Admissions Test Preparation · Practice Set 8
ESAT Advanced Mathematics Mock Module 1 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 | Binomial expansion | Solve for the term number from the power of x — never expand |
| Q2 | Stationary points | Differentiate, factorise, then use the shape of the cubic to identify the maximum |
| Q3 | Definite integrals and symmetry | Odd powers integrate to zero over a symmetric interval |
| Q4 | Logarithmic equations | Combine logs into one, then check the domain before accepting a root |
| Q5 | Harmonic form of a sine wave | $a\sin\theta+b\cos\theta$ is bounded by $\pm\sqrt{a^{2}+b^{2}}$ |
| Q6 | Trigonometric equations | Convert to one trigonometric function, then reject impossible roots on sight |
| Q7 | Geometric series | Identify the first term correctly, then $S_\infty = \dfrac{a}{1-r}$ |
| Q8 | Tangent length to a circle | Substitute the external point into the circle equation and square-root it |
| Q9 | Polynomial roots | Use the product of the roots instead of dividing the cubic |
| Q10 | Area between curves | Integrate (upper − lower) between the intersection points |
| Q11 | Implicit differentiation | Differentiate term by term and substitute the point before rearranging |
| Q12 | Telescoping series | Split into partial fractions and watch the middle terms cancel |
| Q13 | Change of base | Chain logarithms multiplicatively, then invert |
| Q14 | Graph transformations | Complete the square once, then apply each transformation to the vertex only |
| Q15 | Connected rates of change | Chain rule: multiply the rate you want by the derivative that links the variables |
| Q16 | Quotient rule | The numerator of a linear-over-linear derivative is always a constant |
| Q17 | Reverse chain rule | Spot that the outside factor is a multiple of the inner derivative |
| Q18 | Double angle identities | Use the cosine double angle that produces the matching denominator |
| Q19 | Exponential growth | Express the growth factor as a power of the doubling factor |
| Q20 | Term from a sum formula | $u_n = S_n - S_{n-1}$ |
| Q21 | Tangency and the discriminant | Tangent means the intersection quadratic has a repeated root |
| Q22 | Binomial series | Apply the general binomial expansion to the bracket, not the whole expression |
| Q23 | Quadratic inequalities | Rearrange to zero, factorise, then read the sign from the parabola |
| Q24 | Roots of a cubic | Compare the stationary values with the axis instead of solving |
| Q25 | Exponential integration | $\int e^{ax}dx = \tfrac{1}{a}e^{ax}$, and $e^{2\ln 3} = 9$ |
| Q26 | Chord length | Perpendicular distance from the centre, then Pythagoras |
| Q27 | Composite functions | Read fg(x) right to left: g acts first |