A-Level Maths Foundations Course

Algebra and functions are the foundation of A-Level pure maths. Nearly every later topic, from calculus to applied questions, reuses the skills covered here, so weak basics quickly become weak exam results.

For a quadratic ax^2 + bx + c = 0, keep three tools ready: factorising when the roots are simple, the quadratic formula x = (-b ± sqrt(b^2 - 4ac))/(2a), and completing the square. The discriminant b^2 - 4ac decides the nature of the roots: positive means two distinct real roots, zero means one repeated root, and negative means no real roots. Completing the square also gives the turning point, which is needed for sketching and for range questions.

Functions are precise objects. The domain is the set of allowed inputs and the range is the set of outputs. For a composite function fg(x), apply g first, then f. The inverse function f^-1(x) swaps inputs and outputs, so its graph is the reflection of y = f(x) in the line y = x, and it only exists when f is one-to-one.

Coordinate geometry ties algebra to graphs. Use the midpoint, distance and gradient formulas, and remember that perpendicular gradients multiply to -1. A circle with centre (a, b) and radius r has equation (x - a)^2 + (y - b)^2 = r^2. The binomial expansion expands expressions such as (1 + x)^n, and graph transformations such as y = f(x) + a and y = f(x + a) need careful attention to which movement is vertical and which is horizontal.

• Method first: write the formula, then substitute numbers.
• Check the discriminant before deciding how to solve a quadratic.
• Common mistake: in fg(x), g is applied first; reversing the order changes the answer.

Calculus is the engine of A-Level pure maths and connects directly to mechanics and applied questions. Differentiation measures rates of change; integration reverses that process and measures accumulated quantities such as area.

Start with the power rule: if y = x^n, then dy/dx = n x^(n-1). Differentiate linear combinations term by term. Products need the product rule (u'v + uv'), quotients need the quotient rule ((u'v - uv')/v^2), and composites need the chain rule (dy/dx = dy/du × du/dx). Learn the standard results too: d/dx(e^x) = e^x, d/dx(ln x) = 1/x, d/dx(sin x) = cos x and d/dx(cos x) = -sin x.

Derivatives answer geometry questions. A stationary point occurs where dy/dx = 0; decide whether it is a maximum or minimum using the second derivative or a sign table. The tangent at a point shares the gradient of the curve, while the normal is perpendicular to the tangent, so its gradient is the negative reciprocal.

Integration reverses differentiation. For powers, integrate x^n to x^(n+1)/(n+1) + c when n ≠ -1, and always include the constant of integration for indefinite integrals. A definite integral gives the signed area between the curve and the x-axis, so sketch the region when it crosses the axis. In kinematics, velocity is the derivative of displacement, acceleration is the derivative of velocity, and integration moves in the opposite direction.

• Chain rule first: for a bracket raised to a power, differentiate the outside, then multiply by the derivative of the inside.
• Watch the signs: the quotient rule has a minus sign, and d/dx(cos x) = -sin x.
• Common mistake: applying the power rule to products; products need the product rule.

Trigonometry connects angles with ratios and appears throughout A-Level maths, including calculus, vectors, and geometry. Fluency with radians, exact values, graphs, and identities saves time in nearly every paper.

Radians are the natural unit for advanced work: 2π radians = 360°, so π = 180°, π/2 = 90° and π/3 = 60°. Learn the exact values of sine, cosine and tangent for 0, π/6, π/4, π/3 and π/2, and remember the shape of each graph. To solve sin x = 1/2 for 0 ≤ x < 2π, find the principal value x = π/6, then use symmetry: sine is positive in quadrants one and two, so the solutions are x = π/6 and x = 5π/6.

Identities turn complicated expressions into solvable ones. The Pythagorean identity sin^2 x + cos^2 x = 1 is used constantly. The double-angle identities cos 2x = cos^2 x - sin^2 x = 2 cos^2 x - 1 = 1 - 2 sin^2 x and sin 2x = 2 sin x cos x simplify expressions and make integration easier.

For any triangle, the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a^2 = b^2 + c^2 - 2bc cos A handle non-right triangles, while Area = (1/2)ab sin C gives the area from two sides and the included angle. For sectors, arc length = rθ and area = (1/2)r^2 θ, and these formulas require θ in radians.

• Calculator mode: check whether the question uses degrees or radians before calculating.
• Find every solution: use graph symmetry, not just the principal value.
• Common mistake: using the sine rule to find an angle without checking the ambiguous case.

Exponentials, logarithms, and sequences model growth, decay, and repeating patterns. They often appear together in A-Level papers because logarithms undo exponentiation, while series add the terms of sequences.

An exponential function has the form y = a b^x, and e^x is special because its derivative is itself. To solve 3^(x+1) = 20, take logarithms of both sides, use the power law to bring the exponent down, then rearrange: (x + 1) ln 3 = ln 20, so x = ln 20 / ln 3 - 1. The logarithm laws are log(xy) = log x + log y, log(x/y) = log x - log y, and log(x^k) = k log x.

An arithmetic sequence has a constant difference d. Its nth term is a + (n - 1)d and its sum is S_n = n/2 (2a + (n - 1)d). A geometric sequence has a constant ratio r. Its nth term is ar^(n-1) and its sum is S_n = a(1 - r^n)/(1 - r). The sum of an infinite geometric series exists only when |r| < 1 and equals a/(1 - r).

When solving word problems, identify the model first. For growth or decay, find the initial value and the multiplier; for sequences, decide whether the difference or the ratio is constant before choosing a formula. State the formula, substitute, and simplify with exact values where possible.

• Log laws need one log: log(x + y) cannot be split into log x + log y.
• Check the ratio: an infinite sum only exists when |r| < 1.
• Common mistake: answering with the nth term when the question asks for the sum.

A-Level maths exams award method marks as well as final answers, so a clear written solution is part of the working, not just a record. The goal is a routine that earns marks even when the final number is wrong.

Read the whole question before writing, and identify which formula or technique it tests. Write the formula in full, substitute values with correct notation, then simplify line by line. For multi-part questions, later parts usually build on earlier answers, so keep exact values where possible and label each part clearly. Box the final answer and include units when relevant.

Use the marks to plan time: allow roughly one minute per mark, so a 5-mark question deserves five minutes. If a calculation becomes messy, pause and check the method before continuing. Common lost marks come from sign errors, missing constants of integration, forgotten domain restrictions, and answers left in the wrong form.

After each past paper, review every mistake and classify it as a careless error, a misread question, a missing method, or a weak topic. Rebuild weak topics with focused questions, then repeat the cycle. This targeted review converts exam experience into marks faster than doing random papers.

• Show the method: a wrong final answer with the correct method still earns partial marks.
• Read the requirement: underline phrases such as "leave your answer in surd form".
• Common mistake: rushing part (a), then carrying a wrong value through the rest of the question.