CAPE Pure Maths Unit 1 study plan

Classify expressions as polynomials, rational, irrational or modulus-based and state the domain restrictions each imposes

Complete the square on six quadratic expressions and read off vertex, axis of symmetry and range

Solve quadratic inequalities and represent solution sets on a number line and in interval notation

Derive and apply the factor and remainder theorems to factorise cubics with one known root

Sketch graphs of y = f(x) + a, af(x), f(x + a) and |f(x)| from a given base curve without plotting points

Determine domain and range of composite functions and decide when an inverse exists

Find inverse functions algebraically and verify them by composition in both orders

Solve equations and inequalities involving modulus, including cases with two absolute-value terms

Perform polynomial long division and synthetic division and express results in quotient-remainder form

Apply the factor theorem to find unknown coefficients in a cubic given one or two factors

Differentiate from first principles for quadratic, reciprocal and simple surd functions

Use the chain rule on composite functions written in the form (ax + b)^n and (ax + b)^(1/2)

Differentiate products and quotients fluently and simplify answers to a single fraction where required

Find equations of tangents and normals to curves at given points

Locate stationary points, classify them by first-derivative sign change and by second derivative

Sketch curves using intercepts, stationary points and asymptotic behaviour

Integrate standard powers, exponentials and trigonometric functions, including negative and fractional indices

Apply integration by substitution for linear and simple non-linear substitutions

Use integration by parts for products of polynomial and exponential or trigonometric factors

Evaluate definite integrals and apply the fundamental theorem correctly with limits

Find distance, midpoint, gradient and equation of a line through two points, including perpendicular and parallel conditions

Convert between general, gradient-intercept and parametric forms of a line and circle

Solve trigonometric equations in a given interval using identities and the CAST or graph method

Prove and apply compound-angle and double-angle identities to simplify expressions

Compute areas between a curve and the x-axis, including regions below the axis handled with modulus or split limits

Find volumes of revolution about the x-axis and set up the integral from a described region

Apply differentiation to rates of change and connected-rates problems with explicit chain-rule statements

Solve maximum and minimum optimisation problems with a stated constraint and a justified conclusion

Complete two full-length timed papers under exam conditions and record time per question

Categorise every lost mark as conceptual, algebraic or presentation error and rank the causes

Re-teach the three weakest sub-topics from notes and redo the corresponding questions cold

Assemble a final one-page error log of recurring mistakes to review in the last 48 hours