Edexcel IAL P4 Maths Revision: Pure Mathematics 4

Edexcel International A Level P4 (Pure Mathematics 4): every topic with an exam-style question and its mark scheme, and 261 practice questions with mark schemes.

1h 30min written paper (75 marks), calculator permitted
9 topics
261 questions

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Differential Equations · P4 paper · 4 marks · medium

Use the method of separation of variables to find the general solution to the differential equation: $\frac{d y}{d x} = \frac{x}{y}$ Express your answer in the form $y^{2} = f (x)$ .

Worked answer

Separate the variables by multiplying both sides by $y$ and $d x$ : M1
$y d y = x d x$ .
Integrate both sides: $\int y d y = \int x d x$ . M1
$\frac{y^{2}}{2} = \frac{x^{2}}{2} + C$ . A1
Multiply entirely by 2 to isolate $y^{2}$ :
$y^{2} = x^{2} + 2 C$ . (Note: $2 C$ can just be written as a new constant $K$ ). A1

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P4 topics

"Tested in" counts come from our question-by-question mapping of the 4 Pearson P4 papers from June 2024 to June 2025.

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P4 past papers

Official Pearson P4 papers and mark schemes, with a timer: all IAL P4 past papers. Most recent: June 2025 · January 2025 · October 2024 · June 2024.

Examiner Insights for IAL P4: every examiner report since June 2024, the top mistakes and what to focus on next →

Knowledge organiser: Pure Mathematics 4 (P4) on one page (key facts, formulas and mistakes to avoid; printable PDF)

P4 Vectors

Vectors in 2D and 3D; magnitude, unit vectors, scalar product and angles; vector equation of a line; intersecting, parallel and skew lines.

57 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise vectors →

Example question · 7 marks · hard

Line $L_{1}$ has vector equation $r_{1} = (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) + λ (\begin{matrix} 2 \\ 1 \\ 1 \end{matrix})$ and line $L_{2}$ has equation $r_{2} = (\begin{matrix} 5 \\ - 3 \\ 0 \end{matrix}) + μ (\begin{matrix} - 1 \\ 2 \\ 1 \end{matrix})$ .

Show that $L_{1}$ and $L_{2}$ intersect.
Find the exact coordinates of the point of intersection.

Show the mark scheme

M1 for equating the parametric equations: $1 + 2 λ = 5 - μ$ , $λ = - 3 + 2 μ$ , and $1 + λ = μ$ .
M1 for taking two equations to solve for the parameters, e.g., $λ = μ - 1$ into $2 λ + μ = 4$ .
M1 for $2 (μ - 1) + μ = 4 ⟹ 3 μ - 2 = 4 ⟹ 3 μ = 6$ .
A1 for finding $μ = 2$ and $λ = 1$ .
B1 for verifying with the unused equation: $λ - 2 μ = 1 - 4 = - 3$ , which matches the second component equation, showing the lines intersect.
M1 for substituting $λ = 1$ into $L_{1}$ (or $μ = 2$ into $L_{2}$ ).
A1 for the intersection point $(3, 1, 2)$ .

P4 Partial Fractions

Partial fractions with distinct linear and repeated linear factors in the denominator (improper fractions by division first). Irreducible-quadratic denominators are not on the IAL specification.

43 practice questions with mark schemes · Worked examples → · Notes → · Practise partial fractions →

Example question · 7 marks · hard

Consider the rational function $f (x) = \frac{3 x}{(x - 1) (x + 2)}$ .

Express $f (x)$ in partial fractions.
Hence, evaluate exactly the definite integral $\int_{2}^{3} f (x) d x$ , giving your answer in the form $\ln (k)$ where $k$ is a rational number.

Show the mark scheme

M1 for setting up the partial fraction decomposition: $\frac{3 x}{(x - 1) (x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$ .
M1 for clearing the denominators: $3 x = A (x + 2) + B (x - 1)$ .
M1 for substituting strategic values of $x$ (e.g., $x = 1$ and $x = - 2$ ) to solve for constants.
A1 for finding $A = 1$ and $B = 2$ , yielding $\frac{1}{x - 1} + \frac{2}{x + 2}$ .
M1 for setting up the integral and determining the anti-derivatives: ${[\ln | x - 1 | + 2 \ln | x + 2 |]}_{2}^{3}$ .
M1 for substituting the limits: $(\ln 2 + 2 \ln 5) - (\ln 1 + 2 \ln 4)$ .
A1 for using logarithm laws to combine into a single logarithm: $\ln 2 + \ln 25 - 0 - \ln 16 = \ln (\frac{50}{16}) = \ln (\frac{25}{8})$ .

P4 Parametric Equations

Parametric equations of curves; converting between parametric and Cartesian forms; parametric differentiation; tangents and normals.

15 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise parametric equations →

Example question · 8 marks · hard

A curve $C$ has parametric equations $x = t^{2}$ , $y = t^{3} - 3 t$ .
(a) Find $\frac{d y}{d x}$ in terms of $t$ . (3)
(b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
(c) Find the Cartesian equation of the tangent at $t = 2$ . (2)

Show the mark scheme

(a) $\frac{d x}{d t} = 2 t$ , $\frac{d y}{d t} = 3 t^{2} - 3$ M1 A1. $\frac{d y}{d x} = \frac{3 t^{2} - 3}{2 t} = \frac{3 (t^{2} - 1)}{2 t}$ A1.
(b) Horizontal iff $\frac{d y}{d t} = 0$ : $3 t^{2} = 3 \Rightarrow t = \pm 1$ M1. $t = 1$ : $(1, - 2)$ ; $t = - 1$ : $(1, 2)$ A1 A1.
(c) At $t = 2$ : $(x, y) = (4, 2)$ ; gradient $= \frac{3 \cdot 3}{4} = \frac{9}{4}$ ; equation: $y - 2 = \frac{9}{4} (x - 4)$ , i.e. $y = \frac{9}{4} x - 7$ M1 A1.

P4 Differential Equations

First-order separable differential equations, forming DEs from context, general and particular solutions.

13 practice questions with mark schemes · Worked examples → · Notes → · Practise differential equations →

Example question · 6 marks · hard

Find the exact particular solution to the differential equation: $\frac{d y}{d x} = y \cos x$ given the boundary condition that $y (0) = e^{2}$ .

Show the mark scheme

Separate variables: $\frac{1}{y} d y = \cos x d x$ . M1
Integrate both sides: $\int \frac{1}{y} d y = \int \cos x d x$ . M1
$\ln | y | = \sin x + C$ . A1A1
Convert to exponential form: $y = e^{\sin x + C} = A e^{\sin x}$ (where $A = e^{C}$ ). M1
Substitute initial condition $y (0) = e^{2}$ :
$e^{2} = A e^{\sin (0)} ⟹ e^{2} = A e^{0} ⟹ A = e^{2}$ .
Exact particular solution: $y = e^{2} e^{\sin x}$ or $y = e^{\sin x + 2}$ . A1

P4 Proof

Proof by contradiction (e.g. irrationality of √2, infinitely many primes).

13 practice questions with mark schemes · Worked examples → · Notes → · Practise proof →

Example question · 4 marks · hard

Prove by contradiction that there are infinitely many prime numbers.

Show the mark scheme

M1 for assuming that there is a finite number of prime numbers, listing them as $p_{1}, p_{2}, \dots, p_{n}$ .
M1 for constructing a new number $N = (p_{1} \times p_{2} \times \dots \times p_{n}) + 1$ .
M1 for arguing that dividing $N$ by any of the primes $p_{i}$ leaves a remainder of $1$ , meaning $N$ has no prime factors from the finite list.
A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

P4 The Binomial Expansion

Expanding (a + bx)ⁿ for any rational n; validity |bx/a| < 1; using partial fractions with expansions.

30 practice questions with mark schemes · Worked examples → · Notes → · Practise the binomial expansion →

Example question · 5 marks · hard

The binomial expansion of $(a + b x)^{- 2}$ , where $a > 0$ , begins $\frac{1}{9} - \frac{2}{27} x + \dots$ . Find the values of the constants $a$ and $b$ .

Show the mark scheme

M1 for writing $(a + b x)^{- 2} = a^{- 2} (1 + \frac{b}{a} x)^{- 2}$ .
M1 for equating the constant term: $a^{- 2} = \frac{1}{9}$ .
A1 for finding $a = 3$ (since $a > 0$ ).
M1 for equating the $x$ coefficient: $a^{- 2} (- 2) (\frac{b}{a}) = - \frac{2}{27}$ .
A1 for substituting $a = 3$ to get $\frac{1}{9} (\frac{- 2 b}{3}) = - \frac{2}{27} ⟹ - \frac{2 b}{27} = - \frac{2}{27} ⟹ b = 1$ .

P4 Coordinate Geometry (parametric)

Parametric curves and conversion to Cartesian form.

30 practice questions with mark schemes · Worked examples → · Notes → · Practise coordinate geometry (parametric) →

Example question · 5 marks · hard

A curve $C$ has parametric equations $x = \sin θ$ and $y = \cos 2 θ$ for $0 \leq θ \leq π$ . Find the exact coordinates of the points where $C$ intersects the line $y = x$ .

Show the mark scheme

M1 for setting $x = y$ , resulting in $\sin θ = \cos 2 θ$ .
M1 for substituting the double angle formula: $\sin θ = 1 - 2 \sin^{2} θ$ .
M1 for rearranging into a quadratic $2 \sin^{2} θ + \sin θ - 1 = 0$ and factorising $(2 \sin θ - 1) (\sin θ + 1) = 0$ .
A1 for finding valid solutions for $\sin θ$ : $\sin θ = \frac{1}{2}$ (rejecting $\sin θ = - 1$ since $0 \leq θ \leq π$ ).
A1 for calculating the corresponding $(x, y)$ coordinates: $(\frac{1}{2}, \frac{1}{2})$ .

P4 Differentiation

Implicit and parametric differentiation; connected rates of change.

30 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise differentiation →

Example question · 5 marks · hard

A curve has parametric equations $x = \ln t$ and $y = t^{2}$ , $t > 0$ . Find the equation of the normal to the curve at the point where $t = 1$ . Give your answer in the form $a x + b y + c = 0$ .

Show the mark scheme

M1 for finding the coordinates at $t = 1$ : $x = 0$ , $y = 1$ .
M1 for calculating $\frac{d x}{d t} = \frac{1}{t}$ and $\frac{d y}{d t} = 2 t$ , giving $\frac{d y}{d x} = 2 t^{2}$ .
M1 for evaluating the tangent gradient at $t = 1$ to get $m_{t} = 2$ .
M1 for determining the normal gradient $m_{n} = - \frac{1}{2}$ and using $(0, 1)$ to form the equation $y - 1 = - \frac{1}{2} (x - 0)$ .
A1 for rearranging to the required integer format: $x + 2 y - 2 = 0$ .

P4 Integration

Integration by substitution, by parts and using partial fractions; trig identities in integration; volumes of revolution about the x-axis.

30 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise integration →

Example question · 6 marks · hard

Using the substitution $u^{2} = 2 x - 1$ , find $\int x \sqrt{2 x - 1} d x$ .

Show the mark scheme

M1 for implicit differentiation $2 u d u = 2 d x ⟹ d x = u d u$ .
M1 for rearranging the substitution to find $x = \frac{u^{2} + 1}{2}$ .
M1 for substituting everything into the integral: $\int (\frac{u^{2} + 1}{2}) (u) (u) d u$ .
M1 for expanding to $\frac{1}{2} \int (u^{4} + u^{2}) d u$ .
M1 for integrating to $\frac{1}{10} u^{5} + \frac{1}{6} u^{3}$ .
A1 for replacing $u$ with $(2 x - 1)^{\frac{1}{2}}$ to get $\frac{1}{10} (2 x - 1)^{\frac{5}{2}} + \frac{1}{6} (2 x - 1)^{\frac{3}{2}} + c$ .

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FAQ

What is on the Edexcel IAL P4 exam?

P4 Pure Mathematics 4 is assessed by a 1h 30min written paper (75 marks), calculator permitted. It covers Vectors, Partial Fractions, Parametric Equations, Differential Equations, Proof, The Binomial Expansion, Coordinate Geometry (parametric), Differentiation, Integration.

Can I use a calculator in P4?

Yes, a calculator is permitted in P4, as in every IAL maths unit. You still need to show full method to earn the M marks.

How many P4 practice questions are there?

261 exam-style P4 questions across 9 topics, each with an Edexcel-style mark scheme (M, A and B marks).

Which P4 topics come up most?

In our mapping of the 4 Pearson P4 papers from June 2024 to June 2025, Integration (11 questions), Differentiation (8 questions), Vectors (5 questions) were tested most. Every topic can be examined in any series.

Where can I find P4 past papers?

Pearson publishes them. Our IAL P4 past papers page links to each official paper and mark scheme by series, with an exam timer.

How should I revise for P4?

Read the note for a topic, study the worked example and its mark scheme, practise that topic's questions, then sit a timed P4 past paper and check the common mistakes examiners report.

What is free on this page?

Everything here, with no sign-in: one example question with its mark scheme for every P4 topic, the worked examples on the topic pages and a quick quiz.

What does a free account add?

A free account (no card) adds 5 more practice questions on every topic with instant feedback, answers and mark schemes on free questions, 3 AI-marked answers a week, free, and your streak and progress on any device. The rest of the P4 bank with its mark schemes is part of the paid plans.

Other IAL units: IAL P1 Pure Mathematics 1 · IAL P2 Pure Mathematics 2 · IAL P3 Pure Mathematics 3 · IAL S1 Statistics 1 · IAL M1 Mechanics 1

Exam season: Mock exam guides · January 2027 prediction papers · 12-day mock revision plan · Final exam countdown · Results day: what next? · All exam-season plans

Edexcel IAL P4 Maths Revision: Pure Mathematics 4

Try one now

P4 topics

Try it now

P4 past papers

Example question · 7 marks · hard

Example question · 7 marks · hard

Example question · 8 marks · hard

Example question · 6 marks · hard

Example question · 4 marks · hard

Example question · 5 marks · hard

Example question · 5 marks · hard

Example question · 5 marks · hard

Example question · 6 marks · hard

Your P4 revision kit

FAQ

What is on the Edexcel IAL P4 exam?

Can I use a calculator in P4?

How many P4 practice questions are there?

Which P4 topics come up most?

Where can I find P4 past papers?

How should I revise for P4?

What is free on this page?

What does a free account add?