| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | 3D geometry applications |
| Difficulty | Standard +0.3 This is a structured multi-part 3D geometry question with clear scaffolding. Part (i) is verification (routine calculation), parts (ii) and (iii) involve standard techniques (finding plane equations, using dot products for angles), and part (iv) requires position vectors with given ratios. While it has multiple parts and requires careful coordinate work, each step follows standard A-level procedures without requiring novel insight or particularly challenging problem-solving. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04e Line intersections: parallel, skew, or intersecting4.04f Line-plane intersection: find point |
**Question 6:** [2 marks]
P(at least one of 8 visitors goes to top floor) = $1 - \left(\frac{9}{10}\right)^8$6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane $z = 0$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-03_785_1283_424_392}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Verify that the equation of the plane through $\mathrm { A } , \mathrm { B }$ and E is $x + 6 y + 12 = 0$.
Hence, given that F lies in this plane, show that $a = - 2 \frac { 1 } { 3 }$.
\item (A) Show that the vector $\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)$ is normal to the plane DHC.\\
(B) Hence find the cartesian equation of this plane.\\
(C) Given that G lies in the plane DHC , find $b$ and the length FG .
\item Find the angle EFB .
A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that $\mathrm { HQ } = 2 \mathrm { QP }$.
\item Find the height of Q above the ground.
\section*{Question 7 begins on page 4.}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2015 Q6 [18]}}