Admissions Test Preparation · Practice Set 9
ESAT Advanced Mathematics Mock Module 2 of 5
27 a full 27-question module, the same length and shape as one sitting of the real test. Mathematics 2 is pure mathematics throughout - algebra and functions, sequences and series, coordinate geometry, trigonometry, logarithms, differentiation, integration and graphs. 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:
Advanced Mathematics
27 questions, 40 minutes in the real test. Mathematics 2 is pure mathematics throughout - algebra and functions, sequences and series, coordinate geometry, trigonometry, logarithms, differentiation, integration and graphs. 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.
Advanced Mathematics
| # | Topic | Shortcut rehearsed |
|---|---|---|
| Q1 | Partial symmetry in integration | Odd terms vanish over symmetric limits; only the even part survives |
| Q2 | Initial value problems | Integrate, then use the given point to fix the constant |
| Q3 | Quadratic sequences | Second difference is twice the leading coefficient |
| Q4 | Combining trigonometric fractions | Common denominator, then the Pythagorean identity collapses the numerator |
| Q5 | Logarithmic equations with a phantom root | Combine the logs, solve, then reject any root outside the domain |
| Q6 | Binomial coefficients with a negative term | Bracket the whole term so the sign is raised to the power too |
| Q7 | Chain rule on a bracket | Power down, bracket unchanged, times the inner derivative |
| Q8 | Area between a curve and a horizontal line | Use the symmetry: integrate the half and double |
| Q9 | Differentiation from first principles | Expand, cancel the h, then let h tend to zero |
| Q10 | Normals to a curve | Normal gradient is the negative reciprocal of the tangent gradient |
| Q11 | Product rule with exponentials | Factor out $e^{x}$ immediately — it is never zero |
| Q12 | Finding the constant of integration | Integrate, then use the given point to pin down c |
| Q13 | The trapezium rule | $\tfrac{h}{2}\left[\text{ends}+2(\text{middles})\right]$ |
| Q14 | Quadratics in a logarithm | Substitute for the logarithm and solve an ordinary quadratic |
| Q15 | Geometric series sums | Rearrange to a pure power, then match indices |
| Q16 | Arithmetic series | $S_n = \tfrac{n}{2}\left[2a+(n-1)d\right]$ solved for the unknown |
| Q17 | Trigonometric equations with a multiple angle | Solve for the whole angle first, then divide the solutions |
| Q18 | Area of a triangle | $\tfrac12 ab\sin C$ when two sides and the included angle are known |
| Q19 | Circle from a diameter | Centre is the midpoint, radius is half the distance |
| Q20 | Binomial expansion of a binomial with a constant | Track which factor supplies the power of $x$ |
| Q21 | The remainder theorem | Substitute the root of the divisor — no division needed |
| Q22 | Modulus inequalities | $|A| < k$ means $-k < A < k$ |
| Q23 | Half-life | Count halvings rather than solving an exponential equation |
| Q24 | Area between a curve and a line | Integrate the difference between the intersection points |
| Q25 | Chain rule with trigonometry | Differentiate the outside, keep the inside, times the inside's derivative |
| Q26 | Stationary points of a reciprocal function | Write $\tfrac1x$ as $x^{-1}$ and differentiate normally |
| Q27 | Sum to infinity in reverse | Rearrange $S_\infty = \dfrac{a}{1-r}$ for the unknown ratio |