| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Truth tables or Boolean algebra |
| Difficulty | Moderate -0.5 This is a straightforward Boolean logic question requiring standard truth table construction and application of given logical rules. While it has multiple parts, each involves routine mechanical application of logic laws without requiring novel insight or complex problem-solving—easier than average A-level maths. |
| Spec | 4.01a Mathematical induction: construct proofs4.01b Complex proofs: conjecture and demanding proofs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| It doesn't move | B1 | |
| So paint it | B1 | |
| Contrapositive of "if it moves, salute it" / modus tollens reasoning | B1 | Must justify using logical reasoning |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| We do not know / cannot determine | B1 | |
| It could move or not move (painting does not tell us about movement) | B1 | |
| Converse is not necessarily true / painting \(\not\Rightarrow\) doesn't move | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct columns for \(m\), \(s\), \(p\) (8 rows) | B1 | |
| Correct column for \(m \Rightarrow s\) | B1 | |
| Correct column for \({\sim}m \Rightarrow p\) | B1 | |
| Correct column for \((m \Rightarrow s) \land ({\sim}m \Rightarrow p)\) | B1 | |
| Correct column for \(((m \Rightarrow s) \land ({\sim}m \Rightarrow p)) \land {\sim}p\) | B1 | |
| All entries in final column are 1 or match \(s\), proving the result | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Apply rule 1: \({\sim}m \Rightarrow p \Leftrightarrow {\sim}p \Rightarrow m\) | M1 | |
| So have \((m \Rightarrow s) \land ({\sim}p \Rightarrow m) \land {\sim}p\) | A1 | |
| Apply rule 2 to \(({\sim}p \land ({\sim}p \Rightarrow m))\): get \(m\) | M1 | |
| Apply rule 2 to \((m \land (m \Rightarrow s))\): get \(s\) | A1 |
# Question 1:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| It doesn't move | B1 | |
| So paint it | B1 | |
| Contrapositive of "if it moves, salute it" / modus tollens reasoning | B1 | Must justify using logical reasoning |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| We do not know / cannot determine | B1 | |
| It could move or not move (painting does not tell us about movement) | B1 | |
| Converse is not necessarily true / painting $\not\Rightarrow$ doesn't move | B1 | |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct columns for $m$, $s$, $p$ (8 rows) | B1 | |
| Correct column for $m \Rightarrow s$ | B1 | |
| Correct column for ${\sim}m \Rightarrow p$ | B1 | |
| Correct column for $(m \Rightarrow s) \land ({\sim}m \Rightarrow p)$ | B1 | |
| Correct column for $((m \Rightarrow s) \land ({\sim}m \Rightarrow p)) \land {\sim}p$ | B1 | |
| All entries in final column are 1 or match $s$, proving the result | B1 | |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Apply rule 1: ${\sim}m \Rightarrow p \Leftrightarrow {\sim}p \Rightarrow m$ | M1 | |
| So have $(m \Rightarrow s) \land ({\sim}p \Rightarrow m) \land {\sim}p$ | A1 | |
| Apply rule 2 to $({\sim}p \land ({\sim}p \Rightarrow m))$: get $m$ | M1 | |
| Apply rule 2 to $(m \land (m \Rightarrow s))$: get $s$ | A1 | |
---1
\begin{enumerate}[label=(\alph*)]
\item A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it."
Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
\begin{enumerate}[label=(\roman*)]
\item A recruit encounters something which is not painted. What should he do, and why?
\item A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
\end{enumerate}\item Use a truth table to prove $( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s$.
\item You are given the following two rules.
$$\begin{aligned}
& 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a ) \\
& 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y
\end{aligned}$$
Use Boolean algebra to prove that $( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D2 2007 Q1 [16]}}