Admissions Test Preparation · Practice Set 11
ESAT Advanced Mathematics Mock Module 4 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 | Tangent equations | $\tan$ repeats every $180^{\circ}$, not $360^{\circ}$ |
| Q2 | Discriminant conditions with a parameter | No real roots means $b^{2}-4ac < 0$ |
| Q3 | Differentiating fractional powers | Bring the index down, subtract one |
| Q4 | Area under a parabola | Integrate between the roots of the factorised form |
| Q5 | Validity of a binomial expansion | The expansion of $(1+u)^{n}$ needs $|u|<1$ |
| Q6 | Combined transformations | Inside the bracket moves x, outside moves y |
| Q7 | Minimum values via calculus | Differentiate, solve, then substitute back for the value |
| Q8 | Reverse chain rule with a linear bracket | Raise the power, divide by the new power and by the inner coefficient |
| Q9 | The sine rule | $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$ — pair each side with its opposite angle |
| Q10 | Sum to infinity with a negative ratio | $\dfrac{a}{1-r}$ handles negative $r$ unchanged |
| Q11 | Differentiating logarithms | $\ln(kx) = \ln k+\ln x$, so the constant differentiates away |
| Q12 | Exponential equations with different bases | Express both sides as powers of the same prime |
| Q13 | Product rule with trigonometry | Differentiate each factor in turn, then substitute exact values |
| Q14 | Quartic equations in disguise | Substitute $u = x^{2}$ and count the roots at the end |
| Q15 | Rates of change with volume | $\dfrac{dV}{dx}$ for a cube is its total face area |
| Q16 | Integrating negative powers | Write $\dfrac{1}{x^{2}}$ as $x^{-2}$ and apply the power rule |
| Q17 | Telescoping with odd denominators | Partial fractions, then watch the interior cancel |
| Q18 | Telescoping logarithms | A sum of logs of ratios collapses to a single logarithm |
| Q19 | Constant of integration | Integrate, use the point, then evaluate |
| Q20 | Translating a turning point | Track the single known point through each transformation |
| Q21 | Symmetric limits with an even power | Odd terms vanish; double the even part over the half-interval |
| Q22 | Integrating a fractional power | Rewrite the root as a fractional index, then use the ordinary rule |
| Q23 | Cosine rule with an obtuse angle | The cosine rule handles the obtuse case automatically through the sign |
| Q24 | Counting solutions of a multiple-angle equation | Widen the range by the multiple, then count |
| Q25 | Arc length in radians | In radians the arc is just $r\theta$ |
| Q26 | Sector area in radians | $A = \tfrac12 r^{2}\theta$, with no fraction of $360$ |
| Q27 | Area of a segment | A segment is the sector minus the triangle |