| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Equations with conjugate of expressions |
| Difficulty | Standard +0.3 This is a straightforward FP1 question testing basic conjugate manipulation and properties. Part (a) requires routine algebraic manipulation with conjugates (equating real and imaginary parts), while part (b) tests understanding that complex conjugate roots only occur for equations with real coefficients. All steps are standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials |
**(a) It is given that $z = x + yi$, where $x$ and $y$ are real numbers.**
(i) Write down, in terms of $x$ and $y$, an expression for $(z - 2i)^*$. **(1 mark)**
(ii) Solve the equation
$$(z - 2i)^* = 4iz + 3$$
giving your answer in the form $a + bi$. **(5 marks)**
**(b) It is given that $p + qi$, where $p$ and $q$ are real numbers, is a root of the equation**
$$z^2 + 10iz - 29 = 0$$
**Without finding the values of $p$ and $q$, state why $p - qi$ is not a root of the equation $z^2 + 10iz - 29 = 0$. (1 mark)**4
\begin{enumerate}[label=(\alph*)]
\item It is given that $z = x + y \mathrm { i }$, where $x$ and $y$ are real numbers.
\begin{enumerate}[label=(\roman*)]
\item Write down, in terms of $x$ and $y$, an expression for $( z - 2 \mathrm { i } ) ^ { * }$.
\item Solve the equation
$$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$
giving your answer in the form $a + b \mathrm { i }$.
\end{enumerate}\item It is given that $p + q \mathrm { i }$, where $p$ and $q$ are real numbers, is a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$.
Without finding the values of $p$ and $q$, state why $p - q$ i is not a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q4 [7]}}