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 survey team is preparing a map of a coastal reserve. In one triangulation, stations and are apart. The measured angles in triangle are and
A second mapped zone is represented by triangle where and
Determine
Show that correct to three significant figures.
Determine the area of triangle correct to three significant figures.
An architect models a rectangular atrium as a cuboid measuring by on the floor and high. Let be one lower corner and the opposite upper corner, with coordinates in metres and the floor in the plane
Find the coordinates of the midpoint of
Find the angle between and the floor, to the nearest degree.
A mapping team approximates a wetland by triangle The horizontal distance between survey stations and is The measured interior angles are and
The wetland qualifies for a reserve if its mapped area is at least
Find
Show that correct to three significant figures.
Hence, using your answer to part (b), calculate the area of triangle correct to three significant figures, and state whether the wetland qualifies for a reserve.
A surveyor uses triangulation to estimate the area of a triangular wetland Points and lie on an accessible straight bank, and metres. The measured interior angles are and
Find
Show that the area in is approximately given by
The surveyor wants to map a region with an area of at least Determine the required value of to 3 significant figures. Hence state whether a baseline of is sufficient.
Figure 1 shows a line segment on a Cartesian grid.
Write down the coordinates of and
Find the coordinates of the midpoint of
Calculate the gradient of
Hence determine the equation of the perpendicular bisector of giving your answer in the form
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 flat map uses Cartesian coordinates to locate radio beacons. The segment lies on the line and its midpoint is
Determine the equation of the perpendicular bisector of giving your answer in the form
Beacons and have coordinates and An emergency receiver must be equidistant from and and also equidistant from and
Using your answer to (a), determine the exact coordinates of
Two survey boats, A and B, move at constant velocities in a horizontal plane. Coordinates are measured in metres, with the -axis pointing east and the -axis pointing north. At their position vectors are
and their velocity vectors are
in The time is measured in seconds.
Find an expression, in terms of for the relative position vector of B from A,
Using your answer to part (a), determine the time at which the boats are closest and their minimum separation. Give both answers to 3 significant figures.
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.
A courier moves between three depots, and The directed routes leaving each depot are shown below.
| Depot | Depots reachable in one leg |
|---|---|
At each depot, the courier chooses uniformly from the routes leaving that depot. Use the convention that the entry in row column of a transition matrix is the probability of moving from depot to depot
Write down the adjacency matrix for this network, using the order for both rows and columns.
Find the number of two-leg walks from back to using the appropriate entry of
Construct the transition matrix for the courier’s movements, using the order for both rows and columns.
Using your transition matrix, find the probability that the courier is at after exactly two legs, starting from
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