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The TMUA syllabus

The TMUA is two papers and they do not examine the same content, which is worth knowing before you plan any revision. Paper 1 covers the mathematics set out below, which stops at roughly the first year of A level and assumes no Further Maths. Paper 2 has content of its own on top of that, and it also draws on everything in Paper 1's list, so the 43 topics here are examinable across the two papers rather than divided neatly between them.

This is the specification content itself rather than a summary of it, and it is worth reading in both directions: it rules out a good deal of A-level material that is simply not on the specification, and it means every line below is fair game. Each topic links to notes covering it.

Paper 1 — Applications of Mathematical Knowledge

Paper 1 hands you familiar A-level pure content and asks you to apply it quickly. The mathematics is not the hard part: the difficulty is spotting which tool the question wants and reaching the answer by hand. Questions are built so that a well-chosen route takes under a minute and a brute-force route takes five.

36 topics across 8 sections. All Paper 1 notes

Algebra and functions

  • Laws Of Indices For Rational Exponents
    Laws of indices for all rational exponents.
  • Use And Manipulation Of Surds
    Use and manipulation of surds.
    Simplifying expressions that contain surds, including rationalising the denominator.
  • Quadratic Functions And Graphs
    Quadratic functions and their graphs; the discriminant of a quadratic function; completing the square; solution of quadratic equations.
  • Simultaneous Equations Substitution
    Simultaneous equations: analytical solution by substitution, e.g. of one linear and one quadratic equation.
  • Solution Of Linear And Quadratic Inequalities
    Solution of linear and quadratic inequalities.
  • Polynomial Algebra And Theorems
    Algebraic manipulation of polynomials, including:
    a. expanding brackets and collecting like terms
    b. factorisation and simple algebraic division (by a linear polynomial, including those of the form     ax+bax + b, and by quadratics, including those of the form ax2+bx+cax^2 + bx + c)
    c. use of the Factor Theorem and the Remainder Theorem
  • Function Mappings And Common Graphs
    Qualitative understanding that a function is a many-to-one (or sometimes just a one-to-one) mapping.
    Familiarity with the properties of common functions, including     f(x)=xf(x)=\sqrt{x} (which always means the 'positive square root') and f(x)=∣x∣f(x)=|x|

Sequences and series

  • Sequences And Series
    Sequences, including those given by a formula for the nthn^{th} term and those generated by a simple recurrence relation of the form xn+1=f(xn)x_{n+1} = f(x_n)
  • Arithmetic Sequences And Series
    Arithmetic series, including the formula for the sum of the first nn natural numbers.
  • Geometric Series Summation
    The sum of a finite geometric series.
    The sum to infinity of a convergent geometric series, including the use of     ∣r∣<1|r|<1
  • Binomial Expansion Integer Powers
    Binomial expansion of (1+x)n(1 + x)^n for positive integer nn, and for expressions of the form (a+f(x))n(a+f(x))^n for positive integer nn and simple f(x)f(x).
    The notations     n!n! and (nr)\binom{n}{r}.

Coordinate geometry in the (x,y)-plane

  • Coordinate Geometry Straight Lines
    Equation of a straight line, including:
    a.     y−y1=m(x–x1)y-y_1 = m(x – x_1)
    b.     ax+by+c=0ax + by + c = 0
    Conditions for two straight lines to be parallel or perpendicular to each other.
    Finding equations of straight lines given information in various forms.
  • Equation Of A Circle
    Coordinate geometry of the circle, using the equation of a circle in the forms:
    a.     (x–a)2+(y–b)2=r2(x – a)^2 + (y – b)^2 = r^2
    b.     x2+y2+cx+dy+e=0x^2 + y^2 + cx + dy + e = 0
  • Coordinate Geometry Circle Properties
    Use of the following circle properties:

    a. The perpendicular from the centre to a chord bisects the chord.

    b. The tangent at any point on a circle is perpendicular to the radius at that point.

    c. The angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at any point on the circumference.

    d. The angle in a semicircle is a right angle.

    e. Angles in the same segment are equal.

    f. The opposite angles in a cyclic quadrilateral add to 180°.

    g. The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.

Trigonometry

Exponentials and logarithms

  • Exponential Functions And Graphs
    y=axy = a^x and its graph, for simple positive values of aa.
  • Laws Of Logarithms
    Laws of logarithms:
    a.     ab=c⇔b=log⁡aca^b = c \Leftrightarrow b = \log_a c
    b.     log⁡ax+log⁡ay=log⁡a(xy)\log_a x + \log_a y = \log_a (xy)
    c.     log⁡ax−log⁡ay=log⁡a(xy)\log_a x - \log_a y = \log_a (\frac{x}{y})
    d.     klog⁡ax=log⁡a(xk)k\log_a x = \log_a (x^k)
    including the special cases:
    e.     log⁡a(1x)=−log⁡ax\log_a (\frac{1}{x}) = -\log_a x
    f.     log⁡aa=1\log_a a = 1
    Questions requiring knowledge of the change of base formula will not be set.
  • Solving Exponential And Logarithmic Equations
    The solution of equations of the form ax=ba^x = b, and equations which can be reduced to this form, including those that need prior algebraic manipulation.

Differentiation

  • Differentiation Gradient Rate Of Change
    The derivative of f(x)f(x) as the gradient of the tangent to the graph y=f(x)y = f(x) at a point.
    a. Interpretation of a derivative as a rate of change.
    b. Second-order derivatives.
    c. Knowledge of notation:     dydx,d2ydx2,f′(x)\frac{dy}{dx}, \frac{d^2 y}{dx^2}, f'(x), and f′′(x)f''(x)
    Differentiation from first principles is excluded.
  • Differentiation Rational Powers
    Differentiation of xnx^n for rational nn, and related sums and differences. This might require some simplification before differentiating.
  • Applications Of Differentiation
    Applications of differentiation to gradients, tangents, normals, stationary points (maxima and minima only), strictly increasing functions [ if f′(x)>0f'(x) > 0 ] and strictly decreasing functions [ if f′(x)<0f'(x) < 0 ]. Points of inflexion will not be examined, although a qualitative understanding of points of inflexion in the curves of simple polynomial functions is expected.

Integration

  • Definite Integration Area Between Curve Axis
    Definite integration as related to the 'area between a curve and an axis'. The difference between finding a definite integral and finding the area between a curve and an axis is expected to be understood.
  • Integration Basics
    Finding definite and indefinite integrals of xnx^n for nn rational, n≠1n \neq 1, and related sums and differences, including expressions which require simplification prior to integrating.
  • Fundamental Theorem Of Calculus
    An understanding of the Fundamental Theorem of Calculus and its significance to integration. Simple examples of its use may be required in the forms:
    a.     ∫abf(x)dx=F(b)–F(a)\int_a^b f(x)dx = F(b) – F(a), where F′(x)=f(x)F'(x) = f(x)
    b.     ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t)dt = f(x)
  • Combining Definite Integrals
    Combining integrals with either equal or contiguous ranges.
  • Trapezium Rule Approximation
    Approximation of the area under a curve using the trapezium rule; determination of whether this constitutes an overestimate or an underestimate.
  • Solving Simple Differential Equations
    Solving differential equations of the form dydx=f(x)\frac{dy}{dx} = f(x)

Graphs of functions

  • Graphs Of Functions
    Recognise and be able to sketch the graphs of common functions that appear in this specification: these include lines, quadratics, cubics, trigonometric functions, logarithmic functions, exponential functions, square roots, and the modulus function.
  • Graph Transformations
    Knowledge of the effect of simple transformations on the graph of y=f(x)y = f(x) with positive or negative value of aa as represented by:
    a.     y=af(x)y = af(x)
    b.     y=f(x)+ay = f(x) + a
    c.     y=f(x+a)y=f(x+a)
    d.     y=f(ax)y = f(ax)
    Compositions of these transformations. Knowledge and use of the notation     f(g(x))f(g(x)).
  • Linear Graphs M And C
    Understand how altering the values of mm and cc affects the graph of y=mx+cy = mx + c
  • Graphs Of Quadratic Functions Vertex Form
    Understand how altering the values of aa, bb and cc in y=a(x+b)2+cy = a(x + b)^2 + c affects the corresponding graph.

Paper 2 — Mathematical Reasoning

Paper 2 tests how you reason about mathematics rather than how much of it you can do. Questions turn on logic and proof: what follows from what, whether a condition is necessary or sufficient, what a single counterexample destroys, and where a printed argument breaks down. Very few candidates are taught this explicitly at school, which makes it the paper where deliberate preparation moves a score most.

Everything listed under Paper 1 is examinable here as well, and is not repeated below. What follows is the content Paper 2 adds.

7 topics across 3 sections. All Paper 2 notes

The Logic of Arguments

  • Statements And Connectives
    Understand and be able to use mathematical logic in simple situations:
    - The terms true and false;
    - The terms and, or (meaning inclusive or), not;
    - Statements of the form:
     if A then B
     A if B
     A only if B
     A if and only if B
    - The converse of a statement;
    - The contrapositive of a statement;
    - The relationship between the truth of a statement and its converse and its contrapositive.
  • Necessary And Sufficient
    Understand and use the terms necessary and sufficient.
  • Quantifiers
    Understand and use the terms for all, for some (meaning for at least one), and there exists.
  • Negation And Implication
    Be able to negate statements that use any of the above terms.

Mathematical Proof

  • Methods Of Proof
    Follow a proof of the following types, and in simple cases know how to construct such a proof:
    - Direct deductive proof ('Since A, therefore B, therefore C, ..., therefore Z, which is what we wanted to prove.');
    - Proof by cases (for example, by considering even and odd cases separately);
    - Proof by contradiction;
    - Disproof by counterexample.

Identifying Errors in Proofs

  • Spotting Invalid Steps
    Identifying errors in purported proofs.
  • Common Fallacies
    Be aware of common mathematical errors in purported proofs; for example, claiming 'if ab=acab = ac, then b=cb = c' or assuming 'if sin⁡A=sin⁡B\sin A = \sin B, then A=BA = B' neither of which are valid deductions.

How to use the syllabus

Read it once with a pen and mark every line into one of three piles: met it and fluent, met it and rusty, never seen it. The third pile is usually short and is where your time is worth most. The second pile is what past papers are for. The first you can leave alone, however tempting it is to revise what already feels comfortable.

Content alone will not get you the score, though. The TMUA format explains the time limit that turns a familiar topic into a hard question, and it is that constraint, more than the syllabus, that decides most scores.

Work through it topic by topic

Every topic above has notes with worked examples, the mistakes that come up most, and practice questions on exactly that content.

Open the TMUA textbookTMUA past papers