| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -0.5 This is a straightforward logic/Boolean algebra question requiring truth tables and basic logical equivalences. Part (i) is mechanical table construction, part (ii) applies standard Boolean algebra rules, and part (iii) is simple interpretation. While it requires careful symbolic manipulation, it involves no novel problem-solving and is more routine than a typical calculus or mechanics question. |
| Spec | 4.01a Mathematical induction: construct proofs4.01b Complex proofs: conjecture and demanding proofs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Truth table with 4 lines | M1 | |
| Correct columns for T and S | A1 | |
| Correct columns for \(\sim T\) (twice) and \(\sim S\) | A1 | |
| Correct column for \(\Rightarrow\) | A1 | |
| Correct column for \(\wedge\) | A1 | |
| Correct column for \(\sim\) on LHS | A1 | |
| Correct final result column | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Truth table with correct structure | M1 | |
| \(A \Rightarrow B \Leftrightarrow\ \sim A \vee B\) columns correct | A1 | or a correct verbal argument |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sim(\sim T \Rightarrow \sim S) \Leftrightarrow \sim(T \vee \sim S) \Leftrightarrow \sim T \wedge S\) | M1 | Boolean |
| Applying result | A1 | |
| Correct negating | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| "not try" | B1 | |
| "and" | B1 | |
| "succeed" | B1 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Truth table with 4 lines | M1 | |
| Correct columns for T and S | A1 | |
| Correct columns for $\sim T$ (twice) and $\sim S$ | A1 | |
| Correct column for $\Rightarrow$ | A1 | |
| Correct column for $\wedge$ | A1 | |
| Correct column for $\sim$ on LHS | A1 | |
| Correct final result column | M1 A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Truth table with correct structure | M1 | |
| $A \Rightarrow B \Leftrightarrow\ \sim A \vee B$ columns correct | A1 | or a correct verbal argument |
## Boolean simplification
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sim(\sim T \Rightarrow \sim S) \Leftrightarrow \sim(T \vee \sim S) \Leftrightarrow \sim T \wedge S$ | M1 | Boolean |
| Applying result | A1 | |
| Correct negating | A1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| "not try" | B1 | |
| "and" | B1 | |
| "succeed" | B1 | |
---1 (i) Use a truth table to prove $\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )$.\\
(ii) Prove that $( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )$ and hence use Boolean algebra to prove that
$$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
(iii) A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.
\hfill \mbox{\textit{OCR MEI D2 2006 Q1 [16]}}