Admissions Test Preparation · Practice Set 12
ESAT Advanced Mathematics Mock Module 5 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 | The ambiguous case of the sine rule | Two sides and a non-included angle can describe two triangles |
| Q2 | Where a function is increasing | Increasing means the derivative is positive, so solve an inequality |
| Q3 | Differentiating an integral | Differentiating an integral with a variable upper limit returns the integrand |
| Q4 | Combining integrals over the same range | Constants come out and sums split, over a shared range |
| Q5 | Contiguous ranges and reversed limits | Contiguous ranges add, and swapping the limits flips the sign |
| Q6 | The graph of an exponential function | An exponential is always positive and never meets its asymptote |
| Q7 | One-to-one and many-to-one mappings | One-to-one means no horizontal line meets the graph twice |
| Q8 | A recurrence relation | Generate the terms; do not look for a formula |
| Q9 | Which quadrant a line avoids | The signs of m and c decide which quadrants a line can reach |
| Q10 | A negative fractional index | Root, then power, then reciprocal — in that order |
| Q11 | Rationalising a denominator | Multiply top and bottom by the surd in the denominator |
| Q12 | One linear and one quadratic | Substitute the linear equation into the quadratic |
| Q13 | A perpendicular line through a point | Negative reciprocal for the gradient, then the point fixes the rest |
| Q14 | Exact trigonometric values | Read them off the two standard triangles |
| Q15 | Recognising a curve from its features | Turning points and intercepts identify the family |
| Q16 | A definite integral with mixed powers | Rewrite every term as a power of x before integrating |
| Q17 | Solving a trigonometric equation | Find the acute angle, then place it in the right quadrants |
| Q18 | Variance from summary statistics | $\sigma^{2} = \dfrac{\sum x^{2}}{n}-\bar x^{2}$ |
| Q19 | Coding and standard deviation | The mean takes the whole transformation, the spread only the multiplier |
| Q20 | Combining the means of two groups | Combine totals, then divide by the combined count |
| Q21 | Expected value of a discrete random variable | $E(X) = \sum xP(X=x)$ — a probability-weighted mean |
| Q22 | Variance of a discrete random variable | $\text{Var}(X) = E(X^{2})-\left[E(X)\right]^{2}$ |
| Q23 | Expectation of a linear function | $E(aX+b) = aE(X)+b$ |
| Q24 | Expectation from an infinite series | A geometric-style expectation is a series you already know how to sum |
| Q25 | Mean of the first n integers | The mean of a symmetric set is its middle |
| Q26 | Deviations from the mean | The deviations always sum to zero — that is what the mean is |
| Q27 | Effect of an outlier on mean and median | The mean moves by the change divided by n; the median may not move at all |