| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Linear equations in z and z* |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question requiring basic manipulation of complex conjugates and solving simultaneous equations. Part (a) is routine algebraic expansion, and part (b) involves equating real and imaginary parts to solve a simple linear system. While it's Further Maths content, the techniques are mechanical with no conceptual challenges, making it slightly easier than an average A-level question overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Use of \(z^* = x - iy\) | M1 | |
| \(z - 3iz^* = x + iy - 3ix - 3y\) | ml | |
| \(R = x - 3y\), \(I = -3x + y\) | A1 | Condone sign error here; Condone inclusion of \(i\) in \(I\); Allow if correct in (b) |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(x - 3y = 16\), \(-3x + y = 0\) | M1 | |
| Elimination of \(x\) or \(y\) | ml | |
| \(z = -2 - 6i\) | A1F | Accept \(x = -2\), \(y = -6\); ft \(x + 3y\) for \(x - 3y\) |
**(a)** Use of $z^* = x - iy$ | M1 |
$z - 3iz^* = x + iy - 3ix - 3y$ | ml |
$R = x - 3y$, $I = -3x + y$ | A1 | Condone sign error here; Condone inclusion of $i$ in $I$; Allow if correct in (b)
**Total: 3 marks**
**(b)** $x - 3y = 16$, $-3x + y = 0$ | M1 |
Elimination of $x$ or $y$ | ml |
$z = -2 - 6i$ | A1F | Accept $x = -2$, $y = -6$; ft $x + 3y$ for $x - 3y$
**Total: 3 marks**
---3 It is given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real numbers.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $x$ and $y$, the real and imaginary parts of
$$z - 3 \mathbf { i } z ^ { * }$$
where $z ^ { * }$ is the complex conjugate of $z$.
\item Find the complex number $z$ such that
$$z - 3 \mathrm { i } z ^ { * } = 16$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q3 [6]}}