Edexcel FP2 AS 2018 June — Question 2 10 marks

Exam BoardEdexcel
ModuleFP2 AS (Further Pure 2 AS)
Year2018
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGroups
TypeGroups of symmetries
DifficultyStandard +0.3 This is a standard Further Maths group theory question covering the dihedral group Dā‚ƒ. Part (a) requires completing a Cayley table through geometric visualization or systematic composition, which is routine practice. Parts (b)-(d) involve basic group properties (order, cyclic groups, subgroups) that are direct applications of definitions with minimal conceptual challenge. While this is Further Maths content, it's a textbook exercise requiring careful execution rather than insight, making it easier than average overall A-level difficulty.
Spec8.03b Cayley tables: construct for finite sets under binary operation8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{285b6ae9-ca8f-46b7-b4ed-a3310fe4ebe6-04_568_634_248_717} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows an equilateral triangle \(A B C\). The lines \(x , y\) and \(z\) and their point of intersection, \(O\), are fixed in the plane. The triangle \(A B C\) is transformed about these fixed lines and the fixed point \(O\). The lines \(x , y\) and \(z\) each pass through a vertex of the triangle and the midpoint of the opposite side. The transformations \(I , X , Y , Z , R _ { 1 }\) and \(R _ { 2 }\) of the plane containing triangle \(A B C\) are defined as follows:
  • I: Do nothing
  • \(X\) : Reflect in the line \(x\)
  • \(Y\) : Reflect in the line \(y\)
  • \(Z\) : Reflect in the line \(z\)
  • \(R _ { 1 }\) : Rotate \(120 ^ { \circ }\) anticlockwise about \(O\)
  • \(R _ { 2 }\) : Rotate \(240 ^ { \circ }\) anticlockwise about \(O\)
The operation * is defined as 'followed by' on the set \(T = \left\{ I , X , Y , Z , R _ { 1 } , R _ { 2 } \right\}\).
For example, \(X { } ^ { * } Y\) means a reflection in the line \(x\) followed by a reflection in the line \(y\).
    1. Complete the Cayley table on page 5 Given that the associative law is satisfied,
    2. show that \(T\) is a group under the operation *
  2. Show that the element \(R _ { 2 }\) has order 3
  3. Explain why \(T\) is not a cyclic group.
  4. Write down the elements of a subgroup of \(T\) that has order 3
    \multirow{2}{*}{}Second transformation
    *I\(X\)\(Y\)\(Z\)\(R _ { 1 }\)\(R _ { 2 }\)
    \multirow{6}{*}{First Transformation}I
    \(X\)I\(Z\)
    \(Y\)
    \(Z\)
    \(R _ { 1 }\)\(Y\)
    \(R _ { 2 }\)
    \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Only use this grid if you need to re-write your Cayley table}
    \multirow{2}{*}{}Second transformation
    *I\(X\)\(Y\)Z\(R _ { 1 }\)\(R _ { 2 }\)
    \multirow{6}{*}{First Transformation}I
    XIZ
    Y
    Z
    \(R _ { 1 }\)\(Y\)
    \(R _ { 2 }\)
    \end{table}
Question 2:
Part (a)(i)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
First row and first column correctB1 Begins completing the table with at least first row and first column correct
Three rows or three columns correctB1 Mostly correct ā€” demonstrates understanding of using \(*\)
Fully correct Cayley tableB1 Fully correct table
(3 marks)
Part (a)(ii)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(I\) is the identity and closure is shown (by the Cayley table)M1 States closure and identifies the identity as \(I\)
\(X\), \(Y\) and \(Z\) are self-inverse; \(R_1\) and \(R_2\) are inverses (\(I\) is the identity so is self-inverse)M1 States the inverse of each element (reference to Identity not required here). Special case: if inverses not stated explicitly but "all elements have an inverse" is seen, score M1M1A0
(Associative law may be assumed) so \(T\) forms a groupA1 Must see a conclusion
(3 marks)
Part (b)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(R_2*R_2*R_2=(R_2*R_2)*R_2\) or \(R_2*(R_2*R_2)=R_1*R_2\) or \(R_2*R_1\)M1 Clearly begins process to find \(R_2*R_2*R_2\) reaching \(R_1*R_2\) or \(R_2*R_1\)
\(= I\) (the identity) so \(R_2\) has order \(3\)A1 Gives answer as \(I\), states identity, and deduces order is \(3\), e.g. \((R_2)^3=I\)
(2 marks)
Part (c)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(R_1\) (and \(R_2\)) have order \(3\); \(X\), \(Y\) and \(Z\) have order \(2\); so there is no element of \(T\) that generates the group, or there is no element of order \(6\)B1 Demonstrates understanding of the term cyclic by referring to the order of \(R_1\), \(X\), \(Y\) and \(Z\) and makes a suitable conclusion
(1 mark)
Part (d)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\{I,\, R_1,\, R_2\}\)B1 Indicates the set \(\{I, R_1, R_2\}\) (brackets not required)
(1 mark)
(10 marks total)
# Question 2:

## Part (a)(i)

| Answer/Working | Marks | Guidance |
|---|---|---|
| First row and first column correct | B1 | Begins completing the table with at least first row and first column correct |
| Three rows or three columns correct | B1 | Mostly correct ā€” demonstrates understanding of using $*$ |
| Fully correct Cayley table | B1 | Fully correct table |

**(3 marks)**

## Part (a)(ii)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I$ is the identity and closure is shown (by the Cayley table) | M1 | States closure and identifies the identity as $I$ |
| $X$, $Y$ and $Z$ are self-inverse; $R_1$ and $R_2$ are inverses ($I$ is the identity so is self-inverse) | M1 | States the inverse of each element (reference to Identity not required here). Special case: if inverses not stated explicitly but "all elements have an inverse" is seen, score M1M1A0 |
| (Associative law may be assumed) so $T$ forms a group | A1 | Must see a conclusion |

**(3 marks)**

## Part (b)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $R_2*R_2*R_2=(R_2*R_2)*R_2$ or $R_2*(R_2*R_2)=R_1*R_2$ or $R_2*R_1$ | M1 | Clearly begins process to find $R_2*R_2*R_2$ reaching $R_1*R_2$ or $R_2*R_1$ |
| $= I$ (the identity) so $R_2$ has order $3$ | A1 | Gives answer as $I$, states identity, and deduces order is $3$, e.g. $(R_2)^3=I$ |

**(2 marks)**

## Part (c)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $R_1$ (and $R_2$) have order $3$; $X$, $Y$ and $Z$ have order $2$; so there is no element of $T$ that generates the group, **or** there is no element of order $6$ | B1 | Demonstrates understanding of the term cyclic by referring to the order of $R_1$, $X$, $Y$ and $Z$ and makes a suitable conclusion |

**(1 mark)**

## Part (d)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\{I,\, R_1,\, R_2\}$ | B1 | Indicates the set $\{I, R_1, R_2\}$ (brackets not required) |

**(1 mark)**

**(10 marks total)**
2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{285b6ae9-ca8f-46b7-b4ed-a3310fe4ebe6-04_568_634_248_717}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows an equilateral triangle $A B C$. The lines $x , y$ and $z$ and their point of intersection, $O$, are fixed in the plane. The triangle $A B C$ is transformed about these fixed lines and the fixed point $O$. The lines $x , y$ and $z$ each pass through a vertex of the triangle and the midpoint of the opposite side.

The transformations $I , X , Y , Z , R _ { 1 }$ and $R _ { 2 }$ of the plane containing triangle $A B C$ are defined as follows:

\begin{itemize}
  \item I: Do nothing
  \item $X$ : Reflect in the line $x$
  \item $Y$ : Reflect in the line $y$
  \item $Z$ : Reflect in the line $z$
  \item $R _ { 1 }$ : Rotate $120 ^ { \circ }$ anticlockwise about $O$
  \item $R _ { 2 }$ : Rotate $240 ^ { \circ }$ anticlockwise about $O$
\end{itemize}

The operation * is defined as 'followed by' on the set $T = \left\{ I , X , Y , Z , R _ { 1 } , R _ { 2 } \right\}$.\\
For example, $X { } ^ { * } Y$ means a reflection in the line $x$ followed by a reflection in the line $y$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Complete the Cayley table on page 5

Given that the associative law is satisfied,
\item show that $T$ is a group under the operation *
\end{enumerate}\item Show that the element $R _ { 2 }$ has order 3
\item Explain why $T$ is not a cyclic group.
\item Write down the elements of a subgroup of $T$ that has order 3

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} &  & \multicolumn{6}{|c|}{Second transformation} \\
\hline
 & * & I & $X$ & $Y$ & $Z$ & $R _ { 1 }$ & $R _ { 2 }$ \\
\hline
\multirow{6}{*}{First Transformation} & I &  &  &  &  &  &  \\
\hline
 & $X$ &  & I &  &  & $Z$ &  \\
\hline
 & $Y$ &  &  &  &  &  &  \\
\hline
 & $Z$ &  &  &  &  &  &  \\
\hline
 & $R _ { 1 }$ &  & $Y$ &  &  &  &  \\
\hline
 & $R _ { 2 }$ &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Only use this grid if you need to re-write your Cayley table}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} &  & \multicolumn{6}{|c|}{Second transformation} \\
\hline
 & * & I & $X$ & $Y$ & Z & $R _ { 1 }$ & $R _ { 2 }$ \\
\hline
\multirow{6}{*}{First Transformation} & I &  &  &  &  &  &  \\
\hline
 & X &  & I &  &  & Z &  \\
\hline
 & Y &  &  &  &  &  &  \\
\hline
 & Z &  &  &  &  &  &  \\
\hline
 & $R _ { 1 }$ &  & $Y$ &  &  &  &  \\
\hline
 & $R _ { 2 }$ &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{enumerate}

\hfill \mbox{\textit{Edexcel FP2 AS 2018 Q2 [10]}}
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