Edexcel IAL P2 Maths Revision: Pure Mathematics 2

Edexcel International A Level P2 (Pure Mathematics 2): every topic with an exam-style question and its mark scheme, and 305 practice questions with mark schemes.

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

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Proof · P2 paper · 5 marks · medium

(a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]

(b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]

Worked answer

(a) Considers every case: $n = 5 k, 5 k \pm 1, 5 k \pm 2$ (or $5 k, 5 k + 1, 5 k + 2, 5 k + 3, 5 k + 4$ ), $k$ an integer M1

Squares at least two cases correctly, e.g. $(5 k \pm 1)^{2} = 25 k^{2} \pm 10 k + 1$ M1

All cases correct: $(5 k)^{2} = 5 (5 k^{2})$ ; $(5 k \pm 1)^{2} = 5 (5 k^{2} \pm 2 k) + 1$ ; $(5 k \pm 2)^{2} = 5 (5 k^{2} \pm 4 k) + 4$ A1

Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)

(b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1

P2 topics

"Tested in" counts come from our question-by-question mapping of the 4 Pearson P2 papers from June 2024 to June 2025.

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P2 past papers

Official Pearson P2 papers and mark schemes, with a timer: all IAL P2 past papers. Most recent: June 2025 · January 2025 · October 2024 · June 2024.

Examiner Insights for IAL P2: every examiner report since June 2024, the top mistakes and what to focus on next →

Knowledge organiser: Pure Mathematics 2 (P2) on one page (key facts, formulas and mistakes to avoid; printable PDF)

P2 Sequences and Series

Sequences by nth term and recurrence (increasing, decreasing, periodic); arithmetic and geometric series, sum to n terms, sum to infinity (|r| < 1), Σ notation.

35 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise sequences and series →

Example question · 5 marks · hard

The second term of a geometric series is 12 and its sum to infinity is 64. Find the two possible values of the common ratio, $r$ , and the corresponding values of the first term, $a$ .

Show the mark scheme

M1 for setting up the two equations: $a r = 12$ and $\frac{a}{1 - r} = 64$ .
M1 for substituting $a = \frac{12}{r}$ to get $\frac{12 / r}{1 - r} = 64 ⟹ 12 = 64 r (1 - r)$ .
M1 for forming the quadratic $16 r^{2} - 16 r + 3 = 0$ .
A1 for finding $r = 0.25$ and $r = 0.75$ .
A1 for finding the corresponding first terms: $a = 48$ and $a = 16$ .

P2 Logarithms and Exponentials

aˣ and logₐx, laws of logarithms, solving aˣ = b, change of base (natural logs and eˣ are P3).

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

Example question · 5 marks · hard

Solve the equation $\log_{2} x + \log_{x} 2 = 2.5$ for $x > 0, x \neq 1$ .

Show the mark scheme

M1 for applying the reciprocal rule: $\log_{x} 2 = \frac{1}{\log_{2} x}$ .
M1 for making the substitution $u = \log_{2} x$ to form $u + \frac{1}{u} = 2.5$ .
M1 for multiplying by $2 u$ to form the quadratic $2 u^{2} - 5 u + 2 = 0$ and solving it.
M1 for finding the roots $u = 2$ and $u = \frac{1}{2}$ .
A1 for converting back to $x$ and finding both solutions: $x = 4$ and $x = \sqrt{2}$ .

P2 Circles

Equation of a circle in Cartesian form, tangent-radius perpendicularity, geometric problem solving.

45 practice questions with mark schemes · Worked examples → · Practise circles →

Example question · 7 marks · hard

The points $A (1, 2)$ and $B (7, 10)$ lie on a circle $C$ with $A B$ as a diameter.
(a) Find the equation of $C$ . (4)
(b) The tangent to $C$ at $A$ meets the $y$ -axis at $T$ . Find the coordinates of $T$ . (3)

Show the mark scheme

(a) Centre $=$ midpoint $= (4, 6)$ B1. Radius $= \frac{1}{2} | A B | = \frac{1}{2} \sqrt{36 + 64} = 5$ M1 A1. Equation: $(x - 4)^{2} + (y - 6)^{2} = 25$ A1.
(b) Radius from centre to $A$ has gradient $\frac{2 - 6}{1 - 4} = \frac{4}{3}$ B1. Tangent gradient $= - \frac{3}{4}$ ; equation: $y - 2 = - \frac{3}{4} (x - 1)$ . Set $x = 0$ : $y = 2 + \frac{3}{4} = \frac{11}{4}$ M1 A1.

P2 Further Trigonometry

tan θ = sin θ / cos θ and sin²θ + cos²θ = 1; solving trigonometric equations in a given interval (degrees or radians), including quadratics in sin/cos/tan.

46 practice questions with mark schemes · Worked examples → · Practise further trigonometry →

Example question · 5 marks · hard

Solve $2 \sin^{2} x + 3 \cos x - 3 = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Show the mark scheme

M1 for substituting $\sin^{2} x$ with $1 - \cos^{2} x$ to get $2 (1 - \cos^{2} x) + 3 \cos x - 3 = 0$ .
M1 for expanding and rearranging to form the quadratic $2 \cos^{2} x - 3 \cos x + 1 = 0$ .
M1 for factorising the quadratic as $(2 \cos x - 1) (\cos x - 1) = 0$ .
A1 for finding the solutions from $\cos x = 0.5$ giving $x = 60^{\circ}$ and $300^{\circ}$ .
A1 for finding the solutions from $\cos x = 1$ giving $x = 0^{\circ}$ and $360^{\circ}$ .

P2 Algebraic Methods

Algebraic long division; factor and remainder theorems; proof by deduction, exhaustion and counter-example.

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

Example question · 5 marks · hard

Let $f (x) = x^{3} + c x^{2} + d x - 4$ . Given $f (x)$ has a factor $(x + 2)$ and leaves a remainder of $- 10$ when divided by $(x - 3)$ , find the values of $c$ and $d$ .

Show the mark scheme

M1 for applying the Factor Theorem for $(x + 2)$ : $f (- 2) = 0 ⟹ - 8 + 4 c - 2 d - 4 = 0 ⟹ 4 c - 2 d = 12$ .
M1 for applying the Remainder Theorem for $(x - 3)$ : $f (3) = - 10 ⟹ 27 + 9 c + 3 d - 4 = - 10 ⟹ 9 c + 3 d = - 33$ .
M1 for simplifying both equations (e.g., $2 c - d = 6$ and $3 c + d = - 11$ ).
M1 for solving the simultaneous equations (e.g., adding them gives $5 c = - 5$ ).
A1 for finding the correct values $c = - 1$ and $d = - 8$ .

P2 Proof

Proof by deduction, proof by exhaustion and disproof by counter-example, in number, algebra, inequalities and coordinate geometry (proof by contradiction is P4).

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

Example question · 5 marks · medium

(a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]

(b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]

Show the mark scheme

(a) Considers every case: $n = 5 k, 5 k \pm 1, 5 k \pm 2$ (or $5 k, 5 k + 1, 5 k + 2, 5 k + 3, 5 k + 4$ ), $k$ an integer M1

Squares at least two cases correctly, e.g. $(5 k \pm 1)^{2} = 25 k^{2} \pm 10 k + 1$ M1

All cases correct: $(5 k)^{2} = 5 (5 k^{2})$ ; $(5 k \pm 1)^{2} = 5 (5 k^{2} \pm 2 k) + 1$ ; $(5 k \pm 2)^{2} = 5 (5 k^{2} \pm 4 k) + 4$ A1

Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)

(b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1

P2 The Binomial Expansion

Binomial expansion of (a + bx)ⁿ for positive integer n; ⁿCr notation; using expansions for approximations.

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

Example question · 5 marks · hard

The binomial expansion of $(1 + a x)^{b}$ , where $b$ is a positive integer, starts with $1 - 15 x + 90 x^{2}$ . Find the values of $a$ and $b$ .

Show the mark scheme

M1 for equating the $x$ coefficient: $a b = - 15$ .
M1 for equating the $x^{2}$ coefficient: $\frac{b (b - 1)}{2} a^{2} = 90 ⟹ b (b - 1) a^{2} = 180$ .
M1 for substituting $(a b) = - 15$ or expanding to $a^{2} b^{2} - a^{2} b = 180$ and substituting $a^{2} b^{2} = 225$ .
M1 for solving for $a$ : $225 - a (- 15) = 180 ⟹ 15 a = - 45 ⟹ a = - 3$ .
A1 for finding the corresponding value $b = 5$ .

P2 Differentiation (stationary points)

Increasing and decreasing functions; stationary points and their nature; optimisation problems.

33 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise differentiation (stationary points) →

Example question · 4 marks · hard

Find the set of values of $x$ for which $f (x) = x^{4} - 8 x^{2}$ is an increasing function.

Show the mark scheme

M1 for finding the derivative $f^{'} (x) = 4 x^{3} - 16 x$ .
M1 for setting $f^{'} (x) > 0$ and factorising: $4 x (x^{2} - 4) > 0 ⟹ 4 x (x - 2) (x + 2) > 0$ .
M1 for finding the critical values $x = - 2, 0, 2$ and testing regions.
A1 for stating the correct range: $- 2 < x < 0$ or $x > 2$ .

P2 Integration (definite integrals and area)

Definite integrals; area under a curve and between a curve and a line; the trapezium rule.

33 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise integration (definite integrals and area) →

Example question · 7 marks · hard

Find the exact total area of the finite regions bounded by the curves $y = x^{3} + x^{2}$ and $y = 2 x^{2} + 2 x$ .

Show the mark scheme

M1 for setting $x^{3} + x^{2} = 2 x^{2} + 2 x ⟹ x^{3} - x^{2} - 2 x = 0$ .
M1 for factoring $x (x - 2) (x + 1) = 0$ to find the intersection points $x = - 1, 0, 2$ .
M1 for recognising two distinct regions and setting up integrals: $\int_{- 1}^{0} (x^{3} - x^{2} - 2 x) d x$ and $\int_{0}^{2} (2 x^{2} + 2 x - x^{3} - x^{2}) d x$ .
M1 for integrating the generic polynomial to $\pm [\frac{x^{4}}{4} - \frac{x^{3}}{3} - x^{2}]$ .
M1 for evaluating the area of the first region to $\frac{5}{12}$ .
M1 for evaluating the area of the second region to $\frac{8}{3}$ .
A1 for calculating the combined total area: $\frac{5}{12} + \frac{8}{3} = \frac{37}{12}$ .

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FAQ

What is on the Edexcel IAL P2 exam?

P2 Pure Mathematics 2 is assessed by a 1h 30min written paper (75 marks), calculator permitted. It covers Sequences and Series, Logarithms and Exponentials, Circles, Further Trigonometry, Algebraic Methods, Proof, The Binomial Expansion, Differentiation (stationary points), Integration (definite integrals and area).

Can I use a calculator in P2?

Yes, a calculator is permitted in P2, as in every IAL maths unit. You still need to show full method to earn the M marks.

How many P2 practice questions are there?

305 exam-style P2 questions across 9 topics, each with an Edexcel-style mark scheme (M, A and B marks).

Which P2 topics come up most?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, Algebraic Methods (8 questions), Sequences and Series (7 questions), Logarithms and Exponentials (6 questions) were tested most. Every topic can be examined in any series.

Where can I find P2 past papers?

Pearson publishes them. Our IAL P2 past papers page links to each official paper and mark scheme by series, with an exam timer.

How should I revise for P2?

Read the note for a topic, study the worked example and its mark scheme, practise that topic's questions, then sit a timed P2 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 P2 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 P2 bank with its mark schemes is part of the paid plans.

Other IAL units: IAL P1 Pure Mathematics 1 · IAL P3 Pure Mathematics 3 · 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 P2 Maths Revision: Pure Mathematics 2

Try one now

P2 topics

Try it now

P2 past papers

Example question · 5 marks · hard

Example question · 5 marks · hard

Example question · 7 marks · hard

Example question · 5 marks · hard

Example question · 5 marks · hard

Example question · 5 marks · medium

Example question · 5 marks · hard

Example question · 4 marks · hard

Example question · 7 marks · hard

Your P2 revision kit

FAQ

What is on the Edexcel IAL P2 exam?

Can I use a calculator in P2?

How many P2 practice questions are there?

Which P2 topics come up most?

Where can I find P2 past papers?

How should I revise for P2?

What is free on this page?

What does a free account add?