Year: 2021
Paper: 2
Question Number: 4
Course: LFM Stats And Pure
Section: Functions (Transformations and Inverses)
No solution available for this problem.
Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Sketch the curve $y = xe^x$, giving the coordinates of any stationary points.
\item The function $f$ is defined by $f(x) = xe^x$ for $x \geqslant a$, where $a$ is the minimum possible value such that $f$ has an inverse function. What is the value of~$a$?
Let $g$ be the inverse of $f$. Sketch the curve $y = g(x)$.
\item For each of the following equations, find a real root in terms of a value of the function~$g$, or demonstrate that the equation has no real root. If the equation has two real roots, determine whether the root you have found is greater than or less than the other root.
\begin{enumerate}[label=(\alph*)]
\item $e^{-x} = 5x$
\item $2x \ln x + 1 = 0$
\item $3x \ln x + 1 = 0$
\item $x = 3\ln x$
\end{enumerate}
\item Given that the equation $x^x = 10$ has a unique positive root, find this root in terms of a value of the function~$g$.
\end{questionparts}
Part (i) was often successfully answers, with most candidates successfully differentiating the equation of the curve and setting equal to 0 to find the stationary points. In part (ii) some candidates did not link the coordinates of the stationary point found in (i) to the value of a that needed to be stated. In some cases, the graph when sketched extended beyond the point identified even when it had been identified correctly. The sketches of the inverse function were generally well done, although a significant number did not appreciate that the mirror image as the curve approached its stationary point would have a gradient that tends to infinity. In part (iii) some candidates attempted to find a form for the inverse function rather than deducing what was necessary from the information given. In most cases this was not successful, although a small number did successfully reach some of the results. Despite the fact that the question asked candidates to find a real root in the cases where one exists, some candidates did not do this and instead simply stated the number of roots. Those candidates who were successful with (iii)(b) were then usually able to complete the rest of the question successfully.