| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.8 This is a Further Pure Mathematics 3 group theory question requiring formal proofs of group axioms, understanding of group properties (commutativity, element order), and counterexamples. While the algebraic manipulation is straightforward, proving closure, associativity, identity, and inverses systematically, then proving no elements of order 2, requires mathematical maturity beyond standard A-level. The counterexample part is more routine. Overall, moderately above average difficulty for requiring formal abstract algebra reasoning. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups |
\begin{enumerate}[label=(\roman*)]
\item The operation $*$ is defined by $x * y = x + y - a$, where $x$ and $y$ are real numbers and $a$ is a real constant.
\begin{enumerate}[label=(\alph*)]
\item Prove that the set of real numbers, together with the operation $*$, forms a group. [6]
\item State, with a reason, whether the group is commutative. [1]
\item Prove that there are no elements of order 2. [2]
\end{enumerate}
\item The operation $\circ$ is defined by $x \circ y = x + y - 5$, where $x$ and $y$ are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q7 [13]}}