| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Finding specific root values |
| Difficulty | Standard +0.3 This is a straightforward application of Vieta's formulas for polynomials with complex coefficients. Part (a) uses the product of roots formula directly, then requires basic complex number division. Parts (b) and (c) apply sum and sum-of-products formulas. All steps are routine for FP2 students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.05a Roots and coefficients: symmetric functions |
5 The cubic equation
$$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$
where $p$ and $q$ are constants, has three complex roots, $\alpha , \beta$ and $\gamma$.
It is given that $\beta = - 2 + 3 \mathrm { i }$ and $\gamma = 1 + 2 \mathrm { i }$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the value of $\alpha \beta \gamma$.
\item Hence show that $( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }$.
\item Hence find $\alpha$, giving your answer in the form $m + n \mathrm { i }$, where $m$ and $n$ are integers.
\end{enumerate}\item Find the value of $p$.
\item Find the value of the complex number $q$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q5 [9]}}