Year: 2020
Paper: 3
Question Number: 5
Course: LFM Stats And Pure
Section: Partial Fractions
No solution available for this problem.
In spite of the change to criteria for entering the paper, there was still a very healthy number of candidates, and the vast majority handled the protocols for the online testing very well. Just over half the candidates attempted exactly six questions, and whilst about 10% attempted a seventh, hardly any did more than seven. With 20% attempting five questions, and 10% attempting only four, overall, there were very few candidates not attempting the target number. There was a spread of popularity across the questions, with no question attracting more than 90% of candidates and only one less than 10%, but every question received a good number of attempts. Likewise, there was a spread of success on the questions, though every question attracted at least one perfect solution.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Show that for positive integer $n$, $x^n - y^n = (x-y)\displaystyle\sum_{r=1}^{n} x^{n-r} y^{r-1}$.
\begin{questionparts}
\item Let $\mathrm{F}$ be defined by
\[ \mathrm{F}(x) = \frac{1}{x^n(x-k)} \quad \text{for } x \neq 0,\, k \]
where $n$ is a positive integer and $k \neq 0$.
\begin{enumerate}
\item Given that
\[ \mathrm{F}(x) = \frac{A}{x-k} + \frac{\mathrm{f}(x)}{x^n}, \]
where $A$ is a constant and $\mathrm{f}(x)$ is a polynomial, show that
\[ \mathrm{f}(x) = \frac{1}{x-k}\left(1 - \left(\frac{x}{k}\right)^n\right). \]
Deduce that
\[ \mathrm{F}(x) = \frac{1}{k^n(x-k)} - \frac{1}{k}\sum_{r=1}^{n} \frac{1}{k^{n-r}x^r}. \]
\item By differentiating $x^n \mathrm{F}(x)$, prove that
\[ \frac{1}{x^n(x-k)^2} = \frac{1}{k^n(x-k)^2} - \frac{n}{xk^n(x-k)} + \sum_{r=1}^{n} \frac{n-r}{k^{n+1-r}x^{r+1}}. \]
\end{enumerate}
\item Hence evaluate the limit of
\[ \int_2^N \frac{1}{x^3(x-1)^2} \; \mathrm{d}x \]
as $N \to \infty$, justifying your answer.
\end{questionparts}
A popular question, which was well attempted with a fair degree of success: it was marginally less popular than question 2, but marginally more successful. Most submitted quite a large amount of work, and were able to attempt later parts even if earlier parts were not successful as key results (requiring proof) were quoted in each part. The stem was mostly well completed, by a variety of methods, namely, re-summing indices, induction, or geometric series, though there were some candidates who seemed to think it was obvious and produced no working. Part (i) (a) was also well completed though few received full marks. The main problems were finding A and that F(x) is not defined for x = k. The second result in this part was better done, though some candidates struggled with re-summing when changing indices. For (i) (b), many did not realise that they needed to differentiate both sides. Differentiation errors and confusion thwarted many that did differentiate. Part (ii) was well done by candidates that attempted it with most realising that they could use the result of (i) (b). Though many lost marks for failing to show how to take the limit of the logarithm, most realised that they need to use partial fractions to complete the integral. Some candidates sadly left their expressions in terms of k.