| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| 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 standard C4 vectors question testing routine techniques: finding vectors from coordinates, calculating angles using dot product, verifying a plane equation, finding angle between planes, and using coplanarity. All methods are textbook procedures with no novel insight required. The multi-part structure and 18 marks indicate moderate length, but each step follows directly from standard formulas, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is $k$ metres.
\includegraphics{figure_7}
\begin{enumerate}[label=(\roman*)]
\item Write down the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Hence calculate the angle BAC. [6]
\item Verify that the equation of the plane ABC is $x + y - 2z + d = 0$, where $d$ is a constant to be determined.
Calculate the acute angle the plane makes with the horizontal plane. [7]
\item Given that A, B, D and C are coplanar, show that $k = 3$.
Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2011 Q7 [18]}}