Year: 2022
Paper: 2
Question Number: 4
Course: LFM Pure
Section: Modulus function
No solution available for this problem.
Candidates appeared to be generally well prepared for most topics within the examination, but there were a few situations in questions where some did not appear to be as proficient in standard techniques as needed. In particular, the method for finding invariant lines required in question 8 and the manipulation of trigonometric functions that were needed in question 10 caused considerable difficulties for some candidates. An additional issue that occurred at numerous points in the paper relates to the direction in which a deduction is required. It is important that candidates make sure that they know which statement is the one that they should start from as they deduce the other and that it is clear in their solution that the logic has gone in the correct direction. Clarity of solution is also an issue that candidates should be aware of, especially in the situations where the result to be reached has been given. It is important to check that there are no special cases that need to be considered separately, and when dividing by a function it is necessary to confirm that the function cannot be equal to 0 (and in the case of inequalities that the function always has the same sign). When drawing diagrams and sketching graphs it is useful if significant points that need to be clear are not drawn over the lines on the page as these can be difficult to interpret during the marking process.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that the function $\mathrm{f}$, given by the single formula $\mathrm{f}(x) = |x| - |x-5| + 1$, can be written without using modulus signs as
\[\mathrm{f}(x) = \begin{cases} -4 & x \leqslant 0\,,\\ 2x - 4 & 0 \leqslant x \leqslant 5\,,\\ 6 & 5 \leqslant x\,.\end{cases}\]
Sketch the graph with equation $y = \mathrm{f}(x)$.
\item The function $\mathrm{g}$ is given by:
\[\mathrm{g}(x) = \begin{cases} -x & x \leqslant 0\,,\\ 3x & 0 \leqslant x \leqslant 5\,,\\ x + 10 & 5 \leqslant x\,.\end{cases}\]
Use modulus signs to write $\mathrm{g}(x)$ as a single formula.
\item Sketch the graph with equation $y = \mathrm{h}(x)$, where $\mathrm{h}(x) = x^2 - x - 4|x| + |x(x-5)|$.
\item The function $\mathrm{k}$ is given by:
\[\mathrm{k}(x) = \begin{cases} 10x & x \leqslant 0\,,\\ 2x^2 & 0 \leqslant x \leqslant 5\,,\\ 50 & 5 \leqslant x\,.\end{cases}\]
Use modulus signs to write $\mathrm{k}(x)$ as a single formula, explicitly verifying that your formula is correct.
\end{questionparts}
Almost all candidates attempted this question, and many of these did so very successfully. The majority of attempts managed to make some good progress, especially in the first three parts. Overall, the way in which each of the given modulus forms changed sign was generally well grasped, although explanations were often rather lacking in detail. The graphs to be drawn in parts (i) and (iii) were managed well overall, although in some cases the fact that horizontal lines were drawn along the horizontal lines of the answer booklet meant that the intention was not always clear on the scanned script. In part (i) some candidates failed to give sufficient explanation to show that the single formula could be written as the given set of three equations. Although a small number of candidates omitted to sketch the graph, most were able to draw the correct three straight line segments. Part (ii) was well attempted, and many candidates were able to identify that the new function was very closely related to the one in the first part of the question meaning that they were able to write down the correct formula. Those who wrote a general form were almost always able to follow through the process to reach the correct final answer. The introduction of quadratic terms to the function did not cause too much difficulty for candidates and many were able to deduce the correct equations for the sections of the function in part (iii). While many correctly identified the shape of the two quadratic sections, in several cases the graphs presented were symmetric. Part (iv) presented more of a challenge, but many candidates who wrote down a general form were able to work through to achieve the correct final answer. A few candidates were able to write down the answer, but in some cases their answer was not verified even though the question explicitly asked for verification that the formula is correct.