Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.
A mapping team marks two transmitters, and and a straight service path on a coordinate map. The coordinates are measured in kilometres. A receiver must be placed on so that it is the same distance from both transmitters.
Figure 1 shows the positions of and
Find the coordinates of the midpoint of
Find the equation of the perpendicular bisector of Give your answer in the form
Hence, determine the coordinates of the receiver.
A triangular inspection plate is fitted in a three-dimensional frame. From vertex the edges and are represented by the vectors
An engineer uses these edges to assess the size and orientation of the plate.
Determine the area of triangle
Determine the acute angle between the lines and to the nearest
Find the scalar component of in the direction of and the magnitude of the component of perpendicular to Give each answer to 3 significant figures.
Use your answers to part (c) to state whether is more nearly parallel or perpendicular to giving a reason.
For the first 10 hours of a survey, two vessels move in a horizontal plane at constant velocity. Their position vectors, relative to a fixed origin, are
where coordinates are in kilometres and is the time in hours after 08:00.
Write down the initial position vector of B relative to A,
Show that
Hence determine the time, and the distance between the vessels when they are closest during the 10-hour survey. Give both answers to 3 significant figures.
A road-maintenance vehicle starts at depot A, must travel along every road segment at least once, and must return to A. The roads are undirected.
| Road | Length (km) | Road | Length (km) |
|---|---|---|---|
| AB | 4 | AC | 6 |
| AD | 8 | BC | 2 |
| BD | 5 | CD | 3 |
| CE | 7 | DE | 4 |
| DF | 6 | EF | 2 |
The following table gives the least distance, in kilometres, between each pair of locations.
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| A | – | 4 | 6 | 8 | 12 | 14 |
| B | 4 | – | 2 | 5 | 9 | 11 |
| C | 6 | 2 | – | 3 | 7 | 9 |
| D | 8 | 5 | 3 | – | 4 | 6 |
| E | 12 | 9 | 7 | 4 | – | 2 |
| F | 14 | 11 | 9 | 6 | 2 | – |
State all the locations of odd degree.
Using the least-distance table, determine which pairs of odd-degree locations should have their shortest paths repeated to minimize the additional distance.
Hence determine the minimum distance travelled. Explain why the Chinese postman algorithm is appropriate rather than the travelling salesman algorithm, and why the repetitions in part (b) allow the vehicle to return to A after traversing every road.
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