| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on Vieta's formulas and transformed roots. Part (a) is direct recall, part (b) uses standard algebraic identities (e.g., (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα)), and part (c) applies the standard technique of forming an equation from transformed roots. The complex coefficients add minimal difficulty since the algebraic manipulations are identical. This is slightly easier than average as it's a textbook exercise with well-rehearsed methods. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\sum \alpha = -i\) | B1 | 1 mark |
| (ii) \(\sum \alpha\beta = 3\) | B1 | 1 mark |
| (iii) \(\alpha\beta\gamma = 1 + i\) | B1 | 1 mark |
| (b)(i) \(\sum \alpha^2 = (\sum \alpha)^2 - 2\sum \alpha\beta\) used | M1 | Allow if sign error or 2 missing |
| \(= (-i)^2 - 2 \times 3\) | A1F | |
| \(= -7\) | A1F | 3 marks; ft errors in (a) |
| (ii) \(\sum \alpha^2\beta^2 = (\sum \alpha\beta)^2 - 2\alpha\beta\gamma\sum \alpha\) | M1 | Allow if sign error in 2 missing |
| \(= (\sum \alpha\beta)^2 - 2\alpha\beta\gamma\sum \alpha\) | A1 | |
| \(= 9 - 2(1+i)(-i)\) | A1F | ft errors in (a) |
| \(= 7 + 2i\) | A1F | 4 marks; ft errors in (a) |
| (iii) \(\alpha^2\beta^2\gamma^2 = (1+i)^2 = 2i\) | M1, A1F | 2 marks; ft sign error in \(\alpha\beta\gamma\) |
| (c) \(z^3 + 7z^2 + (7 + 2i)z - 2i = 0\) | B1F, B1F | Correct numbers in correct places; Correct signs |
| 2 marks |
**(a)(i)** $\sum \alpha = -i$ | B1 | 1 mark
**(ii)** $\sum \alpha\beta = 3$ | B1 | 1 mark
**(iii)** $\alpha\beta\gamma = 1 + i$ | B1 | 1 mark
**(b)(i)** $\sum \alpha^2 = (\sum \alpha)^2 - 2\sum \alpha\beta$ used | M1 | Allow if sign error or 2 missing
$= (-i)^2 - 2 \times 3$ | A1F |
$= -7$ | A1F | 3 marks; ft errors in (a)
**(ii)** $\sum \alpha^2\beta^2 = (\sum \alpha\beta)^2 - 2\alpha\beta\gamma\sum \alpha$ | M1 | Allow if sign error in 2 missing
$= (\sum \alpha\beta)^2 - 2\alpha\beta\gamma\sum \alpha$ | A1 |
$= 9 - 2(1+i)(-i)$ | A1F | ft errors in (a)
$= 7 + 2i$ | A1F | 4 marks; ft errors in (a)
**(iii)** $\alpha^2\beta^2\gamma^2 = (1+i)^2 = 2i$ | M1, A1F | 2 marks; ft sign error in $\alpha\beta\gamma$
**(c)** $z^3 + 7z^2 + (7 + 2i)z - 2i = 0$ | B1F, B1F | Correct numbers in correct places; Correct signs
| | 2 marks
**Question 4 Total: 14 marks**
---4 The cubic equation
$$z ^ { 3 } + \mathrm { i } z ^ { 2 } + 3 z - ( 1 + \mathrm { i } ) = 0$$
has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of:
\begin{enumerate}[label=(\roman*)]
\item $\alpha + \beta + \gamma$;
\item $\alpha \beta + \beta \gamma + \gamma \alpha$;
\item $\alpha \beta \gamma$.
\end{enumerate}\item Find the value of:
\begin{enumerate}[label=(\roman*)]
\item $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$;
\item $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$;
\item $\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }$.
\end{enumerate}\item Hence write down a cubic equation whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q4 [14]}}