| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | January |
| Marks | 11 |
| Topic | Vectors 3D & Lines |
| Type | Reflection and symmetry |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question testing routine techniques: finding intersection of lines (solving simultaneous equations), checking perpendicularity (dot product), and reflection in a line. All parts follow textbook methods with no novel insight required, though part (c) requires careful application of the reflection formula. Slightly easier than average due to straightforward computational steps. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
With respect to a fixed origin $O$, the lines $l_1$ and $l_2$ are given by the equations
$$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$
$$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$
where $\lambda$ and $\mu$ are scalar parameters.
\begin{enumerate}[label=(\alph*)]
\item Show that $l_1$ and $l_2$ meet and find the position vector of their point of intersection. [6]
\item Show that $l_1$ and $l_2$ are perpendicular to each other. [2]
\end{enumerate}
The point $A$, with position vector $5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}$, lies on $l_1$
The point $B$ is the image of $A$ after reflection in the line $l_2$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the position vector of $B$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q5 [11]}}