| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | Specimen |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Perpendicular distance from point to plane |
| Difficulty | Challenging +1.8 This is a substantial Further Maths question requiring multiple vector techniques: cross product for plane normal, perpendicular distance formula, line-plane intersection, and angle between planes. While each component uses standard methods, the multi-part structure with dependent results and the need to coordinate several vector concepts makes it significantly harder than typical A-level questions, though the calculations themselves are relatively straightforward once the approach is identified. |
| Spec | 1.10c Magnitude and direction: of vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04g Vector product: a x b perpendicular vector |
The points $A , B$ and $C$ have position vectors $\mathbf { i } , 2 \mathbf { j }$ and $4 \mathbf { k }$ respectively, relative to an origin $O$. The point $N$ is the foot of the perpendicular from $O$ to the plane $A B C$. The point $P$ on the line-segment $O N$ is such that $O P = \frac { 3 } { 4 } O N$. The line $A P$ meets the plane $O B C$ at $Q$.\\
(i) Find a vector perpendicular to the plane $A B C$ and show that the length of $O N$ is $\frac { 4 } { \sqrt { } ( 21 ) }$.\\
(ii) Find the position vector of the point $Q$.\\
(iii) Show that the acute angle between the planes $A B C$ and $A B Q$ is $\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q11 EITHER}}