| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Function composition groups |
| Difficulty | Challenging +1.2 This is a Further Maths group theory question requiring function composition and construction of a Cayley table. While the algebraic manipulation is straightforward (composing rational functions, finding inverses), it requires understanding of abstract algebra concepts like group elements, orders, and identity. The composition calculations are routine for FP3 students, and the operation table construction is mechanical once the elements are known. Moderately above average due to the abstract algebra context, but the actual computations are not demanding. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence8.03c Group definition: recall and use, show structure is/isn't a group8.03d Latin square property: for group tables8.03e Order of elements: and order of groups8.03g Cyclic groups: meaning of the term |
The function f is defined by $\text{f} : x \mapsto \frac{1}{2-2x}$ for $x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1$. The function g is defined by $\text{g}(x) = \text{ff}(x)$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\text{g}(x) = \frac{1-x}{1-2x}$ and that $\text{gg}(x) = x$. [4]
\end{enumerate}
It is given that f and g are elements of a group $K$ under the operation of composition of functions. The element e is the identity, where $\text{e} : x \mapsto x$ for $x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item State the orders of the elements f and g. [2]
\item The inverse of the element f is denoted by h. Find $\text{h}(x)$. [2]
\item Construct the operation table for the elements e, f, g, h of the group $K$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2010 Q8 [12]}}