Question 1 - A-Level Maths

OCR MEI D2 2005 June — Question 1 16 marks

Exam Board	OCR MEI
Module	D2 (Decision Mathematics 2)
Year	2005
Session	June
Marks	16
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Groups
Difficulty	Moderate -0.5 This is a straightforward Boolean algebra question requiring interpretation of circuits, truth table construction, and basic algebraic manipulation. While it involves multiple parts, each step uses standard techniques (reading circuits, completing truth tables, applying distributive laws) without requiring novel insight or complex problem-solving. It's slightly easier than average A-level due to its mechanical nature and clear structure.

1 The switching circuit in Fig. 1.1 shows switches, \(s\) for a car's sidelights, \(h\) for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_284_917_404_580} \captionsetup{labelformat=empty} \caption{Fig. 1.1}

\end{figure} (Note: \(s\) and \(h\) are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)

1. Describe in words the conditions under which the foglights will come on. Fig. 1.2 shows a combinatorial circuit. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_367_1235_1183_431} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
  \end{figure}
2. Write the output in terms of a Boolean expression involving \(s , h\) and \(f\).
3. Use a truth table to prove that \(\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }\).
A car's first gear can be engaged ( g ) if either both the road speed is low ( r ) and the clutch is depressed ( d ), or if both the road speed is low ( r ) and the engine speed is the correct multiple of the road speed (m).
1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent \(r\), and single switches to represent each of \(\mathrm { d } , \mathrm { m }\) and g .
2. Draw a combinatorial circuit to represent the Boolean expression \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }\).
3. Use Boolean algebra to prove that \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }\).
4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.

Show mark scheme Show mark scheme source

Question 1:

Part (a)(i)

Answer	Marks	Guidance
Answer	Marks	Guidance
If sidelights and headlights are on, and if the foglights are switched on.	B1 B1	Two separate B1 marks

Part (a)(ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\sim(\sim s \vee \sim h) \wedge f\)	M1 A1

Part (a)(iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Truth table with 8 rows	M1	8 rows required
\(s \wedge h \wedge f\) column correct	A1
\(\sim(\sim s \vee \sim h) \wedge f\) column correct	A1
Accept truth table showing \(s \wedge h = \sim(\sim s \vee \sim h)\)	B1	Comment re \(\wedge f\)
M1 A1	Alternative: 4 lines

Part (b)(i)

Answer	Marks	Guidance
Answer	Marks	Guidance
Correct circuit diagram with \(r\), \(d\), \(m\) inputs giving output \(g\) via multiplexer	M1 A1

Part (b)(ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
Correct logic gate circuit with \(r\), \(d\), \(m\), \(g\) inputs	M1
OR gate correct	A1	"or"
First AND gate correct	A1	"first and"
Second AND gate correct	A1	"second and"

Part (b)(iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(r \wedge (d \vee m) \wedge g = (r \wedge (d \vee m)) \wedge g\) by associativity	M1	Distributive law applied
\(= ((r \wedge d) \vee (r \wedge m)) \wedge g\) by distributivity	A1	Handling brackets correctly; law names not needed

Part (b)(iv)

Answer	Marks	Guidance
Answer	Marks	Guidance
Correct circuit diagram with \(r\) input leading to \(d\), \(m\) multiplexer then \(g\) output	B1

# Question 1:

## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| If sidelights and headlights are on, and if the foglights are switched on. | B1 B1 | Two separate B1 marks |

## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sim(\sim s \vee \sim h) \wedge f$ | M1 A1 | |

## Part (a)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Truth table with 8 rows | M1 | 8 rows required |
| $s \wedge h \wedge f$ column correct | A1 | |
| $\sim(\sim s \vee \sim h) \wedge f$ column correct | A1 | |
| Accept truth table showing $s \wedge h = \sim(\sim s \vee \sim h)$ | B1 | Comment re $\wedge f$ |
| | M1 A1 | Alternative: 4 lines |

## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct circuit diagram with $r$, $d$, $m$ inputs giving output $g$ via multiplexer | M1 A1 | |

## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct logic gate circuit with $r$, $d$, $m$, $g$ inputs | M1 | |
| OR gate correct | A1 | "or" |
| First AND gate correct | A1 | "first and" |
| Second AND gate correct | A1 | "second and" |

## Part (b)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r \wedge (d \vee m) \wedge g = (r \wedge (d \vee m)) \wedge g$ by associativity | M1 | Distributive law applied |
| $= ((r \wedge d) \vee (r \wedge m)) \wedge g$ by distributivity | A1 | Handling brackets correctly; law names not needed |

## Part (b)(iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct circuit diagram with $r$ input leading to $d$, $m$ multiplexer then $g$ output | B1 | |

---

Show LaTeX source

1 The switching circuit in Fig. 1.1 shows switches, $s$ for a car's sidelights, $h$ for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_284_917_404_580}
\captionsetup{labelformat=empty}
\caption{Fig. 1.1}
\end{center}
\end{figure}

(Note: $s$ and $h$ are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Describe in words the conditions under which the foglights will come on.

Fig. 1.2 shows a combinatorial circuit.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_367_1235_1183_431}
\captionsetup{labelformat=empty}
\caption{Fig. 1.2}
\end{center}
\end{figure}
\item Write the output in terms of a Boolean expression involving $s , h$ and $f$.
\item Use a truth table to prove that $\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }$.
\end{enumerate}\item A car's first gear can be engaged ( g ) if either both the road speed is low ( r ) and the clutch is depressed ( d ), or if both the road speed is low ( r ) and the engine speed is the correct multiple of the road speed (m).
\begin{enumerate}[label=(\roman*)]
\item Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent $r$, and single switches to represent each of $\mathrm { d } , \mathrm { m }$ and g .
\item Draw a combinatorial circuit to represent the Boolean expression $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }$.
\item Use Boolean algebra to prove that $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }$.
\item Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI D2 2005 Q1 [16]}}

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