FY13CE Pure Mathematics study guide
FY13CE Pure Mathematics · Work through your exam, one topic at a time.
FY13CE Pure Mathematics is a written, constructed-response examination in pure mathematics at the standard associated with the FY13CE designation. It is aimed at candidates who need a secure, proof-based command of single-variable calculus, algebra, trigonometry, vectors and complex numbers rather than routine procedural fluency alone. The FY13CE Pure Mathematics study guide has 12 sections and about 4,822 words.
01Proof, logic and mathematical reasoning
The Proof, logic and mathematical reasoning domain tests whether you can build a valid deductive argument rather than merely compute an answer. Questions ask you to prove statements about integers, inequalities and algebraic identities, to disprove false claims with a single counterexample, and to identify the logical structure - hypothesis, conclusion and converse - of a given statement.
414 words, about 2 min02Algebra and functions
Algebra and functions tests manipulation of polynomials and rational expressions, and the behaviour of functions as objects: domain, range, inverse, composition and graph transformations. Expect factor and remainder theorem questions, partial fractions, modulus equations and inequalities, and sketching transformed curves.
440 words, about 2 min03Sequences and series
Sequences and series examines arithmetic and geometric progressions, sigma notation, sums to infinity and the binomial expansion. You should be able to derive and apply the standard formulae, decide when an infinite sum converges, and extract a specified coefficient from a binomial expansion without expanding everything.
390 words, about 2 min04Coordinate geometry
Coordinate geometry tests straight lines and circles in the Cartesian plane: equations, gradients, distances, intersections and tangency. Questions typically combine a circle given in expanded form with a line, asking for the centre, radius, points of intersection or a tangent length.
408 words, about 2 min05Trigonometry
Trigonometry tests radian measure, the reciprocal and Pythagorean identities, solving trigonometric equations over a stated interval, and combining sine and cosine into a single harmonic term. You should be fluent in both degrees and radians and know the exact values for the standard angles.
446 words, about 2 min06Exponentials and logarithms
Exponentials and logarithms tests the laws of indices and logs, the graphs of exponential and logarithmic functions, solving equations that mix exponentials, and modelling growth and decay. The natural exponential e and natural logarithm ln are central, and many questions require you to reduce a disguised exponential equation to a quadratic.
428 words, about 2 min07Differentiation and applications
Differentiation and applications tests the technique of differentiating standard, composite, product, quotient, implicit and parametric functions, and the use of derivatives to analyse curves: gradients, tangents and normals, stationary points, rates of change and optimisation. Standard results include d/dx(xⁿ) = nx^(n−1), d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x and d/dx(cos x) = −sin x, with angles in radians.
364 words, about 2 min08Integration and applications
Integration and applications tests indefinite and definite integration, the use of substitution and integration by parts, integration of rational functions via partial fractions, and the geometric applications: area under a curve, area between curves and volumes of revolution. The trapezium rule also appears as a numerical approximation.
378 words, about 2 min09Vectors and three-dimensional geometry
Vectors and three-dimensional geometry tests vector arithmetic in column and i, j, k notation, magnitude and unit vectors, the scalar product, angles between vectors, and the vector equation of a line in three dimensions, including intersections and the classification of line pairs. The magnitude of a = (x, y, z) is a = √(x² + y² + z²), and the unit vector in its direction is a/ a .
375 words, about 2 min10Complex numbers
Complex numbers tests arithmetic in the form a + bi, conjugates, modulus and argument, the Argand diagram, modulus–argument form, and the roots of polynomial equations with real coefficients. De Moivre's theorem and geometric interpretations appear in the more demanding questions.
364 words, about 2 min11Matrices and linear transformations
Matrices and linear transformations tests matrix arithmetic, the determinant and inverse of 2 × 2 matrices, the solution of linear systems, and the geometric interpretation of matrices as transformations of the plane, including invariant points and lines. Matrix addition is elementwise, but multiplication requires matching dimensions and is not commutative, so AB is generally not BA.
412 words, about 2 min12Numerical methods and problem solving
Numerical methods and problem solving tests the approximation of roots and integrals when exact methods are unavailable or awkward. It covers locating a root by sign change, fixed-point iteration, the Newton–Raphson method, and the trapezium rule, together with judging whether an approximation is reliable.
403 words, about 2 min