| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Easy -2.5 This question is from Decision Mathematics 2, not Groups/Further Pure, and contains basic logic/Boolean algebra exercises. Parts (a) and (b) are trivial comprehension and circuit drawing tasks, while part (c) is a mechanical truth table construction requiring no problem-solving insight—well below typical A-level maths difficulty. |
| Spec | 1.01b Logical connectives: congruence, if-then, if and only if |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| "It is hard to overestimate the effect..." or equivalent e.g. "Your contribution will make a significant effect" | B2 | B1 for recognising the double negative; B2 for a clear correct statement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Expression: \((a \wedge b) \vee ((\sim a) \wedge c) \vee ((\sim b) \wedge c)\) identified | B1 | For correct logical expression |
| Correct circuit drawn with AND gates for \(a \wedge b\), \((\sim a) \wedge c\), \((\sim b) \wedge c\) | M1 | For attempting three AND combinations |
| NOT gates on \(a\) and \(b\) | B1 | |
| Three AND gates correctly connected | B1 | |
| Three OR gates combining outputs | B1 | |
| Fully correct circuit | A2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Columns for \(a\), \(b\), \(c\), \(a \wedge b\), \(\sim a\), \(\sim b\), \(\sim c\) | B1 | For correct intermediate columns |
| Column for \((\sim a) \wedge c\) correct | B1 | |
| Column for \((\sim b) \wedge c\) correct | B1 | |
| LHS column correct | B1 | |
| Column for \((\sim a) \wedge (\sim c)\) correct | B1 | |
| Column for \((\sim b) \wedge (\sim c)\) correct | B1 | |
| RHS column correct and matches LHS | B1 | Full verification |
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| "It is hard to **over**estimate the effect..." or equivalent e.g. "Your contribution will make a significant effect" | B2 | B1 for recognising the double negative; B2 for a clear correct statement |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expression: $(a \wedge b) \vee ((\sim a) \wedge c) \vee ((\sim b) \wedge c)$ identified | B1 | For correct logical expression |
| Correct circuit drawn with AND gates for $a \wedge b$, $(\sim a) \wedge c$, $(\sim b) \wedge c$ | M1 | For attempting three AND combinations |
| NOT gates on $a$ and $b$ | B1 | |
| Three AND gates correctly connected | B1 | |
| Three OR gates combining outputs | B1 | |
| Fully correct circuit | A2 | |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Columns for $a$, $b$, $c$, $a \wedge b$, $\sim a$, $\sim b$, $\sim c$ | B1 | For correct intermediate columns |
| Column for $(\sim a) \wedge c$ correct | B1 | |
| Column for $(\sim b) \wedge c$ correct | B1 | |
| LHS column correct | B1 | |
| Column for $(\sim a) \wedge (\sim c)$ correct | B1 | |
| Column for $(\sim b) \wedge (\sim c)$ correct | B1 | |
| RHS column correct and matches LHS | B1 | Full verification |
---1
\begin{enumerate}[label=(\alph*)]
\item The following was said in a charity appeal on Radio 4 in October 2006.\\
"It is hard to underestimate the effect that your contribution will make."\\
Rewrite the comment more simply in your own words without changing its meaning.
\item A machine has three components, A, B and C, each of which is either active or inactive.
\begin{itemize}
\item The machine is active if A and B are both active.
\item The machine is active if A is inactive and C is active.
\item The machine is active if B is inactive and C is active.
\item Otherwise the machine is inactive.
\end{itemize}
The states (active or inactive) of the components and the machine are to be modelled by a combinatorial circuit in which "active" is represented by "true" and "inactive" is represented by "false".
Draw such a circuit.
\item Construct a truth table to show the following.
$$[ ( ( \mathrm { a } \wedge \mathrm {~b} ) \vee ( ( \sim \mathrm { a } ) \wedge \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge \mathrm { c } ) ] \Leftrightarrow \sim [ ( ( \sim \mathrm { a } ) \wedge ( \sim \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge ( \sim \mathrm { c } ) ) ]$$
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D2 2009 Q1 [16]}}