| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with plane |
| Difficulty | Standard +0.3 This is a straightforward Further Maths vector geometry question requiring standard techniques: substituting a line equation into a plane equation to find intersection points, then calculating distance. Part (a) is routine verification (2 marks), and part (b) involves finding parameter λ for Q, then using the distance formula. While it's Further Maths content, the methods are mechanical applications of learned procedures with no novel insight required, making it slightly easier than average even for FM. |
| Spec | 1.10f Distance between points: using position vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04f Line-plane intersection: find point |
The equations of two intersecting lines $l_1$ and $l_2$ are
$$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$
$$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$
where $a$ is a constant.
The equation of the plane $\Pi$ is
$$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$
$l_1$ and $\Pi$ intersect at $Q$.
$l_2$ and $\Pi$ intersect at $R$.
\begin{enumerate}[label=(\alph*)]
\item Verify that the coordinates of $R$ are $(13, 3, -14)$. [2]
\item Determine the exact value of the length of $QR$. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q2 [9]}}