| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 24 |
| 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 standard Further Maths vectors question testing cross products, distance formulas, and scalar triple products. While it requires multiple techniques (cross product, distance formula, volume formula) and involves algebraic manipulation with parameter p, these are all routine applications of learned formulas without requiring novel geometric insight. The multi-part structure and algebraic complexity place it above average, but it remains a textbook-style exercise for FP3 students. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector4.04h Shortest distances: between parallel lines and between skew lines4.04i Shortest distance: between a point and a line |
1 Four points have coordinates
$$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$
where $p$ is a constant.\\
(i) Find the perpendicular distance from C to the line AB .\\
(ii) Find $\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }$ in terms of $p$, and show that the shortest distance between the lines AB and CD is
$$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
(iii) Find, in terms of $p$, the volume of the tetrahedron ABCD .\\
(iv) State the value of $p$ for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case.
Option 2: Multi-variable calculus\\
\hfill \mbox{\textit{OCR MEI FP3 2010 Q1 [24]}}