| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Challenging +1.8 This is a substantial Further Maths question requiring multiple vector techniques (cross product line equation, perpendicular distance, solving quadratic for points on a circle, plane equation, tetrahedron volume, skew line distance). While each individual step uses standard FP1 methods, the multi-part structure, computational complexity, and need to synthesize several concepts makes it significantly harder than average A-level questions but still within expected FP1 territory. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector4.04j Shortest distance: between a point and a plane |
\begin{enumerate}
\item With respect to a fixed origin $O$ the point $A$ has coordinates $( 3,6,5 )$ and the line $l$ has equation
\end{enumerate}
$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$
The points $B$ and $C$ lie on $l$ such that $A B = A C = 15$\\
Given that $A$ does not lie on $l$ and that the $x$ coordinate of $B$ is negative,\\
(a) determine the coordinates of $B$ and the coordinates of $C$\\
(b) Hence determine a Cartesian equation of the plane containing the points $A , B$ and $C$
The point $D$ has coordinates $( - 2,1 , \alpha )$, where $\alpha$ is a constant.\\
Given that the volume of the tetrahedron $A B C D$ is 147\\
(c) determine the possible values of $\alpha$
Given that $\alpha > 0$\\
(d) determine the shortest distance between the line $l$ and the line passing through the points $A$ and $D$, giving your answer to 2 significant figures.\\
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\hfill \mbox{\textit{Edexcel FP1 2023 Q7}}