| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Common perpendicular to two skew lines |
| Difficulty | Challenging +1.8 This is a substantial Further Maths question requiring multiple vector techniques: finding the shortest distance between skew lines using the scalar triple product formula, parametric analysis with trigonometric functions, and calculating angles between planes via normal vectors. While the individual techniques are standard for FP1, the multi-part structure, the parametric circle adding complexity to part (ii), and the three-dimensional geometric reasoning required make this significantly harder than average A-level questions but still within the expected range for Further Maths. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04d Angles: between planes and between line and plane4.04h Shortest distances: between parallel lines and between skew lines |
The line $l_1$ passes through the point $A$ with position vector $\mathbf{i} - \mathbf{j} - 2\mathbf{k}$ and is parallel to the vector $3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}$. The variable line $l_2$ passes through the point $(1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}$, where $0 \leq t < 2\pi$, and is parallel to the vector $15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}$. The points $P$ and $Q$ are on $l_1$ and $l_2$ respectively, and $PQ$ is perpendicular to both $l_1$ and $l_2$.
\begin{enumerate}[label=(\roman*)]
\item Find the length of $PQ$ in terms of $t$. [4]
\item Hence show that the lines $l_1$ and $l_2$ do not intersect, and find the maximum length of $PQ$ as $t$ varies. [3]
\item The plane $\Pi_1$ contains $l_1$ and $PQ$; the plane $\Pi_2$ contains $l_2$ and $PQ$. Find the angle between the planes $\Pi_1$ and $\Pi_2$, correct to the nearest tenth of a degree. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2003 Q9 [11]}}