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These notes are AI-assisted study material. Always cross-check against the official WJEC spec or your teacher before relying on them in an exam.

Unit 1 — Pure Mathematics A

AS Unit 1 · WJEC A Level Mathematics (2017)

1Proof

AS Unit 1 · Methods of mathematical proof

Three proof methods appear at AS level:

  • Proof by deduction — a logical chain of reasoning from known facts to a new statement. Every step must be justified.
  • Proof by exhaustion — testing every possible case. Only works when the number of cases is finite and small enough to check.
  • Disproof by counter-example — finding one example that disproves a universal statement. A single valid counter-example is sufficient.
Exam tip — When asked to "prove", never assume the result. Start from known true statements and work logically towards what is required. A common mistake is to begin with the statement to be proved and rearrange it.

2Algebra & Functions

AS Unit 1 · Surds, indices, quadratics, inequalities, polynomials

Surds and indices form the foundation for algebraic fluency. Learn to rationalise denominators (e.g. rewrite \frac{1}{\sqrt{2}} as \frac{\sqrt{2}}{2}) and simplify expressions with fractional or negative indices.

Quadratic equations — factorisation, completing the square, and the quadratic formula are all required. The discriminant tells you the nature of roots without solving: positive means two real roots, zero means one repeated root, negative means no real roots.

Polynomials — know the Factor Theorem (if f(a)=0 then (x-a) is a factor) and Remainder Theorem. Long division and equating coefficients are standard techniques for factorising cubics and higher polynomials.

Inequalities — remember to reverse the inequality sign when multiplying or dividing by a negative number. Sketching a graph is often the safest way to solve quadratic and rational inequalities.

Common error — Dividing both sides of an inequality by an expression that could be negative (e.g. x) without considering cases. Instead, bring everything to one side and factorise.

3Coordinate Geometry

AS Unit 1 · Straight lines, circles, tangents

Be comfortable switching between forms of a straight line: y=mx+c, ax+by+c=0, and y-y_1=m(x-x_1). The gradient of a perpendicular line is the negative reciprocal.

For circles, know how to find the centre and radius from the expanded equation by completing the square. A tangent to a circle meets the radius at the point of contact at right angles — this property is frequently tested.

The distance formula and midpoint formula are used in many contexts beyond pure coordinate geometry, including vectors and loci problems.

Exam tip — When a question asks you to show that a line is tangent to a circle, solving simultaneously and showing the discriminant is zero is usually the most reliable method.

4Sequences & Series

AS Unit 1 · Arithmetic, geometric, binomial expansion for positive integers

Arithmetic sequences have a common difference; geometric sequences have a common ratio. Be careful with geometric series — the sum to infinity only converges when the common ratio is between -1 and 1.

Binomial expansion at AS is restricted to positive integer powers. Know how to find a specific term (e.g. the coefficient of x^5 in (2+3x)^8) without writing out the whole expansion.

Common error — For geometric series, students often forget to check |r|<1 before using the S_\infty formula. If |r|\geq 1, the series does not converge.

5Trigonometry

AS Unit 1 · Graphs, identities, equations, sine/cosine rule

Know the exact values for 0\text{\textdegree}, 30\text{\textdegree}, 45\text{\textdegree}, 60\text{\textdegree}, 90\text{\textdegree} and their radian equivalents. The sine rule and cosine rule solve any triangle, but choose wisely: sine rule for AAS/ASA, cosine rule for SAS/SSS.

Pythagorean identities connect \sin, \cos, \tan, \sec, \cosec, and \cot. These are essential for simplifying trigonometric expressions and solving equations.

When solving \sin\theta = k or \cos\theta = k, always consider the full range of solutions (use the symmetry of the graph or the CAST diagram). The period of \sin and \cos is 360\text{\textdegree} (or 2\pi rad).

Exam tip — "Solve for 0\text{\textdegree} \leq \theta < 360\text{\textdegree}" means you must find all solutions in that interval. A common mistake is to give only the principal value from a calculator.

6Exponentials & Logarithms

AS Unit 1 · Graphs, laws, solving, modelling

Logarithms undo exponentials: \log_a x is the power you must raise a to in order to get x. The laws of logarithms let you expand and combine log expressions — these mirror the laws of indices.

Exponential growth and decay appear in many modelling contexts: population growth, radioactive decay, cooling, compound interest. The general form is y = Ae^{kt} (growth) or y = Ae^{-kt} (decay).

Linearisation — taking logs of both sides turns relationships like y = ax^n or y = ab^x into straight-line form. This lets you estimate parameters from experimental data using a graph.

Remember — \ln x means \log_e x where e \approx 2.71828. Natural logs appear whenever you differentiate or integrate e^x or \ln x.

7Differentiation

AS Unit 1 · From first principles, polynomials, standard functions, tangents

Differentiation gives the gradient function — the rate of change of y with respect to x. The result is itself a function of x.

At AS you need: polynomials, \sin kx, \cos kx, e^{kx}, \ln x. The chain rule is essential for composite functions (function of a function). The product rule handles products of two functions.

Applications — finding the equation of a tangent or normal to a curve, identifying stationary points (where \frac{dy}{dx}=0), and determining whether they are maxima, minima, or points of inflection using the second derivative.

Stationary points — After finding where \frac{dy}{dx}=0, use the second derivative test: \frac{d^2y}{dx^2}>0 gives a minimum, <0 gives a maximum. If it equals zero, the test is inconclusive and you should examine the sign change of \frac{dy}{dx}.

8Integration

AS Unit 1 · As reverse process, areas under curves

Integration is the reverse of differentiation (with an arbitrary constant c). The definite integral between limits a and b gives the exact area under the curve, provided y \geq 0 throughout the interval.

If the curve dips below the x-axis, the definite integral gives a negative contribution for that portion. The total geometric area requires splitting the integral at the roots and taking absolute values.

Common error — Students often forget the arbitrary constant +c for indefinite integrals, or forget to subtract when evaluating definite integrals ([F(x)]_a^b = F(b) - F(a)).

9Vectors

AS Unit 1 · 2D operations, position vectors, ratio division

A vector has both magnitude and direction. Know how to add and subtract vectors geometrically (triangle/parallelogram laws) and algebraically (component-wise).

The magnitude of a vector is found using Pythagoras. A unit vector has magnitude 1; any vector can be converted to a unit vector by dividing by its magnitude.

The formula for a point dividing a line segment in a given ratio is often tested in coordinate geometry and vector contexts.

These notes are AI-assisted study material. Always cross-check against the official WJEC spec or your teacher before relying on them in an exam.

Unit 2 — Applied Mathematics A

AS Unit 2 · Statistics and Mechanics · WJEC A Level Mathematics (2017)

AStatistics (AS)

AS Unit 2 · Section A · 40 marks

Section A covers data handling and probability:

  • Sampling methods: simple random, systematic, stratified, quota. Know the advantages and limitations of each.
  • Data representation: histograms, box plots, cumulative frequency curves. Be able to read off quartiles, median, and interquartile range.
  • Probability: Venn diagrams, tree diagrams, and the addition/multiplication rules. Mutually exclusive events cannot happen together; independent events do not affect each other.
  • Discrete distributions: binomial and uniform. The binomial models a fixed number of independent trials with constant probability of success.
  • Hypothesis testing at AS uses the binomial distribution. Know the difference between one-tailed and two-tailed tests, and how to interpret a p-value.
Key distinction — "Mutually exclusive" means P(A \cap B) = 0; "independent" means P(A \cap B) = P(A)P(B). These are different concepts and should not be confused.

BMechanics (AS)

AS Unit 2 · Section B · 35 marks

Section B covers basic mechanics:

  • Kinematics: interpreting displacement-time, velocity-time, and acceleration-time graphs. The gradient of displacement gives velocity; the gradient of velocity gives acceleration. The area under a velocity-time graph gives displacement.
  • SUVAT equations apply only to motion with constant acceleration. Identify which quantities are known and which are required, then select the equation that omits the unknown you do not need.
  • Forces: weight (W=mg), normal reaction, tension, friction. Draw clear force diagrams with all forces labelled.
  • Newton's second law (F=ma) is the core principle. Resolve forces parallel and perpendicular to the direction of motion.
  • Friction: the limiting friction is \mu R; the actual friction is whatever is needed to prevent slipping, up to this limit.
Exam tip — Always draw a force diagram. Marks are often awarded for correctly identifying the forces acting on a body, even if the subsequent algebra goes wrong.
These notes are AI-assisted study material. Always cross-check against the official WJEC spec or your teacher before relying on them in an exam.

Unit 3 — Pure Mathematics B

A2 Unit 3 · WJEC A Level Mathematics (2017)

1Proof by Contradiction

A2 Unit 3 · Advanced proof techniques

Proof by contradiction follows a three-step structure:

  1. Assume the opposite of what you want to prove.
  2. Show, through logical deduction, that this assumption leads to an impossible or contradictory statement.
  3. Conclude that the original statement must be true.

The classic example is proving \sqrt{2} is irrational. Assume it is rational, write it as \frac{p}{q} in lowest terms, and deduce that both p and q must be even — contradicting the assumption that the fraction was in lowest terms.

Common structure — "Suppose, for contradiction, that... Then... This is a contradiction. Therefore..." Make the logical structure explicit in your written proof.

2Advanced Algebra

A2 Unit 3 · Rational expressions, modulus, composite/inverse functions

Modulus (|x|) turns any input non-negative. Equations and inequalities involving modulus often require case analysis or graphical interpretation. Sketching y=|f(x)| from y=f(x): reflect any part below the x-axis above it.

Inverse functions — swap x and y, then rearrange. The domain of the inverse is the range of the original function. A function must be one-to-one to have an inverse over its entire domain.

Composite functions — fg(x) means "apply g first, then f". The domain of fg requires x to be in the domain of g, AND g(x) to be in the domain of f.

3Parametric Equations

A2 Unit 3 · Graphs, modelling, differentiation, area

Parametric equations define x and y separately in terms of a third variable (parameter) t. They are useful for describing curves that fail the vertical line test, such as circles and ellipses described anticlockwise.

To find \frac{dy}{dx}, use the chain rule: \frac{dy}{dx} = \frac{dy/dt}{dx/dt}. For the second derivative, differentiate \frac{dy}{dx} with respect to t, then divide by \frac{dx}{dt} again.

The area under a parametric curve is found by substituting x = x(t) and dx = \frac{dx}{dt}\,dt, so the integral becomes \int y \frac{dx}{dt}\,dt with t-limits.

Common error — When finding the second derivative, students often forget the final division by \frac{dx}{dt}. The result must be with respect to x, not t.

4Sequences & Series (A2)

A2 Unit 3 · Sigma notation, summation formulas, infinite series

The binomial expansion extends to any real power n (not just positive integers), provided |x| < 1. This is used for approximations and for expanding expressions like (1+2x)^{-3}.

Summation formulas for \sum r, \sum r^2, and \sum r^3 are given in the formula booklet and are used in method-of-differences problems. Know how to split a complex summation into simpler parts.

Approximation — The binomial series for (1+x)^n is valid only when |x|<1. When using it for approximation, choose x small enough that higher powers become negligible.

5Trigonometry (A2)

A2 Unit 3 · Radians, compound/double angle, sec/cosec/cot, small angles

At A2, all trigonometry uses radians. Know the exact values and be able to convert between degrees and radians (\pi rad = 180\text{\textdegree}).

Compound angle formulas expand \sin(A\pm B), \cos(A\pm B), and \tan(A\pm B). Double angle formulas are special cases where A=B. These are used to solve equations, prove identities, and simplify integrals.

Small angle approximations (\sin x \approx x, \cos x \approx 1-\frac{x^2}{2}, \tan x \approx x) are valid for x in radians near zero. They are used to approximate trigonometric expressions and to derive limits.

Identity technique — When proving identities, start with the more complicated side and manipulate it towards the simpler side. Never treat an identity like an equation (moving terms across the equals sign).

6Advanced Differentiation

A2 Unit 3 · Product, quotient, chain, implicit, parametric

At A2 you need the quotient rule in addition to the product and chain rules. Implicit differentiation handles equations where y is not isolated, such as x^2 + y^2 = 25 or e^{xy} = x+y.

When differentiating implicitly, every time you differentiate a y-term, multiply by \frac{dy}{dx}. Then collect terms and solve for \frac{dy}{dx}.

The derivatives of \tan x, \sec x, \cot x, and \cosec x are required and can all be derived from \sin and \cos using the quotient rule — but memorising them saves time in exams.

7Advanced Integration

A2 Unit 3 · By parts, substitution, partial fractions, areas

Integration by parts is the reverse of the product rule. Choose u using LIATE (Log, Inverse trig, Algebraic, Trig, Exponential) — the function that appears earlier in the list is usually your u.

Partial fractions split a complicated rational function into simpler terms that can be integrated directly. The standard cases are: linear factors, repeated linear factors, and irreducible quadratics.

Substitution simplifies integrals by changing the variable. Always remember to change the limits (for definite integrals) and to substitute back if finding an indefinite integral.

Integration by parts — repeated application — For \int x^2 e^x\,dx you must apply parts twice. The tabular method (DI method) can speed this up and reduce errors.

8Numerical Methods

A2 Unit 3 · Iteration, Newton-Raphson, trapezium rule

When exact solutions are impossible or impractical, numerical methods give approximate answers to a desired accuracy.

Change of sign — If a continuous function f has f(a) and f(b) with opposite signs, there is at least one root in (a,b). This gives an interval, not a precise value.

Newton-Raphson iteratively improves an estimate using the formula x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. It converges quickly if the initial guess is close to the root, but can diverge or cycle if the guess is poor or near a stationary point.

Trapezium rule approximates a definite integral by dividing the area into trapezoids. It overestimates when the curve is concave up and underestimates when concave down.

Exam tip — Questions often ask you to show that a root lies in a given interval. Always explicitly state that f is continuous (usually true for polynomials) and show the sign change with calculated values.
These notes are AI-assisted study material. Always cross-check against the official WJEC spec or your teacher before relying on them in an exam.

Unit 4 — Applied Mathematics B

A2 Unit 4 · Statistics, Differential Equations, and Mechanics · WJEC A Level Mathematics (2017)

AStatistics (A2)

A2 Unit 4 · Section A · 40 marks

A2 Statistics builds on AS with continuous distributions and more advanced inference:

  • Conditional probability and Bayes' theorem — update beliefs in light of new evidence. Tree diagrams are often the clearest way to organise the information.
  • Normal distribution — the most important continuous distribution. Standardise using Z = \frac{X-\mu}{\sigma} to use standard normal tables. The normal approximation to the binomial is useful when n is large and p is near 0.5.
  • Hypothesis testing — tests for correlation (using the pmcc) and for a population mean (using the normal distribution or t-distribution). Know how to state hypotheses, find critical values, and draw conclusions in context.
Hypothesis testing wording — Your conclusion must refer to the original claim and use the phrase "significant evidence" or "insufficient evidence". Never say "prove" or "the probability that the null is true".

BDifferential Equations

A2 Unit 4 · Section B · First-order differential equations

A differential equation relates a function to its derivatives. At A2 you deal with first-order equations that can be solved by separation of variables.

The method: rearrange so all y terms (including dy) are on one side and all x terms on the other, then integrate both sides. The result is a general solution containing an arbitrary constant.

Use initial conditions (e.g. y=3 when x=0) to find the value of the constant and obtain the particular solution.

Many modelling questions set up a differential equation from a verbal description (e.g. "the rate of cooling is proportional to the temperature difference") and then ask you to solve it.

Modelling tip — Always define your variables at the start. If the question asks when a quantity reaches a certain value, solve for t and check that the answer makes sense in the real-world context.

CMechanics (A2)

A2 Unit 4 · Section B · Vector kinematics, circular motion, moments

A2 Mechanics extends to two-dimensional motion and more complex force situations:

  • Vector kinematics — position \mathbf{r}, velocity \mathbf{v} = \frac{d\mathbf{r}}{dt}, acceleration \mathbf{a} = \frac{d\mathbf{v}}{dt}. Integration reverses the process. In 2D, resolve into horizontal and vertical components.
  • Circular motion — an object moving in a circle at constant speed is accelerating towards the centre (centripetal acceleration). The force required is directed towards the centre.
  • Moments — the turning effect of a force: moment = force \times perpendicular distance from the pivot. For equilibrium, the sum of moments about any point must be zero.
Projectile motion — Resolve initial velocity into horizontal (u\cos\alpha) and vertical (u\sin\alpha) components. Horizontal motion has constant velocity; vertical motion has constant acceleration -g. Time is the common link between the two components.

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