| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Line of intersection of two planes |
| Difficulty | Standard +0.3 This is a standard FP3 vectors question requiring routine techniques: finding a normal vector via cross product, converting between plane forms, and finding the line of intersection of two planes. While it involves multiple steps and vector manipulation, these are well-practiced procedures in Further Maths with no novel insight required, making it slightly easier than average for an A-level question overall but typical for FP3. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
The plane $\Pi_1$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -5 \\ -2 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Express the equation of $\Pi_1$ in the form $\mathbf{r} \cdot \mathbf{n} = p$. [4]
The plane $\Pi_2$ has equation $\mathbf{r} \cdot \begin{pmatrix} 7 \\ 1 \\ -3 \end{pmatrix} = 21$.
\item Find an equation of the line of intersection of $\Pi_1$ and $\Pi_2$, giving your answer in the form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q6 [9]}}