| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Moderate -0.8 This is a straightforward group theory question testing basic definitions (identity and inverse elements) with routine calculations. Part (a) requires finding the complex inverse using standard methods (multiply by conjugate), and part (b) is even simpler as it involves matrix addition. While FP3 content is more advanced, this particular question demands only recall of definitions and mechanical computation with no problem-solving or insight required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument8.03c Group definition: recall and use, show structure is/isn't a group |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Identity = \(1 + 0i\); Inverse = \(\frac{1}{1+2i}\) = \(\frac{1}{1+2i} \times \frac{1-2i}{1-2i}\) = \(\frac{1-2i}{5}\) = \(\frac{1}{5} - \frac{2}{5}i\) | B1; B1; B1 3 | For correct identity. Allow 1; For \(\frac{1}{1+2i}\) seen or implied; For correct inverse AEFcartesian |
| (b) Identity = \(\begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}\); Inverse = \(\begin{pmatrix}-3 & 0\\0 & 0\end{pmatrix}\) | B1; B1 2 [5] | For correct identity; For correct inverse |
**(a)** Identity = $1 + 0i$; Inverse = $\frac{1}{1+2i}$ = $\frac{1}{1+2i} \times \frac{1-2i}{1-2i}$ = $\frac{1-2i}{5}$ = $\frac{1}{5} - \frac{2}{5}i$ | B1; B1; B1 3 | For correct identity. Allow 1; For $\frac{1}{1+2i}$ seen or implied; For correct inverse **AEF**cartesian
**(b)** Identity = $\begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}$; Inverse = $\begin{pmatrix}-3 & 0\\0 & 0\end{pmatrix}$ | B1; B1 2 [5] | For correct identity; For correct inverse
---\begin{enumerate}[label=(\alph*)]
\item For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of $1 + 2\mathrm{i}$, giving your answers in the form $a + ib$. [3]
\item For the group of matrices of the form $\begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}$ under matrix addition, where $a \in \mathbb{R}$, state the identity element and the inverse of $\begin{pmatrix} 3 & 0 \\ 0 & 0 \end{pmatrix}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q1 [5]}}