| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Complete or analyse Cayley table |
| Difficulty | Challenging +1.2 This is a Further Maths group theory question requiring Cayley table completion and isomorphism identification, which are standard exercises once the topic is learned. Part (b)(ii) involves modular arithmetic proof requiring systematic case analysis of odd quadratic residues, but follows a predictable pattern. While conceptually more advanced than single maths, it's a routine application of learned techniques rather than requiring novel insight. |
| Spec | 8.02g Quadratic residues: calculate and solve equations involving them8.02l Fermat's little theorem: both forms8.03b Cayley tables: construct for finite sets under binary operation8.03l Isomorphism: determine using informal methods |
5
\begin{enumerate}[label=(\alph*)]
\item The group $G$ consists of the set $S = \{ 1,9,17,25 \}$ under $\times _ { 32 }$, the operation of multiplication modulo 32.
\begin{enumerate}[label=(\roman*)]
\item Complete the Cayley table for $G$ given in the Printed Answer Booklet.
\item Up to isomorphisms, there are only two groups of order 4.
\begin{itemize}
\end{enumerate}\item $C _ { 4 }$, the cyclic group of order 4
\item $K _ { 4 }$, the non-cyclic (Klein) group of order 4
\end{itemize}
State, with justification, to which of these two groups $G$ is isomorphic.
\item \begin{enumerate}[label=(\roman*)]
\item List the odd quadratic residues modulo 32.
\item Given that $n$ is an odd integer, prove that $n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2023 Q5 [10]}}