TMUA Formulae to Memorise (No Formula Sheet)

Unlike A-level, the TMUA hands you no formula booklet and no calculator. Whatever you need has to come out of your own memory, instantly, while the clock is running at under four minutes a question. That makes a clean mental list of must-know formulae one of the highest-return things you can revise. This guide groups every formula worth memorising by topic, with a one-line note on when each one earns its place, and finishes with a real specimen question so you can check that you can actually use them, not just recite them.

Indices and surds

These power almost every algebraic simplification on Paper 1, and they are the first thing examiners hide a fast route behind.

• $a^m \times a^n = a^{m+n}$ and $a^m \div a^n = a^{m-n}$: combine powers with the same base.
• $(a^m)^n = a^{mn}$: a power raised to a power.
• $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$: zero and negative indices.
• $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m}$: fractional indices as roots.
• $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$: splitting surds.
• Rationalise with $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$, and clear $a + \sqrt{b}$ by multiplying by the conjugate $a - \sqrt{b}$.

Knowing these as reflexes is half of what the calculator-free techniques guide calls "slickness": the exam rewards the student who never writes a decimal.

Logarithms

Logs appear in equations, growth problems and disguised index questions. The three laws plus the change of base cover almost everything.

• $\log_a (xy) = \log_a x + \log_a y$: product becomes a sum.
• $\log_a \frac{x}{y} = \log_a x - \log_a y$: quotient becomes a difference.
• $\log_a (x^k) = k \log_a x$: bring the power down to the front.
• $\log_a a = 1$ and $\log_a 1 = 0$: the two anchor values.
• $a^x = b \iff x = \log_a b$: the definition, used to escape an exponent.
• Change of base: $\log_a x = \frac{\log_b x}{\log_b a}$, which lets you switch to a base you can evaluate.

Sequences and series

Arithmetic and geometric progressions are a reliable source of marks if you have the four core formulae memorised cold.

For an arithmetic progression with first term $a$ and common difference $d$:

• $n$th term: $u_n = a + (n-1)d$.
• Sum: $S_n = \frac{n}{2}(2a + (n-1)d)$, or equivalently $S_n = \frac{n}{2}(a + l)$ where $l$ is the last term.

For a geometric progression with first term $a$ and common ratio $r$:

• $n$th term: $u_n = a r^{n-1}$.
• Sum of $n$ terms: $S_n = \frac{a(1 - r^n)}{1 - r}$.
• Sum to infinity: $S_\infty = \frac{a}{1 - r}$, valid only when $|r| < 1$.

The single most common slip here is forgetting the convergence condition $|r| < 1$ before quoting the sum to infinity, so tie the two together in your memory. It is also worth being fluent at moving between the two progressions: a question that looks arithmetic on the surface can hide a geometric ratio, and spotting which you are dealing with is usually half the work.

Coordinate geometry

Lines and circles turn up constantly, often dressed as something else. These are the formulae that let you set up the algebra without thinking.

Two facts pair with these and are worth memorising as reflexes: perpendicular gradients satisfy $m_1 m_2 = -1$, and a tangent to a circle meets the radius at the point of contact at a right angle. Together they unlock most circle questions. It also pays to remember that a circle equation often arrives in the expanded form $x^2 + y^2 + 2gx + 2fy + c = 0$, and completing the square on both variables recovers the centre and radius, so the standard form above is always within reach.

Trigonometry

Trig on the TMUA is mostly about identities and exact values, since you have no calculator to find $\sin 30^\circ$ for you.

The identities to know cold:

• $\sin^2\theta + \cos^2\theta = 1$: the Pythagorean identity, the workhorse.
• $\tan\theta = \frac{\sin\theta}{\cos\theta}$: converting between ratios.

The exact values you must recall without hesitation, because no calculator will give them to you:

Also keep the area of a triangle $\frac{1}{2} a b \sin C$ and the sine and cosine rules to hand, but the table above is the part that catches people out under pressure. Reaching for these instantly is exactly the kind of habit the calculator-free techniques reward.

Try a real one

Theory only goes so far. Here is a genuine specimen Paper 1 question that leans on the formulae above. Attempt it fully before revealing the solution, and you will quickly learn which formulae you actually have at your fingertips:

Differentiation and integration of powers

Basic calculus is firmly on the syllabus, and the two power rules cover the overwhelming majority of what Paper 1 asks.

• Differentiate: if $y = x^n$ then $\frac{dy}{dx} = n x^{n-1}$: multiply by the power, drop it by one.
• Integrate: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$, valid for $n \neq -1$: add one to the power, divide by the new power.
• A constant differentiates to $0$, and $\frac{d}{dx}(k x) = k$.
• A stationary point occurs where $\frac{dy}{dx} = 0$; the sign of $\frac{d^2y}{dx^2}$ tells you maximum (negative) or minimum (positive).
• A definite integral $\int_a^b f(x)\, dx$ gives the area under the curve between $x = a$ and $x = b$.

These show up in optimisation, tangents and area problems. If your calculus is rusty, the wider preparation guide sets out where it fits in a revision plan.

The discriminant and quadratic formula

Quadratics are everywhere, and the discriminant is one of the most heavily tested single ideas on the whole test, because it answers questions about the number of roots without you ever solving the equation.

For $a x^2 + b x + c = 0$:

• Solve with $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: the quadratic formula.
• The discriminant is $\Delta = b^2 - 4ac$, and its sign decides everything:
  • $b^2 - 4ac > 0$ gives two distinct real roots,
  • $b^2 - 4ac = 0$ gives one repeated root,
  • $b^2 - 4ac < 0$ gives no real roots.
• Completing the square turns $a x^2 + b x + c$ into $a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$, which is how you find a turning point by hand.

Many "find the values of $k$ for which..." questions are pure discriminant problems in disguise, so train yourself to spot them. For how this kind of pattern-spotting splits across the two papers, see Paper 1 vs Paper 2.

Turning the list into reflexes

A list you have read once is not a list you have memorised. The fastest way to make these automatic is to use them, not to stare at them: drill questions that force you to reach for each group until you stop pausing. Reproduce the whole sheet from a blank page, then check it; the gaps you find are exactly the formulae you would have fumbled in the exam. A useful test is to time yourself writing out every group in under ten minutes, because if you cannot recall a formula calmly at your desk you certainly will not under exam pressure.

The split across the two papers matters too. Paper 1 leans hardest on the algebraic and calculus formulae above, while Paper 2 uses fewer of them but expects you to deploy each one inside a logical argument, so it helps to know where each formula tends to surface; the Paper 1 vs Paper 2 breakdown covers that division in detail. None of this is harder than A-level content, which is the point made in is the TMUA hard: the difficulty is speed and recall, and recall is the part you can simply remove with practice.

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