| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a straightforward application of standard Further Maths techniques: using Vieta's formulas and Newton's identities to find coefficients from sum of roots and sum of squares, then using the complex conjugate root theorem. All steps are routine for FP2 students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(p = -4\) | B1 | |
| \((\alpha + \beta + \gamma)^2 = \sum\alpha^2 + 2\sum\alpha\beta\) | M1 | |
| \(16 = 20 + 2\sum\alpha\beta\) | A1 | |
| \(\sum\alpha\beta = -2\) | A1F | |
| \(q = -2\) | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3 - i\) is a root | B1 | |
| Third root is \(-2\) | B1F | |
| \(\alpha\beta\gamma = (3+i)(3-i)(-2)\) | M1 | |
| \(= -20\) | A1F | Real \(\alpha\beta\gamma\) |
| \(r = +20\) | A1F | Real \(r\) |
| Alternative: Substitute \(3+i\): \((3+i)^2 = 8+6i\) | M1, B1 | |
| \((3+i)^3 = 18 + 26i\) | B1 | |
| \(r = 20\) | A2,1,0 | Provided \(r\) is real |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p = -4$ | B1 | |
| $(\alpha + \beta + \gamma)^2 = \sum\alpha^2 + 2\sum\alpha\beta$ | M1 | |
| $16 = 20 + 2\sum\alpha\beta$ | A1 | |
| $\sum\alpha\beta = -2$ | A1F | |
| $q = -2$ | A1F | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3 - i$ is a root | B1 | |
| Third root is $-2$ | B1F | |
| $\alpha\beta\gamma = (3+i)(3-i)(-2)$ | M1 | |
| $= -20$ | A1F | Real $\alpha\beta\gamma$ |
| $r = +20$ | A1F | Real $r$ |
| **Alternative:** Substitute $3+i$: $(3+i)^2 = 8+6i$ | M1, B1 | |
| $(3+i)^3 = 18 + 26i$ | B1 | |
| $r = 20$ | A2,1,0 | Provided $r$ is real |
---2 The cubic equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
where $p , q$ and $r$ are real, has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Given that
$$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$
find the values of $p$ and $q$.
\item Given further that one root is $3 + \mathrm { i }$, find the value of $r$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q2 [10]}}