Edexcel IAL P3 Maths Revision: Pure Mathematics 3

Edexcel International A Level P3 (Pure Mathematics 3): every topic with an exam-style question and its mark scheme, and 268 practice questions with mark schemes.

1h 30min written paper (75 marks), calculator permitted
8 topics
268 questions

Try one now

Free · no sign-in

Trigonometric Addition Formulae · P3 paper · 6 marks · hard

Solve the equation $\cos 2 x = \sin x$ for $- π \leq x \leq π$ . Give your answers as exact radian values.

Worked answer

Substitute the double angle identity $\cos 2 x = 1 - 2 \sin^{2} x$ . M1
$1 - 2 \sin^{2} x = \sin x ⟹ 2 \sin^{2} x + \sin x - 1 = 0$ . A1
Factorise the quadratic: $(2 \sin x - 1) (\sin x + 1) = 0$ . M1A1
Case 1: $2 \sin x - 1 = 0 ⟹ \sin x = \frac{1}{2}$ .
Within the domain $- π \leq x \leq π$ , solutions are $x = \frac{π}{6}$ and $x = \frac{5 π}{6}$ . A1
Case 2: $\sin x + 1 = 0 ⟹ \sin x = - 1$ .
Within the domain $- π \leq x \leq π$ , the solution is $x = - \frac{π}{2}$ . A1

More P3 trigonometric addition formulae questions →

Start practising P3 →

P3 topics

"Tested in" counts come from our question-by-question mapping of the 4 Pearson P3 papers from June 2024 to June 2025.

Practise next: P3 questions by topic · IAL Skill Builders, short sets with instant feedback · P3 past papers with a timer · P3 mock exam guide · every IAL unit's topics

5 more P3 practice questions on every topic, with instant feedback. Free account, no card. Join free

P3 past papers

Official Pearson P3 papers and mark schemes, with a timer: all IAL P3 past papers. Most recent: June 2025 · January 2025 · October 2024 · June 2024.

Examiner Insights for IAL P3: every examiner report since June 2024, the top mistakes and what to focus on next →

Knowledge organiser: Pure Mathematics 3 (P3) on one page (key facts, formulas and mistakes to avoid; printable PDF)

P3 Functions

Domain, range, composite and inverse functions; the modulus function y = |f(x)| and y = f(|x|); combinations of transformations.

35 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise functions →

Example question · 5 marks · hard

Solve the inequality $| 2 x + 1 | \leq | x - 5 |$ .

Show the mark scheme

M1 for squaring both sides to remove the modulus signs: $(2 x + 1)^{2} \leq (x - 5)^{2}$ .
M1 for expanding the brackets: $4 x^{2} + 4 x + 1 \leq x^{2} - 10 x + 25$ .
M1 for rearranging to form a quadratic inequality: $3 x^{2} + 14 x - 24 \leq 0$ .
M1 for factorising to find the critical values: $(3 x - 4) (x + 6) \leq 0$ , yielding critical values $- 6$ and $\frac{4}{3}$ .
A1 for stating the correct closed interval $- 6 \leq x \leq \frac{4}{3}$ .

P3 Numerical Methods

Locating roots of f(x) = 0 by a change of sign; approximate solutions using iteration xₙ₊₁ = g(xₙ), including cobweb/staircase diagrams. (Newton–Raphson is Further Pure 1, not P3.)

39 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise numerical methods →

Example question · 7 marks · hard

$f (x) = x^{3} - 3 x - 4$ .
(a) Show that $f (x) = 0$ has a root in $(2, 3)$ . (2)
(b) Rewrite the equation in the form $x = \sqrt[3]{3 x + 4}$ and use this iteration with $x_{0} = 2$ to find $x_{1}, x_{2}, x_{3}, x_{4}$ each to 4 d.p. State the value of the root to 3 d.p. (5)

Show the mark scheme

(a) $f (2) = 8 - 6 - 4 = - 2 < 0$ ; $f (3) = 27 - 9 - 4 = 14 > 0$ M1 A1.
(b) $x_{1} = \sqrt[3]{10} = 2.1544$ ; $x_{2} = \sqrt[3]{10.4633} = 2.1872$ ; $x_{3} = \sqrt[3]{10.5616} = 2.1940$ ; $x_{4} = \sqrt[3]{10.5821} = 2.1955$ M1 A1 A1 A1. Root $\approx 2.196$ A1.

P3 Further Differentiation

Chain, product and quotient rules; differentiating eˣ, ln x, sin x, cos x, tan x (and sec, cosec, cot); dy/dx = 1/(dx/dy).

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

Example question · 5 marks · hard

A curve has the equation $y = e^{- x} \sin x$ . Find the exact x-coordinate of the first positive turning point of the curve.

Show the mark scheme

M1 for using the product rule to differentiate: $\frac{d y}{d x} = - e^{- x} \sin x + e^{- x} \cos x$ .
M1 for setting the derivative equal to 0 to find turning points: $e^{- x} (\cos x - \sin x) = 0$ .
M1 for dividing by $e^{- x}$ (since $e^{- x} \neq 0$ ) to get $\cos x = \sin x$ .
M1 for dividing by $\cos x$ to obtain $\tan x = 1$ .
A1 for identifying the first positive solution as $x = \frac{π}{4}$ .

P3 Further Integration

Integrating eˣ, 1/x, sin x and cos x, including f(ax + b) forms (substitution and parts are P4).

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

Example question · 4 marks · hard

Find $\int (\sin x + \cos x)^{2} d x$ .

Show the mark scheme

M1 for expanding the bracket to $\sin^{2} x + 2 \sin x \cos x + \cos^{2} x$ .
M1 for simplifying using $\sin^{2} x + \cos^{2} x \equiv 1$ and $2 \sin x \cos x \equiv \sin 2 x$ .
M1 for integrating $1 + \sin 2 x$ .
A1 for $x - \frac{1}{2} \cos 2 x + c$ .

P3 Algebraic Methods

Simplifying rational expressions (factorising, cancelling, algebraic division).

30 practice questions with mark schemes · Worked examples → · Notes → · Practise algebraic methods →

Example question · 5 marks · hard

Express $\frac{x + 2}{x^{2} - 5 x + 6} - \frac{x - 1}{x^{2} - 4 x + 3}$ as a single fraction in its simplest form.

Show the mark scheme

M1 for factorising $x^{2} - 5 x + 6$ to $(x - 2) (x - 3)$ .
M1 for factorising $x^{2} - 4 x + 3$ to $(x - 1) (x - 3)$ .
M1 for simplifying the second fraction: $\frac{x - 1}{(x - 1) (x - 3)}$ cancels to $\frac{1}{x - 3}$ .
M1 for creating a common denominator $(x - 2) (x - 3)$ and subtracting: $\frac{x + 2}{(x - 2) (x - 3)} - \frac{x - 2}{(x - 2) (x - 3)}$ .
A1 for reaching the final simplified answer $\frac{4}{(x - 2) (x - 3)}$ .

P3 Trigonometric Functions

sec, cosec and cot; arcsin, arccos, arctan and their domains/ranges; sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.

30 practice questions with mark schemes · Worked examples → · Notes → · Practise trigonometric functions →

Example question · 6 marks · hard

Solve $3 \cot^{2} θ + 5 cosec θ + 1 = 0$ for $0^{\circ} \leq θ \leq 360^{\circ}$ .

Show the mark scheme

M1 for substituting $\cot^{2} θ = {cosec}^{2} θ - 1$ to form $3 ({cosec}^{2} θ - 1) + 5 cosec θ + 1 = 0$ .
M1 for simplifying to the quadratic $3 {cosec}^{2} θ + 5 cosec θ - 2 = 0$ .
M1 for factorising to $(3 cosec θ - 1) (cosec θ + 2) = 0$ .
M1 for showing $cosec θ = \frac{1}{3} ⟹ \sin θ = 3$ yields no real solutions.
M1 for solving $cosec θ = - 2 ⟹ \sin θ = - 0.5$ .
A1 for finding the two correct angles: $210^{\circ}$ and $330^{\circ}$ .

P3 Trigonometric Addition Formulae

Addition and double-angle formulae; a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α).

34 practice questions with mark schemes · Worked examples → · Notes → · Practise trigonometric addition formulae →

Example question · 6 marks · hard

Solve the equation $\cos 2 x = \sin x$ for $- π \leq x \leq π$ . Give your answers as exact radian values.

Show the mark scheme

P3 Exponentials and Logarithms

eˣ and ln x as inverse functions; exponential growth/decay; log-linear graphs for y = axⁿ and y = kbˣ.

30 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise exponentials and logarithms →

Example question · 5 marks · hard

The variables $x$ and $y$ satisfy the equation $y = a b^{x}$ . A graph of $\ln y$ against $x$ is plotted and forms a straight line passing through $(0, 2)$ and $(4, 10)$ . Find the exact values of the constants $a$ and $b$ .

Show the mark scheme

M1 for taking natural logarithms of both sides to get $\ln y = \ln a + x \ln b$ .
M1 for calculating the gradient of the line: $\ln b = \frac{10 - 2}{4 - 0} = 2$ .
A1 for solving to find exactly $b = e^{2}$ .
M1 for identifying the y-intercept from the coordinate $(0, 2)$ , so $\ln a = 2$ .
A1 for solving to find exactly $a = e^{2}$ .

Your P3 revision kit

Save your P3 answers and track your progressA free account remembers the questions you have done and brings you back to this page.

Join free

Already have an account? Sign in

FAQ

What is on the Edexcel IAL P3 exam?

P3 Pure Mathematics 3 is assessed by a 1h 30min written paper (75 marks), calculator permitted. It covers Functions, Numerical Methods, Further Differentiation, Further Integration, Algebraic Methods, Trigonometric Functions, Trigonometric Addition Formulae, Exponentials and Logarithms.

Can I use a calculator in P3?

Yes, a calculator is permitted in P3, as in every IAL maths unit. You still need to show full method to earn the M marks.

How many P3 practice questions are there?

268 exam-style P3 questions across 8 topics, each with an Edexcel-style mark scheme (M, A and B marks).

Which P3 topics come up most?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, Functions (9 questions), Exponentials and Logarithms (9 questions), Further Differentiation (7 questions) were tested most. Every topic can be examined in any series.

Where can I find P3 past papers?

Pearson publishes them. Our IAL P3 past papers page links to each official paper and mark scheme by series, with an exam timer.

How should I revise for P3?

Read the note for a topic, study the worked example and its mark scheme, practise that topic's questions, then sit a timed P3 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 P3 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 P3 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 P4 Pure Mathematics 4 · 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 P3 Maths Revision: Pure Mathematics 3

Try one now

P3 topics

Try it now

P3 past papers

Example question · 5 marks · hard

Example question · 7 marks · hard

Example question · 5 marks · hard

Example question · 4 marks · hard

Example question · 5 marks · hard

Example question · 6 marks · hard

Example question · 6 marks · hard

Example question · 5 marks · hard

Your P3 revision kit

FAQ

What is on the Edexcel IAL P3 exam?

Can I use a calculator in P3?

How many P3 practice questions are there?

Which P3 topics come up most?

Where can I find P3 past papers?

How should I revise for P3?

What is free on this page?

What does a free account add?