Year: 2007
Paper: 1
Question Number: 5
Course: LFM Pure and Mechanics
Section: Vectors
There were significantly more candidates attempting this paper this year (an increase of nearly 50%), but many found it to be very difficult and only achieved low scores. In particular, the level of algebraic skill required by the questions was often lacking. The examiners' express their concern that this was the case despite a conscious effort to make the paper more accessible than last year's. At this level, the fluent, confident and correct handling of mathematical symbols (and numbers) is necessary and is expected; many good starts to questions soon became unstuck after a simple slip. Graph sketching was usually poor: if future candidates wanted to improve one particular skill, they would be well advised to develop this. There were of course some excellent scripts, full of logical clarity and perceptive insight. It was pleasing to note that the applied questions were more popular this year, and many candidates scored well on at least one of these. It was however surprising how rarely answers to questions such as 5, 9, 10, 11 and 12 began with a diagram. However, the examiners were left with the overall feeling that some candidates had not prepared themselves well for the examination. The use of past papers to ensure adequate preparation is strongly recommended. A student's first exposure to STEP questions can be a daunting, demanding experience; it is a shame if that takes place during a public examination on which so much rides. Further, and fuller, discussion of the solutions to these questions can be found in the Hints and Answers document.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
\textit{ Note: a regular octahedron is a polyhedron with eight faces each of which is an equilateral triangle.}
\begin{questionparts}
\item Show that the angle between any two faces of a regular octahedron is $\arccos \left( -{\frac1 3} \right)$.
\item Find the ratio of the volume of a regular octahedron to the volume of the cube whose vertices are the centres of the faces of the octahedron.
\end{questionparts}\begin{questionparts}
\item Suppose the vertices are $(\pm1, 0,0), (0,\pm1,0), (0,0,\pm1)$, then clearly this is an octahedron.
We can measure the angle between the faces, by looking at vectors in the same plane and also in two of the faces: $\langle \frac12, \frac12, - 1\rangle$ and $\langle \frac12, \frac12, 1\rangle$, then by considering the dot product:
\begin{align*}
&& \cos \theta &= \frac{\frac14+\frac14-1}{\frac14+\frac14+1} \\
&&&= \frac{-2}{6} = -\frac13
\end{align*}
\item The volume of our octahedron is $2 \cdot \frac13 \cdot \underbrace{\sqrt{2}^2}_{\text{base}} \cdot \underbrace{1}_{\text{height}} = \frac43$.
The centre of two touching faces are $\langle \frac13, \frac13, \frac13 \rangle$ and $\langle \frac13, \frac13, -\frac13 \rangle$ and so the length of the side of the cube is $\frac23$ and so the volume of the cube is $\frac8{27}$. Therefore the ratio is $\frac{2}{9}$
\end{questionparts}
Only a few candidates made much progress with this question, even though it only required GCSE Mathematics. Basic properties of triangles (for example, the sine and cosine rule, and the location of the centroid, the circumcentre and the incentre) are assumed knowledge at this level. It was surprising how many candidates tried to answer this question without a diagram.