Edexcel IAL P3 Maths Revision: Pure Mathematics 3
Everything for Edexcel International A Level P3 (Pure Mathematics 3) in one place: what each topic covers, an exam-style question with its mark scheme for each topic, and 268 practice questions with instant marking.
- 1h 30min written paper (75 marks), calculator permitted
- 8 topics
- 268 questions
P3 topics
P3 Functions
Domain, range, composite and inverse functions; the modulus function y = |f(x)| and y = f(|x|); combinations of transformations.
Example question
Sketch the graph of $y = |2x + 3| - 4$. State the coordinates of the turning point and the points where the graph intersects the coordinate axes.
Show the mark scheme
- M1 for finding the vertex (turning point) of the V-shape at $x = -1.5$, giving $y = -4$. Point is $(-1.5, -4)$.
- M1 for setting $x=0$ to find the y-intercept at $y = |3| - 4 = -1$. Point is $(0, -1)$.
- M1 for setting $y=0$ and solving $|2x+3| = 4$ to find x-intercepts.
- A1 for correctly stating the x-intercepts at $(0.5, 0)$ and $(-3.5, 0)$.
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.)
Example question
The speed of light is approximately \(3.0 \times 10^8 \text{ m s}^{-1}\). The distance from Earth to a specific star is \(4.5 \times 10^{12} \text{ m}\).
- Calculate the time it takes for light from the star to reach Earth. Give your answer in standard form.
- Convert this time into hours, giving your answer to the nearest whole hour.
Show the mark scheme
- Time \(= \frac{\text{Distance}}{\text{Speed}} = \frac{4.5 \times 10^{12}}{3.0 \times 10^8}\) [M1]
\(= 1.5 \times 10^4 \text{ seconds}\) [A1] - \(\frac{1.5 \times 10^4}{3600}\) [M1]
\(= 4.166\dots \approx 4 \text{ hours}\) [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).
Example question
Find the equation of the tangent to the curve $y = \ln(3x - 2)$ at the point where $x = 1$. Give your answer in the form $y = mx + c$.
Show the mark scheme
- M1 for substituting $x = 1$ to find the y-coordinate: $y = \ln(1) = 0$.
- M1 for differentiating to find the gradient function: $\frac{dy}{dx} = \frac{3}{3x-2}$.
- M1 for substituting $x = 1$ into the derivative to find the gradient of the tangent, $m = 3$.
- A1 for substituting into the line equation to find $y - 0 = 3(x - 1)$, which simplifies to $y = 3x - 3$.
P3 Further Integration
Integrating eˣ, 1/x, sin x and cos x, including f(ax + b) forms (substitution and parts are P4).
P3 Algebraic Methods
Simplifying rational expressions (factorising, cancelling, algebraic division).
Example question
Simplify fully $\frac{2x^2 - x - 1}{x^2 - x} \div \frac{4x^2 - 1}{x}$.
Show the mark scheme
- M1 for rewriting the division as multiplication by the reciprocal: $\times \frac{x}{4x^2-1}$.
- M1 for factorising $2x^2-x-1$ to $(2x+1)(x-1)$ and $x^2-x$ to $x(x-1)$.
- M1 for factorising $4x^2-1$ to $(2x-1)(2x+1)$.
- A1 for cancelling all common factors to yield $\frac{1}{2x-1}$.
P3 Trigonometric Functions
sec, cosec and cot; arcsin, arccos, arctan and their domains/ranges; sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.
Example question
Prove that $\text{cosec} \theta - \sin \theta \equiv \cos \theta \cot \theta$.
Show the mark scheme
- M1 for rewriting the left-hand side as $\frac{1}{\sin \theta} - \sin \theta$.
- M1 for finding a common denominator: $\frac{1 - \sin^2 \theta}{\sin \theta}$.
- M1 for substituting $1 - \sin^2 \theta = \cos^2 \theta$ to get $\frac{\cos^2 \theta}{\sin \theta}$.
- A1 for factoring out $\cos \theta$ to show $\cos \theta (\frac{\cos \theta}{\sin \theta}) \equiv \cos \theta \cot \theta$.
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).
Example question
Using the double angle identities, prove the trigonometric identity:
\[3\sin 2\theta + \cos 2\theta - 1 = 2\sin\theta(3\cos\theta - \sin\theta)\]
Show the mark scheme
Substitute the double angle identities: \(\sin 2\theta = 2\sin\theta\cos\theta\) and \(\cos 2\theta = 1 - 2\sin^2\theta\). M1
\(\text{LHS} = 3(2\sin\theta\cos\theta) + (1 - 2\sin^2\theta) - 1\). A1
\(\text{LHS} = 6\sin\theta\cos\theta + 1 - 2\sin^2\theta - 1\).
\(\text{LHS} = 6\sin\theta\cos\theta - 2\sin^2\theta\). A1
Factorise out \(2\sin\theta\):
\(\text{LHS} = 2\sin\theta(3\cos\theta - \sin\theta) = \text{RHS}\). A1 *
P3 Exponentials and Logarithms
eˣ and ln x as inverse functions; exponential growth/decay; log-linear graphs for y = axⁿ and y = kbˣ.
Example question
Solve the equation $e^{2x} + 4e^x - 12 = 0$, giving your answer(s) in exact form.
Show the mark scheme
- M1 for identifying the hidden quadratic and factorising: $(e^x + 6)(e^x - 2) = 0$.
- M1 for extracting the roots $e^x = 2$ and $e^x = -6$.
- M1 for rejecting $e^x = -6$ since an exponential function must be positive.
- A1 for the exact valid solution $x = \ln 2$.
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. 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).