Admissions Test Preparation · Recall & Speed
ESAT Essentials: Formulas, Shortcuts and Exam Tactics
No formula booklet is provided in the ESAT and no calculator is permitted, so everything below has to come from memory and mental arithmetic. This page collects what you must recall on sight, plus the shortcuts that recur most often across the mock papers on this site.
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The pacing arithmetic
Each module gives you 27 multiple-choice questions in 40 minutes. That is about 89 seconds per question: and that average already includes the time you spend reading.
Key Idea (💡): You cannot afford a full derivation on every question. Budget roughly 60 seconds for questions you recognise, which buys you two to three minutes for the handful that are genuinely hard. If a question has consumed 2 minutes with no route in sight, mark it and move on, there is no negative marking, so an unanswered question is strictly worse than a guess.
Fastest Approach (🚀): Read the answer options before you start solving. They tell you the required form (exact surd or decimal, units, sign), and they often let you eliminate two or three choices on dimensions or magnitude alone, sometimes without any algebra at all.
Seven tactics that recur across every paper
- Turn means into sums. Whenever items are added to or removed from a group, convert every average using $\text{Sum} = \text{Mean} \times \text{Count}$, do the arithmetic on sums, then convert back at the end.
- Multiply scale factors; never add percentages. For $z \propto \dfrac{w x^3}{v^2}$, a change multiplies by $\dfrac{k_w\,k_x^{\,3}}{k_v^{\,2}}$. A 20% decrease is a factor of $0.8$, not $0.2$, and it must be cubed if the variable is cubed.
- Square and cube your scale factors. If lengths scale by $k$, areas scale by $k^2$ and volumes by $k^3$. This single idea accounts for a large share of the geometry questions.
- Use the complement for "at least" and "not". $P(\text{at least one}) = 1 - P(\text{none})$ is almost always faster than enumerating cases.
- Filter roots against physical reality. Algebra may hand you three roots; lengths, masses and radii must be positive, so discard the rest. A question whose answer is "0 valid values" is testing exactly this.
- Check what was actually asked. Having found $x$, confirm the question did not ask for $x^3$, or for $RS = x + 5$. This is the single most common way to lose a mark you had already earned.
- Use dimensional analysis as a free check. If an unfamiliar formula appears, reduce every candidate to base units ($\text{kg}$, $\text{m}$, $\text{s}$). Usually only one option has the right dimensions.
Mathematics, recall list
| Quadratic roots (Vieta) | For $ax^2+bx+c=0$: sum $=-\dfrac{b}{a}$, product $=\dfrac{c}{a}$. Discriminant $\Delta=b^2-4ac$ gives 2, 1 or 0 real roots. |
|---|---|
| Quadratic formula | $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$. With $a>0$ and $b$ fixed, a more negative $c$ gives a larger positive root. |
| Arithmetic series | $S_n=\dfrac{n}{2}(a+l)$. Terms from $A$ to $B$ inclusive with common difference $d$: $n=\dfrac{B-A}{d}+1$, which is $n=B-A+1$ for consecutive integers (the "+1" is a classic trap). |
| Geometric series | $u_n=ar^{n-1}$; $S_\infty=\dfrac{a}{1-r}$ valid only for $|r|<1$. Apply the ratio $n-1$ times, not $n$. |
| Circle and sector | $A=\pi r^2$, $C=2\pi r$. Sector $=\tfrac12 r^2\theta$; segment $=\tfrac12 r^2(\theta-\sin\theta)$, with $\theta$ in radians. |
| Triangles | Area $=\tfrac12 ab\sin C$. Equilateral height $=\tfrac{\sqrt3}{2}l$, area $=\tfrac{\sqrt3}{4}l^2$. Right-angled triangle inradius $=\dfrac{a+b-h}{2}$ for shorter sides $a$, $b$ and hypotenuse $h$. |
| Exact trig values | $\sin30^\circ=\tfrac12$, $\sin45^\circ=\tfrac{1}{\sqrt2}$, $\sin60^\circ=\tfrac{\sqrt3}{2}$, $\tan30^\circ=\tfrac{1}{\sqrt3}$, $\tan60^\circ=\sqrt3$. If $\tan\theta=m$, sides are $1:m:\sqrt{1+m^2}$. |
| Clock geometry | Minute hand $6^\circ$/min; hour hand $0.5^\circ$/min. Relative speed $5.5^\circ$/min. Each hour mark is $30^\circ$. |
| Probability | Total probability $P(A)=\sum P(A\mid B_i)P(B_i)$; Bayes $P(B\mid A)=\dfrac{P(A\mid B)P(B)}{P(A)}$. Binomial $P(X=k)=\binom{n}{k}p^k q^{n-k}$. |
| Counting | Handshakes / unordered pairs $=\dbinom{n}{2}=\dfrac{n(n-1)}{2}$. Multiset arrangements $=\dfrac{n!}{n_1!\,n_2!\cdots}$. |
| Surds and indices | $x\sqrt{x}=x^{3/2}$; rationalise with the conjugate. Convert every base to its smallest prime before comparing powers. |
| Polygons | Exterior angle of a regular $n$-gon $=\dfrac{360^\circ}{n}$; interior $=180^\circ-\dfrac{360^\circ}{n}$. Prefer the exterior form, the algebra is far cleaner. |
Common Mistake (⚠️): Forgetting the $\pm$ when you square-root, dropping the "+1" in an inclusive count, and adding standard-form numbers without first matching their powers of ten.
Advanced Mathematics, recall list
| Differentiation | $\dfrac{\text{d}}{\text{d}x}x^n=nx^{n-1}$; product, quotient and chain rules. $\dfrac{\text{d}}{\text{d}x}e^x=e^x$, $\dfrac{\text{d}}{\text{d}x}\ln x=\dfrac1x$. |
|---|---|
| Stationary points | Set $f'(x)=0$. Then $f''>0$ is a minimum (concave up), $f''<0$ a maximum. For a cubic $f$, roots of $f'$ at $x_1,x_2$ mean $f'(x)=k(x-x_1)(x-x_2)$. |
| Integration | $\displaystyle\int x^n\,\text{d}x=\dfrac{x^{n+1}}{n+1}+C$ for $n\neq-1$. Split a quotient with a monomial denominator into separate powers of $x$ before integrating; there is no quotient rule for integration. |
| Definite integrals | $\displaystyle\int_a^c=\int_a^b+\int_b^c$. Linear substitution $u=mx+k$ introduces a factor $\tfrac{1}{m}$ outside the integral, and the limits must be converted to values of $u$. |
| Binomial expansion | General term $T_{k}=\dbinom{n}{k}a^{n-k}b^{k}$. Set the exponent equal to your target power and solve for $k$, do not expand everything. |
| Logs and indices | $b^{\log_b x}=x$; $a^{\log_b c}=c^{\log_b a}$; $\log\dfrac{A}{B}=\log A-\log B$. Dividing by a negative $\log$ flips an inequality. |
| Trig identities | $\sin2x=2\sin x\cos x$; $\cos2x=2\cos^2x-1$; $\sin^2x+\cos^2x=1$. For a multiple angle, scale the domain before listing solutions. |
| Polynomials | Rational root theorem: test factors of the constant term divided by factors of the leading coefficient. $(x-y)$ always divides $x^n-y^n$. After extracting a root, check the leftover discriminant. |
| Logic | Only the contrapositive $\lnot Q\Rightarrow\lnot P$ is equivalent to $P\Rightarrow Q$. Concluding the converse or the inverse from $P\Rightarrow Q$ is a fallacy: either may be true or false. A counterexample must satisfy the premise and fail the conclusion. |
| Recurrence relations | If a recurrence looks intractable, compute the first four or five terms, it is very often periodic. |
Physics, recall list
| Kinematics (SUVAT) | $v=u+at$; $s=ut+\tfrac12at^2$; $v^2=u^2+2as$. On a $v$-$t$ graph, gradient is acceleration and signed area is displacement (count area below the axis as positive for distance). |
|---|---|
| Projectiles | Max height $h=\dfrac{u_y^2}{2g}$ ($u_y$ the initial vertical velocity), independent of mass. Check whether the angle is given to the horizontal or the vertical, it swaps $\sin$ and $\cos$. |
| Momentum | $p=mv$; total momentum is conserved in any collision with no external force; kinetic energy only in elastic ones. Impulse $F\Delta t=m\Delta v$. On a bounce, $\Delta v$ is the sum of the speeds. |
| Energy and work | $E_k=\tfrac12mv^2$, $E_p=mgh$, $W=Fd$, $P=\dfrac{W}{t}$. The bridge between momentum and energy is $E_k=\dfrac{p^2}{2m}$. |
| Circuits | $V=IR$; series $R=R_1+R_2$; parallel $\dfrac1R=\dfrac1{R_1}+\dfrac1{R_2}$. Use $P=\dfrac{V^2}{R}$ in parallel (shared $V$) and $P=I^2R$ in series (shared $I$). $Q=It$, $W=VQ$. |
| Springs | $F=kx$. Springs behave oppositely to resistors: in series stiffness adds reciprocally ($k/2$), in parallel it adds directly ($2k$). |
| Waves | $v=f\lambda$, $T=\dfrac1f$. A particle travels $4A$ per cycle, independent of wavelength. Compression to rarefaction is half a period. |
| Refraction | $n_1\sin\theta_1=n_2\sin\theta_2$. Dense to rare bends away from the normal. Critical angle $\sin c=\dfrac{n_2}{n_1}$, and total internal reflection is only possible high-to-low. |
| Radioactivity | $N=N_0\left(\tfrac12\right)^{t/t_{1/2}}$. Alpha drops $A$ by 4 and $Z$ by 2; beta leaves $A$ unchanged and raises $Z$ by 1. Mass number and charge are both conserved. |
| Moments and density | Moment $=F\times$ perpendicular distance. $\rho=\dfrac{m}{V}$; a floating body displaces its own mass of fluid. |
| Useful approximation | $2^{10}=1024\approx10^3$, indispensable for half-life and order-of-magnitude estimates without a calculator. |
Takeaway (📌): Drag and sliding friction always oppose the current direction of motion relative to the fluid or surface, so the force equation flips sign the moment an object reverses. Watch for it in terminal-velocity questions.
Chemistry, recall list
| Moles | $n=\dfrac{m}{M_r}$; $n=cV$ with $V$ in $\text{dm}^3$; $1\,\text{dm}^3=1000\,\text{cm}^3$. |
|---|---|
| Yield and economy | Percentage yield $=\dfrac{\text{actual}}{\text{theoretical}}\times100$. Atom economy $=\dfrac{M_r(\text{desired})}{\text{total }M_r(\text{all reactants})}\times100$, with each $M_r$ multiplied by its coefficient in the balanced equation. |
| Equilibrium | For $aA+bB\rightleftharpoons cC+dD$, $K_c=\dfrac{[C]^c[D]^d}{[A]^a[B]^b}$. Derive the units from the powers, they are frequently the discriminating part of the answer. |
| Le Chatelier | The system shifts to oppose the change. Raising the pressure favours the side with fewer gas moles; temperature favours the endothermic direction when heated. A catalyst changes rate only, never position or yield. |
| Energetics | $q=mc\Delta T$. Hess's law: reversing an equation flips the sign of $\Delta H$; multiplying scales it. |
| Redox | Oxidation is loss of electrons. Disproportionation is one element both oxidised and reduced, and is always a redox reaction. Balance by equating electrons transferred. |
| Electron configuration | Shells fill 2, 8, 8, 2 up to calcium ($Z=20$). Beyond that follow the Aufbau order $4s$ before $3d$. |
| Organic reactions | Alkene + HBr is electrophilic addition; Markovnikov puts the halogen on the carbon giving the more stable (tertiary > secondary > primary) carbocation. C-F is the strongest carbon-halogen bond and so the least reactive. |
| Ion tests | With silver nitrate: chloride white, bromide cream, iodide yellow. In mass spectra, chlorine gives a 3:1 and bromine a 1:1 $M$/$M{+}2$ pattern. |
Biology, recall list
| Hardy-Weinberg | $p+q=1$ and $p^2+2pq+q^2=1$. Homozygous recessives are $q^2$; heterozygotes $2pq$. Multiply by population size for counts. |
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| Genetics | Monohybrid $Aa\times Aa$ gives 3:1 phenotypic, 1:2:1 genotypic. Two recessive parents can never produce a dominant offspring, the fastest pedigree elimination there is. |
| DNA | A pairs with T (2 hydrogen bonds), C with G (3). So $A=T$ and $C=G$. Insertions and deletions cause frameshifts unless the number of bases is a multiple of three; substitutions may be silent. |
| Transport | Diffusion and osmosis are passive and go down a gradient. Movement against a gradient, or of large particles by endo/exocytosis, requires ATP, and therefore a living cell. |
| Enzymes | Rate rises with temperature to an optimum, then falls sharply on denaturation. Competitive inhibitors are overcome by more substrate. Pepsin works at pH 1.5 to 2; intestinal proteases near pH 8. |
| Respiration | Aerobic: glucose $+$ oxygen $\rightarrow$ carbon dioxide $+$ water. Anaerobic in animals gives lactic acid only, no $\text{CO}_2$, no oxygen used, no water formed. |
| Photosynthesis | Light-dependent reactions occur on the thylakoid membranes (stacked into grana); the light-independent stage occurs in the stroma. |
| Ecology | Roughly 10% of energy transfers between trophic levels; the rest is lost as heat from respiration (including movement), in parts not eaten, and in faeces and excretory waste. Kidneys remove metabolic waste from blood, faeces are digestive waste and never enter it. |
The traps these papers set most often
- Extraneous data. Wall heights, wavelengths and masses are routinely supplied and not needed. Extra numbers are not a signal that you have missed a step.
- Unit conversion of areas and volumes. $1\,\text{cm}^2=10^{-4}\,\text{m}^2$ and $1\,\text{cm}^3=10^{-6}\,\text{m}^3$, square or cube the linear factor, don't divide by 100 once.
- Round trips. In echo, sonar and ultrasound questions the pulse travels twice the depth, so depth $=\tfrac12vt$.
- Energy barriers. Before computing a final speed from a height drop, check the object can clear any intermediate peak at all.
- Domain restrictions. Cancelling a factor from an algebraic fraction can silently introduce a root, and dividing both sides of an equation by a factor that may be zero can silently lose one. State the excluded values first, then check your answers against them.
- Order of operations. Scan every option for multiplication and brackets before evaluating anything left to right.
- "None of the above" is a real answer. If your derivation is sound and the value is absent, trust the derivation.
Key Idea (💡): Every item on this page appears somewhere in the four mock papers on this site. The most efficient revision loop is to sit a module under timed conditions, then return here and re-read only the rows that cost you marks.
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