D2 June 2006 Q4
4. During the school holidays four building tasks, rebuilding a wall (\(W\)), repairing the roof (\(R\)), repainting the hall (\(H\)) and relaying the playground (\(P\)), need to be carried out at a Junior School.
Four builders, \(A\), \(B\), \(C\) and \(D\) will be hired for these tasks. Each builder must be assigned to one task. Builder \(B\) is not able to rebuild the wall and therefore cannot be assigned to this task.
The cost, in thousands of pounds, of using each builder for each task is given in the table below.
| Cost | \(H\) | \(P\) | \(R\) | \(W\) |
|---|---|---|---|---|
| \(A\) | 3 | 5 | 11 | 9 |
| \(B\) | 3 | 7 | 8 | – |
| \(C\) | 2 | 5 | 10 | 7 |
| \(D\) | 8 | 3 | 7 | 6 |
| Scheme | Marks | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Adding \(n \geqslant 20\) to table to give
| B1 | |||||||||||||||||||||||||
| Reducing rows first \(\begin{bmatrix}0&2&8&6\\0&4&5&n-3\\0&3&8&5\\5&0&4&3\end{bmatrix}\) then columns \(\begin{bmatrix}0&2&4&3\\0&4&1&n-6\\0&3&4&2\\5&0&0&0\end{bmatrix}\) (lines through column H and row D) | M1 A1 (3) | |||||||||||||||||||||||||
| Either \(\begin{bmatrix}0&1&3&2\\0&3&0&n-7\\0&2&3&1\\6&0&0&0\end{bmatrix}\) (lines through column H and rows B and D) or \(\begin{bmatrix}0&1&3&2\\0&3&0&n-7\\0&2&3&1\\6&0&0&0\end{bmatrix}\) (lines through columns H and R and row D) | M1 A1ft | |||||||||||||||||||||||||
| \(\downarrow\) \(\begin{bmatrix}0&0&2&1\\1&3&0&n-7\\0&1&2&0\\7&0&0&0\end{bmatrix}\) \(\downarrow\) \(\begin{bmatrix}0&0&3&1\\0&2&0&n-8\\0&1&3&0\\7&0&1&0\end{bmatrix}\) | M1 A1ft (4) | |||||||||||||||||||||||||
| A – H, B – R, C – W, D – P or A – P, B – R, C – H, D – W cost £21 000 | A1 A1 (2) | |||||||||||||||||||||||||
| (9) |
Notes
The scheme shows the covering lines as dashed lines on the matrices; they are described in words here (columns H, P, R, W; rows A, B, C, D).
The scheme prints \(n\) as N in the table; the part marks are printed as “M1 A13” (M1 A1, column total 3).
| Scheme | Marks |
|---|---|
| Not unique – gives the other solution | M1 A1ft |
| (2) | |
| (11 marks) |