| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Challenging +1.8 This is a Further Maths group theory question requiring systematic computation of a Cayley table with complex number arithmetic, verification of group axioms, identification of element orders, subgroups, and generators. While the individual complex multiplications are routine, the multi-part structure demands sustained abstract reasoning about algebraic structures, making it significantly harder than typical A-level questions but not exceptionally difficult for Further Maths students who have studied group theory. |
| Spec | 8.03a Binary operations: and their properties on given sets8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03g Cyclic groups: meaning of the term8.03h Generators: of cyclic and non-cyclic groups |
6 The set $S$ consists of the following four complex numbers.\\
$\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}$\\
For $z _ { 1 } , z _ { 2 } \in S$, the binary operation $\bigcirc$ is defined by $z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Complete the Cayley table for $( S , \bigcirc )$ given in the Printed Answer Booklet.
\item Verify that ( $S , \bigcirc$ ) is a group.
\item State the order of each element of $( S , \bigcirc )$.
\end{enumerate}\item Write down the only proper subgroup of ( $S , \bigcirc$ ).
\item \begin{enumerate}[label=(\roman*)]
\item Explain why ( $S , \bigcirc$ ) is a cyclic group.
\item List all possible generators of $( S , \bigcirc )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2021 Q6 [11]}}