These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
P1 — Pure Mathematics 1
Unit WMA11 — Edexcel International A Level Mathematics (2018)
Authors: Henrik Gu, Yuchen Wang
1Algebra & Functions
Indices, surds, expansion, factorisation.
Laws of indices
For any real numbers a>0 and rationals m,n: a^m a^n = a^{m+n}, \dfrac{a^m}{a^n} = a^{m-n}, (a^m)^n = a^{mn}, a^0 = 1, a^{-n} = \dfrac{1}{a^n}, a^{1/n} = \sqrt[n]{a}.
Surds
A surd is an irrational root such as \sqrt{2}. Simplify by extracting square factors: \sqrt{50} = 5\sqrt{2}. Rationalise the denominator by multiplying top and bottom by the conjugate: \dfrac{1}{a+\sqrt{b}} = \dfrac{a-\sqrt{b}}{a^2 - b}.
Expanding brackets and factorising
Expand using distributivity; factorise by pulling out common factors or grouping. Recognise: a^2 - b^2 = (a-b)(a+b); a^2 \pm 2ab + b^2 = (a \pm b)^2.
2Quadratics
Solving, completing the square, discriminant, sketching.
Quadratic formula
Roots of ax^2+bx+c=0x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Completing the square
Rewrite ax^2 + bx + c as a(x + p)^2 + q where p = \dfrac{b}{2a} and q = c - ap^2. Useful for finding the vertex (-p,\; q) and solving without the formula.
Discriminant
\Delta = b^2 - 4ac. If \Delta > 0: two real roots. If \Delta = 0: one repeated root. If \Delta < 0: no real roots (curve doesn't cross the x-axis).
3Equations & Inequalities
Simultaneous equations, linear and quadratic inequalities.
Simultaneous equations
Two linear equations — solve by elimination or substitution. One linear, one quadratic — substitute the linear into the quadratic and solve.
Linear inequalities
Solve like equations, but flip the inequality sign when multiplying/dividing by a negative number.
Quadratic inequalities
Factorise, sketch the parabola, and read off which region satisfies the inequality. E.g. x^2 - 5x + 6 > 0 \;\Rightarrow\; (x-2)(x-3) > 0 \;\Rightarrow\; x < 2 or x > 3.
4Graphs & Transformations
Sketching cubics, reciprocals, and applying transformations.
Standard shapes
Cubic y = ax^3 + \ldots: 1-2 turning points, opposite end behaviours. Reciprocal y = \dfrac{k}{x}: two branches, asymptotes at x=0 and y=0. Modulus y = |f(x)|: reflect any negative y-values in the x-axis.
The coefficients come from Pascal's triangle. For (1 + x)^n: 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \ldots
9Trigonometry
Ratios, graphs, identities, equations.
Definitions and identities
In a right triangle: \sin\theta = \dfrac{\text{opp}}{\text{hyp}}, \cos\theta = \dfrac{\text{adj}}{\text{hyp}}, \tan\theta = \dfrac{\text{opp}}{\text{adj}}.
Gradient of curve at point = value of derivative at that point. Tangent has same gradient; normal has gradient -1/f'(x_0). Stationary point: f'(x) = 0; second derivative or sign analysis classifies max/min/inflection.
\displaystyle \int_a^b f(x)\,dx = F(b) - F(a), the (signed) area between the curve and x-axis from x=a to x=b. Area below the axis counts as negative.
12Vectors (2D)
Component form, magnitude, unit vectors, position vectors.
A 2D vector \mathbf{a} = \binom{a_1}{a_2} has magnitude |\mathbf{a}| = \sqrt{a_1^2 + a_2^2}. Unit vector: \hat{\mathbf{a}} = \dfrac{1}{|\mathbf{a}|}\mathbf{a}. Position vector of P from origin: \overrightarrow{OP} = \mathbf{p}. Vector from A to B: \overrightarrow{AB} = \mathbf{b} - \mathbf{a}.
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
P2 — Pure Mathematics 2
Unit WMA12 — Edexcel International A Level Mathematics (2018)
1Algebra & Functions
Partial fractions, algebraic division extended.
Partial fractions
Split a rational function into simpler pieces. For distinct linear factors: \dfrac{P(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}. For repeated: include \dfrac{C}{(x-a)^2}. Multiply and equate coefficients (or substitute convenient values) to find A,B,C.
Improper fractions
If degree of numerator \ge degree of denominator, divide first to get a polynomial plus a proper fractional remainder before splitting.
2Coordinate Geometry
Extending straight-line and circle work.
Locus problems: describe the set of points satisfying a geometric condition, e.g. equidistance from two points \Rightarrow perpendicular bisector. Use x, y algebra to derive the locus equation.
Recurrence relations: u_{n+1} = f(u_n). Iterate to find terms.
4Binomial Expansion
Beyond positive integers.
Still positive-integer expansions in P2. Extended to any rational n (with |x|<1) in P4.
5Trigonometry
Radians, arc length, sector area, general solutions.
Radian measure
Convert \theta_\text{rad} = \theta_\text{deg} \times \dfrac{\pi}{180}. Arc length s = r\theta; sector area A = \tfrac{1}{2}r^2\theta (with \theta in radians).
Solving equations over given intervals
Rewrite using identities where needed, find all solutions in the interval; check for extraneous ones after squaring or dividing by trig functions.
6Exponentials & Logarithms
Growth models, laws of logs, natural log \ln and e^x.
Laws of logs
\log_a xy = \log_a x + \log_a y \quad \log_a \tfrac{x}{y} = \log_a x - \log_a y \quad \log_a x^n = n\log_a x
Change of base: \log_a x = \dfrac{\log_b x}{\log_b a}. Special: \log_a 1 = 0, \log_a a = 1.
Solving exponential equations
Take logs both sides, apply the power law. E.g. 3^x = 20 \Rightarrow x = \dfrac{\ln 20}{\ln 3}.
Standard integrals\int e^x \, dx = e^x + C \qquad \int \dfrac{1}{x} \, dx = \ln|x| + C \qquad \int \sin x \, dx = -\cos x + C \qquad \int \cos x \, dx = \sin x + C
9Numerical Methods
Locating roots, iteration, trapezium rule.
Change of sign
If f is continuous and f(a), f(b) have opposite signs, a root lies in [a,b].
Iteration
Rearrange f(x)=0 as x = g(x) then iterate x_{n+1} = g(x_n). Converges if |g'(\alpha)| < 1 near the root \alpha.
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
P3 — Pure Mathematics 3
Unit WMA13 — Edexcel International A Level Mathematics (2018)
1Algebraic Fractions
Simplifying, adding, multiplying, dividing, and partial fractions.
Factorising and cancelling
Factorise everything first, then cancel or find a common denominator. Watch domain restrictions when cancelling.
Polynomial long division
When the numerator has degree greater than or equal to the denominator, divide first to obtain a polynomial plus a proper fraction: \dfrac{P(x)}{Q(x)} = A(x) + \dfrac{R(x)}{Q(x)} where \deg(R) < \deg(Q).
Partial fractions
Split a proper rational function into simpler pieces. The form depends on the denominator:
Distinct linear factors\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}
Repeated linear factor\frac{P(x)}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}
Multiply through by the denominator, then equate coefficients or substitute convenient values to find the constants.
2Functions
Domain, range, composite, inverse, modulus, and graph transformations.
Notation f:x \mapsto \ldots or f(x) = \ldots. Domain: the set of allowed inputs. Range: the set of all output values.
Domain restrictions
Denominator cannot be zero: f(x) = \frac{1}{x-2} has domain x \neq 2.
Even root must be non-negative: f(x) = \sqrt{x-3} has domain x \geq 3.
Logarithm must be positive: f(x) = \ln(x+1) has domain x > -1.
Composite functions
(g \circ f)(x) = g(f(x)); order matters. The domain of g \circ f is the set of x in domain of f such that f(x) is in domain of g.
Inverse functions
A function has an inverse iff it is one-to-one (passes the horizontal line test). To find f^{-1}(x): swap x and y, then solve for y. Domain of f^{-1} = range of f; range of f^{-1} = domain of f.
Modulus function
|x| = distance from 0. Sketch y = |f(x)|: reflect any part below x-axis above it. Sketch y = f(|x|): keep right side, reflect in y-axis.
Solve modulus equations/inequalities by splitting piecewise linear cases. For |f(x)| = a (a > 0): f(x) = a or f(x) = -a. For |f(x)| < a: -a < f(x) < a. For |f(x)| > a: f(x) < -a or f(x) > a.
Graph transformations
Form
Effect on y = f(x)
y = f(x) + a
translate by \binom{0}{a}
y = f(x + a)
translate by \binom{-a}{0}
y = af(x)
stretch parallel to y-axis by factor a
y = f(ax)
stretch parallel to x-axis by factor \frac{1}{a}
y = -f(x)
reflect in x-axis
y = f(-x)
reflect in y-axis
Roots of equations
Sign change method: if f is continuous and f(a) \cdot f(b) < 0, a root lies in (a, b). Iterative rearrangement: x_{n+1} = g(x_n). Bounds: verify root to given decimal places by upper/lower bound sign test.
\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A
Power reduction\sin^2 A = \frac{1 - \cos 2A}{2}, \qquad \cos^2 A = \frac{1 + \cos 2A}{2}
Addition & subtraction formulas
\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
Harmonic form
a\sin\theta + b\cos\theta = R\sin(\theta + \alpha) \text{ with } R = \sqrt{a^2 + b^2}, \; \tan\alpha = \frac{b}{a}
Alternatively: a\sin\theta + b\cos\theta = R\cos(\theta - \alpha) with R = \sqrt{a^2 + b^2}, \tan\alpha = \frac{a}{b}. Choose the form that matches the required phase.
Inverse trig functions
Function
Domain
Range
\arcsin x
[-1, 1]
-\frac{\pi}{2} \le \arcsin x \le \frac{\pi}{2}
\arccos x
[-1, 1]
0 \le \arccos x \le \pi
\arctan x
all reals
-\frac{\pi}{2} < \arctan x < \frac{\pi}{2}
Solving trigonometric equations
General method: use identities to write in terms of a single trig function, then solve for the angle. Find the principal value first, then add/subtract the period (2\pi for \sin/\cos, \pi for \tan) to find all solutions in the given interval.
Strategy for different forms:
\sin\theta = k or \cos\theta = k: find \arcsin(k) or \arccos(k), use symmetry (CAST diagram or graph).
Stationary points: solve \frac{dy}{dx} = 0, classify via second derivative test.
Implicit differentiation
Differentiate both sides w.r.t. x, treating y as a function of x (chain rule adds \frac{dy}{dx} factor). Rearrange to isolate \frac{dy}{dx}. E.g. for x^2 + y^2 = 25: 2x + 2y\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}.
Parametric differentiation
If x = f(t) and y = g(t), then \frac{dy}{dx} = \frac{dy/dt}{dx/dt}. For second derivative: \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \div \frac{dx}{dt}.
Second derivative
f'' > 0: concave up (minimum). f'' < 0: concave down (maximum). Point of inflection: f'' = 0 and sign changes (or f'' changes sign through the point).
6Integration
Standard integrals, reverse chain rule, substitution, integration by parts, and partial fractions.
Choose u = g(x) so that du = g'(x)\,dx appears in the integrand. Change limits for definite integrals. Common substitutions: u = f(x) for \int f'(x)h(f(x))\,dx.
Integration by parts
Integration by parts formula\int u \frac{dv}{dx} \, dx = uv - \int v \frac{du}{dx} \, dx
Choose u using the LIATE heuristic (Log, Inverse trig, Algebraic, Trig, Exponential — pick first on the list). For definite integrals: \int_a^b u \frac{dv}{dx}\,dx = [uv]_a^b - \int_a^b v \frac{du}{dx}\,dx.
Integrate powers of sin/cos
Use double-angle identities: \sin^2 A = \frac{1-\cos 2A}{2}, \cos^2 A = \frac{1+\cos 2A}{2}. For higher even powers, apply repeatedly.
Integration using partial fractions
Split rational functions into partial fractions first, then integrate each term. Typical forms:
Interval bisection, linear interpolation, and Newton-Raphson.
Interval bisection
If f is continuous on [a, b] and f(a) \cdot f(b) < 0, a root lies in (a, b). Take midpoint c = \frac{a+b}{2}; evaluate f(c). Replace the endpoint with the same sign as f(c). Repeat until the interval width is smaller than the required precision.
Linear interpolation
False position formulax = a - \frac{f(a)(b-a)}{f(b) - f(a)}
Join (a, f(a)) and (b, f(b)) with a straight line; the root estimate is where this line crosses the x-axis. Evaluate f(x) and replace the endpoint with the same sign.
Requires f' to be computable. Converges quadratically near a simple root when f' is non-zero and f is smooth. Fails if f'(x_n)=0 or a stationary point is near the root.
Verifying accuracy
To verify a root to n decimal places: show the function changes sign between x = a and x = b where a and b round to the same n-decimal value but bracket the root.
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
P4 — Pure Mathematics 4
Unit WMA14 — Edexcel International A Level Mathematics (2018)
1Series — Binomial Extended
Expansion of (1+x)^n for any rational n, valid for |x| < 1.
For (a + bx)^n factor out a^n: (a + bx)^n = a^n\!\left(1 + \tfrac{bx}{a}\right)^n, valid for \left|\tfrac{bx}{a}\right| < 1.
2Parametric Equations
x = f(t),\; y = g(t); sketching, converting, differentiating, and area under a curve.
Eliminate the parameter to get a Cartesian equation (often by expressing t in one and substituting). Chain rule for slope: \dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}.
Area under a parametric curve
A = \int_a^b y \, dx = \int_{t_1}^{t_2} y \frac{dx}{dt} \, dt
Change the limits to match the parameter values.
3Trigonometry (further)
Compound angles, double angles, half-angle formulas, and factor formulas.
Useful half-angle identities: \sin^2 A = \tfrac{1 - \cos 2A}{2}, \cos^2 A = \tfrac{1 + \cos 2A}{2} — often needed to integrate \sin^2, \cos^2.
Factor formulas (sum-to-product)
\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}
\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}
\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}
\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}
Related rates: connect \frac{dV}{dt} to \frac{dr}{dt} via \frac{dV}{dr} (chain rule). If variables are connected by an equation, differentiate implicitly with respect to time t.
5Integration by Substitution
Change of variable u = g(x) and trigonometric substitutions.
Choose u so du = g'(x) \, dx appears (up to a constant) in the integrand. Change limits when definite. Trigonometric substitutions:
\sqrt{a^2 - x^2}: let x = a\sin\theta or x = a\cos\theta
\int u \dfrac{dv}{dx} \, dx = uv - \int v \dfrac{du}{dx} \, dx
Choose u using the LIATE heuristic (Log, Inverse trig, Algebraic, Trig, Exponential — pick the first one on the list). Some problems reduce to themselves (recursive) and can be solved algebraically.
7Differential Equations
Separable first-order and exponential growth/decay.
If \dfrac{dy}{dx} = f(x)g(y), separate: \dfrac{dy}{g(y)} = f(x)\,dx; integrate both sides. Apply initial condition to find the constant of integration.
Exponential growth/decay
\frac{dN}{dt} = kN \quad \Rightarrow \quad N = N_0 e^{kt}
k > 0: growth; k < 0: decay. Half-life: time for N to halve.
Newton's law of cooling
\frac{dT}{dt} = -k(T - T_0) where T_0 is ambient temperature. Solution: T = T_0 + (T_{initial} - T_0)e^{-kt}.
8Vectors (3D)
Dot product, angle between vectors, vector equation of a line, and distance from a point to a line.
Perpendicular iff \mathbf{a} \cdot \mathbf{b} = 0. Vector equation of a line: \mathbf{r} = \mathbf{a} + t\mathbf{d}, position of a point plus a scalar multiple of the direction.
Point P, line \mathbf{r} = \mathbf{a} + t\mathbf{d}. Distance = \dfrac{|\overrightarrow{AP} \times \mathbf{d}|}{|\mathbf{d}|} where A is a point on the line.
Angle between two lines
Use the direction vectors: \cos\theta = \dfrac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}.
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
S1 — Statistics 1
Unit WST01 — Edexcel International A Level Mathematics (2018)
1Mathematical Models
Model-building cycle in statistics.
Simplification — apply mathematics — interpret — compare with reality — refine. A model isn't reality; it's a useful approximation.
2Data Representation
Data classification, histograms, stem-and-leaf, box plots.
Data classification
Type
Examples
Discrete
Separate distinct values (shoe sizes)
Continuous
Range of values (time, height)
Qualitative
Non-numerical (colour)
Quantitative
Numerical data
Histogram: area of each bar equals the frequency; frequency density = frequency / class width. Use for continuous data with unequal class widths.
Stem & leaf: preserves raw data values. Frequency polygon: plot midpoint of each class frequency.
Box plot: min, Q1, median, Q3, max — shows spread and skew.
3Location & Spread
Mean, median, mode, variance, standard deviation, outlier test, coding.
Mean and variance\bar{x} = \frac{\sum x}{n} \qquad \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2
Standard deviation is the (positive) square root of variance. IQR = Q3 − Q1 (resistant to outliers).
Outlier test
Value is an outlier if: Value < Q_1 - 1.5 \times \text{IQR} or Value > Q_3 + 1.5 \times \text{IQR}.
Coding transformation
Linear transformation y = mx + c: Mean of y = m \times mean of x + c. Variance unchanged by addition constant: \text{Var}(y) = m^2 \text{Var}(x).
Quartile calculation: use interpolation for grouped continuous data.
4Probability
Events, Venn diagrams, conditional probability, two-way tables, tree diagrams.
P(A \cup B) = P(A) + P(B) - P(A \cap B) \qquad P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Key rules
Mutually exclusive: P(A \cap B) = 0, so P(A \cup B) = P(A) + P(B).
Independent events: P(A \cap B) = P(A) \times P(B).
Calculation tools
Two-way tables: organise data by two categories; fill in row/column totals. Tree diagrams: show sequential events; multiply along branches for "and", add across branches for "or".
Sample space: all possible outcomes of an experiment.
Look up \Phi(z) = P(Z \le z) in the standard normal table. Symmetry: \Phi(-z) = 1 - \Phi(z).
Calculate: P(Z < z), P(Z > z), P(a < Z < b).
These notes are AI-assisted study material. Always cross-check against the official Pearson Edexcel spec or your teacher before relying on them in an exam.
M1 — Mechanics 1
Unit WME01 — Edexcel International A Level Mathematics (2018)
1Mathematical Models
Common simplifying assumptions.
Particle: object treated as a point; no rotation or size.
Light string/rod: mass negligible.
Inextensible string: doesn't stretch.
Smooth surface/pulley: no friction.
Rigid body: shape doesn't deform.
2Vectors (2D)
Position, velocity, acceleration as vectors.
Add/subtract componentwise; scalar multiply to scale; magnitude via Pythagoras. Unit vectors \mathbf{i}, \mathbf{j} along x, y axes.
3Kinematics (straight line)
Constant acceleration and calculus.
SUVAT (constant a)v = u + at \quad s = ut + \tfrac{1}{2}at^2 \quad s = \tfrac{1}{2}(u+v)t \quad v^2 = u^2 + 2as
Variable acceleration: v = \tfrac{ds}{dt}, a = \tfrac{dv}{dt}; s = \int v\,dt, v = \int a\,dt.
4Dynamics of a Particle
Newton's laws, weight, friction.
\mathbf{F} = m\mathbf{a} \qquad W = mg \qquad F_{\max} = \mu R
Weight acts downward with magnitude mg (take g = 9.8\,\text{m s}^{-2} unless told). Friction: static friction adjusts to prevent motion up to F_{\max} = \mu R; kinetic friction is \mu R opposite the motion.
5Connected Particles
Ropes, pulleys, coupled motion.
Two particles joined by a taut, inextensible string share magnitude of acceleration. Apply F = ma to each separately along the direction of motion; solve simultaneously. For a smooth pulley: tension is the same throughout the string.
6Statics
Equilibrium of a particle.
Resolve forces in perpendicular directions; both totals equal zero. On an inclined plane: resolve along and perpendicular to the slope for a cleaner solution.
7Moments
Turning effect of a force.
M = Fd
Moment about a point = force \times perpendicular distance from the point to the line of action. In equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.