| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Challenging +1.2 This is a Further Maths FP3 group theory question requiring proof of group axioms and analysis of a finite group. Part (a) is a standard group axioms verification (closure, identity, inverses given associativity). Part (b) involves modular arithmetic with polynomials but asks straightforward questions: stating order (counting elements), finding an inverse (routine calculation), and determining if cyclic (checking element orders). While this requires understanding of abstract algebra concepts beyond standard A-level, the actual execution is mechanical and follows standard patterns taught in FP3. The modulo 3 arithmetic keeps calculations simple. This is moderately above average difficulty due to the abstract nature and Further Maths content, but not exceptionally challenging within FP3. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03g Cyclic groups: meaning of the term |
| Answer | Marks | Guidance |
|---|---|---|
| Closure \((ax+b)+(cx+d) = (a+c)x+(b+d)\) | B1 | For obtaining correct sum from 2 distinct elements |
| \(\in P\) | B1 | For stating result is in \(P\) OR is of the correct form |
| Answer | Marks | Guidance |
|---|---|---|
| Identity \(0x+0\) | B1 | For stating identity (allow 0) |
| Inverse \(-ax-b\) | B1 4 | For stating inverse |
| Answer | Marks | Guidance |
|---|---|---|
| Order 9 | B1* 1 | For correct order |
| Answer | Marks | Guidance |
|---|---|---|
| \(a^2+2^2 = (a+a)+(b+b) + t(a+b)+(b+b) = 3ax+3b\) | B1 1 | For correct inverse element |
| Answer | Marks | Guidance |
|---|---|---|
| \((ax+b)+(ax+b) + t(ax+b) + (a+b) = 3ax+3b\) | M1 | For considering sums of \(ax+p\) and obtaining \(3ax+3b\) and equating to \(0x+0\) OR 0 |
| \(= 0x+0 \Rightarrow ax+b\) has order 3 \(\forall a, b\) (except \(a=b=0\)) | A1 | For equating to \(0x+0\) OR 0 and obtaining order 3; SR For order 3 stated only OR found from incomplete consideration of numerical cases award B1 |
| Cyclic group of order 9 has element(s) of order 9 | M1 | For reference to element(s) of order 9 |
| \(\Rightarrow (\mathbb{Q}, +(mod 3))\) is not cyclic | A1 4 | For correct conclusion |
## (a)
Closure $(ax+b)+(cx+d) = (a+c)x+(b+d)$ | B1 | For obtaining correct sum from 2 distinct elements
$\in P$ | B1 | For stating result is in $P$ OR is of the correct form
SR award this mark if any of the closure result, the identity or the inverse element is stated to be in $P$ OR of the correct form
Identity $0x+0$ | B1 | For stating identity (allow 0)
Inverse $-ax-b$ | B1 4 | For stating inverse
## (b(i))
Order 9 | B1* 1 | For correct order
## (b(ii))
$a^2+2^2 = (a+a)+(b+b) + t(a+b)+(b+b) = 3ax+3b$ | B1 1 | For correct inverse element
## (b(iii))
$(ax+b)+(ax+b) + t(ax+b) + (a+b) = 3ax+3b$ | M1 | For considering sums of $ax+p$ and obtaining $3ax+3b$ and equating to $0x+0$ OR 0
$= 0x+0 \Rightarrow ax+b$ has order 3 $\forall a, b$ (except $a=b=0$) | A1 | For equating to $0x+0$ OR 0 and obtaining order 3; SR For order 3 stated only OR found from incomplete consideration of numerical cases award B1
Cyclic group of order 9 has element(s) of order 9 | M1 | For reference to element(s) of order 9
$\Rightarrow (\mathbb{Q}, +(mod 3))$ is not cyclic | A1 4 | For correct conclusion
---\begin{enumerate}[label=(\alph*)]
\item The set of polynomials $\{ax + b\}$, where $a, b \in \mathbb{R}$, is denoted by $P$. Assuming that the associativity property holds, prove that $P$, under addition, is a group. [4]
\item The set of polynomials $\{ax + b\}$, where $a, b \in \{0, 1, 2\}$, is denoted by $Q$. It is given that $Q$, under addition modulo 3, is a group, denoted by $(Q, +(\text{mod}3))$.
\begin{enumerate}[label=(\roman*)]
\item State the order of the group. [1]
\item Write down the inverse of the element $2x + 1$. [1]
\item $q(x) = ax + b$ is any element of $Q$ other than the identity. Find the order of $q(x)$ and hence determine whether $(Q, +(\text{mod}3))$ is a cyclic group. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2011 Q6 [10]}}