| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| 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 Pure 1 question requiring basic manipulation of complex numbers and their conjugates. Part (a) is routine substitution and simplification, while part (b) involves solving simultaneous equations by equating real and imaginary parts—a standard technique with no conceptual challenges beyond the mechanics. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Use of \(z^* = x - iy\); Use of \(i^2 = -1\); \(3iz + 2z^* = (2x - 3y) + i(3x - 2y)\) | M1, M1, A1 | 3 marks |
| (b) Equating R and I parts; \(2x - 3y = 7, 3x - 2y = 8\); \(z = 2 - i\) | M1, m1, A1 | 3 marks |
**(a)** Use of $z^* = x - iy$; Use of $i^2 = -1$; $3iz + 2z^* = (2x - 3y) + i(3x - 2y)$ | M1, M1, A1 | 3 marks | Condone inclusion of i in 1 part
**(b)** Equating R and I parts; $2x - 3y = 7, 3x - 2y = 8$; $z = 2 - i$ | M1, m1, A1 | 3 marks | with attempt to solve; Allow $x = 2, y = -1$
**Total: 6 marks**2 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
$$3 \mathrm { i } z + 2 z ^ { * }$$
where $z ^ { * }$ is the complex conjugate of $z$.
\item Find the complex number $z$ such that
$$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q2 [6]}}