| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 18 |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Challenging +1.2 This is a structured multi-part 3D vectors question requiring standard techniques: finding plane equations from normal vectors and points, calculating distances between parallel planes, finding parametric equations, and identifying a geometric locus (circle). While it has many parts (18 marks total), each step follows routine procedures with clear guidance. The most challenging aspect is part (iv)(b) requiring combination of earlier results, but the question structure provides significant scaffolding. Slightly above average difficulty due to length and the locus identification, but well within standard Further Maths expectations. |
| Spec | 1.10c Magnitude and direction: of vectors4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles |
The planes $\Pi_1$ and $\Pi_2$ are both perpendicular to $\mathbf{n}$, where $\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}$. The points $A(0, -9, 13)$ and $B(8, 7, -3)$ lie in $\Pi_1$ and $\Pi_2$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Find the equations of $\Pi_1$ and $\Pi_2$ in the form $\mathbf{r} \cdot \mathbf{n} = d$ and show that $\overrightarrow{AB}$ is parallel to $\mathbf{n}$. [4]
\item Calculate the perpendicular distance between $\Pi_1$ and $\Pi_2$. [2]
\item Write down two vectors which are perpendicular to $\mathbf{n}$ and hence find, in the form
$$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$
an equation for the plane $\Pi_3$ which is parallel to $\Pi_1$ and $\Pi_2$ and exactly half-way between them. [4]
\item The locus of all points $P$ such that $AP = BP = 12\sqrt{2}$ is denoted by $L$.
\begin{enumerate}[label=(\alph*)]
\item Give a full geometrical description of $L$. [4]
\item Using the result of part (iii), or otherwise, find a point on $L$ which has integer coordinates. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q13 [18]}}