# Question 7:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = e^{\frac{k\pi}{3}i}$, $k = 0, 1, 2, 3, 4, 5$ | M1 | For sight of $e^{\frac{k\pi}{3}i}$, accept any value of $k$ |
| All six roots fully defined | A1 | All six roots listed with $\theta$ in given range, no incorrect/extra values; condone $e^0$ and/or $-1$ for $e^{\pi i}$ |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| 6 points forming a hexagon with point on positive real axis and point on negative real axis, one point in each quadrant | B1 | Points form hexagon; do not concern about distance from centre |
| Hexagon centred at origin, axes as lines of symmetry | dB1 | Centre at origin; axes need not be labelled |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\sqrt{3}+i)^6 = \left(2e^{\frac{\pi}{6}i}\right)^6 = 64e^{i\pi} = -64$ | M1 | Valid method shown |
| $-64$ | A1* | Correct answer with no errors, printed answer confirmed |

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = 2$ | B1 | Correct modulus |
| $z = 2e^{\frac{\pi}{6}i} \times e^{\frac{k\pi}{3}i}$, $k = 0, 1, 2, 3, 4, 5$ | M1 | Multiplies their roots from (a) by $2e^{\frac{\pi}{6}i}$ |
| $z = 2e^{\left(\frac{\pi}{6}+\frac{k\pi}{3}\right)i}$, $k = 0, 1, 2, 3, 4, 5$ | A1 | All six roots correct |

# Question 7 (Complex Numbers - parts c and d):

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Converts $\sqrt{3}+i$ to polar form $re^{i\theta}$ with $r=2$ or $\theta=\frac{\pi}{6}$, applies power of 6 to obtain $r^6e^{6\theta i}$ | M1 | |
| Obtains given answer; minimum: $2^6e^{\frac{6\pi i}{6}}=-64$ or $2^6e^{\pi i}=-64$ | A1* | If $r=-2$ seen in workings, withhold this mark |
| **OR:** Converts to modulus-argument form $r(\cos\theta+i\sin\theta)$ with $r=2$ or $\theta=\frac{\pi}{6}$, applies power of 6 to obtain $r^6(\cos 6\theta+i\sin 6\theta)$ | M1 | |
| **OR:** Attempts full expansion of $(\sqrt{3}+i)^6$ using binomial expansion with 7 terms, correct coefficients, $a=\sqrt{3}$, $b=i$, $n=6$ | M1 | |
| Obtains given answer with at least one intermediate line | A1* | No brackets, no irrational numbers, no $i$ terms in simplified answer |

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Deduces $r=2$ (only) | B1 | |
| Obtains at least one value of $z$ in form $re^{i\theta}$ with consistent $r$, and $\theta$ taking one of $\left\{\frac{\pi}{6},\frac{\pi}{2},\frac{5\pi}{6},\frac{7\pi}{6},\frac{3\pi}{2},\frac{11\pi}{6}\right\}$ | M1 | |
| $2e^{\frac{\pi}{6}i},\ 2e^{\frac{\pi}{2}i},\ 2e^{\frac{5\pi}{6}i},\ 2e^{\frac{7\pi}{6}i},\ 2e^{\frac{3\pi}{2}i},\ 2e^{\frac{11\pi}{6}i}$ with no incorrect or extra values | A1 | Accept unsimplified arguments e.g. $2e^{\frac{9\pi}{6}i}$; ensure $i$ and $\pi$ present; accept $2e^{\frac{\pi}{2}i}$ as $2i$ and $2e^{\frac{3\pi}{2}i}$ as $-2i$ |

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