| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Real and imaginary part expressions |
| Difficulty | Easy -1.2 This is a straightforward Further Pure 1 question testing basic complex number manipulation: writing the conjugate, expanding expressions, and solving a simple equation. All steps are routine algebraic manipulation with no problem-solving insight required, making it easier than average despite being from Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks |
|---|---|
| \(z^* = x - iy\) | B1 (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = 2x - y\) | B1 | \(i^2 = -1\) must be used |
| \(I = -x + 2y\) | B1 (2 marks) | Condone \(I = i(x + 2y)\); Answers may appear in (c) |
| Answer | Marks | Guidance |
|---|---|---|
| Equating R and/or I parts | M1 | |
| Attempt to solve simultaneous equations | m1 | |
| \(z = 1 + 2i\) | A1 (3 marks) | Allow \(x = 1\), \(y = 2\) |
## Question 3:
**Part (a)**
$z^* = x - iy$ | B1 (1 mark) |
**Part (b)**
$R = 2x - y$ | B1 | $i^2 = -1$ must be used
$I = -x + 2y$ | B1 (2 marks) | Condone $I = i(x + 2y)$; Answers may appear in (c)
**Part (c)**
Equating R and/or I parts | M1 |
Attempt to solve simultaneous equations | m1 |
$z = 1 + 2i$ | A1 (3 marks) | Allow $x = 1$, $y = 2$
---3 It is given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real numbers.
\begin{enumerate}[label=(\alph*)]
\item Write down, in terms of $x$ and $y$, an expression for $z ^ { * }$, the complex conjugate of $z$.
\item Find, in terms of $x$ and $y$, the real and imaginary parts of
$$2 z - \mathrm { i } z ^ { * }$$
\item Find the complex number $z$ such that
$$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q3 [6]}}