| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Equations with conjugate of expressions |
| Difficulty | Moderate -0.3 This is a straightforward Further Pure 1 question testing basic conjugate manipulation and solving linear equations in complex form. Part (a) is routine substitution, and part (b) requires equating real and imaginary parts to solve a simple system—standard textbook material with no conceptual challenges beyond the definitions. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((z+i)^* = x - iy - i\) | B2 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\ldots = 2ix - 2y + 1\) | M1 | \(i^2 = -1\) used at some stage |
| Equating R and I parts | M1 | involving at least 5 terms in all |
| \(x = -2y+1,\ -y-1=2x\) | A1\(\checkmark\) | ft one sign error in (a) |
| \(z = -1 + i\) | m1A1\(\checkmark\) (5) | ditto; allow \(x=-1,\ y=1\) |
## Question 6:
### Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $(z+i)^* = x - iy - i$ | B2 (2) | |
### Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\ldots = 2ix - 2y + 1$ | M1 | $i^2 = -1$ used at some stage |
| Equating R and I parts | M1 | involving at least 5 terms in all |
| $x = -2y+1,\ -y-1=2x$ | A1$\checkmark$ | ft one sign error in (a) |
| $z = -1 + i$ | m1A1$\checkmark$ (5) | ditto; allow $x=-1,\ y=1$ |
---6 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
$$( z + \mathrm { i } ) ^ { * }$$
where $( z + \mathrm { i } ) ^ { * }$ denotes the complex conjugate of $( z + \mathrm { i } )$.
\item Solve the equation
$$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$
giving your answer in the form $a + b \mathrm { i }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q6 [7]}}