Admissions Test Preparation · Practice Set 2
ESAT Mathematics Mock Module 2 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 | Complementary counting with digits | Count the complement: total minus the arrangements that avoid the digit |
| Q2 | Two-set counting | Inclusion-exclusion: add the sets, subtract the overlap once |
| Q3 | Difference of two squares in geometry | An annulus area $\pi(R^2-r^2)$ factorises into $\pi(R-r)(R+r)$ |
| Q4 | Multipliers with powers | Chain the multipliers, each raised to the power it carries in the formula |
| Q5 | Sum to infinity | $S_\infty = \dfrac{a}{1-r}$ once $|r|<1$ is confirmed |
| Q6 | Arrangements with repeated letters | Divide by the factorial of each repeat |
| Q7 | Combined filling rates | Add the rates, then invert |
| Q8 | Sum of an even-number series | Factor the common multiple out, then use the triangular sum |
| Q9 | Two-way tables | Fill the four regions; each row and column must total |
| Q10 | Modal class with unequal widths | With unequal classes the mode follows frequency density, not frequency |
| Q11 | Simplifying surds | Pull out the largest square factor so every surd shares a root |
| Q12 | Fractional and negative indices | Root first, then power, then reciprocal |
| Q13 | Completing the square | The completed square reads off the minimum immediately |
| Q14 | Ratio | Convert the stated difference into the value of one part |
| Q15 | Successive percentages | Multiply the fractions rather than computing each stage |
| Q16 | Consecutive integers | For an odd run of consecutive numbers, the mean is the middle term |
| Q17 | Parallel lines | Parallel means identical gradient — rearrange only far enough to read it |
| Q18 | Sectors of a circle | Take the fraction of the full circle given by the angle |
| Q19 | Distance between two points | Look for a Pythagorean triple in the coordinate differences |
| Q20 | Two dice | Count the favourable pairs out of 36 |
| Q21 | Combinations | Cancel the factorials before multiplying anything |
| Q22 | Geometric sequences | Match powers instead of dividing repeatedly |
| Q23 | Weighted means | Combine totals, not means |
| Q24 | Closing speed | Add the speeds when two objects approach each other |
| Q25 | Density | Mass = density × volume, with units checked first |
| Q26 | Difference of two squares in algebra | $A^2-B^2 = (A-B)(A+B)$ works for whole brackets too |
| Q27 | Linear inequalities | Collect the variable on the side that keeps it positive |