Year: 2022
Paper: 3
Question Number: 8
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
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 Use De Moivre's theorem to prove that for any positive integer $k > 1$,
\[ \sin(k\theta) = \sin\theta\cos^{k-1}\theta \left( k - \binom{k}{3}(\sec^2\theta - 1) + \binom{k}{5}(\sec^2\theta - 1)^2 - \cdots \right) \]
and find a similar expression for $\cos(k\theta)$.
\item Let $\theta = \cos^{-1}(\frac{1}{a})$, where $\theta$ is measured in degrees, and $a$ is an odd integer greater than $1$.
Suppose that there is a positive integer $k$ such that $\sin(k\theta) = 0$ and $\sin(m\theta) \neq 0$ for all integers $m$ with $0 < m < k$.
Show that it would be necessary to have $k$ even and $\cos(\frac{1}{2}k\theta) = 0$.
Deduce that $\theta$ is irrational.
\item Show that if $\phi = \cot^{-1}(\frac{1}{b})$, where $\phi$ is measured in degrees, and $b$ is an even integer greater than $1$, then $\phi$ is irrational.
\end{questionparts}
This was the least popular Pure question, being attempted by marginally fewer than question 7, but by more than any of the Mechanics or Probability and Statistics. The mean mark was 6/20. Generally, part (i) was done well and candidates used binomial expansions accurately, manipulating their results to find the two required expressions. A few did not gain full credit through providing insufficient working for the result given in the question. More than half the candidates progressed no further than attempting part (ii) and, of those who did attempt it, often stopped part of the way through, although there were some very well-reasoned attempts. Most candidates attempting part (ii) substituted a = sec(θ) into their sin expansion but found it difficult to complete the argument to explain why k had to be even. Of those who got further and successfully managed to show the given results, often the relevance of those results was not appreciated, and some candidates attempted to prove irrationality by quoting the irrationality of π, despite the fact that the question stated θ was measured in degrees. Very few candidates gained full credit for this part. Those candidates who gained full credit in part (ii) also did well in (iii).