Year: 2020
Paper: 3
Question Number: 6
Course: UFM Pure
Section: Polar coordinates
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
\begin{questionparts}
\item Sketch the curve $y = \cos x + \sqrt{\cos 2x}$ for $-\frac{1}{4}\pi \leqslant x \leqslant \frac{1}{4}\pi$.
\item The equation of curve $C_1$ in polar co-ordinates is
\[ r = \cos\theta + \sqrt{\cos 2\theta} \qquad -\tfrac{1}{4}\pi \leqslant \theta \leqslant \tfrac{1}{4}\pi. \]
Sketch the curve $C_1$.
\item The equation of curve $C_2$ in polar co-ordinates is
\[ r^2 - 2r\cos\theta + \sin^2\theta = 0 \qquad -\tfrac{1}{4}\pi \leqslant \theta \leqslant \tfrac{1}{4}\pi. \]
Find the value of $r$ when $\theta = \pm\frac{1}{4}\pi$.
Show that, when $r$ is small, $r \approx \frac{1}{2}\theta^2$.
Sketch the curve $C_2$, indicating clearly the behaviour of the curve near $r=0$ and near $\theta = \pm\frac{1}{4}\pi$.
Show that the area enclosed by curve $C_2$ and above the line $\theta = 0$ is $\dfrac{\pi}{2\sqrt{2}}$.
\end{questionparts}
This was quite a popular question, being attempted by about three fifths of the candidates, but on average scoring only a bit better than one third marks. Most candidates were broadly successful at sketching the first graph in part (i), but though they had differentiated, many did not consider the gradients at the endpoints. Attempts to draw the sketch for part (ii) were usually less successful, and few dealt well with the behaviour near the endpoints. Few candidates gave a completely accurate justification of the small r approximation in (iii). Many candidates did not solve the equation of curve C2 for r and thus did not realise that C1 was one branch of C2. Most only drew one or other branch, and very few considered how to join the branches. Most candidates did not know how to compute areas in polar coordinates: successful ones realised the area was a difference of two polar integrals and used trigonometric substitutions to perform the integral.