coefficients, and use partial expansions for approximations.

Binomial expansion: Full expansions use powers no greater than 6. Larger powers appear only when students need a few terms or a particular coefficient, so unnecessary expansion is avoided.

Statistics → Data presentation & interpretation

Central tendency and variation

Work with mean, population variance and standard deviation, quartiles, outliers, box plots and histograms.

Read cumulative-frequency graphs

Estimate medians, quartiles, percentiles and interquartile ranges from a supplied graph

Cumulative frequency: counts and probabilities

Read numbers and proportions above, below or between given values

Central tendency and variation: Population standard deviation is used throughout. Grouped data use class midpoints for estimated mean and variation; median and quartile calculations are never requested from grouped data.

Statistics

Statistical sampling

Recognise, describe, calculate with and evaluate the main A-Level sampling methods.

Choose a sampling method

Recommend and justify a practical method for a stated population and purpose

Statistical sampling: Convenience and opportunity are treated as equivalent names, as are self-selection and volunteer sampling. Quota and convenience sampling are non-random. Simple random sampling is used within strata and to select clusters.

Pure → Differentiation

Basic differentiation

Build the power rule from simple gradients, extend it to negative and fractional indices, rewrite roots and reciprocals in index form, and then find gradients at specified points.

Basic differentiation: Worked answers use the power rule explicitly. Roots, reciprocals and quotients with monomial denominators are rewritten in index form before differentiation, exact fractional indices are retained, and gradient-at-a-point cards differentiate before substituting.

Pure → Differentiation

Differentiation from first principles

Build the derivative from the limit definition, beginning with the algebra of f(x+h), then extend the method from polynomials to reciprocal, square-root and exponential functions.

Find and expand f(x+h)

Substitute x+h into procedurally generated linear, quadratic and cubic functions, then expand fully.

Simplify the difference quotient

Form and simplify [f(x+h)-f(x)]/h before taking any limit.

Differentiate a linear or quadratic function from first principles

Show the full limiting process for functions of the form ax²+bx+c, including straight-line cases.

Differentiate a cubic function from first principles

Expand (x+h)³, cancel the original function and take the limit.

Find a gradient at a point from first principles

Use lim [f(a+h)-f(a)]/h directly for linear, quadratic and cubic functions.

✦ Optional extension

Differentiate a reciprocal function from first principles

Combine the fractions in the difference quotient and cancel h before taking the limit.

Available

✦ Optional extension

Differentiate a square-root function from first principles

Multiply by the conjugate to simplify the difference quotient exactly.

Available

✦ Optional extension

Differentiate e^x from first principles

Factor out e^x and use lim (e^h-1)/h=1.

Available

First principles: Every worked answer begins from the limit definition. Algebraic cancellation is shown before the limit is evaluated; reciprocal and square-root cases do not use the power or chain rule.

Pure → Differentiation

Differentiating exponentials

Differentiate functions involving e^(kx), both alone and alongside algebraic terms, simplifying first where needed.

Differentiate ae^(kx)

Start with e^x, then mix coefficients and positive or negative values of k in one procedural skill.

Rewrite an exponential expression, then differentiate

Simplify products and quotients such as 4/e^x=4e^-x before differentiating.

Differentiate exponential and algebraic terms together

Differentiate functions of the form y=ae^(bx)+cx^d, including later questions where the algebraic term must be rewritten first.

Find a gradient involving e^(kx)

Differentiate first, then substitute a stated value of x, keeping exact answers in terms of e.

Differentiating exponentials: Worked answers rewrite exponential or algebraic terms where needed, use upright e and derivative notation with upright d, and retain exact values when finding gradients.

Pure → Differentiation

Standard derivatives

Differentiate sine, cosine, tangent, natural-logarithmic and arbitrary-base exponential functions, then mix them in longer expressions.

Differentiate a sin(kx)

Start with sin x, then introduce integer and fractional values of k and an outside coefficient.

Differentiate a cos(kx)

Build carefully from d/dx cos x = −sin x, retaining the correct sign as coefficients are introduced.

Differentiate a tan(kx)

Use d/dx tan(kx) = k sec²(kx), with a gradual introduction of coefficients.

Differentiate a ln(kx)

Recognise that d/dx ln(kx) = 1/x, then include an outside coefficient.

Differentiate ca^x

Apply d/dx a^x = ln(a)a^x, with positive integer bases and later outside coefficients.

Differentiate a mixture of standard functions

Begin with two familiar terms, then build to expressions such as 4 sin(8x) + 5 tan(3x) + ln(200x) + 20^x.

Standard derivatives: Trigonometric arguments are in radians, sec²(kx) is retained for tangent derivatives, logarithmic answers are simplified, and factors such as ln(a) remain exact.

Pure → Differentiation

Function family

Pure → Differentiation

Tangents and normals

Use differentiation to find equations of tangents and normals, then use the sign of the derivative to determine where a curve is increasing or decreasing.

Find the equation of a tangent to a polynomial curve

Differentiate a polynomial, find the gradient at a given point and use the point-gradient form of a straight line.

Find the equation of a normal to a polynomial curve

Find the tangent gradient first, then use the negative reciprocal to form the normal through the stated point.

Find tangents to mixed algebraic functions

Include roots, reciprocals and quotients that require preparing the vegetables before differentiating.

Find normals to mixed algebraic functions

Prepare the function, find the tangent gradient and use its negative reciprocal.

Find tangents involving e^(kx)

Form exact tangent equations, including later examples away from x=0.

Find normals involving e^(kx)

Find the exact tangent gradient, then use its negative reciprocal.

Determine whether a curve is increasing or decreasing at a point

Begin with simple polynomials, then introduce fuller polynomials, prepared algebraic terms and exponentials before mixing function types.

Find where a curve is increasing or decreasing

Start with quadratic curves, then move to cubics; solve dy/dx>0 or dy/dx<0 and give the result in set or interval notation.

Tangents and normals: Worked answers always differentiate before substituting. Tangent gradients match the curve; normal gradients are negative reciprocals. Roots, reciprocals and quotients are rewritten in index form first, and increasing/decreasing conclusions are justified from the sign of the derivative.

Pure → Differentiation

Stationary points

Connect a curve to its gradient function, solve dy/dx=0, and find the full coordinates of stationary points.

Sketch the gradient function of a curve

Use zero gradients at stationary points and the signs of the gradients between them to sketch the corresponding gradient function.

Find the stationary points of a polynomial curve

Start with quadratic curves with one stationary point, then move to cubics with two stationary points.

Find stationary points involving e^(kx)

Solve the exponential equation exactly and give the full stationary-point coordinates in logarithmic form.

Find stationary points of trigonometric curves on a stated interval

Mix exact stationary points for single sine or cosine curves with x-coordinates to 3 significant figures for y=p sin(kx)+q cos(kx).

Find stationary points involving natural logarithms

Mix a ln(kx) with a linear term and give the exact coordinate.

Find stationary points involving a^x

Mix an arbitrary-base exponential with a linear term and retain the logarithmic coordinate exactly.

Find stationary points involving mixed standard functions

Consolidate solvable logarithmic, natural-exponential, arbitrary-base exponential and trigonometric cases.

Stationary points: Worked answers always solve dy/dx=0 first. Complete coordinates are found where requested; numerical trigonometric questions may ask only for x-coordinates. Classification is taught in the following second-derivatives topic.

Pure → Differentiation

Second derivatives & points of inflection

Differentiate twice, use the sign of d²y/dx² to study shape and stationary points, and test candidates carefully before declaring a point of inflection.

Find second derivatives

Find both dy/dx and d²y/dx², beginning with polynomials before introducing e^(kx).

Determine whether a curve is convex or concave at a point

Evaluate the second derivative at the stated value: positive means convex and negative means concave.

Find intervals of convexity and concavity

Begin with cubic curves, then move to quartics; solve second-derivative inequalities and mix set and interval notation.

Find and classify stationary points using the second derivative

Find complete coordinates, then use a positive second derivative for a local minimum and a negative one for a local maximum.

Handle an inconclusive second-derivative test

When d²y/dx²=0 at a stationary point, test the first derivative 0.1 either side to classify it.

Find non-stationary points of inflection

Check that the second derivative changes sign and that the first derivative is non-zero at the point.

Find and identify all points of inflection

Test every candidate for a sign change, find its coordinates and state whether each point of inflection is stationary or non-stationary.

Second derivatives: A positive second derivative means convex and a negative one means concave. At a stationary point this gives a local minimum or maximum respectively. A zero second derivative is inconclusive, and a point is only a point of inflection when the second derivative changes sign.

Pure

Integration

Reverse differentiation, evaluate definite integrals exactly, and use integration to find signed integrals and areas.

Rectangle estimates

Estimate a definite integral using left or right endpoint rectangles

Judge rectangle estimates

Use increasing or decreasing behaviour to explain an overestimate or underestimate

Integration as the limit of a sum

Translate a limiting rectangle sum into a definite integral and evaluate it

Integration conventions: Every indefinite integral includes +c, but a definite integral does not. A definite integral is signed; an area is always positive and must be split at any crossing of the x-axis.

Pure

Numerical Methods

Use numerical methods to approximate values that cannot be obtained conveniently by exact methods.

Trapezium Rule conventions: Answers are explicitly identified as estimates and are normally given to 3 significant figures. There are n+1 ordinates for n strips.

Pure → Differentiation

Chain Rule

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