Year: 2022
Paper: 3
Question Number: 6
Course: LFM Pure
Section: Small angle approximation
No solution available for this problem.
One question was attempted by well over 90% of the candidates two others by about 90%, and a fourth by over 80%. Two questions were attempted by about half the candidates and a further three questions by about a third of the candidates. Even the other three received attempts from a sixth of the candidates or more, meaning that even the least popular questions were markedly more popular than their counterparts in previous years. Nearly 90% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that when $\alpha$ is small, $\cos(\theta + \alpha) - \cos\theta \approx -\alpha\sin\theta - \frac{1}{2}\alpha^2\cos\theta$.
Find the limit as $\alpha \to 0$ of
\[ \frac{\sin(\theta+\alpha) - \sin\theta}{\cos(\theta+\alpha) - \cos\theta} \qquad (*) \]
in the case $\sin\theta \neq 0$.
In the case $\sin\theta = 0$, what happens to the value of expression $(*)$ when $\alpha \to 0$?
\item A circle $C_1$ of radius $a$ rolls without slipping in an anti-clockwise direction on a fixed circle $C_2$ with centre at the origin $O$ and radius $(n-1)a$, where $n$ is an integer greater than $2$. The point $P$ is fixed on $C_1$. Initially the centre of $C_1$ is at $(na, 0)$ and $P$ is at $\big((n+1)a, 0\big)$.
\begin{enumerate}
\item Let $Q$ be the point of contact of $C_1$ and $C_2$ at any time in the rolling motion. Show that when $OQ$ makes an angle $\theta$, measured anticlockwise, with the positive $x$-axis, the $x$-coordinate of $P$ is $x(\theta) = a(n\cos\theta + \cos n\theta)$, and find the corresponding expression for the $y$-coordinate, $y(\theta)$, of $P$.
\item Find the values of $\theta$ for which the distance $OP$ is $(n-1)a$.
\item Let $\theta_0 = \dfrac{1}{n-1}\pi$. Find the limit as $\alpha \to 0$ of
\[ \frac{y(\theta_0 + \alpha) - y(\theta_0)}{x(\theta_0 + \alpha) - x(\theta_0)}\,. \]
Hence show that, at the point $\big(x(\theta_0),\, y(\theta_0)\big)$, the tangent to the curve traced out by $P$ is parallel to $OP$.
\end{enumerate}
\end{questionparts}
About half the candidates attempted this, but it was one of the least successful with a mean score of one quarter marks. Many candidates managed the opening 'show that' in part (i) but the limit attempt had varying levels of success, and a common error was division by a quantity that was not necessarily nonzero. In part (ii), diagrams were regularly lacking, often being drawn extremely small with the most salient details omitted. In part (a), very few indicated from where the second term in the expression for x arose. Most attempts appealed to a diagram but did not indicate the pertinent angles. Many formed the correct equation in (b), but a large number forgot to account for the periodicity; those that remembered to do so largely did so correctly. Many who got to (c), erroneously evaluated a 0/0 limit and then argued that the cotangent was the answer they wanted. However, pleasingly others did spot the zeros and manipulated the trigonometry effectively.