FY13CE Pure Mathematics study plan

Write out the truth table for implication, converse, contrapositive and negation, and use them to negate four quantified statements correctly

Construct direct proofs of five standard statements (sum of two odd integers is even, product of rational and irrational, divisibility of consecutive integers) without referring to notes

Classify five given arguments as valid or invalid by counterexample, stating the counterexample explicitly

State the first-principles definition of the derivative and use it to differentiate three simple polynomials and one reciprocal function

Apply the chain, product and quotient rules to ten composite expressions and record every algebraic slip in an error log

Complete the square on six quadratic expressions and read off vertex, range and discriminant conditions

Solve polynomial and rational inequalities, expressing solution sets in interval notation

Sketch five rational functions showing asymptotes, intercepts and behaviour near excluded values

Differentiate to locate and classify stationary points of four curves, justifying each classification with the second or first derivative test

Solve three optimisation problems end to end, including defining the variable and stating the domain of validity

Integrate standard forms including powers, exponentials, trigonometric functions and 1/x, checking each answer by differentiating back

Apply substitution to eight integrals, choosing the substitution and changing limits for definite cases

Use integration by parts on four products, deciding which factor to differentiate

Compute definite integrals to find areas between curves and volumes of revolution about the x-axis

Solve three differential equations by separating variables and apply an initial condition to fix the constant

Derive and use the identities for secant, cosecant and cotangent, and prove four identities by transforming one side only

Solve trigonometric equations in a given interval, listing all solutions and rejecting extraneous ones

Apply the double-angle and addition formulae to rewrite expressions and evaluate exact values

Use the binomial expansion for positive integer powers and find a named coefficient without expanding fully

Sum arithmetic and geometric series, including to infinity where convergent, and decide convergence from the common ratio

Find equations of lines and circles, including centre and radius from the general form by completing the square

Determine intersections of lines with circles and use the discriminant to test tangency

Perform vector addition, scalar multiplication, dot product and cross product, and interpret each geometrically

Find the angle between two vectors and the shortest distance from a point to a line

Write equations of lines and planes in vector form and find intersections of a line with a plane

Apply the laws of logarithms to solve exponential equations and equations of the form a^x = b

Sketch exponential and logarithmic graphs, stating domain, range, intercepts and asymptotes

Convert between Cartesian, modulus-argument and exponential forms of a complex number for eight examples

Multiply, divide and find powers and roots of complex numbers using modulus-argument form, including the nth roots of unity

Solve quadratic equations with complex coefficients and locate roots on an Argand diagram

Carry out matrix multiplication, find determinants and inverses of 2x2 and 3x3 matrices, and solve linear systems by elimination

Describe the geometric effect of a given 2x2 transformation on the unit square and identify invariant lines

Apply the Newton-Raphson method and fixed-point iteration for three equations, choosing a suitable starting value

Use the sign-change criterion to locate a root and state why it guarantees a root only under continuity

Estimate the error in a numerical approximation and explain when an iteration fails to converge

Sit two full-length mixed papers under timed conditions and mark them against a written scheme

Re-derive every formula used in the papers from memory to confirm recall

Reattempt all previously logged errors and confirm each is now solved correctly

Write a condensed formula and method sheet covering all twelve domains from memory, then fill gaps

Rank remaining weak areas and schedule targeted practice for the final days