Year: 2020
Paper: 3
Question Number: 4
Course: LFM Pure
Section: 3x3 Matrices
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
The plane $\Pi$ has equation $\mathbf{r} \cdot \mathbf{n} = 0$ where $\mathbf{n}$ is a unit vector. Let $P$ be a point with position vector $\mathbf{x}$ which does not lie on the plane $\Pi$. Show that the point $Q$ with position vector $\mathbf{x} - (\mathbf{x} \cdot \mathbf{n})\mathbf{n}$ lies on $\Pi$ and that $PQ$ is perpendicular to $\Pi$.
\begin{questionparts}
\item Let transformation $T$ be a reflection in the plane $ax+by+cz=0$, where $a^2+b^2+c^2=1$. Show that the image of $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ under $T$ is $\begin{pmatrix} b^2+c^2-a^2 \\ -2ab \\ -2ac \end{pmatrix}$, and find the images of $\mathbf{j}$ and $\mathbf{k}$ under $T$.
Write down the matrix $\mathbf{M}$ which represents transformation $T$.
\item The matrix
\[ \begin{pmatrix} 0.64 & 0.48 & 0.6 \\ 0.48 & 0.36 & -0.8 \\ 0.6 & -0.8 & 0 \end{pmatrix} \]
represents a reflection in a plane. Find the cartesian equation of the plane.
\item The matrix $\mathbf{N}$ represents a rotation through angle $\pi$ about the line through the origin parallel to $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a^2+b^2+c^2=1$. Find the matrix $\mathbf{N}$.
\item Identify the single transformation which is represented by the matrix $\mathbf{NM}$.
\end{questionparts}
This was not a popular question but it received a respectable number of attempts with about one sixth trying it. The average score was a little under half marks, but on each part of the question, if the part was attempted it was generally fully correct. Most candidates had no problem demonstrating the desired properties, and if they used this in part (i) they had little problem obtaining full marks. Even if they could not apply the stem in (i), they nearly all found the images of j and k correctly using symmetry and hence the matrix M. In part (ii), almost all the candidates could solve the equations, though some lost marks by working inaccurately. The few that attempted part (iii) either got it completely correct or scored nothing: those getting it correct generally drew a parallel with the technique used in (i). As a consequence, only a small number attempted part (iv), and few scored both marks, either losing a mark for insufficient justification, or for describing the transformation as a rotation about the origin.