| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Topic | Groups |
| Type | Matrix groups |
| Difficulty | Challenging +1.2 This is a group theory question requiring verification of group axioms and subgroup analysis. While the topic (abstract algebra) is advanced for A-level, the actual verification is straightforward: closure follows from simple matrix multiplication, identity and inverses are easily found by inspection. Finding a subgroup of order 2 requires minimal insight (p=1/2 works), and showing no order-3 subgroups uses basic group theory (elements would need to satisfy g³=e). The matrices have a special simple form that makes calculations routine. This is harder than typical A-level due to the abstract algebra content, but easier than many Further Maths proof questions requiring deeper insight. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar8.03c Group definition: recall and use, show structure is/isn't a group8.03g Cyclic groups: meaning of the term |
8 Consider the set $S$ of all matrices of the form $\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)$, where $p$ is a non-zero rational number.\\
(i) Show that $S$, under the operation of matrix multiplication, forms a group, $G$. (You may assume that matrix multiplication is associative.)\\
(ii) Find a subgroup of $G$ of order 2 and show that $G$ contains no subgroups of order 3.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q8}}