| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Topic | Groups |
| Type | Matrix groups |
| Difficulty | Challenging +1.2 This is a structured group theory question with standard verification steps. Part (i) requires routine checking of group axioms (closure, identity, inverses) with straightforward matrix multiplication, recognizing commutativity, and identifying an obvious isomorphism to (ℤ,+). Part (ii) asks for geometric interpretation of a shear transformation. While group theory is Further Maths content, the question follows a predictable template with no novel insights required—harder than average due to abstract algebra content but well within standard Further Maths difficulty. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear8.03c Group definition: recall and use, show structure is/isn't a group8.03l Isomorphism: determine using informal methods |
**(i)(a)**
$\begin{pmatrix}1&a\\0&1\end{pmatrix}\begin{pmatrix}1&b\\0&1\end{pmatrix} = \begin{pmatrix}1&a+b\\0&1\end{pmatrix} \Rightarrow$ Closure **B1**
Associativity given
$\begin{pmatrix}1&0\\0&1\end{pmatrix}$ is the identity (in $S$) **B1**
$\begin{pmatrix}1&a\\0&1\end{pmatrix}^{-1} = \begin{pmatrix}1&-a\\0&1\end{pmatrix}$ is in $S$, since $-a$ is an integer **B1** [3]
**(b)** Yes, abelian, since $\begin{pmatrix}1&a\\0&1\end{pmatrix}\begin{pmatrix}1&b\\0&1\end{pmatrix} = \begin{pmatrix}1&b\\0&1\end{pmatrix}\begin{pmatrix}1&a\\0&1\end{pmatrix} = \begin{pmatrix}1&a+b\\0&1\end{pmatrix}$ **B1** [1]
**(c)** $f: n \to \begin{pmatrix}1&n\\0&1\end{pmatrix}$ (or v.v.) gives $G \cong H$ **B1** (allow just this... but should also mention that operation $(a)(b) \to a+b$ preserved) [1]
**(ii)** Shear **M1** parallel to the $x$-axis **A1**
mapping (e.g.) $\begin{pmatrix}0\\1\end{pmatrix}\to\begin{pmatrix}n\\1\end{pmatrix}$ **A1** (any point not on $x$-axis and its image) [3]6 (i) The set $S$ consists of all $2 \times 2$ matrices of the form $\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)$, where $n \in \mathbb { Z }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $S$, under the operation of matrix multiplication, forms a group $G$. [You may assume that matrix multiplication is associative.]
\item State, giving a reason, whether $G$ is abelian.
\item The group $H$ is the set $\mathbb { Z }$ together with the operation of addition. Explain why $G$ is isomorphic to $H$.\\
(ii) The plane transformation $T$ is given by the matrix $\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)$, where $n$ is a non-zero integer. Describe $T$ fully.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q6 [8]}}