| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a substantial FP2 question requiring complex conjugate roots, Vieta's formulas, conversion to exponential form, De Moivre's theorem, and algebraic manipulation of powers. While the techniques are standard for Further Maths, the multi-part structure with the final algebraic proof involving three roots and simplification to the given form requires careful execution across multiple steps, placing it moderately above average difficulty. |
| Spec | 4.01a Mathematical induction: construct proofs4.02d Exponential form: re^(i*theta)4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(\beta = 2 - 2\sqrt{3}i\) | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha\beta = -8\) | M1 | Allow for +8 but not ±16 |
| \(\alpha\beta = 16\) | B1 | |
| \(\gamma = -\frac{1}{2}\) | A1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Either \(-\frac{p}{2} = \alpha + \beta + \gamma\) or \(\frac{q}{2} = \alpha\beta + \beta\gamma + \gamma\alpha\) | M1 | SC if failure to divide by 2 throughout, allow M1A1 for either p or q correct ft |
| \(p = -7, q = 28\) | A1F, A1F | 3 marks |
| Answer | Marks |
|---|---|
| \((z^2 - 4z + 16)(az + b)\) | M1 |
| \(\alpha\beta = 16\) | B1 |
| \(a = 2, b = +1, \gamma = -\frac{1}{2}\) | A1 |
| Equating coefficients | M1 |
| \(p = -7\) | A1F |
| \(q = 28\) | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 4, \theta = \frac{\pi}{3}\) | B1, B1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((2 + 2\sqrt{3}i) = \left(4e^{i\frac{\pi}{3}}\right)\) | M1 | |
| \(= 4^n\left(\cos\frac{n\pi}{3} + i\sin\frac{n\pi}{3}\right)\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((2 - 2\sqrt{3}i)^n = 4^n\left(\cos\frac{n\pi}{3} - i\sin\frac{n\pi}{3}\right)\) | B1 | |
| \(\alpha^n + \beta^n + \gamma^n = 4^n\left(\cos\frac{n\pi}{3} + i\sin\frac{n\pi}{3}\right) + 4^n\left(\cos\frac{n\pi}{3} - i\sin\frac{n\pi}{3}\right) + \left(-\frac{1}{2}\right)^n\) | M1 | |
| \(= 2^{2n+1}\cos\frac{n\pi}{3} + \left(-\frac{1}{2}\right)^n\) | A1 | 3 marks |
### Part (a)(i)
$\beta = 2 - 2\sqrt{3}i$ | B1 | 1 mark
### Part (a)(ii)
$\alpha\beta = -8$ | M1 | Allow for +8 but not ±16
$\alpha\beta = 16$ | B1 |
$\gamma = -\frac{1}{2}$ | A1 | 3 marks total
### Part (a)(iii)
Either $-\frac{p}{2} = \alpha + \beta + \gamma$ or $\frac{q}{2} = \alpha\beta + \beta\gamma + \gamma\alpha$ | M1 | SC if failure to divide by 2 throughout, allow M1A1 for either p or q correct ft
$p = -7, q = 28$ | A1F, A1F | 3 marks | ft incorrect $\gamma$
**Alternative to (a)(i) and (a)(iii):**
$(z^2 - 4z + 16)(az + b)$ | M1 |
$\alpha\beta = 16$ | B1 |
$a = 2, b = +1, \gamma = -\frac{1}{2}$ | A1 |
Equating coefficients | M1 |
$p = -7$ | A1F |
$q = 28$ | A1F |
### Part (b)(i)
$r = 4, \theta = \frac{\pi}{3}$ | B1, B1 | 2 marks
### Part (b)(ii)
$(2 + 2\sqrt{3}i) = \left(4e^{i\frac{\pi}{3}}\right)$ | M1 |
$= 4^n\left(\cos\frac{n\pi}{3} + i\sin\frac{n\pi}{3}\right)$ | A1 | 2 marks | AG
### Part (b)(iii)
$(2 - 2\sqrt{3}i)^n = 4^n\left(\cos\frac{n\pi}{3} - i\sin\frac{n\pi}{3}\right)$ | B1 |
$\alpha^n + \beta^n + \gamma^n = 4^n\left(\cos\frac{n\pi}{3} + i\sin\frac{n\pi}{3}\right) + 4^n\left(\cos\frac{n\pi}{3} - i\sin\frac{n\pi}{3}\right) + \left(-\frac{1}{2}\right)^n$ | M1 |
$= 2^{2n+1}\cos\frac{n\pi}{3} + \left(-\frac{1}{2}\right)^n$ | A1 | 3 marks | AG3 The cubic equation
$$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$
where $p$ and $q$ are real, has roots $\alpha , \beta$ and $\gamma$.\\
It is given that $\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }$.\\
(a) (i) Write down another root, $\beta$, of the equation.\\
(ii) Find the third root, $\gamma$.\\
(iii) Find the values of $p$ and $q$.\\
(b) (i) Express $\alpha$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\
(ii) Show that
$$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
(iii) Show that
$$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$
where $n$ is an integer.
\hfill \mbox{\textit{AQA FP2 2010 Q3 [14]}}