| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line-plane intersection and related angle/perpendicularity |
| Difficulty | Standard +0.3 This is a structured multi-part 3D coordinate geometry question with clear guidance at each step. Part (i) involves routine vector perpendicularity checks and midpoint substitution. Part (ii) requires finding intersection of line with plane (standard λ-substitution) and writing a line equation. Parts (iii) and (iv) follow naturally from established results. While it involves multiple techniques (vectors, planes, lines in 3D), each step is straightforward with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04f Line-plane intersection: find point |
With respect to cartesian coordinates $Oxyz$, a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation $x + 2y - 3z = 0$, as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Show that AA' is perpendicular to the plane $x + 2y - 3z = 0$, and that M lies in the plane. [4]
\end{enumerate}
The vector equation of the line AB is $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the coordinates of B, and a vector equation of the line A'B. [6]
\item Given that A'BC is a straight line, find the angle $\theta$. [4]
\item Find the coordinates of the point where BC crosses the $Oxz$ plane (the plane containing the $x$- and $z$-axes). [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2012 Q8 [17]}}