| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic complex number arithmetic. Part (a) requires simple multiplication of (1+i)², part (b) uses repeated squaring to reach z^8=16, and part (c) verifies a conjugate property. All parts are routine calculations with no problem-solving insight needed, making it easier than average despite being Further Maths content. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| 2(a) | \(z^2 = 1 + 2i + i^2 = 2i\) | M1A1 |
| 2 marks | ||
| 2(b) | \(z^8 = (2i)^4\) | M1 |
| \(... = 16i^4 = 16\) | A1 | convincingly shown (AG) |
| 2 marks | ||
| 2(c) | \((z^*)^2 = (1 - i)^2\) | M1 |
| \(... = -2i = -z^2\) | A1 | convincingly shown (AG) |
| 2 marks | ||
| Total for Q2 | 6 marks |
2(a) | $z^2 = 1 + 2i + i^2 = 2i$ | M1A1 | M1 for use of $i^2 = -1$ |
| | | 2 marks |
2(b) | $z^8 = (2i)^4$ | M1 | or equivalent complete method |
| $... = 16i^4 = 16$ | A1 | convincingly shown (AG) |
| | | 2 marks |
2(c) | $(z^*)^2 = (1 - i)^2$ | M1 | for use of $z^* = 1 - i$ |
| $... = -2i = -z^2$ | A1 | convincingly shown (AG) |
| | | 2 marks |
| **Total for Q2** | **6 marks** |
---2 The complex number $z$ is defined by
$$z = 1 + \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $z ^ { 2 }$, giving your answer in its simplest form.
\item Hence show that $z ^ { 8 } = 16$.
\item Show that $\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q2 [6]}}