| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Standard +0.3 This is a straightforward vector geometry question testing standard techniques: cross product calculation, understanding that the cross product gives a normal to a plane, and using the scalar triple product for volume. All parts follow directly from knowing these facts with minimal problem-solving required. While it's Further Maths content, the methods are routine applications of formulas. |
| Spec | 1.10b Vectors in 3D: i,j,k notation4.04f Line-plane intersection: find point4.04g Vector product: a x b perpendicular vector4.04i Shortest distance: between a point and a line |
Relative to an origin $O$, the points $P$, $Q$ and $R$ have position vectors
$$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$
respectively.
\begin{enumerate}[label=(\roman*)]
\item Determine $\mathbf{p} \times \mathbf{q}$. [2]
\item Deduce the value of $\alpha$ for which
\begin{enumerate}[label=(\alph*)]
\item $OR$ is normal to the plane $OPQ$, [1]
\item the volume of tetrahedron $OPQR$ is 50, [3]
\item $R$ lies in the plane $OPQ$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q7 [8]}}