Year: 2023
Paper: 3
Question Number: 2
Course: UFM Pure
Section: Polar coordinates
No solution available for this problem.
The total entry was a marginal increase on that of 2022 (by just over 1%). Two questions were attempted by more than 90% of candidates, another two by 80%, and another two by about two thirds. The least popular questions were attempted by more than a sixth of candidates. All the questions were perfectly answered by at least three candidates (but mostly more than this), with one being perfectly answered by eighty candidates. Very nearly 90% of candidates attempted no more than 7 questions. One general comment regarding all the questions is that candidates need to make sure that they read the question carefully, paying particular attention to command words such as "hence" and "show that".
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The polar curves $C_1$ and $C_2$ are defined for $0 \leqslant \theta \leqslant \pi$ by
\[r = k(1 + \sin\theta)\]
\[r = k + \cos\theta\]
respectively, where $k$ is a constant greater than $1$.
\begin{questionparts}
\item Sketch the curves on the same diagram. Show that if $\theta = \alpha$ at the point where the curves intersect, $\tan\alpha = \dfrac{1}{k}$.
\item The region A is defined by the inequalities
\[0 \leqslant \theta \leqslant \alpha \quad \text{and} \quad r \leqslant k(1+\sin\theta)\,.\]
Show that the area of A can be written as
\[\frac{k^2}{4}(3\alpha - \sin\alpha\cos\alpha) + k^2(1 - \cos\alpha)\,.\]
\item The region B is defined by the inequalities
\[\alpha \leqslant \theta \leqslant \pi \quad \text{and} \quad r \leqslant k + \cos\theta\,.\]
Find an expression in terms of $k$ and $\alpha$ for the area of B.
\item The total area of regions A and B is denoted by $R$. The area of the region enclosed by $C_1$ and the lines $\theta = 0$ and $\theta = \pi$ is denoted by $S$. The area of the region enclosed by $C_2$ and the lines $\theta = 0$ and $\theta = \pi$ is denoted by $T$.
Show that as $k \to \infty$,
\[\frac{R}{T} \to 1\]
and find the limit of
\[\frac{R}{S}\]
as $k \to \infty$.
\end{questionparts}
This was the fifth most popular question on the paper, being attempted by about two thirds of the candidates, just a few more than question 8. However, it was the most successfully attempted with a mean score of over 12/20. Many candidates produced excellent responses to this question, and a number scored a perfect 20/20. The two curves were generally well sketched in part (i), with the commonest fault being failure to obtain values for r at the points where θ = 0, ½π, π. The derivation of the required result in (i) for the point of intersection as well as the result for the area of A in part (ii) were generally well done. Similarly, in part (iii) the area of B was well attempted, although algebraic errors were more common here with the required result not being given in the question, unlike the area of A in part (ii). Part iv) was found to be the most difficult part of the question, though marks for finding expressions for S and T were generally obtained. After this, the first challenge was to notice that α → 0, and it was important to justify this observation using the expression tan(α) = 1/k from part (i). A small number of candidates made heuristic arguments that α → 0, using their sketches - however, this was not acceptable for the mark without further justification. The next challenge was to compute the limits rigorously, and candidates found this to be the most challenging aspect of the question. Common mistakes included not noticing that α depended on k, and substituting α = 0 prematurely, which, for example, led to the erroneous conclusion that k cos(α) → 0.