## Question 3:

### Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^6=1\Rightarrow z=e^{2k\pi i/6}$ | M1 | |
| $k=0,1,2,3,4,5$ | A1 | Oe exactly 6 roots. Accept roots $1,-1$ given as integers |
| Diagram | | |
| 6 roots in right quadrant | B1 | |
| Correct angles and moduli | B1 | As evidenced by labels, circles, or accurate diagram, or by co-ordinates |
| **[4]** | | |

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### Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+i)^6=\left(\sqrt{2}\,e^{\frac{1}{4}\pi i}\right)^6$ | M1 | Attempts modulus-argument form, getting at least 1 correct |
| $8e^{\frac{6}{4}\pi i}$ | M1 | For $(\text{mod})^6$ and $\arg\times 6$ |
| $=-8i$ | A1 | **ag**. Complete argument including start line |

**Alternative:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+i)^6=1+6i+15i^2+20i^3+15i^4+6i^5+i^6$ | M1 | |
| $=1+6i-15-20i+15+6i-1$ | M1 | No more than 1 term wrong. Sc 2 for only lines 2 & 3 correct |
| $=-8i$ | A1 | **ag** |

**Alternative:** $(1+i)^2=2i$

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+i)^2=2i$ | M1 | |
| $(1+i)^6=(2i)^3$ | M1 | |
| $=-8i$ | A1 | **ag** |
| **[3]** | | |

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### Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^6=-8i\Rightarrow z=(1+i)e^{2k\pi i/6}$ | M1 | |
| $=\sqrt{2}\,e^{i\frac{\pi}{4}}\,e^{2k\pi i/6}$ | M1 | |
| $\sqrt{2}\,e^{i\pi(1/4+k/3)}$, $k=0,1,2,3,4,5$ | A1 | Or equivalent $k$ |

**Alternative:** $z^6=8e^{i\pi(\frac{3}{2}+2k)}$

| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^6=8e^{i\pi(\frac{3}{2}+2k)}$ | M1 | |
| $\sqrt{2}\,e^{i\pi(1/4+k/3)}$, $k=0,1,2,3,4,5$ | M1 A1 | Or equivalent e.g. $\sqrt{2}\,e^{i\pi(-1/12+k/3)}$. Accept unsimplified modulus |
| **[3]** | | |