Year: 2022
Paper: 3
Question Number: 7
Course: UFM Additional Further Pure
Section: Vector Product and Surfaces
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
Let $\mathbf{n}$ be a vector of unit length in three dimensions. For each vector $\mathbf{r}$, $\mathrm{f}(\mathbf{r})$ is defined by
\[ \mathrm{f}(\mathbf{r}) = \mathbf{n} \times \mathbf{r}\,. \]
\begin{questionparts}
\item Given that
\[ \mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \]
show that the $x$-component of $\mathrm{f}(\mathrm{f}(\mathbf{r}))$ is $-x(b^2+c^2)+aby+acz$. Show further that
\[ \mathrm{f}(\mathrm{f}(\mathbf{r})) = (\mathbf{n}.\mathbf{r})\mathbf{n} - \mathbf{r}\,. \]
Explain, by means of a diagram, how $\mathrm{f}(\mathrm{f}(\mathbf{r}))$ is related to $\mathbf{n}$ and $\mathbf{r}$.
\item Let $R$ be the point with position vector $\mathbf{r}$ and $P$ be the point with position vector $\mathrm{g}(\mathbf{r})$, where $\mathrm{g}$ is defined by
\[ \mathrm{g}(\mathbf{s}) = \mathbf{s} + \sin\theta\,\mathrm{f}(\mathbf{s}) + (1-\cos\theta)\,\mathrm{f}(\mathrm{f}(\mathbf{s}))\,. \]
By considering $\mathrm{g}(\mathbf{n})$ and $\mathrm{g}(\mathbf{r})$ when $\mathbf{r}$ is perpendicular to $\mathbf{n}$, state, with justification, the geometric transformation which maps $R$ onto $P$.
\item Let $R$ be the point with position vector $\mathbf{r}$ and $Q$ be the point with position vector $\mathrm{h}(\mathbf{r})$, where $\mathrm{h}$ is defined by
\[ \mathrm{h}(\mathbf{s}) = -\mathbf{s} - 2\,\mathrm{f}(\mathrm{f}(\mathbf{s}))\,. \]
State, with justification, the geometric transformation which maps $R$ onto $Q$.
\end{questionparts}
More than a third attempted this, marginally more successfully than questions 3 and 6. Many attempts were restricted to part (i). The first result was generally achieved, and whilst the second result was often obtained, quite a few had difficulties doing so because they overlooked that n was a unit vector and what this implied. Far fewer correctly drew and labelled the diagram required in part (i) because they failed to appreciate the magnitudes of the three vectors and that two were perpendicular. Parts (ii) and (iii), when attempted, saw candidates fall into two camps. A small number could see what both transformations were and using the considerations suggested in (ii) in part (iii) as well, could justify their answers. However, a larger number had some idea what the transformations might be, but often failed to define them precisely, and likewise failed to justify their conclusions, even given the approach to use in (ii).