| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Pigeonhole principle applications |
| Difficulty | Moderate -0.5 This is a straightforward application of binary search/divide-and-conquer logic with clear instructions. Students must trace through a given algorithm rather than devise one themselves. Part (i) requires following 3-4 steps of the procedure, and part (ii) adds minimal complexity by having two poisoned flagons. The question tests careful reading and systematic working rather than mathematical insight or problem-solving creativity. |
| Spec | 7.03a Algorithm definition: input, output, deterministic, finite |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Test 1: flags 1,2,3,4 → Result A | B1 | cao |
| Test 2: flags 5,6 → Result A | B1 | cao ... allow extra second line of 5678 D, but with \(-1\) |
| Test 3: flag 7 → Result D | B1 | cao |
| Test 4: flag 8 → Result A | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Test 1: flags 1,2,3,4 → Result D | B1 | cao |
| Test 2: flags 5,6,7,8 → Result D | B1 | cao |
| Test 3: flags 1,2 → Result A | B1 | award the last two B1s only for contiguous blocks of 3 tests |
| Test 4: flag 3 → Result D | ||
| Test 5: flag 4 → Result A | ||
| Tests 6: flags 5,6 → Result A | B1 | from line 3 allow extraneous lines but \(-1\) once only, and only from the last two B1s |
| Test 7: flag 7 → Result D | ||
| Test 8: flag 8 → Result A |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Test 1: flags 1,2,3,4 → Result A | B1 | cao |
| Test 2: flags 5,6 → Result A | B1 | cao ... allow extra second line of 5678 D, but with $-1$ |
| Test 3: flag 7 → Result D | B1 | cao |
| Test 4: flag 8 → Result A | B1 | cao |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Test 1: flags 1,2,3,4 → Result D | B1 | cao |
| Test 2: flags 5,6,7,8 → Result D | B1 | cao |
| Test 3: flags 1,2 → Result A | B1 | award the last two B1s only for contiguous blocks of 3 tests |
| Test 4: flag 3 → Result D | | |
| Test 5: flag 4 → Result A | | |
| Tests 6: flags 5,6 → Result A | B1 | from line 3 allow extraneous lines but $-1$ once only, and only from the last two B1s |
| Test 7: flag 7 → Result D | | |
| Test 8: flag 8 → Result A | | |
---2 King Elyias has been presented with eight flagons of fine wine. Intelligence reports indicate that at least one of the eight flagons has been poisoned. King Elyias will have the wine tasted by the royal wine tasters to establish which flagons are poisoned.
Samples for testing are made by using wine from one or more flagons. If a royal wine taster tastes a sample of wine which includes wine from a poisoned flagon, the taster will die. The king has to make a very generous payment for each sample tasted.
To minimise payments, the royal mathematicians have devised the following scheme:\\
Test a sample made by mixing wine from flagons $1,2,3$ and 4.\\
If the taster dies, then test a sample made by mixing wine from flagons $5,6,7$ and 8 .\\
If the taster lives, then there is no poison in flagons $1,2,3$ or 4 . So there is poison in at least one of flagons 5, 6, 7 and 8, and there is no need to test a sample made by mixing wine from all four of them.
If the sample from flagons $1,2,3$ and 4 contains poison, then test a fresh sample made by mixing wine from flagons 1 and 2, and proceed similarly, testing a sample from flagons 3 and 4 only if the taster of the sample from flagons 1 and 2 dies.
Continue, testing new samples made from wine drawn from half of the flagons corresponding to a poisoned sample, and testing only when necessary.\\
(i) Record what happens using the mathematicians' scheme when flagon number 7 is poisoned, and no others.\\
(ii) Record what happens using the mathematicians' scheme when two flagons, numbers 3 and 7, are poisoned.
\hfill \mbox{\textit{OCR MEI D1 2011 Q2 [8]}}