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Edexcel IAL P2 Maths Revision: Pure Mathematics 2

Everything for Edexcel International A Level P2 (Pure Mathematics 2) in one place: what each topic covers, an exam-style question with its mark scheme for each topic, and 287 practice questions with instant marking.

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P2 topics

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 · Practise sequences and series →

Example question · 4 marks · medium

Find the least number of terms required for the sum of the arithmetic series $5 + 9 + 13 + 17 + ...$ to exceed 1000.

Show the mark scheme
  1. M1 for setting up the inequality: $S_n = \frac{n}{2}(2(5) + (n-1)4) > 1000$.
  2. M1 for simplifying to a quadratic inequality: $n(10 + 4n - 4) > 2000 \implies 4n^2 + 6n - 2000 > 0$.
  3. M1 for solving the quadratic equation $2n^2 + 3n - 1000 = 0$ to find the critical value $n \approx 21.6$.
  4. A1 for concluding that since $n$ must be an integer, the least number of terms is $n = 22$.

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 · Practise logarithms and exponentials →

Example question · 4 marks · medium

A curve has the equation $y = ab^x$, where $a$ and $b$ are constants and $b > 0$. The curve passes through the points $(1, 10)$ and $(3, 40)$. Find the values of $a$ and $b$.

Show the mark scheme
  1. M1 for setting up the equations $ab = 10$ and $ab^3 = 40$.
  2. M1 for dividing the equations to eliminate $a$: $b^2 = 4$.
  3. A1 for stating $b = 2$ (since $b > 0$).
  4. A1 for finding $a = 5$.

P2 Circles

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

45 practice questions with mark schemes · Practise circles →

Example question · 4 marks · medium


Let \(f(x) = \sin(e^x)\) for \(0 \le x \le 1.5\).

  1. Use your graphic display calculator to sketch the graph of \(f\), clearly showing its shape.

  2. Find the exact value of the \(x\)-intercept of the graph of \(f\) within this domain. (Hint: recall the definition of the natural logarithm).

Show the mark scheme

  1. CG50: MENU 5 (Graph). Y1 = sin(e^x). Set V-Window X: \(0 \to 1.5\), Y: \(-1 \to 1\). DRAW. (M1)
    Sketch should show a wave that begins to compress (increase in frequency) as \(x\) increases, crossing the \(x\)-axis once and dipping down. A1

  2. The \(x\)-intercept occurs where \(\sin(e^x) = 0\). (M1)
    This means \(e^x = k\pi\) for \(k \in \mathbb{Z}\).
    Since \(0 \le x \le 1.5\), the range of \(e^x\) is \(e^0 \le e^x \le e^{1.5} \implies 1 \le e^x \le 4.48\).
    The only multiple of \(\pi\) in this range is \(1\pi \approx 3.14159\). (R1)
    Therefore, \(e^x = \pi \implies x = \ln \pi\). A1
    (Using GDC G-Solv ROOT gives \(x \approx 1.1447\), which confirms the exact value \(\ln\pi\)).

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 · Practise further trigonometry →

Example question · 4 marks · medium

Given that $\sin x = \frac{3}{5}$ and $x$ is an acute angle ($0 < x < \frac{\pi}{2}$), find the exact values of $\cos x$ and $\tan x$.

Show the mark scheme

M1 for using the Pythagorean identity $\cos^2 x = 1 - \sin^2 x$ or drawing a right-angled triangle.
A1 for finding $\cos x = \frac{4}{5}$ (taking the positive root for an acute angle).
M1 for using the ratio identity $\tan x = \frac{\sin x}{\cos x}$.
A1 for finding $\tan x = \frac{3/5}{4/5} = \frac{3}{4}$.

P2 Algebraic Methods

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

30 practice questions with mark schemes · Practise algebraic methods →

Example question · 4 marks · medium

Find the quotient and the remainder when $3x^3 + 8x^2 - 4x + 5$ is divided by $x + 3$.

Show the mark scheme
  1. M1 for setting up long division and establishing the first term $3x^2$.
  2. M1 for finding the next quotient term $-x$.
  3. A1 for the exact quotient $3x^2 - x - 1$.
  4. A1 for finding the correct remainder of $8$.

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 · Practise the binomial expansion →

Example question · 4 marks · medium

Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(3-x)^5$.

Show the mark scheme
  1. M1 for setting up the unsimplified terms: $3^5 + \binom{5}{1}3^4(-x) + \binom{5}{2}3^3(-x)^2 + \binom{5}{3}3^2(-x)^3$.
  2. M1 for evaluating the combinations as 1, 5, 10, 10.
  3. M1 for evaluating the powers of 3 and taking care with the negative signs: $243 - 5(81)x + 10(27)x^2 - 10(9)x^3$.
  4. A1 for the completely simplified answer: $243 - 405x + 270x^2 - 90x^3$.

P2 Differentiation (stationary points)

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

33 practice questions with mark schemes · Practise differentiation (stationary points) →

Example question · 4 marks · medium

Find the x-coordinates of the stationary points on the curve with equation $y = x^3 - 3x^2 - 9x + 2$.

Show the mark scheme
  1. M1 for finding $\frac{dy}{dx} = 3x^2 - 6x - 9$.
  2. M1 for setting the gradient to zero: $3x^2 - 6x - 9 = 0$.
  3. M1 for dividing by 3 and factorising or solving the quadratic: $x^2 - 2x - 3 = 0 \implies (x-3)(x+1) = 0$.
  4. A1 for the correct x-coordinates $x = 3$ and $x = -1$.

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 · Practise integration (definite integrals and area) →

Example question · 4 marks · medium

Find the exact value of $\int_{4}^{9} \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) \, dx$.

Show the mark scheme
  1. M1 for writing as fractional indices: $\int (x^{\frac{1}{2}} + x^{-\frac{1}{2}}) \, dx$.
  2. M1 for integrating to get $[\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}}]$.
  3. M1 for substituting the limits: $(\frac{2}{3}(27) + 2(3)) - (\frac{2}{3}(8) + 2(2))$.
  4. A1 for evaluating to get exactly $14\frac{2}{3}$ (or $\frac{44}{3}$).

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, 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. You still need to show full method to earn the M marks.

How many P2 practice questions are there?

287 exam-style P2 questions across 8 topics, each with an Edexcel-style mark scheme (M, A and B marks).

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