Admissions Test Preparation · Practice Set 1
ESAT Mathematics Mock Module 1 of 7
27 a full 27-question module, the same length and shape as one sitting of the real test. Mathematics 1 is compulsory for every ESAT candidate. Number, ratio, algebra, geometry, statistics and probability, drawn from across the whole specification. Every question states the specification sub-topic it was written against.
How to use this set
Every question carries a target time, and no question needs a calculator. Attempt each one against the clock with the solution collapsed. If you are still setting up algebra when the target time passes, stop and open the solution: the shortcut is what the question is testing, and the arithmetic is not.
Read the Key Idea before the full breakdown. If the key idea alone lets you redo the question in under a minute, the method has landed and you can move on. If it does not, work the step-by-step breakdown and then return to the question a day later.
The Common Mistake section on each question is the slower route that still reaches the right answer. Under exam conditions that route is usually the difference between finishing the module and running out of time.
Modules in this set
Select a module below to view the questions and their step-by-step worked solutions:
Mathematics
27 questions, 40 minutes in the real test. Mathematics 1 is compulsory for every ESAT candidate. Number, ratio, algebra, geometry, statistics and probability, drawn from across the whole specification. 27 worked solutions.
Shortcut index
Use this as a revision checklist. If you can state the shortcut before opening the question, you already have the mark.
Mathematics
| # | Topic | Shortcut rehearsed |
|---|---|---|
| Q1 | Surds and conjugates | Rationalise by pairing conjugates before expanding |
| Q2 | Difference of two squares | Factorise $a^2-b^2$ instead of squaring large numbers |
| Q3 | Sum and product of roots | Use Vieta's formulas instead of solving the quadratic |
| Q4 | Exponential equations | Rewrite both sides to a common base and equate indices |
| Q5 | Percentage change | Chain multipliers rather than tracking amounts |
| Q6 | Circles and distance | Read $x^2+y^2$ as a squared distance from the origin |
| Q7 | Exact trigonometric values | Reduce every angle to its acute reference angle and fix the sign by quadrant |
| Q8 | Series and counting | Sum everything, then subtract the unwanted subset |
| Q9 | Probability | Use the complement for 'at least one' |
| Q10 | Combined rates of work | Add rates, not times — and subtract for anything working against you |
| Q11 | Perpendicular bisectors | Midpoint plus negative reciprocal gradient — no simultaneous equations |
| Q12 | Average speed | Equal distances mean the harmonic mean, never the arithmetic mean |
| Q13 | Arrangements with restrictions | Total arrangements minus the glued-together case |
| Q14 | Symmetric sums | Expand $(x+y)^3$ to reach $x^3+y^3$ without finding $x$ or $y$ |
| Q15 | Similar solids | Length ratio k means area ratio k² and volume ratio k³ |
| Q16 | Inverse proportion | Scale the variables directly instead of solving for the constant |
| Q17 | Symmetric simultaneous equations | Add the equations when the coefficients are swapped |
| Q18 | Reverse percentages | Divide by the multiplier to undo a percentage change |
| Q19 | Standard form | Split the calculation into digits and powers of ten |
| Q20 | Arithmetic series | Pair first and last terms rather than summing term by term |
| Q21 | Polygon angles | Work with exterior angles — they always sum to 360° |
| Q22 | Three-dimensional Pythagoras | Apply $d^2=a^2+b^2+c^2$ in one step |
| Q23 | Independent events | Exactly one = P(A)P(not B) + P(not A)P(B) |
| Q24 | Algebraic fractions | Factorise both parts before cancelling anything |
| Q25 | Unit conversion | Convert per-second to per-hour by multiplying by 3600 |
| Q26 | Compound interest | Use the multiplier raised to a power, then subtract the principal |
| Q27 | Means and totals | Work with totals, not averages |