| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with linearly transformed roots |
| Difficulty | Standard +0.8 This is a substantial Further Maths question requiring multiple techniques: Vieta's formulas, complex number arithmetic, repeated roots analysis, and transformation of roots to form a new equation. Part (d) especially requires careful algebraic manipulation to find the equation with transformed roots. While systematic, it demands fluency across several FP2 topics and extended multi-step reasoning beyond standard exercises. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.05a Roots and coefficients: symmetric functions |
The cubic equation $27z^3 + kz^2 + 4 = 0$ has roots $\alpha$, $\beta$ and $\gamma$.
**(a) [2 marks]**
Write down the values of $\alpha\beta + \beta\gamma + \gamma\alpha$ and $\alpha\beta\gamma$.
**(b) [6 marks total]**
**(i) [5 marks]**
In the case where $\beta = \gamma$, find the roots of the equation.
**(ii) [1 mark]**
Find the value of $k$ in this case.
**(c) [4 marks total]**
**(i) [2 marks]**
In the case where $\alpha = 1 - i$, find $\alpha^2$ and $\alpha^3$.
**(ii) [2 marks]**
Hence find the value of $k$ in this case.
**(d) [5 marks]**
In the case where $k = -12$, find a cubic equation with integer coefficients which has roots $\frac{1}{\alpha} + 1$, $\frac{1}{\beta} + 1$ and $\frac{1}{\gamma} + 1$.7 The cubic equation $27 z ^ { 3 } + k z ^ { 2 } + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.
\item \begin{enumerate}[label=(\roman*)]
\item In the case where $\beta = \gamma$, find the roots of the equation.
\item Find the value of $k$ in this case.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item In the case where $\alpha = 1 - \mathrm { i }$, find $\alpha ^ { 2 }$ and $\alpha ^ { 3 }$.
\item Hence find the value of $k$ in this case.
\end{enumerate}\item In the case where $k = - 12$, find a cubic equation with integer coefficients which has roots $\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1$ and $\frac { 1 } { \gamma } + 1$.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q7 [17]}}