| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Solve polynomial equations with complex roots |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard techniques: completing the square or quadratic formula to find complex roots, converting to polar form, and applying De Moivre's theorem. While it involves multiple steps, each is routine for FP3 students with no novel problem-solving required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02i Quadratic equations: with complex roots4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z=\dfrac{6\pm\sqrt{36-144}}{2}\) | M1 | For using quadratic equation formula or completing the square |
| \(z=3\pm3\sqrt{3}\,i\) | A1 | For obtaining cartesian values AEF |
| Obtain \((r=)\ 6\) | A1 | For correct modulus |
| Obtain \((\theta=)\ \frac{1}{3}\pi\) | A1 4 | For correct argument |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: \(6^{-3}\) OR \(\frac{1}{216}\) seen | B1\(\sqrt{}\) | f.t. from their \(r^{-3}\) |
| \(Z^{-3}=6^{-3}(\cos(-\pi)\pm i\sin(-\pi))\) | M1 | For using de Moivre with \(n=\pm3\) |
| Obtain \(-\frac{1}{216}\) | A1 | For correct value |
| OR: \(z^3=6z^2-36z=6(6z-36)-36z\) | M1 | For using equation to find \(z^3\) |
| \(216\) seen | B1 | Ignore any remaining \(z\) terms |
| Obtain \(-\frac{1}{216}\) | A1 3 | For correct value |
## Question 3:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z=\dfrac{6\pm\sqrt{36-144}}{2}$ | M1 | For using quadratic equation formula or completing the square |
| $z=3\pm3\sqrt{3}\,i$ | A1 | For obtaining cartesian values AEF |
| Obtain $(r=)\ 6$ | A1 | For correct modulus |
| Obtain $(\theta=)\ \frac{1}{3}\pi$ | A1 **4** | For correct argument |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: $6^{-3}$ OR $\frac{1}{216}$ seen | B1$\sqrt{}$ | f.t. from their $r^{-3}$ |
| $Z^{-3}=6^{-3}(\cos(-\pi)\pm i\sin(-\pi))$ | M1 | For using de Moivre with $n=\pm3$ |
| Obtain $-\frac{1}{216}$ | A1 | For correct value |
| OR: $z^3=6z^2-36z=6(6z-36)-36z$ | M1 | For using equation to find $z^3$ |
| $216$ seen | B1 | Ignore any remaining $z$ terms |
| Obtain $-\frac{1}{216}$ | A1 **3** | For correct value |
---3 (i) Solve the equation $z ^ { 2 } - 6 z + 36 = 0$, and give your answers in the form $r ( \cos \theta \pm \mathrm { i } \sin \theta )$, where $r > 0$ and $0 \leqslant \theta \leqslant \pi$.\\
(ii) Given that $Z$ is either of the roots found in part (i), deduce the exact value of $Z ^ { - 3 }$.
\hfill \mbox{\textit{OCR FP3 2007 Q3 [7]}}