| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Real and imaginary part expressions |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question requiring basic complex number manipulation: expanding z², applying complex conjugate, and solving a simple equation. While it's Further Maths content, the algebraic steps are routine with no conceptual challenges—easier than average A-level questions overall. |
| 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 |
|---|---|
| \(z^2 = (x+2i)^2 = x^2 + 4xi + 4i^2 = x^2 - 4 + 4xi\) | M1 A1 |
| Real part: \(x^2 - 4\), Imaginary part: \(4x\) | A1 |
| Answer | Marks |
|---|---|
| \(z^* = x - 2i\) | B1 |
| \(z^2 + 2z^* = (x^2 - 4 + 4xi) + 2(x-2i) = (x^2 + 2x - 4) + (4x-4)i\) | M1 A1 |
| Real part: \(x^2 + 2x - 4\), Imaginary part: \(4x - 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| For \(z^2 + 2z^*\) to be real, imaginary part \(= 0\) | M1 | |
| \(4x - 4 = 0 \Rightarrow x = 1\) | A1 | Exactly one solution shown |
# Question 3:
## Part (a)(i)
| $z^2 = (x+2i)^2 = x^2 + 4xi + 4i^2 = x^2 - 4 + 4xi$ | M1 A1 | |
| Real part: $x^2 - 4$, Imaginary part: $4x$ | A1 | |
## Part (a)(ii)
| $z^* = x - 2i$ | B1 | |
| $z^2 + 2z^* = (x^2 - 4 + 4xi) + 2(x-2i) = (x^2 + 2x - 4) + (4x-4)i$ | M1 A1 | |
| Real part: $x^2 + 2x - 4$, Imaginary part: $4x - 4$ | | |
## Part (b)
| For $z^2 + 2z^*$ to be real, imaginary part $= 0$ | M1 | |
| $4x - 4 = 0 \Rightarrow x = 1$ | A1 | Exactly one solution shown |
---3 The complex number $z$ is defined by
$$z = x + 2 \mathrm { i }$$
where $x$ is real.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $x$, the real and imaginary parts of:
\begin{enumerate}[label=(\roman*)]
\item $z ^ { 2 }$;
\item $z ^ { 2 } + 2 z ^ { * }$.
\end{enumerate}\item Show that there is exactly one value of $x$ for which $z ^ { 2 } + 2 z ^ { * }$ is real.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q3 [7]}}