sierra2WMA13 Pure Mathematics 3, exam Wed 21 Oct, afternoon
Contents

P3 in 22 moves

Every question in the 19 P3 papers from January 2020 to June 2026 is built from the same 22 moves. Each one is below as a recipe, with the marks it earns written in the margin the way the mark scheme hands them out, and a worked example you reveal one step at a time.

57/75needed for an A in June 2026 (A* was 63)
1 h 3075 marks, so about 70 seconds a mark
5½ hoursto learn the 22 moves, then timed papers

Where the 75 marks come from

Differentiation
12.3
Functions & transformations
9.3
Trig identities & equations
9.3
Modulus graphs
8.6
Integration & division
8.3
Exponential models
7.2
Log graphs
5.7
x = f(y), dy/dx
5.3
Iteration & sign change
5.0
R form
4.1
0510152025

Bars are the average marks per paper; lines run from the lowest to the highest of the 19 papers. Each figure is roughly ±2, since mixed questions were split by marking point.

Your plan

For each move: read the recipe, cover it, then work the example with “First step” so you check yourself one line at a time. Finish with one drill question from a 2020–2024 paper, marked with its mark scheme. The 2025–2026 papers stay unseen for timed mocks.

Session 1: learn (about 5½ hours, plus breaks)

  1. Toolkit (25 min)The basics every move leans on. Skim what you already know.
  2. Block 1: moves 1–5, differentiation (75 min)About 18 marks a paper. Then do the Differentiate cards in Quick-fire.
  3. Block 2: moves 6–9, exponentials, logs, iteration (60 min)About 18 marks, and the most predictable ones. Then the Logs and Iteration cards.
  4. Block 3: moves 10–15, functions and modulus (60 min)About 18 marks. Then the Functions and Modulus cards.
  5. Block 4: moves 16–19, trigonometry (75 min)About 13 marks. Learn the six identities under Formulas first. Then the Trig cards.
  6. Block 5: moves 20–22, integration (40 min)About 8 marks. Then the Integrate cards.
  7. Mock 1: June 2026 (1 h 30, timed)Formula booklet only. Mark it strictly and note which moves lost marks.

Then: papers

  1. Fix list (45 min)Redo every quick-fire card marked “Missed it”, and one drill for each move you dropped marks on in mock 1.
  2. Mock 2: October 2025, timed, then markAn October paper with the least structured questions recently (Q7 and Q8).
  3. Mock 3: January 2026, timed, then markSpares if you have time: June 2025, then January 2025.
  4. Last 20 minutes before the examFormulas, Spot the pattern, and your “To redo” cards.

October 2026: what’s most likely

These are patterns from 19 papers, not leaks. Learn all 22 moves; use this list to decide what to over-prepare.

  1. Near-certainInverse and composite functions (moves 10, 11). An inverse in 18 of 19 papers, and 9 of the last 13 then asked you to solve an equation built from fg, gf or f−1.
  2. Near-certainIteration (move 9). In all 19 papers, usually after a sign-change “show there’s a root” (14 of 19). The easiest 5–7 marks on the paper.
  3. Near-certainA trig “show that”, then “hence solve” (moves 16, 17, 18). In 18 of 19 papers. Usually it ends in a quadratic in one trig function, or tan θ = k.
  4. Near-certainProduct or quotient rule with stationary points (moves 1, 2). Stationary points in 17 of 19 papers.
  5. Near-certainA modulus graph (moves 13, 14, 15). In all 19 papers. June 2026 only gave it 5 marks, so expect a full 8–10 mark question.
  6. Very likelyAn exponential model (moves 6, 7). In 18 of 19 papers: a constant, a time, a rate of change and a limit. The fraction-shaped kind was last seen in October 2025.
  7. Very likelyA log graph turned into a model (move 8). In 18 of 19 papers, with a constant to interpret in context.
  8. LikelyThe R form (move 19). In 11 of 19 papers. It was in June 2025, October 2025 and January 2026, then missing from June 2026.
  9. Likelyx in terms of y, then dy/dx (move 5). In 14 of 19 papers, including both 2026 papers, often ending with a tangent or normal.
  10. LikelyDivide, then an exact ln integral (move 21). In 10 of 19 papers, latest January 2026.
  11. Worth 20 minutessin 3x = 3 sin x − 4 sin3x (move 18). Three of the six October papers used it (2020, 2022 and 2024), and June 2025 used tan 3x.

Expect one question near the end with fewer steps than you’d like, as in October 2025 (Q7: a range of k for two turning points, 7 marks and no hints; Q8: a 13-mark modulus and cubic question). The marks still go to the standard moves, so write down the first move even if you can’t finish.

Paper by paper: marks per topic in all 19 papers

Darker means more marks. The outlined column is June 2026, the paper just before yours. On a phone, swipe the table sideways.

TopicJan
20		Oct
20		Jan
21		Jun
21		Oct
21		Jan
22		Jun
22		Oct
22		Jan
23		Jun
23		Oct
23		Jan
24		Jun
24		Oct
24		Jan
25		Jun
25		Oct
25		Jan
26		Jun
26
Differentiation1717441782513141115181217126969
Functions & transformations8414106119768101491210119811
Trig identities & equations899991389915103111398979
Modulus graphs94981010812899114781013105
Integration & division101378116109125441189451011
Exponential models63111367–49877329812912
Log graphs5677687–56676635567
x = f(y), dy/dx–410577–5–89–105911–46
Iteration & sign change5643353765563565765
R form79–8––596––56––769–

Toolkit

P3 assumes P1 and P2. These are the bits of them that come up in almost every question.

  • Log laws (the same for ln and ): , , , , .
  • e and ln undo each other: and . So gives , and gives .
  • Hidden quadratics: is with . Trig works the same way with .
  • Exact values: , , , , , . In radians , , , .
  • Partners in a range: for sin use , for cos use , for tan use ; then add or subtract () to cover the range.
  • Multiple angles: for with , solve for in , then halve.
  • Calculator mode: radians whenever you see or “radians”, degrees when you see °. The wrong mode loses a whole question.
  • From P1 and P2: , , stationary points have , and the discriminant is .

The 22 moves

M1A method mark: for the right method, even with a slip in it.A1An accuracy mark: needs the M mark before it.B1An independent mark, for a fact or a value on its own.dM1A method mark that needs the method mark before it.A1*The answer is printed, so every step has to be shown with no slips.ftFollow-through: a wrong earlier answer used correctly still scores.

The margin shows the usual pattern from the mark schemes. The exact split changes from paper to paper.

Block 1: Differentiation

about 75 min
1

Product rule, then stationary points

Stationary points in 17 of 19 papers, usually 6–9 marks

Looks like“Show that . Hence find the exact coordinates of the stationary points.”

The recipemarks
  1. Name the two parts and , and differentiate each (chain rule inside them).
  2. Write .
    M1 A1
  3. Take out the common factors at their lowest powers (, shared brackets) and tidy what’s left.
    dM1 A1
  4. Set each factor to 0. is never 0, so ignore it.
    M1
  5. Put each back into for . Keep e, ln and surds exact.
    A1 A1

Try it, then reveal one step at a time. Find in factorised form and the exact stationary points.

  1. and
  2. M1 A1
  3. dM1 A1
  4. or
    M1
  5. and
    A1 A1
  • A minus sign in the product rule. It’s ; only the quotient rule has a minus.
  • Taking out the wrong power: with and in the same line, take out .

Drill: Jun 2023 Q8, Oct 2022 Q3, Oct 2023 Q7

2

Quotient rule, then increasing or decreasing

The quotient rule turns up in almost every paper; “increasing” in 6 of 19

Looks like“Use calculus to find the range of values of for which is increasing.”

The recipemarks
  1. Top is , bottom is . Find and .
  2. (printed in the booklet).
    M1 A1
  3. Expand the top, simplify, factorise.
    dM1 A1
  4. The bottom is always positive, so the top decides: increasing when top , decreasing when top .
    M1 A1
  5. “Show is decreasing”: write for all and say why, e.g. a negative number over a square.
    A1

Try it, then reveal one step at a time. Find the values of for which is increasing.

  1. and
  2. M1 A1
  3. dM1 A1
  4. , so increasing when : or
    M1 A1
  • Writing on top. The order matters.
  • Forgetting to square the bottom.

Drill: Jan 2020 Q4, Jan 2024 Q4b, Oct 2020 Q3

3

Tangents and normals

In 11 of 19 papers, 3–5 marks

Looks like“Find an equation of the normal to at , giving your answer in the form .”

The recipemarks
  1. Differentiate (often done in an earlier part).
    M1
  2. Put the point’s into : that’s the tangent gradient .
    M1
  3. For a normal, use .
    dM1
  4. , then rearrange to the form they ask for, with integers if told.
    A1

Try it, then reveal one step at a timeFind the normal to at the point where .

  1. M1
  2. At : and the gradient is
    M1
  3. Normal gradient :
    dM1
  4. A1
  • Using the tangent gradient in a normal.
  • Leaving fractions when they asked for integers , , .

Drill: Jun 2022 Q1, Jun 2024 Q6a, Oct 2024 Q9b

4

A trig “show that” for a derivative

June 2022, October 2022 and January 2024, 4–6 marks

Looks like“Show that .”

The recipemarks
  1. Quotient or product rule as usual.
    M1 A1
  2. Expand, then swap for 1 (or use ).
    M1
  3. Multiply top and bottom by whatever turns it into the printed form, often or .
    M1
  4. Arrive at exactly the printed form, with no slips anywhere.
    A1*

Try it, then reveal one step at a timeShow that .

  1. M1 A1
  2. M1
  3. A1*
  • Skipping the line where becomes 1. That line is a mark.

Drill: Oct 2022 Q6, Jun 2022 Q9a, Jan 2024 Q6

5

in terms of , then

In 14 of 19 papers, including both 2026 papers, 4–10 marks

Looks like“. Find , hence show that .”

The recipemarks
  1. Differentiate with respect to .
    M1 A1
  2. Flip it: .
    M1
  3. Replace the terms with using an identity (, ). Pick the root’s sign from the range of .
    M1
  4. Simplify to the printed form.
    A1
  5. Tangent or normal: get from the given , then as in move 3. Vertical tangent: .
    M1 A1

Try it, then reveal one step at a time. Show , then find the tangent where .

  1. M1 A1
  2. M1
  3. A1
  4. At : , gradient
    M1
  5. A1
  • Leaving a in an answer that should be in terms of .
  • The wrong sign on the square root. Check it against the range of .

Drill: Oct 2023 Q10, Jun 2024 Q9, Oct 2024 Q2, Jan 2021 Q10

Block 2: Exponentials, logs and iteration

about 60 min
6

Exponential models:

In 18 of 19 papers, 6–12 marks

Looks like“Find the initial temperature. Show that . Find the rate of change when . Explain why can never be 15.”

The recipemarks
  1. Initial value: put , and .
    B1
  2. A constant from a data point: substitute, get on its own, take ln.
    M1 dM1 A1
  3. The time when is some value: the same moves, solved for .
    M1 dM1 A1
  4. Rate of change: differentiate, , then substitute . Give units.
    M1 A1
  5. Long term: , so . Use that to explain any limit.
    B1

Try it, then reveal one step at a time (°C, in minutes), and when .

  1. : °C
    B1
  2. M1
  3. dM1 A1
  4. at : cooling at 2.94 °C per minute
    M1 A1
  5. for all , so and it never reaches 15 °C
    B1
  • Rounding and reusing the rounded value. Store it in the calculator.
  • Losing the minus: .
  • Units: if is in thousands, 100 means 100 000.

Drill: Jan 2024 Q5, Oct 2023 Q4, Jan 2021 Q5, Oct 2021 Q3

7

Fraction models and two models together

Fraction models in 5 papers (latest October 2025); two models in June 2023, January 2026 and June 2026

Looks like“. Find at the start, its long-term value, and when .”

The recipemarks
  1. Start: .
    B1
  2. Long term: divide top and bottom by the exponential, then let .
    B1
  3. Solve : cross-multiply, collect the exponential terms, take ln.
    M1 dM1 A1
  4. Maximum or rate: quotient rule for ; set it to 0 for a maximum.
    M1 A1
  5. Two models equal: set them equal. With and , put to get a quadratic.
    M1 dM1 A1

Try it, then reveal one step at a time

  1. :
    B1
  2. B1
  3. M1
  4. dM1 A1
  • Not multiplying every term when you cross-multiply.
  • Throwing away a root of the quadratic without checking it: just has to be positive.

Drill: Jan 2023 Q10, Jan 2022 Q4, Jun 2023 Q7, Jan 2020 Q1

8

Log graphs to or

In 18 of 19 papers, 5–8 marks

Looks like“The line of against passes through and . Find and in .”

The recipemarks
  1. Read the horizontal axis. or means . means .
  2. Line: gradient from the two points, intercept where it crosses the vertical axis.
    M1 A1
  3. Undo the logs: and (or for the power model).
    M1 dM1 A1
  4. Interpret: is the value at the start; is the factor per unit, so 1.19 means up 19% each year.
    B1
  5. Rate of change: .
    M1 A1

Try it, then reveal one step at a timeA line of against passes through and .

  1. , so
    M1 A1
  2. M1 dM1 A1
  3. At :
    M1 A1

Try it, then reveal one step at a timePower version: a line of against through and .

  1. M1 A1
  2. M1 A1
  • Giving as the gradient itself instead of .
  • Using rounded and later: gives 28.5, not 29.1.
  • Using ln when the axis says .

Drill: Jun 2022 Q4, Oct 2024 Q4, Oct 2021 Q7, Jan 2020 Q3

9

Sign change and iteration

Iteration in all 19 papers, sign change in 14, usually 5–7 marks

Looks like“Show that lies between 1 and 2. Show the equation can be written as . Using , find and to 4 d.p.”

The recipemarks
  1. Work out and and show their signs.
    M1
  2. Write: “sign change, and is continuous on , so there is a root”.
    A1
  3. Rearrange into the printed , every step shown.
    B1
  4. Iterate with the Ans key: type , press =, type , keep pressing =. Write (and ) to 4 d.p.
    M1 A1
  5. Keep pressing until the 4 d.p. value stops changing: that’s .
    A1
  6. “Show to 4 d.p.”: test and for a sign change, plus continuity and a conclusion.
    M1 A1

Try it, then reveal one step at a time

  1. and
    M1
  2. Sign change and continuous on , so a root lies in
    A1
  3. B1
  4. : ,
    M1 A1
  5. (4 d.p.)
    A1
  6. and : sign change, continuous, so
    M1 A1
  • Degrees mode when the formula has trig in it. Iterations are in radians.
  • Writing only . Show , or a calculator-only answer can score nothing.
  • Leaving out the word “continuous”.

Drill: Oct 2023 Q1, Jan 2024 Q2, Jan 2023 Q9, Jan 2021 Q6

Block 3: Functions and modulus

about 60 min
10

Range, composite values and inverses

An inverse in 18 of 19 papers, 6–10 marks with the rest of the question

Looks like“State the range of . Find and state its domain. Find .”

The recipemarks
  1. Range: use the turning point, the asymptote or the end of the domain. Write it with , not .
    M1 A1
  2. : work out first, then put that into .
    M1 A1
  3. Inverse: write , make the subject, then swap the letters.
    M1 A1
  4. Domain of = range of .
    B1

Try it, then reveal one step at a time and

  1. and , so
    B1
  2. M1 A1
  3. M1 A1
  4. B1
  • Doing in the wrong order.
  • Writing the range as .
  • No domain for .

Drill: Oct 2023 Q2, Jun 2022 Q2, Oct 2024 Q6, Jan 2023 Q1

11

Equations with , and

In 9 of the last 13 papers, 3–4 marks

Looks like“Solve .” “Find the exact value of for which .” “Solve .”

The recipemarks
  1. Build the composite as one expression (the inner function goes inside the outer one).
    M1
  2. Clear fractions and collect terms: usually a linear or quadratic equation.
    dM1
  3. Solve, then check each answer is allowed by the domain.
    A1
  4. for an increasing : solve instead, since they meet on .
    M1 A1
  5. If is two fractions, combine first: factorise the denominators, common denominator, cancel.
    M1 A1

Try it, then reveal one step at a timeWith and , solve .

  1. M1
  2. dM1
  3. A1
  • Keeping a solution that the domain rules out, especially after squaring.
  • Reading as . It means: solve .

Drill: Jun 2022 Q2c, Jan 2024 Q4d, Oct 2024 Q6c, Jan 2020 Q2c

12

Transformations of points and graphs

In 11 of 19 papers, 2–4 marks

Looks like“The point lies on . Find where goes on .”

The recipemarks
  1. Inside the bracket changes , the opposite way to how it looks: moves right 1, halves every .
    B1
  2. Outside changes , exactly as it looks: triples , adds 8.
    B1
  3. : negative values become positive. : keep the part with and mirror it in the -axis.
    B1
  4. A vertex of goes to on .
    M1 A1

Try it, then reveal one step at a time lies on . Find its image on and on .

  1. divides by 3:
    B1
  2. , so goes to
    B1
  3. On :
    B1
  • Moving left instead of right.
  • Applying the outside change before the inside one.

Drill: Jan 2024 Q1, Jan 2021 Q2, Oct 2021 Q2d

13

Modulus graphs: vertex, intercepts, sketch

A modulus graph in all 19 papers; this part is 2–5 marks

Looks like“. Find the coordinates of the vertex and of the points where the graph meets the axes.”

The recipemarks
  1. The vertex of is : a minimum if , a maximum if .
    B1 B1
  2. -intercept: put .
    B1
  3. -intercepts: solve , which gives two values.
    M1 A1
  4. Sketch the V and label all four points.
    B1 B1

Try it, then reveal one step at a time

  1. Vertex , a minimum
    B1 B1
  2. : , so
    B1
  3. or
    M1 A1
  • Vertex coordinates the wrong way round.
  • An unlabelled sketch: the labelled points carry the marks.

Drill: Jun 2023 Q6, Oct 2021 Q2, Jan 2024 Q8a

14

Modulus equations and inequalities

In almost every modulus question, 3–4 marks

Looks like“Solve .” “Find the values of for which .”

The recipemarks
  1. Sketch both graphs roughly to see which arms meet.
  2. Right arm (inside ): drop the bars. Left arm: put a minus in front of the inside. Solve each.
    M1 A1 M1 A1
  3. Check each answer really lies on the arm you used. One often doesn’t.
  4. Inequality: the “=” answers are the critical values. Read “between” or “outside” off the sketch.
    dM1 A1
  5. Two moduli, like : square both sides, then solve the quadratic inequality.
    M1

Try it, then reveal one step at a time. Solve , then .

  1. Right arm: (it’s , so valid)
    M1 A1
  2. Left arm: (it’s , so valid)
    M1 A1
  3. The line is above the V between them:
    dM1 A1
  • Keeping a fake solution from the wrong arm.
  • Writing an “outside” answer with “and” or . Use “or”.

Drill: Jun 2022 Q5b, Jan 2022 Q7b, Jun 2024 Q1b, Oct 2024 Q3

15

How many solutions? Ranges of

In 8 of 19 papers, 2–4 marks

Looks like“Given that has exactly two roots, find the range of .” “The line meets the graph at exactly two points. Find .”

The recipemarks
  1. Sketch the graph with its turning points, vertex and asymptotes.
  2. Horizontal line : slide it up and down and count crossings. The boundaries are turning-point -values and asymptotes.
    M1 A1
  3. Line through a fixed point (like ): the boundaries are the line through the vertex and lines parallel to the arms.
    M1 A1
  4. At each boundary, test the boundary case itself to decide between and .
    A1

Try it, then reveal one step at a timeFor which does meet at exactly two points?

  1. Every line passes through . The V has vertex and arms of gradient .
  2. Through the vertex: (one point)
    M1 A1
  3. Parallel to the right arm: (one point)
    M1
  4. Between them the line cuts both arms:
    A1
  • Including a boundary where the count is actually one.
  • Forgetting an asymptote is never reached, so its end is strict.

Drill: Jun 2022 Q5c, Jan 2020 Q6c, Oct 2023 Q7c, Oct 2022 Q3b

Block 4: Trigonometry

about 75 min
16

A quadratic in one trig function

In most papers, 4–5 marks

Looks like“Solve, for , the equation .”

The recipemarks
  1. Use the identity that leaves one function: , , or or .
    M1
  2. Collect into a quadratic and factorise (or use the formula).
    M1 A1
  3. Reject impossible values: and .
  4. Solve each: calculator value, then its partner in the range.
    dM1
  5. Every answer in the range, none extra, in the right units and accuracy.
    A1

Try it, then reveal one step at a timeSolve for .

  1. M1
  2. M1 A1
  3. or
    dM1
  4. A1
  • Dividing by or and losing the solutions where it’s 0. Factorise instead.
  • One extra answer inside the range costs the last mark.

Drill: Oct 2024 Q1, Jun 2021 Q2b, Oct 2023 Q8b, Oct 2020 Q1

17

Compound angles to

October 2021, June 2024, January 2025 and January 2026, about 7 marks

Looks like“Show that can be written as . Hence solve .”

The recipemarks
  1. Expand with the booklet formulas and put in exact values like .
    M1
  2. Collect terms on one side and terms on the other.
    M1
  3. Divide by to get , in exact form if asked.
    A1*
  4. “Hence solve” a shifted version: spot what plays the part of , solve for it in its shifted range, then convert back.
    M1 A1 A1

Try it, then reveal one step at a timeSolve for .

  1. M1
  2. M1
  3. A1
  4. or
    M1 A1
  • The sign in .
  • Solving the shifted equation in the unshifted range.

Drill: Oct 2021 Q4, Jun 2024 Q7

18

Proving identities, including

A trig “show that” in 18 of 19 papers, 3–4 marks

Looks like“Prove that .” “Show that can be written as .”

The recipemarks
  1. Start from the messier side.
  2. Turn sec, cosec, cot and tan into sin and cos, and expand any double angles.
    M1
  3. Combine into one fraction and use (or a variant).
    M1
  4. Cancel and land exactly on the other side. Finish with “= RHS”.
    A1*

Try it, then reveal one step at a timeProve that .

  1. M1
  2. M1
  3. A1*

Try it, then reveal one step at a timeShow that .

  1. M1
  2. M1
  3. dM1
  4. A1*
  • Using the identity you’re proving as a step in its own proof.
  • Working on both sides at once without a clear conclusion.

Drill: Jun 2021 Q2a, Oct 2023 Q8a, Oct 2024 Q5a, Oct 2020 Q5a

19

The R form:

In 11 of 19 papers but missing from June 2026, 6–9 marks

Looks like“Express in the form . Hence find the minimum value of … and the value of where it occurs.”

The recipemarks
  1. Expand the form you’re given and match: is one coefficient, is the other.
  2. , left exact.
    B1
  3. . Give in radians to 3 d.p. (radian mode).
    M1 A1
  4. Max and min: and run from to . For a fraction like , the biggest value comes from the smallest bottom.
    M1 A1
  5. Where: set the bracket to the angle that gives it (0 or for a cos max, for a sin max) and solve for .
    M1 A1
  6. Solving : in the shifted range, then add .
    M1 A1 A1

Try it, then reveal one step at a time. Write it as and use it.

  1. and
  2. B1
  3. M1 A1
  4. Max of is 13, when , so
    M1 A1
  5. has max
    M1 A1
  6. : , so or , giving or
    M1 A1 A1
  • in degrees when the question wants radians scores A0.
  • as a decimal when it says exact.
  • Not shifting the range before solving for .

Drill: Jan 2023 Q2, Jun 2021 Q9, Oct 2022 Q8, Jun 2024 Q4

Block 5: Integration

about 40 min
20

Standard integrals and spotting the derivative

Integration in all 19 papers, 2–5 marks a part

Looks like“Find .” “Find the exact value of .”

The recipemarks
  1. Rewrite into standard pieces: split fractions, use negative powers.
  2. , , ,
    M1 A1
  3. M1 A1
  4. Top is a multiple of the bottom’s derivative: .
    M1 A1
  5. A bracket times its inside’s derivative: .
    M1 A1
  6. Definite: top limit minus bottom limit, then combine logs. Indefinite: add .
    dM1 A1

Try it, then reveal one step at a timeFind exactly.

  1. The top is 3 times the derivative of
  2. M1 A1
  3. dM1 A1
  • Forgetting the : .
  • No modulus in ln when the inside can be negative, e.g. limits to in .
  • No on an indefinite integral.

Drill: Jan 2022 Q3, Jan 2021 Q9, Oct 2021 Q5, Jun 2022 Q3

21

Divide first, then an exact ln answer

In 10 of 19 papers (latest January 2026), 7–9 marks

Looks like“Find , and such that . Hence find the integral from 3 to 5 in the form .”

The recipemarks
  1. Divide (long division), or write an identity and compare coefficients.
    M1 A1 A1
  2. Integrate each piece. The remainder over the denominator becomes ln.
    M1 A1ft
  3. Put in the limits and subtract.
    dM1
  4. Combine the logs into the form asked for.
    M1 A1

Try it, then reveal one step at a timeFind exactly.

  1. M1 A1
  2. So the fraction is
    A1
  3. M1 A1 dM1
  4. M1 A1
  • Trying to integrate a top-heavy fraction without dividing.
  • Log slips: : subtracting logs gives the log of a quotient, never a quotient of logs.

Drill: Oct 2022 Q1, Jun 2024 Q2, Jan 2023 Q4, Oct 2024 Q9

22

Trig powers: an identity first

October 2020, October 2021 and October 2023 among others, 4–7 marks

Looks like“Show that , then find the exact area.”

The recipemarks
  1. Replace the power: , , , .
    M1
  2. Integrate term by term, dividing by the number in front of .
    dM1 A1
  3. Put in the limits using exact values.
    A1

Try it, then reveal one step at a timeFind .

  1. M1
  2. dM1 A1
  3. A1
  • The wrong sign: it’s and .
  • Dropping the after the identity doubles the angle.

Drill: Oct 2023 Q3, Oct 2021 Q10, Oct 2020 Q5b, Jun 2023 Q9c

Spot the pattern

The fastest revision there is. Cover the right-hand column and say the first move out loud.

If you seeYour first move
and (or and ) in one equation, then a quadratic in one function (move 16)
and together (move 16)
next to . Next to :
next to or , then factorise. Never divide.
, or similarExpand, collect, divide by to get (move 17)
with “max”, “min” or “”R form (move 19)
something in (move 5)
“Show ” with a factorised answerProduct or quotient rule, then take out common factors (moves 1–2)
“Show that lies between and ”, , signs, “continuous”, conclusion (move 9)
and “4 d.p.”Ans-key iteration, write (move 9)
A graph of against : , (move 8)
A graph of against : , gradient (move 8)
“Initial” or “at the start”Put
“In the long term” or “can never reach”, so (move 6)
“Rate of change” in a modelDifferentiate the model, then substitute
A top-heavy fraction to integrateDivide first (move 21)
The top is a multiple of the bottom’s derivative (move 20)
or to integrateDouble-angle identity first (move 22)
and “solve”Two equations, one per arm, then check each (move 14)
“Range of ”Turning point or asymptote; write , not (move 10)
Make the subject, swap letters; domain of = range of (move 10)

Quick-fire

44 short questions. Try each in your head or on paper, show the answer, then mark yourself honestly. The “To redo” filter collects everything you’ve missed.

Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
Differentiate
. Find
Integrate
Integrate
Integrate
Integrate
Integrate
Integrate
Integrate
Integrate
Logs and exponentials
Solve
Logs and exponentials
Solve
Logs and exponentials
Solve
Logs and exponentials
. Write it as
Logs and exponentials
. Write it as
Logs and exponentials
. Initial and long-term values?
Trig
Write in terms of
Trig
Write in terms of
Trig
Simplify
Trig
Solve for
Trig
Solve for
Trig
Write as
Trig
Max and min of
Functions
, . Find and
Functions
Find for
Functions
Range of
Functions
lies on . Its image on ?
Modulus
Vertex of
Modulus
Solve
Modulus
Solve
Modulus
Solve
Iteration
with . Find
Iteration
Show has a root in
Iteration
To show a root is 1.710 to 3 d.p., which interval do you test?

Formulas

The P3 specification lists these as formulas you’re expected to know, so they’re not in the booklet. Write the highlighted ones from memory twice before mock 1.

Not in the booklet: learn these

Not formulas, but learn them anyway

In the booklet: find them in the first minute

, , the quotient rule, ,

That’s the usual P3 page of the booklet. Check your own copy once so you know exactly where everything sits.

Exam rules

  • Show the method, always. Most marks are method marks, and the accuracy marks hang off them. When a question says “solutions relying entirely on calculator technology are not acceptable”, a bare answer can score zero.
  • “Show that” answers are marked for the steps. Write every line. If you can’t get there, use the printed result in the next part anyway: those marks are still yours.
  • Follow-through. A wrong number from an earlier part, used correctly, still earns the later method marks. Keep going.
  • Accuracy. “Exact” means keep surds, π, e and ln. “3 s.f.” and “4 d.p.” mean exactly that. Keep full values in the calculator until the end.
  • Radians or degrees. If the range uses π or the question says radians, answer in radians.
  • Solutions in a range. All of them and no extras: one extra inside the range costs the final mark.
  • Notation. Ranges are written with f(x) or y, never x. Every f−1 needs its domain.
  • Pace. About 70 seconds a mark. Move on after 3 stuck minutes, and keep 10 minutes at the end to check calculator mode, ranges and accuracy.

Papers

Every drill on this page comes from a 2020–2024 paper, so the 2025–2026 papers stay unseen for timed mocks. Sit them with the formula booklet only, then mark strictly.

OrderPaperWhy
1June 2026The paper just before yours, so it shows the current style.
2October 2025An October paper, with the least structured recent questions (Q7 and Q8).
3January 2026R form, x in terms of y, and two exponential models.
SpareJune 2025R form, a normal and a tangent from x in terms of y, and a 10-mark modulus question.
SpareJanuary 2025A clean all-rounder.