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.

• 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

• 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

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

• 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

• 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 $-f(x)$, $f(-x)$

• 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

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

• 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 $A = \frac{1}{2}r^2\theta$

• 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

• 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

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

• 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

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.

Pure mathematics is the language of pattern and structure underlying all quantitative reasoning. Algebra provides the grammar for expressing relationships, while calculus gives the tools to analyse change. Functions are machines that transform inputs to outputs, and their inverses reverse the process. Sequences and series build complexity from repetition, and proof ensures every claim rests on unshakeable logical foundations. The discipline trains your mind to decompose complex problems into manageable steps, a skill that transfers far beyond mathematics.

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.

• Mechanics Index — Mechanics applies the algebraic, calculus, and vector skills developed in Pure Mathematics to physical problems.
• Statistics Index — Statistics uses algebraic manipulation, functions, and numerical methods to analyse data and probability.
• Further Pure Mathematics 1 — Extends the core Pure content into complex numbers, matrices, and advanced calculus.
• Proof — Proof techniques unify all pure mathematics topics through rigorous logical argument.

Confusing $f'(x)$ with $f(x)$: The derivative $f'(x)$ gives the gradient of the tangent at any point, not the value of the function. Students often evaluate $f'(x)$ when asked for $f(x)$, or vice versa. Always check whether the question asks for the function value or the rate of change.

Forgetting the chain rule: When differentiating composite functions like $\sin(2x)$ or $e^{3x}$, the chain rule gives $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$. Students often forget the inner derivative, writing the derivative of $\sin(2x)$ as $\cos(2x)$ instead of $2\cos(2x)$.

Losing the constant of integration: Every indefinite integral requires $+ C$. Forgetting it loses a mark even if the rest of the integration is correct. The constant represents the family of all antiderivatives.