| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Equations with z/z* or zz* terms |
| Difficulty | Standard +0.3 This is a straightforward Further Maths FP1 question requiring expansion of complex expressions and solving simultaneous equations. Part (a) is routine algebraic manipulation with z* (conjugate), while part (b) involves equating real and imaginary parts to solve a system—standard techniques with no novel insight required. Slightly above average difficulty due to being Further Maths content, but still a textbook exercise. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use of \(z^* = x - iy\) | M1 | |
| \((z - i)(z^* - i) = (x^2 + y^2 - 1) - 2ix\) | m1A1 | A1 may be earned in (b) |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Equating R and I parts | M1 | |
| \(-2x = -8\) so \(x = 4\) | A1 | |
| \(16 + y^2 - 1 = 24\) so \(y = \pm 3\) \((z = 4 \pm 3i)\) | m1A1 | A0 if \(x = -4\) used |
| Total | 4 |
## Question 3:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use of $z^* = x - iy$ | M1 | |
| $(z - i)(z^* - i) = (x^2 + y^2 - 1) - 2ix$ | m1A1 | A1 may be earned in (b) |
| **Total** | **3** | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Equating R and I parts | M1 | |
| $-2x = -8$ so $x = 4$ | A1 | |
| $16 + y^2 - 1 = 24$ so $y = \pm 3$ $(z = 4 \pm 3i)$ | m1A1 | A0 if $x = -4$ used |
| **Total** | **4** | |
---3 It is given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $x$ and $y$, the real and imaginary parts of
$$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right)$$
\item Given that
$$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right) = 24 - 8 \mathrm { i }$$
find the two possible values of $z$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q3 [7]}}