| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This FP2 question requires multiple sophisticated techniques: Vieta's formulas, complex conjugate root theorem, and algebraic manipulation of symmetric functions. Part (b)(i) requires conceptual understanding of why α²+β²+γ²<0 implies complex roots, while parts (b)(ii) and (c) involve non-trivial algebraic steps connecting the given constraint to finding p and q. The multi-step reasoning and combination of complex number theory with polynomial relationships places this above average difficulty for Further Maths. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\sum \alpha\beta = 6\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Sum of squares \(< 0\) \(\therefore\) not all real | E1 | |
| Coefficients real \(\therefore\) conjugate pair | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((\sum\alpha)^2 = \sum\alpha^2 + 2\sum\alpha\beta\) | M1A1 | A1 for numerical values inserted |
| \((\sum\alpha)^2 = 0\) | A1F | |
| \(p = 0\) | A1F | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(-1-3i\) is a root | B1 | |
| Use of appropriate relationship e.g. \(\sum\alpha = 0\) | M1 | M0 if \(\sum\alpha^2\) used unless root 2 is checked |
| Third root \(= 2\) | A1F | incorrect \(p\checkmark\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(q = -(-1-3i)(-1+3i)(2)\) | M1 | allow even if sign error |
| \(= -20\) | A1F | ft incorrect 3rd root |
## Question 2:
### Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\sum \alpha\beta = 6$ | B1 | |
### Part (b)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| Sum of squares $< 0$ $\therefore$ not all real | E1 | |
| Coefficients real $\therefore$ conjugate pair | E1 | |
### Part (b)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $(\sum\alpha)^2 = \sum\alpha^2 + 2\sum\alpha\beta$ | M1A1 | A1 for numerical values inserted |
| $(\sum\alpha)^2 = 0$ | A1F | |
| $p = 0$ | A1F | cao |
### Part (c)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $-1-3i$ is a root | B1 | |
| Use of appropriate relationship e.g. $\sum\alpha = 0$ | M1 | M0 if $\sum\alpha^2$ used unless root 2 is checked |
| Third root $= 2$ | A1F | incorrect $p\checkmark$ |
### Part (c)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $q = -(-1-3i)(-1+3i)(2)$ | M1 | allow even if sign error |
| $= -20$ | A1F | ft incorrect 3rd root |
---2 The cubic equation
$$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$
has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\alpha \beta + \beta \gamma + \gamma \alpha$.
\item Given that $p$ and $q$ are real and that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12$ :
\begin{enumerate}[label=(\roman*)]
\item explain why the cubic equation has two non-real roots and one real root;
\item find the value of $p$.
\end{enumerate}\item One root of the cubic equation is $- 1 + 3 \mathrm { i }$.
Find:
\begin{enumerate}[label=(\roman*)]
\item the other two roots;
\item the value of $q$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2007 Q2 [12]}}