| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Equation of plane containing line and point/parallel to vector |
| Difficulty | Standard +0.8 This FP3 question requires parametric representation of 3D lines, solving simultaneous equations to find intersection conditions, and computing a plane equation from two direction vectors. While systematic, it demands careful algebraic manipulation across multiple parameters and understanding of 3D vector geometry—more challenging than standard C3/C4 but typical for Further Maths content. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
Two lines have equations
$$\frac{x-k}{2} = \frac{y+1}{-5} = \frac{z-1}{-3} \quad \text{and} \quad \frac{x-k}{1} = \frac{y+4}{-4} = \frac{z}{-2},$$
where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that, for all values of $k$, the lines intersect, and find their point of intersection in terms of $k$. [6]
\item For the case $k = 1$, find the equation of the plane in which the lines lie, giving your answer in the form $ax + by + cz = d$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q5 [10]}}