| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | June |
| Marks | 18 |
| 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 18 marks total. While it involves several techniques (distance formula, dot product, angle calculation, plane equations, line-plane intersection), each part follows standard procedures with clear signposting. Part (ii)(a) even tells students the normal vector to verify. The question requires methodical application of learned techniques rather than problem-solving insight, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry4.04a 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 |
Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5).
\includegraphics{figure_7}
\begin{enumerate}[label=(\roman*)]
\item Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
\item \begin{enumerate}[label=(\alph*)]
\item Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
\item Hence find the equation of this plane. [2]
\end{enumerate}
\item Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]
\end{enumerate}
The volume of a tetrahedron is $\frac{1}{3} \times \text{area of base} \times \text{height}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find the volume of the tetrahedron ABCD. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2014 Q7 [18]}}