| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 13 |
| Topic | Groups |
| Type | Matrix groups |
| Difficulty | Standard +0.3 This is a guided, multi-part question on matrix groups that systematically builds understanding through verification steps. Part (i) involves routine matrix multiplication and determinant calculation. Part (ii) requires proving group axioms but the hardest part (associativity) is given. Part (iii) involves computing powers of a specific matrix to find its order (which is 8), then identifying an isomorphic subgroup. While this covers Further Maths content (group theory), the question provides substantial scaffolding and requires mainly careful calculation rather than deep insight. It's slightly easier than average A-level difficulty overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03h Determinant 2x2: calculation8.03c Group definition: recall and use, show structure is/isn't a group8.03g Cyclic groups: meaning of the term |
**Question 11(i)(a)**
$\mathbf{AB} = \begin{pmatrix}ae+bg & af+bh \\ ce+dg & cf+dh\end{pmatrix}$ **B1**
$\det\mathbf{A} = ad-bc$ and $\det\mathbf{B} = eh-fg$ **B1**
**Question 11(i)(b)**
$\det(\mathbf{AB}) = (ae+bg)(cf+dh)-(af+bh)(ce+dg)$ and some attempt to multiply out **M1**
$= acef + adeh + bcfg + bdgh - acef - bceh - adfg - bdgh$
$= adeh - bceh - adfg + bcfg$
$= (ad-bc)(eh-fg)$ **A1** Legitimately shown
**Question 11(ii)**
*CLOSURE*: $\mathbf{A}, \mathbf{B} \in S \Rightarrow \det\mathbf{A} = \det\mathbf{B} = 1$ **M1** Attempted
and above result $\Rightarrow \det\mathbf{AB} = 1 \Rightarrow \mathbf{AB} \in S$ **A1** Convincing
(*ASSOCIATIVITY*: given)
*IDENTITY*: $\mathbf{I} = \begin{pmatrix}1&0\\0&1\end{pmatrix} \in S$ since $\det\mathbf{I} = 1.1 - 0.0 = 1$ **B1** Must show why $\mathbf{I}\in S$ and not just say that $\mathbf{I}$ is the identity
*INVERSES*: $\mathbf{A} = \begin{pmatrix}a&b\\c&d\end{pmatrix}\in S \Rightarrow \mathbf{A}^{-1} = \begin{pmatrix}d&-b\\-c&a\end{pmatrix}\in S$ **B1** for stating $\mathbf{A}^{-1}$ (or explaining that it exists)
Since $da-(-b)(-c) = ad-bc = 1$ Hence $(S,\times_\mathbf{M})$ is a group, $G$. **B1** for justifying its membership of $S$
**Question 11(iii)(a)**
$\det\mathbf{K} = 1.0 - \text{i.i} = -i^2 = 1$ (so $\mathbf{K}\in S$) **B1**
**Question 11(iii)(b)**
Attempt at powers of $\mathbf{K}$; $\mathbf{K}^2$ & $\mathbf{K}^3$ **M1**
$\mathbf{K}^2 = \begin{pmatrix}0&i\\i&-1\end{pmatrix}$ and $\mathbf{K}^3 = \begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ **A1**
NB $\mathbf{K}^4 = \begin{pmatrix}-1&-i\\-i&0\end{pmatrix}$ and $\mathbf{K}^5 = \begin{pmatrix}0&-i\\-i&1\end{pmatrix}$
$\Rightarrow \mathbf{K}^6 = \mathbf{I}$ and $H$ has order $n=6$ **A1**
**Question 11(iii)(c)**
e.g. The set of rotations about $O$ through multiples of $60°$ **B1** FT for any $n$
OR $(\mathbf{K}^*)=$ group generated by $\begin{pmatrix}1&-i\\-i&0\end{pmatrix}$
Justifying the two are isomorphic **B1** e.g. stating both are cyclic, etc.
**Total: 13 marks**11
\begin{enumerate}[label=(\roman*)]
\item (a) Given $\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)$, work out the matrix $\mathbf { A B }$ and write down expressions for $\operatorname { det } \mathbf { A }$ and $\operatorname { det } \mathbf { B }$.\\
(b) Verify, by direct calculation, that $\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }$.
Let $S$ be the set of all $2 \times 2$ matrices with determinant equal to 1 .
\item Show that $\left( S , \times _ { \mathrm { M } } \right)$ forms a group, $G$, where $\times _ { \mathrm { M } }$ is the operation of matrix multiplication. [You may assume that $\mathrm { X } _ { \mathrm { M } }$ is associative.]
\item (a) Show that $\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)$ is an element of $G$.
Let $H$ be the smallest subgroup of $G$ that contains $\mathbf { K }$ and let $n$ be the order of $H$.\\
(b) Determine the value of $n$.\\
(c) Give a second subgroup of $G$, also of order $n$, which is isomorphic to $H$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2017 Q11 [13]}}