Pure Mathematics

Pure Mathematics is the backbone of A-Level Mathematics. It develops the algebraic fluency, calculus techniques, and proof skills that underpin every applied topic in the course. This section covers all pure mathematics content from algebraic manipulation through to numerical methods.

Topics Covered

Algebraic Expressions

• Indices and surds — laws of indices, rationalising denominators, simplifying $\sqrt{a} + \sqrt{b}$ expressions
• Algebraic fractions — simplifying, adding, subtracting, multiplying, dividing
• Partial fractions — decomposition for integration and series expansion

Quadratics

• Solving quadratics — factorising, quadratic formula, completing the square
• The discriminant — $b^2 - 4ac > 0$ (two real roots), $= 0$ (repeated root), $< 0$ (no real roots)
• Graphs of quadratics — vertex form $y = a(x - h)^2 + k$, transformations, roots and intercepts
• Modelling with quadratics — projectile paths, optimisation problems

Equations and Inequalities

• Simult equations — substitution and elimination methods
• Quadratic inequalities — solving and representing on a number line
• Graphical solutions — intersection points as solutions to simultaneous equations

Coordinate Geometry

• Straight lines — gradient $m = \frac{y_2 - y_1}{x_2 - x_1}$, equation $y - y_1 = m(x - x_1)$
• Parallel and perpendicular lines — $m_1 = m_2$ and $m_1 m_2 = -1$
• Circles — $(x-a)^2 + (y-b)^2 = r^2$; tangent and radius properties

Functions

• Domain and range — restrictions (denominators, square roots, logarithms)
• Composite functions — $fg(x)$; order of composition matters
• Inverse functions — reflection in $y = x$; finding $f^{-1}(x)$ algebraically
• Transformations — translations $f(x - a) + b$, stretches $pf(qx)$, reflections $-f(x)$, $f(-x)$

Sequences and Series

• Arithmetic sequences — $a_n = a + (n-1)d$; $S_n = \frac{n}{2}(2a + (n-1)d)$
• Geometric sequences — $a_n = ar^{n-1}$; $S_n = \frac{a(1-r^n)}{1-r}$; sum to infinity for $|r| < 1$
• Sigma notation — $\sum$ notation and its evaluation

Binomial Expansion

• $(a+b)^n$ — Pascal’s triangle, binomial coefficients $\binom{n}{r} = \frac{n!}{r!(n-r)!}$
• Approximations — using binomial expansion for $(1+x)^n$ when $|x| < 1$

Trigonometry

• Ratios and graphs — $\sin$, $\cos$, $\tan$; periods, amplitudes, transformations
• Identities — $\sin^2\theta + \cos^2\theta = 1$, $\tan\theta = \frac{\sin\theta}{\cos\theta}$, double angle formulae
• Solving trigonometric equations — finding all solutions in a given range; CAST diagram
• Radians — arc length $s = r\theta$, sector area $A = \frac{1}{2}r^2\theta$

Exponentials and Logarithms

• Exponential functions — $y = e^x$; growth and decay models
• Logarithms — $\log_a b = c \iff a^c = b$; laws of logarithms
• Solving equations — using logarithms to solve $a^x = b$; change of base formula

Calculus

• Differentiation — power rule, chain rule, product rule, quotient rule; finding gradients, tangents, normals, stationary points, maxima and minima
• Integration — reverse of differentiation; definite and indefinite integrals; area under a curve; integration by substitution and by parts
• Differential equations — forming and solving first-order separable equations
• Connected rates of change — related rates problems using the chain rule

Vectors

• 2D vectors — magnitude, direction, addition, subtraction, scalar multiplication
• Position vectors — $\vec{AB} = \mathbf{b} - \mathbf{a}$
• Geometric problems — parallel vectors, collinear points, dividing a line in a ratio

Proof

• Proof by deduction — direct logical argument
• Proof by exhaustion — checking all cases
• Proof by contradiction — assuming the negation and deriving a contradiction

Numerical Methods

• Location of roots — sign change between $f(a)$ and $f(b)$
• Iteration — fixed-point iteration $x_{n+1} = g(x_n)$; staircase and cobweb diagrams
• Newton-Raphson — $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$; derivation and limitations

Study Tips

1. Practise algebraic manipulation daily — fluency with indices, fractions, and factorisation is non-negotiable. Every question requires it.
2. Show every step in proofs — examiners mark each logical step. Skipping steps loses marks even if the conclusion is correct.
3. Sketch graphs — always sketch before solving. Understanding the geometry of a function prevents errors in finding solutions.
4. Learn derivative and integral rules — chain, product, and quotient rules must be automatic. Derive them once to understand them, then practise until they’re fast.
5. Check calculus answers — differentiate your integral (or integrate your derivative) to verify.

How to Use These Notes

Follow the sidebar order. Each page provides rigorous definitions, proofs, worked examples with full working, and exam-style problems. The material is cumulative — master each topic before moving to the next.