Admissions Test Preparation · Practice Set 10
ESAT Advanced Mathematics Mock Module 3 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 | Combining logarithms | Collapse to a single logarithm before evaluating |
| Q2 | Horizontal stretches | $y = f(2x)$ halves every $x$-coordinate |
| Q3 | Line meeting a circle | Substitute, then use the discriminant to count intersections |
| Q4 | Tangent to a circle at a point | The tangent is perpendicular to the radius at the point of contact |
| Q5 | Quadratic sequences | Second differences give twice the leading coefficient |
| Q6 | Integrating trigonometric functions | $\int\cos(kx)\,dx = \tfrac1k\sin(kx)$ |
| Q7 | Optimisation | Express the quantity in one variable, then differentiate |
| Q8 | Sum of consecutive odd numbers | The first $n$ odd numbers sum to $n^{2}$ |
| Q9 | Remainder theorem with a non-unit divisor | The remainder theorem uses the root of the divisor, not its coefficients |
| Q10 | Factor theorem for an unknown coefficient | A stated factor gives one equation for the unknown |
| Q11 | Inverse of a linear function | Swap and solve, or just undo the operations in reverse |
| Q12 | Inverse of a cubic function | Undo the operations in reverse order |
| Q13 | Trapezium rule with three strips | Ends once, middles twice, all times half the strip width |
| Q14 | Trapezium rule on a reciprocal curve | Same weighting, fractional ordinates |
| Q15 | Equation of a tangent | Point from the curve, gradient from the derivative |
| Q16 | Normal to a root curve | Differentiate the fractional power, then take the negative reciprocal |
| Q17 | Points of inflection | Set the second derivative to zero |
| Q18 | Integrating fractional powers | Write $\sqrt{x}$ as $x^{1/2}$ and use the standard rule |
| Q19 | Equation of a tangent | Gradient from the derivative, point from the curve, then one line equation |
| Q20 | Logarithms with a fractional value | Convert to exponential form immediately |
| Q21 | Simplifying with the Pythagorean identity | Replace $1-\cos^{2}\theta$ with $\sin^{2}\theta$ on sight |
| Q22 | Summing multiples | Factor out the common multiple, then use the standard sum |
| Q23 | Disguised quadratics in an exponential | Substitute $u = 2^{x}$ and note $4^{x} = u^{2}$ |
| Q24 | Signed areas | A definite integral below the axis is negative — take the modulus for area |
| Q25 | Inverse functions | Swap and solve |
| Q26 | Sum of a finite geometric series | $S_n = \dfrac{a\left(1-r^{n}\right)}{1-r}$ when $r<1$ |
| Q27 | Roots of a cubic | Use Vieta on the coefficients rather than solving |