| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Linear equations in z and z* |
| Difficulty | Moderate -0.3 This is a straightforward FP1 question requiring basic manipulation of complex numbers and their conjugates. Part (a) involves routine algebraic expansion and separation into real/imaginary parts, while part (b) requires solving two simultaneous linear equations. The techniques are standard and the question is well-scaffolded, making it slightly easier than average for A-level. |
| 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 |
| PART | REFERENCE |
| \(\_\_\_\_\) | \(\_\_\_\_\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1-2i)(x+iy) - (x-iy)\) | M1 | Substituting \(z = x+iy\), \(z^* = x-iy\) |
| \(= x + iy - 2ix - 2i^2y - x + iy\) | M1 | Expanding correctly |
| Real part: \(2y\) | A1 | Correct real part |
| Imaginary part: \(2y - 2x\) (i.e. \((2y-2x)i\)) | A1 | Correct imaginary part |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2y = 20\) and \(2y - 2x = 1\) | M1 | Equating real and imaginary parts |
| \(y = 10\), \(x = \frac{19}{2}\) so \(z = \frac{19}{2} + 10i\) | A1 | Correct answer |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1-2i)(x+iy) - (x-iy)$ | M1 | Substituting $z = x+iy$, $z^* = x-iy$ |
| $= x + iy - 2ix - 2i^2y - x + iy$ | M1 | Expanding correctly |
| Real part: $2y$ | A1 | Correct real part |
| Imaginary part: $2y - 2x$ (i.e. $(2y-2x)i$) | A1 | Correct imaginary part |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2y = 20$ and $2y - 2x = 1$ | M1 | Equating real and imaginary parts |
| $y = 10$, $x = \frac{19}{2}$ so $z = \frac{19}{2} + 10i$ | A1 | Correct answer |
---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
$$( 1 - 2 i ) z - z ^ { * }$$
\item Hence find the complex number $z$ such that
$$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$
\begin{center}
\begin{tabular}{|l|l|}
\hline
PART & REFERENCE \\
\hline
& \\
\hline
$\_\_\_\_$ & $\_\_\_\_$ \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q2 [6]}}