| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Shortest distance between two skew lines |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question on skew lines requiring cross products and the standard formula for shortest distance between skew lines. Part (i) involves setting up and solving a quadratic equation, parts (ii) and (iii) are routine applications of point-to-line distance and angle between planes. While it requires multiple techniques and careful calculation, all methods are standard FM1 procedures with no novel insight needed, making it moderately above average difficulty. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04h Shortest distances: between parallel lines and between skew lines4.04i Shortest distance: between a point and a line |
The position vectors of the points $A , B , C , D$ are
$$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad 5 \mathbf { i } - 5 \mathbf { j } + \alpha \mathbf { k } ,$$
respectively, where $\alpha$ is a positive integer. It is given that the shortest distance between the line $A B$ and the line $C D$ is equal to $2 \sqrt { } 2$.\\
(i) Show that the possible values of $\alpha$ are 3 and 5 .\\
(ii) Using $\alpha = 3$, find the shortest distance of the point $D$ from the line $A C$, giving your answer correct to 3 significant figures.\\
(iii) Using $\alpha = 3$, find the acute angle between the planes $A B C$ and $A B D$, giving your answer in degrees.\\
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{CAIE FP1 2017 Q12 OR}}